A1 SAS EXAMPLES

SAS is general-purpose software for a wide variety of statistical analyses The mainprocedures (PROCs) for categorical data analyses are FREQ GENMOD LOGISTICNLMIXED GLIMMIX and CATMOD PROC FREQ performs basic analyses fortwo-way and three-way contingency tables PROC GENMOD fits generalized linearmodels using ML or Bayesian methods cumulative link models for ordinal responseszero-inflated Poisson regression models for count data and GEE analyses for marginalmodels PROC LOGISTIC gives ML fitting of binary response models cumulative linkmodels for ordinal responses and baseline-category logit models for nominal responses(PROC SURVEYLOGISTIC fits binary and multi-category regression models to sur-vey data by incorporating the sample design into the analysis and using the method ofpseudo ML) PROC CATMOD fits baseline-category logit models and can fit a varietyof other models using weighted least squares PROC NLMIXED gives ML fitting ofgeneralized linear mixed models using adaptive GaussndashHermite quadrature PROCGLIMMIX also fits such models with a variety of fitting methods

The examples in this appendix show SAS code for version 93 We focus on basicmodel fitting rather than the great variety of options For more detail see

Stokes Davis and Koch (2012) Categorical Data Analysis Using SAS 3rd edCary NC SAS Institute

Allison (2012) Logistic Regression Using SAS Theory and Application 2nd editionCary NC SAS Institute

For examples of categorical data analyses with SAS for many data sets in my textAn Introduction to Categorical Data Analysis see the useful site

wwwatsuclaedustatexamplesicda

set up by the UCLA Statistical Computing Center A useful SAS site on-line withdetails about the options as well as many examples for each PROC is at

supportsascomrndappdastatproceduresCategoricalDataAnalysishtml

In the SAS code below The symbol in an input line indicates that each lineof data contains more than one observation Input of a variable as characters ratherthan numbers requires an accompanying $ label in the INPUT statement (But ofcourse if you are already a SAS user you know this and much more)

Chapter 1 Introduction

With PROC FREQ for a 1times 2 table of counts of successes and failures for a binomialvariate confidence limits for the binomial proportion include AgrestindashCoull Jeffreys(ie Bayes with beta(05 05) prior) score (Wilson) and ClopperndashPearson exactmethod The keyword BINOMIAL and the EXACT statement yields binomial testsTable 1 shows code for confidence intervals for the example in the text Section 143about estimating the proportion of people who are vegetarians when 0 of 25 peoplein a sample are vegetarian The AC option gives the AgrestindashCoull interval and theWILSON option gives the score-test-based interval

1

Table 1 SAS Code for Confidence Intervals for a Proportion

----------------------------------------------------------------------data veginput response $ countdatalinesno 25yes 0proc freq data=veg weight counttables response binomial(ac wilson exact jeffreys) alpha=05run-----------------------------------------------------------------------

Chapters 2ndash3 Two-Way Contingency Tables

Table 2 uses SAS to analyze Table 32 in Categorical Data Analysis on education andbelief in God PROC FREQ forms the table with the TABLES statement orderingrow and column categories alphanumerically To use instead the order in which thecategories appear in the data set (eg to treat the variable properly in an ordinal anal-ysis) use the ORDER = DATA option in the PROC statement The WEIGHT state-ment is needed when you enter the cell counts from the contingency table instead ofsubject-level data PROC FREQ can conduct Pearson and likelihood-ratio chi-squaredtests of independence (CHISQ option) show its estimated expected frequencies (EX-PECTED) provide a wide assortment of measures of association and their standarderrors (MEASURES) and provide ordinal statistic (316) with a ldquononzero correlationrdquotest (CMH1) You can also perform chi-squared tests using PROC GENMOD (usingloglinear models discussed in Chapters 9-10) as shown Its RESIDUALS option pro-vides cell residuals The output labeled ldquoStd Pearson Residualrdquo is the standardizedresidual

For creating mosaic plots in SAS see wwwdatavisca and wwwdataviscabooks

vcd

With PROC FREQ for 2 times 2 tables the MEASURES option in the TABLESstatement provides confidence intervals for the odds ratio and the relative risk andthe RISKDIFF option provides intervals for the proportions and their difference UsingRISKDIFF(CL=(MN)) gives the interval based on inverting a score test as suggestedby Miettinen and Nurminen (1985) which is much preferred over a Wald interval SeeTable 3 for an example

For tables having small cell counts the EXACT statement can provide variousexact analyses These include Fisherrsquos exact test (with its two-sided P -value based onthe sum of the probabilities that are no greater than the probability of the observedtable) and its generalization for I times J tables treating variables as nominal withkeyword FISHER Table 4 analyzes the tea tasting data in Table 39 of the textbookTable 4 also uses PROC LOGISTIC to get a profile-likelihood confidence interval forthe odds ratio (CLODDS = PL) viewing the odds ratio as a parameter in a simplelogistic regression model with a binary indicator as a predictor PROC LOGISTIC

2

Table 2 SAS Code for Chi-Squared Measures of Association andResiduals for Data on Education and Belief in God in Table 32

-------------------------------------------------------------------data table

input degree belief $ count datalines1 1 9 1 2 8 1 3 27 1 4 8 1 5 47 1 6 2362 1 23 2 2 39 2 3 88 2 4 49 2 5 179 2 6 7063 1 28 3 2 48 3 3 89 3 4 19 3 5 104 3 6 293

proc freq order=data weight counttables degreebelief chisq expected measures cmh1

proc genmod order=data class degree beliefmodel count = degree belief dist=poi link=log residuals

run-------------------------------------------------------------------

Table 3 SAS Code for Confidence Intervals for 2times2 Table

-------------------------------------------------------------------------data exampleinput group $ outcome $ count datalinesplacebo yes 2 placebo no 18 active yes 7 active no 13proc freq order=data weight count

tables groupoutcome riskdiff(CL=(WALD MN)) measures MN = Miettinen and Nurminen inverted score test

run-------------------------------------------------------------------------

uses FREQ to weight counts serving the same purpose for which PROC FREQ usesWEIGHT The BARNARD option in the EXACT statement provides an unconditionalexact test for the difference of proportions for 2times 2 tables

The OR keyword gives the odds ratio and its large-sample Wald confidence intervalbased on (32) and the small-sample interval based on the noncentral hypergeometricdistribution (1628) Other EXACT statement keywords include unconditional exactconfidence limits for the difference of proportions (for the keyword RISKDIFF) exacttrend tests for I times 2 tables (TREND) and exact chi-squared tests (CHISQ) and exactcorrelation tests for I times J tables (MHCHI) With keyword RISKDIFF in the EXACTstatement SAS seems to construct an exact unconditional interval due to Santner andSnell in 1980 (JASA) that is very conservative Version 93 includes the option RISKD-

3

Table 4 SAS Code for Fisherrsquos Exact Test and Confidence Intervalsfor Odds Ratio for Tea-Tasting Data in Table 39

-------------------------------------------------------------------data fisherinput poured guess count datalines1 1 3 1 2 1 2 1 1 2 2 3proc freq weight count

tables pouredguess measures riskdiffexact fisher or alpha=05

proc logistic descending freq countmodel guess = poured clodds=pl

run-------------------------------------------------------------------

IFF(METHOD=SCORE) which is based on the Chan and Zhang (1999) approach ofinverting two separate one-sided score tests which is less conservative See

httpsupportsascomdocumentationcdlenprocstat67528HTMLdefaultviewer

htmprocstat_freq_details53htmprocstatfreqfreqrdiffexact

for details and Table 5 for an example for a 2times2 table (The software StatXact alsoprovides the Agresti and Min (2001) method of inverting a single two-sided score testwhich is less conservative yet) You can use Monte Carlo simulation (option MC) toestimate exact P -values when the exact calculation is too time-consuming

Chapter 4 Generalized Linear Models

PROC GENMOD fits GLMs It specifies the response distribution in the DIST option(ldquopoirdquo for Poisson ldquobinrdquo for binomial ldquomultrdquo for multinomial ldquonegbinrdquo for negativebinomial) and specifies the link function in the LINK option For binomial models withgrouped data the response in the model statements takes the form of the number ofldquosuccessesrdquo divided by the number of cases Table 6 illustrates for the snoring datain Table 42 of the textbook Profile likelihood confidence intervals are provided inPROC GENMOD with the LRCI option

Table 7 uses PROC GENMOD for count modeling of the horseshoe crab data inTable 43 of the textbook (Note that the complete data set is in the Datasets linkwwwstatufledu~aacdadatahtml at this website) The variable SATELL inthe data set refers to the number of satellites that a female horseshoe crab has Eachobservation refers to a single crab Using width as the predictor the first two modelsuse Poisson regression and the third model assumes a negative binomial distribution

Table 8 uses PROC GENMOD for the overdispersed teratology-study data of Ta-ble 47 of the textbook (Note that the complete data set is in the Datasets linkwwwstatufledu~aacdadatahtml at this website) A CLASS statement requests

4

Table 4 SAS Code for Fisherrsquos Exact Test and Confidence Intervalsfor Odds Ratio for Tea-Tasting Data in Table 39

-------------------------------------------------------------------data fisherinput poured guess count datalines1 1 3 1 2 1 2 1 1 2 2 3proc freq weight count

tables pouredguess measures riskdiffexact fisher or alpha=05

proc logistic descending freq countmodel guess = poured clodds=pl

run-------------------------------------------------------------------

IFF(METHOD=SCORE) which is based on the Chan and Zhang (1999) approach ofinverting two separate one-sided score tests which is less conservative See

httpsupportsascomdocumentationcdlenprocstat67528HTMLdefaultviewer

htmprocstat_freq_details53htmprocstatfreqfreqrdiffexact

for details and Table 5 for an example for a 2times2 table (The software StatXact alsoprovides the Agresti and Min (2001) method of inverting a single two-sided score testwhich is less conservative yet) You can use Monte Carlo simulation (option MC) toestimate exact P -values when the exact calculation is too time-consuming

Chapter 4 Generalized Linear Models

PROC GENMOD fits GLMs It specifies the response distribution in the DIST option(ldquopoirdquo for Poisson ldquobinrdquo for binomial ldquomultrdquo for multinomial ldquonegbinrdquo for negativebinomial) and specifies the link function in the LINK option For binomial models withgrouped data the response in the model statements takes the form of the number ofldquosuccessesrdquo divided by the number of cases Table 6 illustrates for the snoring datain Table 42 of the textbook Profile likelihood confidence intervals are provided inPROC GENMOD with the LRCI option

Table 7 uses PROC GENMOD for count modeling of the horseshoe crab data inTable 43 of the textbook (Note that the complete data set is in the Datasets linkwwwstatufledu~aacdadatahtml at this website) The variable SATELL inthe data set refers to the number of satellites that a female horseshoe crab has Eachobservation refers to a single crab Using width as the predictor the first two modelsuse Poisson regression and the third model assumes a negative binomial distribution

Table 8 uses PROC GENMOD for the overdispersed teratology-study data of Ta-ble 47 of the textbook (Note that the complete data set is in the Datasets linkwwwstatufledu~aacdadatahtml at this website) A CLASS statement requests

4

Table 5 SAS Code for ldquoExactrdquo Confidence Intervals for 2times2 Table

-------------------------------------------------------------------------data exampleinput group $ outcome $ count datalinesplacebo yes 2 placebo no 18 active yes 7 active no 13proc freq order=data weight count

tables groupoutcome exact or riskdiff(CL=(MN)) runproc freq order=data weight count

tables groupoutcome exact riskdiff(method=score)

exact unconditional inverting two one-sided score testsrun-------------------------------------------------------------------------

Table 6 SAS Code for Binary GLMs for Snoring Data in Table 42

-------------------------------------------------------------------------data glminput snoring disease total datalines0 24 1379 2 35 638 4 21 213 5 30 254proc genmod model diseasetotal = snoring dist=bin link=identityproc genmod model diseasetotal = snoring dist=bin link=logit LRCIproc genmod model diseasetotal = snoring dist=bin link=probitrun-------------------------------------------------------------------------

indicator (dummy) variables for the groups With no intercept in the model (optionNOINT) for the identity link the estimated parameters are the four group probabili-ties The ESTIMATE statement provides an estimate confidence interval and test fora contrast of model parameters in this case the difference in probabilities for the firstand second groups The second analysis uses the Pearson statistic to scale standarderrors to adjust for overdispersion PROC LOGISTIC can also provide overdispersionmodeling of binary responses see Table 37 in the Chapter 14 part of this appendix forSAS

The final PROC GENMOD run in Table 10 fits the Poisson regression model withlog link for the grouped data of Tables 44 and 52 It models the total number of

5

Table 7 SAS Code for Poisson and Negative Binomial GLMs forHorseshoe Crab Data in Table 43

--------------------------------------------------------------data crabinput color spine width satell weightdatalines3 3 283 8 3054 3 225 0 1553 2 245 0 200proc genmod

model satell = width dist=poi link=log proc genmod

model satell = width dist=poi link=identity proc genmod

model satell = width dist=negbin link=identity run---------------------------------------------------------------

Table 8 SAS Code for Overdispersion Modeling of Teratology Datain Table 47

----------------------------------------------------------------------data mooreinput litter group n y

datalines1 1 10 1 2 1 11 4 3 1 12 9 4 1 4 4 5 1 10 1055 4 14 1 56 4 8 0 57 4 6 0 58 4 17 0proc genmod class groupmodel yn = group dist=bin link=identity noint

estimate rsquopi1-pi2rsquo group 1 -1 0 0proc genmod class groupmodel yn = group dist=bin link=identity noint scale=pearson

run----------------------------------------------------------------------

satellites at each width level (variable ldquosatellrdquo) using the log of the number of casesas offset

6

Chapters 5ndash7 Logistic Regression and Binary ResponseAnalyses

You can fit logistic regression models using either software for GLMs or specializedsoftware for logistic regression PROC GENMOD uses Newton-Raphson whereasPROC LOGISTIC uses Fisher scoring Both yield ML estimates but the SE valuesuse the inverted observed information matrix in PROC GENMOD and the invertedexpected information matrix in PROC LOGISTIC These are the same for the logitlink because it is the canonical link function for the binomial but differ for other linksWith PROC LOGISTIC logistic regression is the default for binary data PROCLOGISTIC has a built-in check of whether logistic regression ML estimates exist Itcan detect complete separation of data points with 0 and 1 outcomes in which caseat least one estimate is infinite PROC LOGISTIC can also apply other links such asthe probit

Table 9 shows logistic regression analyses for the horseshoe crab data of Table 43The variable SATELL in the data set at wwwstatufledu~aacdadatahtml refersto the number of satellites that a female horseshoe crab has The logistic modelsrefer to a constructed binary variable Y that equals 1 when a horseshoe crab hassatellites and 0 otherwise With binary data entry PROC GENMOD and PROCLOGISTIC order the levels alphanumerically forming the logit with (1 0) responsesas log[P (Y = 0)P (Y = 1)] Invoking the procedure with DESCENDING followingthe PROC name reverses the order The first two PROC GENMOD statements useboth color and width as predictors color is qualitative in the first model (by theCLASS statement) and quantitative in the second A CONTRAST statement testscontrasts of parameters such as whether parameters for two levels of a factor areidentical The statement shown contrasts the first and fourth color levels The thirdPROC GENMOD statement uses an indicator variable for color indicating whethera crab is light or dark (color = 4) The fourth PROC GENMOD statement fits themain effects model using all the predictors PROC LOGISTIC has options for stepwiseselection of variables as the final model statement shows The LACKFIT option yieldsthe HosmerndashLemeshow statistic Predicted probabilities and lower and upper 95confidence limits for the probabilities are shown in a plot with the PLOTS=ALL optionor with PLOTS=EFFECT The ROC curve is produced using the PLOTS=ROCoption

Table 10 applies PROC GENMOD and PROC LOGISTIC to the grouped dataas shown in Table 52 of the textbook when ldquoyrdquo out of ldquonrdquo crabs had satellites ata given width level Profile likelihood confidence intervals are provided in PROCGENMOD with the LRCI option and in PROC LOGISTIC with the PLCL optionIn PROC GENMOD the ALPHA = option can specify an error probability otherthan the default of 005 and the TYPE3 option provides likelihood-ratio tests foreach parameter In PROC LOGISTIC the INFLUENCE option provides Pearson anddeviance residuals and diagnostic measures (Pregibon 1981) The STB option providesstandardized estimates by multiplying by sxj

radic3π (text Section 547) Following

the model statement Table 10 requests predicted probabilities and lower and upper95 confidence limits for the probabilities In the PLOTS option the standardizedresiduals are plotted against the linear predictor values (In the Chapter 9ndash10 sectionwe discuss the second GENMOD analysis of a loglinear model)

Table 11 uses PROC GENMOD and PROC LOGISTIC to fit a logistic model with

7

Table 7 SAS Code for Poisson and Negative Binomial GLMs forHorseshoe Crab Data in Table 43

--------------------------------------------------------------data crabinput color spine width satell weightdatalines3 3 283 8 3054 3 225 0 1553 2 245 0 200proc genmod

model satell = width dist=poi link=log proc genmod

model satell = width dist=poi link=identity proc genmod

model satell = width dist=negbin link=identity run---------------------------------------------------------------

Table 8 SAS Code for Overdispersion Modeling of Teratology Datain Table 47

----------------------------------------------------------------------data mooreinput litter group n y

datalines1 1 10 1 2 1 11 4 3 1 12 9 4 1 4 4 5 1 10 1055 4 14 1 56 4 8 0 57 4 6 0 58 4 17 0proc genmod class groupmodel yn = group dist=bin link=identity noint

estimate rsquopi1-pi2rsquo group 1 -1 0 0proc genmod class groupmodel yn = group dist=bin link=identity noint scale=pearson

run----------------------------------------------------------------------

satellites at each width level (variable ldquosatellrdquo) using the log of the number of casesas offset

6

Chapters 5ndash7 Logistic Regression and Binary ResponseAnalyses

You can fit logistic regression models using either software for GLMs or specializedsoftware for logistic regression PROC GENMOD uses Newton-Raphson whereasPROC LOGISTIC uses Fisher scoring Both yield ML estimates but the SE valuesuse the inverted observed information matrix in PROC GENMOD and the invertedexpected information matrix in PROC LOGISTIC These are the same for the logitlink because it is the canonical link function for the binomial but differ for other linksWith PROC LOGISTIC logistic regression is the default for binary data PROCLOGISTIC has a built-in check of whether logistic regression ML estimates exist Itcan detect complete separation of data points with 0 and 1 outcomes in which caseat least one estimate is infinite PROC LOGISTIC can also apply other links such asthe probit

Table 9 shows logistic regression analyses for the horseshoe crab data of Table 43The variable SATELL in the data set at wwwstatufledu~aacdadatahtml refersto the number of satellites that a female horseshoe crab has The logistic modelsrefer to a constructed binary variable Y that equals 1 when a horseshoe crab hassatellites and 0 otherwise With binary data entry PROC GENMOD and PROCLOGISTIC order the levels alphanumerically forming the logit with (1 0) responsesas log[P (Y = 0)P (Y = 1)] Invoking the procedure with DESCENDING followingthe PROC name reverses the order The first two PROC GENMOD statements useboth color and width as predictors color is qualitative in the first model (by theCLASS statement) and quantitative in the second A CONTRAST statement testscontrasts of parameters such as whether parameters for two levels of a factor areidentical The statement shown contrasts the first and fourth color levels The thirdPROC GENMOD statement uses an indicator variable for color indicating whethera crab is light or dark (color = 4) The fourth PROC GENMOD statement fits themain effects model using all the predictors PROC LOGISTIC has options for stepwiseselection of variables as the final model statement shows The LACKFIT option yieldsthe HosmerndashLemeshow statistic Predicted probabilities and lower and upper 95confidence limits for the probabilities are shown in a plot with the PLOTS=ALL optionor with PLOTS=EFFECT The ROC curve is produced using the PLOTS=ROCoption

Table 10 applies PROC GENMOD and PROC LOGISTIC to the grouped dataas shown in Table 52 of the textbook when ldquoyrdquo out of ldquonrdquo crabs had satellites ata given width level Profile likelihood confidence intervals are provided in PROCGENMOD with the LRCI option and in PROC LOGISTIC with the PLCL optionIn PROC GENMOD the ALPHA = option can specify an error probability otherthan the default of 005 and the TYPE3 option provides likelihood-ratio tests foreach parameter In PROC LOGISTIC the INFLUENCE option provides Pearson anddeviance residuals and diagnostic measures (Pregibon 1981) The STB option providesstandardized estimates by multiplying by sxj

radic3π (text Section 547) Following

the model statement Table 10 requests predicted probabilities and lower and upper95 confidence limits for the probabilities In the PLOTS option the standardizedresiduals are plotted against the linear predictor values (In the Chapter 9ndash10 sectionwe discuss the second GENMOD analysis of a loglinear model)

Table 11 uses PROC GENMOD and PROC LOGISTIC to fit a logistic model with

7

Chapters 5ndash7 Logistic Regression and Binary ResponseAnalyses

You can fit logistic regression models using either software for GLMs or specializedsoftware for logistic regression PROC GENMOD uses Newton-Raphson whereasPROC LOGISTIC uses Fisher scoring Both yield ML estimates but the SE valuesuse the inverted observed information matrix in PROC GENMOD and the invertedexpected information matrix in PROC LOGISTIC These are the same for the logitlink because it is the canonical link function for the binomial but differ for other linksWith PROC LOGISTIC logistic regression is the default for binary data PROCLOGISTIC has a built-in check of whether logistic regression ML estimates exist Itcan detect complete separation of data points with 0 and 1 outcomes in which caseat least one estimate is infinite PROC LOGISTIC can also apply other links such asthe probit

Table 9 shows logistic regression analyses for the horseshoe crab data of Table 43The variable SATELL in the data set at wwwstatufledu~aacdadatahtml refersto the number of satellites that a female horseshoe crab has The logistic modelsrefer to a constructed binary variable Y that equals 1 when a horseshoe crab hassatellites and 0 otherwise With binary data entry PROC GENMOD and PROCLOGISTIC order the levels alphanumerically forming the logit with (1 0) responsesas log[P (Y = 0)P (Y = 1)] Invoking the procedure with DESCENDING followingthe PROC name reverses the order The first two PROC GENMOD statements useboth color and width as predictors color is qualitative in the first model (by theCLASS statement) and quantitative in the second A CONTRAST statement testscontrasts of parameters such as whether parameters for two levels of a factor areidentical The statement shown contrasts the first and fourth color levels The thirdPROC GENMOD statement uses an indicator variable for color indicating whethera crab is light or dark (color = 4) The fourth PROC GENMOD statement fits themain effects model using all the predictors PROC LOGISTIC has options for stepwiseselection of variables as the final model statement shows The LACKFIT option yieldsthe HosmerndashLemeshow statistic Predicted probabilities and lower and upper 95confidence limits for the probabilities are shown in a plot with the PLOTS=ALL optionor with PLOTS=EFFECT The ROC curve is produced using the PLOTS=ROCoption

Table 10 applies PROC GENMOD and PROC LOGISTIC to the grouped dataas shown in Table 52 of the textbook when ldquoyrdquo out of ldquonrdquo crabs had satellites ata given width level Profile likelihood confidence intervals are provided in PROCGENMOD with the LRCI option and in PROC LOGISTIC with the PLCL optionIn PROC GENMOD the ALPHA = option can specify an error probability otherthan the default of 005 and the TYPE3 option provides likelihood-ratio tests foreach parameter In PROC LOGISTIC the INFLUENCE option provides Pearson anddeviance residuals and diagnostic measures (Pregibon 1981) The STB option providesstandardized estimates by multiplying by sxj

radic3π (text Section 547) Following

the model statement Table 10 requests predicted probabilities and lower and upper95 confidence limits for the probabilities In the PLOTS option the standardizedresiduals are plotted against the linear predictor values (In the Chapter 9ndash10 sectionwe discuss the second GENMOD analysis of a loglinear model)

Table 11 uses PROC GENMOD and PROC LOGISTIC to fit a logistic model with

7

Table 9 SAS Code for Logistic Regression Models with HorseshoeCrab Data in Table 43

------------------------------------------------------------------------data crabinput color spine width satell weightif satellgt0 then y=1 if satell=0 then y=0if color=4 then light=0 if color lt 4 then light=1datalines2 3 283 8 3054 3 225 0 1552 2 245 0 200proc genmod descending class colormodel y = width color dist=bin link=logit lrci type3 obstatscontrast rsquoa-drsquo color 1 0 0 -1

proc genmod descendingmodel y = width color dist=bin link=logit

proc genmod descendingmodel y = width light dist=bin link=logit

proc genmod descending class color spinemodel y = width weight color spine dist=bin link=logit type3

ods graphics onods htmlproc logistic descending plots=allclass color spine param=refmodel y = width weight color spine selection=backward lackfit

proc gam plots=components(clm commonaxes) generalized additive modelmodel y (event=rsquo1rsquo) = spline(width) dist=binary smooth data

runods html closeods graphics offrun------------------------------------------------------------------------

8

Table 10 SAS Code for Modeling Grouped Crab Data in Tables 44and 52

---------------------------------------------------------------------data crabinput width y n satell logcases=log(n)datalines2269 5 14 143041 14 14 72ods graphics onods htmlproc genmodmodel yn = width dist=bin link=logit lrci alpha=01 type3

proc genmod plots=stdreschi(xbeta)model satell = width dist=poi link=log offset=logcases residuals

proc logistic data=crab plots=allmodel yn = width influence stboutput out=predict p=pi_hat lower=LCL upper=UCL

proc print data=predictrunods html closeods graphics offrun---------------------------------------------------------------------

9

Table 11 SAS Code for Logistic Modeling of AIDS Data in Table 56

-----------------------------------------------------------------------data aidsinput race $ azt $ y n datalinesWhite Yes 14 107 White No 32 113 Black Yes 11 63 Black No 12 55

proc genmod class race aztmodel yn = azt race dist=bin type3 lrci residuals obstats

proc logistic class race azt param=referencemodel yn = azt race aggregate scale=none clparm=both clodds=bothoutput out=predict p=pi_hat lower=lower upper=upper

proc print data=predictproc logistic class race azt (ref=first) param=refmodel yn = azt aggregate=(azt race) scale=none

run-----------------------------------------------------------------------

qualitative predictors to the AIDS and AZT study of Table 56 In PROC GENMODthe OBSTATS option provides various ldquoobservation statisticsrdquo including predictedvalues and their confidence limits The RESIDUALS option requests residuals suchas the Pearson residuals and standardized residuals (labeled ldquoStd Pearson Residualrdquo)A CLASS statement requests indicator variables for the factor By default in PROCGENMOD the parameter estimate for the last level of each factor equals 0 In PROCLOGISTIC estimates sum to zero That is dummies take the effect coding (1minus1)with values of 1 when in the category and minus1 when not for which parameters sumto 0 In the CLASS statement in PROC LOGISTIC the option PARAM = REF re-quests (1 0) indicator variables with the last category as the reference level PuttingREF = FIRST next to a variable name requests its first category as the referencelevel The CLPARM = BOTH and CLODDS = BOTH options provide Wald andprofile likelihood confidence intervals for parameters and odds ratio effects of explana-tory variables With AGGREGATE SCALE = NONE in the model statement PROCLOGISTIC reports Pearson and deviance tests of fit it forms groups by aggregatingdata into the possible combinations of explanatory variable values without overdis-persion adjustments Adding variables in parentheses after AGGREGATE (as in thesecond use of PROC LOGISTIC in Table 11) specifies the predictors used for formingthe table on which to test fit even when some predictors may have no effect in themodel

Table 12 analyzes the clinical trial data of Table 69 of the textbook The CMHoption in PROC FREQ specifies the CMH statistic the MantelndashHaenszel estimate of acommon odds ratio and its confidence interval and the BreslowndashDay statistic FREQuses the two rightmost variables in the TABLES statement as the rows and columns foreach partial table the CHISQ option yields chi-square tests of independence for eachpartial table For I times 2 tables the TREND keyword in the TABLES statement pro-vides the CochranndashArmitage trend test The EQOR option in an EXACT statement

10

Table 12 SAS Code for CochranndashMantelndashHaenszel Test and RelatedAnalyses of Clinical Trial Data in Table 69

-------------------------------------------------------------------data cmhinput center $ treat response count datalinesa 1 1 11 a 1 2 25 a 2 1 10 a 2 2 27h 1 1 4 h 1 2 2 h 2 1 6 h 2 2 1proc freq weight counttables centertreatresponse cmh chisq exact eqor

run-------------------------------------------------------------------

provides an exact test for equal odds ratios proposed by Zelen (1971) OrsquoBrien (1986)gave a SAS macro for computing powers using the noncentral chi-squared distribution

Models with probit and complementary log-log (CLOGLOG) links are availablewith PROC GENMOD PROC LOGISTIC or PROC PROBIT PROC SURVEYL-OGISTIC fits binary regression models to survey data by incorporating the sampledesign into the analysis and using the method of pseudo ML (with a Taylor series ap-proximation) It can use the logit probit and complementary log-log link functions

For the logit link PROC GENMOD can perform exact conditional logistic anal-yses with the EXACT statement It is also possible to implement the small-sampletests with mid-P -values and confidence intervals based on inverting tests using mid-P -values The option CLTYPE = EXACT | MIDP requests either the exact or mid-Pconfidence intervals for the parameter estimates By default the exact intervals areproduced

Exact conditional logistic regression is also available in PROC LOGISTIC withthe EXACT statement

PROC GAM fits generalized additive models such as shown in Table 9

Table 13 shows the use of Bayesian methods for the analysis described in Section722 (Note that the complete data set is in the Datasets link wwwstatufledu

~aacdadatahtml at this website) Variances can be set for normal priors (egVAR = 100) the length of the Markov chain can be set by NMC and DIAGNOS-TICS=MCERROR gives the amount of Monte Carlo error in the parameter estimatesat the end of the MCMC fitting process

The Firth penalized likelihood approach to reducing bias in estimation of logisticregression parameters is available with the FIRTH option in PROC LOGISTIC asshown in Table 13

11

Table 10 SAS Code for Modeling Grouped Crab Data in Tables 44and 52

---------------------------------------------------------------------data crabinput width y n satell logcases=log(n)datalines2269 5 14 143041 14 14 72ods graphics onods htmlproc genmodmodel yn = width dist=bin link=logit lrci alpha=01 type3

proc genmod plots=stdreschi(xbeta)model satell = width dist=poi link=log offset=logcases residuals

proc logistic data=crab plots=allmodel yn = width influence stboutput out=predict p=pi_hat lower=LCL upper=UCL

proc print data=predictrunods html closeods graphics offrun---------------------------------------------------------------------

9

Table 11 SAS Code for Logistic Modeling of AIDS Data in Table 56

-----------------------------------------------------------------------data aidsinput race $ azt $ y n datalinesWhite Yes 14 107 White No 32 113 Black Yes 11 63 Black No 12 55

proc genmod class race aztmodel yn = azt race dist=bin type3 lrci residuals obstats

proc logistic class race azt param=referencemodel yn = azt race aggregate scale=none clparm=both clodds=bothoutput out=predict p=pi_hat lower=lower upper=upper

proc print data=predictproc logistic class race azt (ref=first) param=refmodel yn = azt aggregate=(azt race) scale=none

run-----------------------------------------------------------------------

qualitative predictors to the AIDS and AZT study of Table 56 In PROC GENMODthe OBSTATS option provides various ldquoobservation statisticsrdquo including predictedvalues and their confidence limits The RESIDUALS option requests residuals suchas the Pearson residuals and standardized residuals (labeled ldquoStd Pearson Residualrdquo)A CLASS statement requests indicator variables for the factor By default in PROCGENMOD the parameter estimate for the last level of each factor equals 0 In PROCLOGISTIC estimates sum to zero That is dummies take the effect coding (1minus1)with values of 1 when in the category and minus1 when not for which parameters sumto 0 In the CLASS statement in PROC LOGISTIC the option PARAM = REF re-quests (1 0) indicator variables with the last category as the reference level PuttingREF = FIRST next to a variable name requests its first category as the referencelevel The CLPARM = BOTH and CLODDS = BOTH options provide Wald andprofile likelihood confidence intervals for parameters and odds ratio effects of explana-tory variables With AGGREGATE SCALE = NONE in the model statement PROCLOGISTIC reports Pearson and deviance tests of fit it forms groups by aggregatingdata into the possible combinations of explanatory variable values without overdis-persion adjustments Adding variables in parentheses after AGGREGATE (as in thesecond use of PROC LOGISTIC in Table 11) specifies the predictors used for formingthe table on which to test fit even when some predictors may have no effect in themodel

Table 12 analyzes the clinical trial data of Table 69 of the textbook The CMHoption in PROC FREQ specifies the CMH statistic the MantelndashHaenszel estimate of acommon odds ratio and its confidence interval and the BreslowndashDay statistic FREQuses the two rightmost variables in the TABLES statement as the rows and columns foreach partial table the CHISQ option yields chi-square tests of independence for eachpartial table For I times 2 tables the TREND keyword in the TABLES statement pro-vides the CochranndashArmitage trend test The EQOR option in an EXACT statement

10

Table 12 SAS Code for CochranndashMantelndashHaenszel Test and RelatedAnalyses of Clinical Trial Data in Table 69

-------------------------------------------------------------------data cmhinput center $ treat response count datalinesa 1 1 11 a 1 2 25 a 2 1 10 a 2 2 27h 1 1 4 h 1 2 2 h 2 1 6 h 2 2 1proc freq weight counttables centertreatresponse cmh chisq exact eqor

run-------------------------------------------------------------------

provides an exact test for equal odds ratios proposed by Zelen (1971) OrsquoBrien (1986)gave a SAS macro for computing powers using the noncentral chi-squared distribution

Models with probit and complementary log-log (CLOGLOG) links are availablewith PROC GENMOD PROC LOGISTIC or PROC PROBIT PROC SURVEYL-OGISTIC fits binary regression models to survey data by incorporating the sampledesign into the analysis and using the method of pseudo ML (with a Taylor series ap-proximation) It can use the logit probit and complementary log-log link functions

For the logit link PROC GENMOD can perform exact conditional logistic anal-yses with the EXACT statement It is also possible to implement the small-sampletests with mid-P -values and confidence intervals based on inverting tests using mid-P -values The option CLTYPE = EXACT | MIDP requests either the exact or mid-Pconfidence intervals for the parameter estimates By default the exact intervals areproduced

Exact conditional logistic regression is also available in PROC LOGISTIC withthe EXACT statement

PROC GAM fits generalized additive models such as shown in Table 9

Table 13 shows the use of Bayesian methods for the analysis described in Section722 (Note that the complete data set is in the Datasets link wwwstatufledu

~aacdadatahtml at this website) Variances can be set for normal priors (egVAR = 100) the length of the Markov chain can be set by NMC and DIAGNOS-TICS=MCERROR gives the amount of Monte Carlo error in the parameter estimatesat the end of the MCMC fitting process

The Firth penalized likelihood approach to reducing bias in estimation of logisticregression parameters is available with the FIRTH option in PROC LOGISTIC asshown in Table 13

11

Table 11 SAS Code for Logistic Modeling of AIDS Data in Table 56

-----------------------------------------------------------------------data aidsinput race $ azt $ y n datalinesWhite Yes 14 107 White No 32 113 Black Yes 11 63 Black No 12 55

proc genmod class race aztmodel yn = azt race dist=bin type3 lrci residuals obstats

proc logistic class race azt param=referencemodel yn = azt race aggregate scale=none clparm=both clodds=bothoutput out=predict p=pi_hat lower=lower upper=upper

proc print data=predictproc logistic class race azt (ref=first) param=refmodel yn = azt aggregate=(azt race) scale=none

run-----------------------------------------------------------------------

qualitative predictors to the AIDS and AZT study of Table 56 In PROC GENMODthe OBSTATS option provides various ldquoobservation statisticsrdquo including predictedvalues and their confidence limits The RESIDUALS option requests residuals suchas the Pearson residuals and standardized residuals (labeled ldquoStd Pearson Residualrdquo)A CLASS statement requests indicator variables for the factor By default in PROCGENMOD the parameter estimate for the last level of each factor equals 0 In PROCLOGISTIC estimates sum to zero That is dummies take the effect coding (1minus1)with values of 1 when in the category and minus1 when not for which parameters sumto 0 In the CLASS statement in PROC LOGISTIC the option PARAM = REF re-quests (1 0) indicator variables with the last category as the reference level PuttingREF = FIRST next to a variable name requests its first category as the referencelevel The CLPARM = BOTH and CLODDS = BOTH options provide Wald andprofile likelihood confidence intervals for parameters and odds ratio effects of explana-tory variables With AGGREGATE SCALE = NONE in the model statement PROCLOGISTIC reports Pearson and deviance tests of fit it forms groups by aggregatingdata into the possible combinations of explanatory variable values without overdis-persion adjustments Adding variables in parentheses after AGGREGATE (as in thesecond use of PROC LOGISTIC in Table 11) specifies the predictors used for formingthe table on which to test fit even when some predictors may have no effect in themodel

Table 12 analyzes the clinical trial data of Table 69 of the textbook The CMHoption in PROC FREQ specifies the CMH statistic the MantelndashHaenszel estimate of acommon odds ratio and its confidence interval and the BreslowndashDay statistic FREQuses the two rightmost variables in the TABLES statement as the rows and columns foreach partial table the CHISQ option yields chi-square tests of independence for eachpartial table For I times 2 tables the TREND keyword in the TABLES statement pro-vides the CochranndashArmitage trend test The EQOR option in an EXACT statement

10

Table 12 SAS Code for CochranndashMantelndashHaenszel Test and RelatedAnalyses of Clinical Trial Data in Table 69

-------------------------------------------------------------------data cmhinput center $ treat response count datalinesa 1 1 11 a 1 2 25 a 2 1 10 a 2 2 27h 1 1 4 h 1 2 2 h 2 1 6 h 2 2 1proc freq weight counttables centertreatresponse cmh chisq exact eqor

run-------------------------------------------------------------------

provides an exact test for equal odds ratios proposed by Zelen (1971) OrsquoBrien (1986)gave a SAS macro for computing powers using the noncentral chi-squared distribution

Models with probit and complementary log-log (CLOGLOG) links are availablewith PROC GENMOD PROC LOGISTIC or PROC PROBIT PROC SURVEYL-OGISTIC fits binary regression models to survey data by incorporating the sampledesign into the analysis and using the method of pseudo ML (with a Taylor series ap-proximation) It can use the logit probit and complementary log-log link functions

For the logit link PROC GENMOD can perform exact conditional logistic anal-yses with the EXACT statement It is also possible to implement the small-sampletests with mid-P -values and confidence intervals based on inverting tests using mid-P -values The option CLTYPE = EXACT | MIDP requests either the exact or mid-Pconfidence intervals for the parameter estimates By default the exact intervals areproduced

Exact conditional logistic regression is also available in PROC LOGISTIC withthe EXACT statement

PROC GAM fits generalized additive models such as shown in Table 9

Table 13 shows the use of Bayesian methods for the analysis described in Section722 (Note that the complete data set is in the Datasets link wwwstatufledu

~aacdadatahtml at this website) Variances can be set for normal priors (egVAR = 100) the length of the Markov chain can be set by NMC and DIAGNOS-TICS=MCERROR gives the amount of Monte Carlo error in the parameter estimatesat the end of the MCMC fitting process

The Firth penalized likelihood approach to reducing bias in estimation of logisticregression parameters is available with the FIRTH option in PROC LOGISTIC asshown in Table 13

11

Table 12 SAS Code for CochranndashMantelndashHaenszel Test and RelatedAnalyses of Clinical Trial Data in Table 69

-------------------------------------------------------------------data cmhinput center $ treat response count datalinesa 1 1 11 a 1 2 25 a 2 1 10 a 2 2 27h 1 1 4 h 1 2 2 h 2 1 6 h 2 2 1proc freq weight counttables centertreatresponse cmh chisq exact eqor

run-------------------------------------------------------------------

provides an exact test for equal odds ratios proposed by Zelen (1971) OrsquoBrien (1986)gave a SAS macro for computing powers using the noncentral chi-squared distribution

Models with probit and complementary log-log (CLOGLOG) links are availablewith PROC GENMOD PROC LOGISTIC or PROC PROBIT PROC SURVEYL-OGISTIC fits binary regression models to survey data by incorporating the sampledesign into the analysis and using the method of pseudo ML (with a Taylor series ap-proximation) It can use the logit probit and complementary log-log link functions

For the logit link PROC GENMOD can perform exact conditional logistic anal-yses with the EXACT statement It is also possible to implement the small-sampletests with mid-P -values and confidence intervals based on inverting tests using mid-P -values The option CLTYPE = EXACT | MIDP requests either the exact or mid-Pconfidence intervals for the parameter estimates By default the exact intervals areproduced

Exact conditional logistic regression is also available in PROC LOGISTIC withthe EXACT statement

PROC GAM fits generalized additive models such as shown in Table 9

Table 13 shows the use of Bayesian methods for the analysis described in Section722 (Note that the complete data set is in the Datasets link wwwstatufledu

~aacdadatahtml at this website) Variances can be set for normal priors (egVAR = 100) the length of the Markov chain can be set by NMC and DIAGNOS-TICS=MCERROR gives the amount of Monte Carlo error in the parameter estimatesat the end of the MCMC fitting process

The Firth penalized likelihood approach to reducing bias in estimation of logisticregression parameters is available with the FIRTH option in PROC LOGISTIC asshown in Table 13

11

Table 13 SAS Code for Bayesian Modeling Example in Section 722of Data on Endometrial Cancer in Table 72

---------------------------------------------------------------------data endometrialinput nv pi eh hg nv2 = nv - 05 pi2 = (pi-173797)99978 eh2 = (eh-16616)06621datalines

0 13 164 00 16 226 00 8 314 00 34 268 00 20 128 00 5 231 00 17 180 00 10 168 0

1 11 101 11 21 098 10 5 035 11 19 102 10 33 085 1

proc genmod descendingmodel hg = nv2 pi2 eh2 dist=bin link=logit

bayes coeffprior=normal (var=10) diagnostics=mcerror nmc=1000000runproc genmod descendingmodel hg = nv2 pi2 eh2 dist=bin link=logit

bayes coeffprior=normal (var=100) diagnostics=mcerror nmc=1000000runproc logistic descendingmodel hg = nv2 pi2 eh2 firth clparm=pl

run-----------------------------------------------------------------------

12

Table 14 SAS Code for Baseline-Category Logit Models with AlligatorData in Table 81

----------------------------------------------------------------------data gatorinput lake gender size food countdatalines1 1 1 1 71 1 1 2 11 1 1 3 01 1 1 4 01 1 1 5 54 2 2 1 84 2 2 2 14 2 2 3 04 2 2 4 04 2 2 5 1proc logistic freq countclass lake size param=refmodel food(ref=rsquo1rsquo) = lake size link=glogit aggregate scale=noneproc catmod weight countpopulation lake size gendermodel food = lake size pred=freq pred=probrun----------------------------------------------------------------------

Chapter 8 Multinomial Response Models

PROC LOGISTIC fits baseline-category logit models using the LINK = GLOGIT op-tion The final response category is the default baseline for the logits Exact inferenceis also available using the conditional distribution to eliminate nuisance parametersPROC CATMOD also fits baseline-category logit models as Table 14 shows for thetext example on alligator food choice (Table 81) CATMOD codes estimates for afactor so that they sum to zero The PRED = PROB and PRED = FREQ optionsprovide predicted probabilities and fitted values and their standard errors The POP-ULATION statement provides the variables that define the predictor settings Forinstance with ldquogenderrdquo in that statement the model with lake and size effects isfitted to the full table also classified by gender

PROC GENMOD can fit the proportional odds version of cumulative logit modelsusing the DIST = MULTINOMIAL and LINK = CLOGIT options Table 15 fits itto the data shown in Table 85 on happiness number of traumatic events and raceWhen the number of response categories exceeds 2 by default PROC LOGISTIC fitsthis model It also gives a score test of the proportional odds assumption of identical

13

effect parameters for each cutpoint Both procedures use the αj + βx form of themodel Both procedures now have graphic capabilities of displaying the cumulativeprobabilities as a function of a predictor (with PLOTS = EFFECT) at fixed valuesof other predictors Cox (1995) used PROC NLIN for the more general model havinga scale parameter

With the UNEQUALSLOPES option in PROC LOGISTIC one can fit the modelwithout proportional odds structure For example here it is for the 4times5 table on4 doses with an ordinal outcome for data on p 207 of the 2nd edition of my bookrdquoAnalysis of Ordinal Categorical Datardquo

-------------------------------------------------------------------------

proc logistic data=trauma non-proportional odds cumulative logit model

model outcome = dose unequalslopes aggregate scale=none

Standard Wald

Parameter outcome DF Estimate Error Chi-Square Pr gt ChiSq

Intercept 1 1 -08646 01969 192793 lt0001

Intercept 2 1 -00937 01803 02705 06030

Intercept 3 1 07063 01766 159919 lt0001

Intercept 4 1 19087 02388 638771 lt0001

dose 1 1 -01129 00741 23235 01274

dose 2 1 -02689 00690 152000 lt0001

dose 3 1 -01823 00643 80366 00046

dose 4 1 -01193 00850 19704 01604

-------------------------------------------------------------------------

Both PROC GENMOD and PROC LOGISTIC can use other links in cumulativelink models PROC GENMOD uses LINK = CPROBIT for the cumulative probitmodel and LINK = CCLL for the cumulative complementary log-log model PROCLOGISTIC fits a cumulative probit model using LINK = PROBIT

PROC SURVEYLOGISTIC described above for incorporating the sample designinto the analysis can also fit multicategory regression models to survey data with linkssuch as the baseline-category logit and cumulative logit

You can fit adjacent-categories logit models in CATMOD by fitting equivalentbaseline-category logit models Table 16 uses it for Table 85 from the textbook onhappiness number of traumatic events and race Each line of code in the modelstatement specifies the predictor values (for the two intercepts trauma and race) forthe two logits The trauma and race predictor values are multiplied by 2 for the firstlogit and 1 for the second logit to make effects comparable in the two models PROCCATMOD has options (CLOGITS and ALOGITS) for fitting cumulative logit andadjacent-categories logit models to ordinal responses however those options provideweighted least squares (WLS) rather than ML fits A constant must be added toempty cells for WLS to run CATMOD treats zero counts as structural zeros so theymust be replaced by small constants when they are actually sampling zeros

With the CMH option PROC FREQ provides the generalized CMH tests of con-ditional independence The statistic for the ldquogeneral associationrdquo alternative treatsX and Y as nominal [statistic (818) in the text] the statistic for the ldquorow meanscores differrdquo alternative treats X as nominal and Y as ordinal and the statistic for

14

Table 15 SAS Code for Cumulative Logit and Probit Models withHappiness Data in Table 85

------------------------------------------------------------------------data gss

input race $ trauma happy race is 0=white 1=blackdatalines

0 0 10 0 10 0 10 0 10 0 10 0 10 0 10 0 2

1 3 3

ods graphics onods htmlproc genmod class racemodel happy = trauma race dist=multinomial link=clogit lrci type3effectplot slicefitrunproc logistic data=gss plots=effect(at(race=all)) class racemodel happy = trauma race output out=predict p=hi_hat lower=LCL upper=UCLproc print data=predict

runproc logistic data=gss plots=all class racemodel happy = trauma racerunods html closeods graphics offrun------------------------------------------------------------------------

15

Table 16 SAS Code for Adjacent-Categories Logit Model Fitted toTable 85 on Happiness Traumatic Events and Race

-------------------------------------------------------------------data gssinput race trauma happy countcount2 = count + 000000001datalines0 0 1 70 0 2 150 1 1 80 1 2 120 1 3 10 2 1 50 2 2 160 2 3 10 3 1 10 3 2 90 3 3 10 4 1 10 4 2 40 4 3 00 5 1 00 5 2 00 5 3 21 0 1 01 0 2 11 0 3 11 1 1 01 1 2 31 1 3 11 2 1 01 2 2 31 2 3 11 3 1 01 3 2 21 3 3 11 4 1 01 4 2 01 4 3 01 5 1 01 5 2 01 5 3 0

proc catmod order=data weight count2population race traumamodel happy = (1 0 0 0 0 1 0 0

1 0 2 0 0 1 1 01 0 4 0 0 1 2 01 0 6 0 0 1 3 01 0 8 0 0 1 4 01 0 10 0 0 1 5 01 0 0 2 0 1 0 11 0 2 2 0 1 1 11 0 4 2 0 1 2 11 0 6 2 0 1 3 11 0 8 2 0 1 4 11 0 10 2 0 1 5 1) ML NOGLS

run-------------------------------------------------------------------

16

Table 17 SAS Code for Fitting Loglinear Models to High School DrugSurvey Data in Table 93

-------------------------------------------------------------------------data drugsinput a c m count datalines1 1 1 911 1 1 2 538 1 2 1 44 1 2 2 4562 1 1 3 2 1 2 43 2 2 1 2 2 2 2 279proc genmod class a c mmodel count = a c m am ac cm dist=poi link=log lrci type3 obstats

run--------------------------------------------------------------------------

the ldquononzero correlationrdquo alternative treats X and Y as ordinal [statistic (819)] SeeStokes et al (2012 Sec 62) for a detailed discussion of various generalized CMHanalyses

PROC MDC fits multinomial discrete choice models with logit and probit links(including multinomial probit models) See

supportsascomdocumentationcdlenetsug60372HTMLdefaultviewerhtm

etsug_mdc_sect021htm

For such models one can also use PROC PHREG which is designed for the Coxproportional hazards model for survival analysis because the partial likelihood forthat analysis has the same form as the likelihood for the multinomial model (Allison1999 Chap 7 Chen and Kuo 2001)

Chapters 9ndash10 Loglinear Models

For details on the use of SAS (mainly with PROC GENMOD) for loglinear modelingof contingency tables and discrete response variables see Advanced Log-Linear ModelsUsing SAS by D Zelterman (published by SAS 2002)

Table 17 uses PROC GENMOD to fit loglinear model (AC AM CM ) to Table 93from the survey of high school students about using alcohol cigarettes and marijuana

Table 18 uses PROC GENMOD for table raking of Table 915 from the textbookNote the artificial pseudo counts used for the response to ensure the smoothed mar-gins

Table 19 uses PROC GENMOD to fit the linear-by-linear association model (105)and the row effects model (107) to Table 103 in the textbook (with column scores 12 4 5) The defined variable ldquoassocrdquo represents the cross-product of row and columnscores which has β parameter as coefficient in model (105)

Correspondence analysis is available in SAS with PROC CORRESP

17

Table 14 SAS Code for Baseline-Category Logit Models with AlligatorData in Table 81

----------------------------------------------------------------------data gatorinput lake gender size food countdatalines1 1 1 1 71 1 1 2 11 1 1 3 01 1 1 4 01 1 1 5 54 2 2 1 84 2 2 2 14 2 2 3 04 2 2 4 04 2 2 5 1proc logistic freq countclass lake size param=refmodel food(ref=rsquo1rsquo) = lake size link=glogit aggregate scale=noneproc catmod weight countpopulation lake size gendermodel food = lake size pred=freq pred=probrun----------------------------------------------------------------------

Chapter 8 Multinomial Response Models

PROC LOGISTIC fits baseline-category logit models using the LINK = GLOGIT op-tion The final response category is the default baseline for the logits Exact inferenceis also available using the conditional distribution to eliminate nuisance parametersPROC CATMOD also fits baseline-category logit models as Table 14 shows for thetext example on alligator food choice (Table 81) CATMOD codes estimates for afactor so that they sum to zero The PRED = PROB and PRED = FREQ optionsprovide predicted probabilities and fitted values and their standard errors The POP-ULATION statement provides the variables that define the predictor settings Forinstance with ldquogenderrdquo in that statement the model with lake and size effects isfitted to the full table also classified by gender

PROC GENMOD can fit the proportional odds version of cumulative logit modelsusing the DIST = MULTINOMIAL and LINK = CLOGIT options Table 15 fits itto the data shown in Table 85 on happiness number of traumatic events and raceWhen the number of response categories exceeds 2 by default PROC LOGISTIC fitsthis model It also gives a score test of the proportional odds assumption of identical

13

effect parameters for each cutpoint Both procedures use the αj + βx form of themodel Both procedures now have graphic capabilities of displaying the cumulativeprobabilities as a function of a predictor (with PLOTS = EFFECT) at fixed valuesof other predictors Cox (1995) used PROC NLIN for the more general model havinga scale parameter

With the UNEQUALSLOPES option in PROC LOGISTIC one can fit the modelwithout proportional odds structure For example here it is for the 4times5 table on4 doses with an ordinal outcome for data on p 207 of the 2nd edition of my bookrdquoAnalysis of Ordinal Categorical Datardquo

-------------------------------------------------------------------------

proc logistic data=trauma non-proportional odds cumulative logit model

model outcome = dose unequalslopes aggregate scale=none

Standard Wald

Parameter outcome DF Estimate Error Chi-Square Pr gt ChiSq

Intercept 1 1 -08646 01969 192793 lt0001

Intercept 2 1 -00937 01803 02705 06030

Intercept 3 1 07063 01766 159919 lt0001

Intercept 4 1 19087 02388 638771 lt0001

dose 1 1 -01129 00741 23235 01274

dose 2 1 -02689 00690 152000 lt0001

dose 3 1 -01823 00643 80366 00046

dose 4 1 -01193 00850 19704 01604

-------------------------------------------------------------------------

Both PROC GENMOD and PROC LOGISTIC can use other links in cumulativelink models PROC GENMOD uses LINK = CPROBIT for the cumulative probitmodel and LINK = CCLL for the cumulative complementary log-log model PROCLOGISTIC fits a cumulative probit model using LINK = PROBIT

PROC SURVEYLOGISTIC described above for incorporating the sample designinto the analysis can also fit multicategory regression models to survey data with linkssuch as the baseline-category logit and cumulative logit

You can fit adjacent-categories logit models in CATMOD by fitting equivalentbaseline-category logit models Table 16 uses it for Table 85 from the textbook onhappiness number of traumatic events and race Each line of code in the modelstatement specifies the predictor values (for the two intercepts trauma and race) forthe two logits The trauma and race predictor values are multiplied by 2 for the firstlogit and 1 for the second logit to make effects comparable in the two models PROCCATMOD has options (CLOGITS and ALOGITS) for fitting cumulative logit andadjacent-categories logit models to ordinal responses however those options provideweighted least squares (WLS) rather than ML fits A constant must be added toempty cells for WLS to run CATMOD treats zero counts as structural zeros so theymust be replaced by small constants when they are actually sampling zeros

With the CMH option PROC FREQ provides the generalized CMH tests of con-ditional independence The statistic for the ldquogeneral associationrdquo alternative treatsX and Y as nominal [statistic (818) in the text] the statistic for the ldquorow meanscores differrdquo alternative treats X as nominal and Y as ordinal and the statistic for

14

Table 15 SAS Code for Cumulative Logit and Probit Models withHappiness Data in Table 85

------------------------------------------------------------------------data gss

input race $ trauma happy race is 0=white 1=blackdatalines

0 0 10 0 10 0 10 0 10 0 10 0 10 0 10 0 2

1 3 3

ods graphics onods htmlproc genmod class racemodel happy = trauma race dist=multinomial link=clogit lrci type3effectplot slicefitrunproc logistic data=gss plots=effect(at(race=all)) class racemodel happy = trauma race output out=predict p=hi_hat lower=LCL upper=UCLproc print data=predict

runproc logistic data=gss plots=all class racemodel happy = trauma racerunods html closeods graphics offrun------------------------------------------------------------------------

15

Table 16 SAS Code for Adjacent-Categories Logit Model Fitted toTable 85 on Happiness Traumatic Events and Race

-------------------------------------------------------------------data gssinput race trauma happy countcount2 = count + 000000001datalines0 0 1 70 0 2 150 1 1 80 1 2 120 1 3 10 2 1 50 2 2 160 2 3 10 3 1 10 3 2 90 3 3 10 4 1 10 4 2 40 4 3 00 5 1 00 5 2 00 5 3 21 0 1 01 0 2 11 0 3 11 1 1 01 1 2 31 1 3 11 2 1 01 2 2 31 2 3 11 3 1 01 3 2 21 3 3 11 4 1 01 4 2 01 4 3 01 5 1 01 5 2 01 5 3 0

proc catmod order=data weight count2population race traumamodel happy = (1 0 0 0 0 1 0 0

1 0 2 0 0 1 1 01 0 4 0 0 1 2 01 0 6 0 0 1 3 01 0 8 0 0 1 4 01 0 10 0 0 1 5 01 0 0 2 0 1 0 11 0 2 2 0 1 1 11 0 4 2 0 1 2 11 0 6 2 0 1 3 11 0 8 2 0 1 4 11 0 10 2 0 1 5 1) ML NOGLS

run-------------------------------------------------------------------

16

Table 17 SAS Code for Fitting Loglinear Models to High School DrugSurvey Data in Table 93

-------------------------------------------------------------------------data drugsinput a c m count datalines1 1 1 911 1 1 2 538 1 2 1 44 1 2 2 4562 1 1 3 2 1 2 43 2 2 1 2 2 2 2 279proc genmod class a c mmodel count = a c m am ac cm dist=poi link=log lrci type3 obstats

run--------------------------------------------------------------------------

the ldquononzero correlationrdquo alternative treats X and Y as ordinal [statistic (819)] SeeStokes et al (2012 Sec 62) for a detailed discussion of various generalized CMHanalyses

PROC MDC fits multinomial discrete choice models with logit and probit links(including multinomial probit models) See

supportsascomdocumentationcdlenetsug60372HTMLdefaultviewerhtm

etsug_mdc_sect021htm

For such models one can also use PROC PHREG which is designed for the Coxproportional hazards model for survival analysis because the partial likelihood forthat analysis has the same form as the likelihood for the multinomial model (Allison1999 Chap 7 Chen and Kuo 2001)

Chapters 9ndash10 Loglinear Models

For details on the use of SAS (mainly with PROC GENMOD) for loglinear modelingof contingency tables and discrete response variables see Advanced Log-Linear ModelsUsing SAS by D Zelterman (published by SAS 2002)

Table 17 uses PROC GENMOD to fit loglinear model (AC AM CM ) to Table 93from the survey of high school students about using alcohol cigarettes and marijuana

Table 18 uses PROC GENMOD for table raking of Table 915 from the textbookNote the artificial pseudo counts used for the response to ensure the smoothed mar-gins

Table 19 uses PROC GENMOD to fit the linear-by-linear association model (105)and the row effects model (107) to Table 103 in the textbook (with column scores 12 4 5) The defined variable ldquoassocrdquo represents the cross-product of row and columnscores which has β parameter as coefficient in model (105)

Correspondence analysis is available in SAS with PROC CORRESP

17

effect parameters for each cutpoint Both procedures use the αj + βx form of themodel Both procedures now have graphic capabilities of displaying the cumulativeprobabilities as a function of a predictor (with PLOTS = EFFECT) at fixed valuesof other predictors Cox (1995) used PROC NLIN for the more general model havinga scale parameter

With the UNEQUALSLOPES option in PROC LOGISTIC one can fit the modelwithout proportional odds structure For example here it is for the 4times5 table on4 doses with an ordinal outcome for data on p 207 of the 2nd edition of my bookrdquoAnalysis of Ordinal Categorical Datardquo

-------------------------------------------------------------------------

proc logistic data=trauma non-proportional odds cumulative logit model

model outcome = dose unequalslopes aggregate scale=none

Standard Wald

Parameter outcome DF Estimate Error Chi-Square Pr gt ChiSq

Intercept 1 1 -08646 01969 192793 lt0001

Intercept 2 1 -00937 01803 02705 06030

Intercept 3 1 07063 01766 159919 lt0001

Intercept 4 1 19087 02388 638771 lt0001

dose 1 1 -01129 00741 23235 01274

dose 2 1 -02689 00690 152000 lt0001

dose 3 1 -01823 00643 80366 00046

dose 4 1 -01193 00850 19704 01604

-------------------------------------------------------------------------

Both PROC GENMOD and PROC LOGISTIC can use other links in cumulativelink models PROC GENMOD uses LINK = CPROBIT for the cumulative probitmodel and LINK = CCLL for the cumulative complementary log-log model PROCLOGISTIC fits a cumulative probit model using LINK = PROBIT

PROC SURVEYLOGISTIC described above for incorporating the sample designinto the analysis can also fit multicategory regression models to survey data with linkssuch as the baseline-category logit and cumulative logit

You can fit adjacent-categories logit models in CATMOD by fitting equivalentbaseline-category logit models Table 16 uses it for Table 85 from the textbook onhappiness number of traumatic events and race Each line of code in the modelstatement specifies the predictor values (for the two intercepts trauma and race) forthe two logits The trauma and race predictor values are multiplied by 2 for the firstlogit and 1 for the second logit to make effects comparable in the two models PROCCATMOD has options (CLOGITS and ALOGITS) for fitting cumulative logit andadjacent-categories logit models to ordinal responses however those options provideweighted least squares (WLS) rather than ML fits A constant must be added toempty cells for WLS to run CATMOD treats zero counts as structural zeros so theymust be replaced by small constants when they are actually sampling zeros

With the CMH option PROC FREQ provides the generalized CMH tests of con-ditional independence The statistic for the ldquogeneral associationrdquo alternative treatsX and Y as nominal [statistic (818) in the text] the statistic for the ldquorow meanscores differrdquo alternative treats X as nominal and Y as ordinal and the statistic for

14

Table 15 SAS Code for Cumulative Logit and Probit Models withHappiness Data in Table 85

------------------------------------------------------------------------data gss

input race $ trauma happy race is 0=white 1=blackdatalines

0 0 10 0 10 0 10 0 10 0 10 0 10 0 10 0 2

1 3 3

ods graphics onods htmlproc genmod class racemodel happy = trauma race dist=multinomial link=clogit lrci type3effectplot slicefitrunproc logistic data=gss plots=effect(at(race=all)) class racemodel happy = trauma race output out=predict p=hi_hat lower=LCL upper=UCLproc print data=predict

runproc logistic data=gss plots=all class racemodel happy = trauma racerunods html closeods graphics offrun------------------------------------------------------------------------

15

Table 16 SAS Code for Adjacent-Categories Logit Model Fitted toTable 85 on Happiness Traumatic Events and Race

-------------------------------------------------------------------data gssinput race trauma happy countcount2 = count + 000000001datalines0 0 1 70 0 2 150 1 1 80 1 2 120 1 3 10 2 1 50 2 2 160 2 3 10 3 1 10 3 2 90 3 3 10 4 1 10 4 2 40 4 3 00 5 1 00 5 2 00 5 3 21 0 1 01 0 2 11 0 3 11 1 1 01 1 2 31 1 3 11 2 1 01 2 2 31 2 3 11 3 1 01 3 2 21 3 3 11 4 1 01 4 2 01 4 3 01 5 1 01 5 2 01 5 3 0

proc catmod order=data weight count2population race traumamodel happy = (1 0 0 0 0 1 0 0

1 0 2 0 0 1 1 01 0 4 0 0 1 2 01 0 6 0 0 1 3 01 0 8 0 0 1 4 01 0 10 0 0 1 5 01 0 0 2 0 1 0 11 0 2 2 0 1 1 11 0 4 2 0 1 2 11 0 6 2 0 1 3 11 0 8 2 0 1 4 11 0 10 2 0 1 5 1) ML NOGLS

run-------------------------------------------------------------------

16

Table 17 SAS Code for Fitting Loglinear Models to High School DrugSurvey Data in Table 93

-------------------------------------------------------------------------data drugsinput a c m count datalines1 1 1 911 1 1 2 538 1 2 1 44 1 2 2 4562 1 1 3 2 1 2 43 2 2 1 2 2 2 2 279proc genmod class a c mmodel count = a c m am ac cm dist=poi link=log lrci type3 obstats

run--------------------------------------------------------------------------

the ldquononzero correlationrdquo alternative treats X and Y as ordinal [statistic (819)] SeeStokes et al (2012 Sec 62) for a detailed discussion of various generalized CMHanalyses

PROC MDC fits multinomial discrete choice models with logit and probit links(including multinomial probit models) See

supportsascomdocumentationcdlenetsug60372HTMLdefaultviewerhtm

etsug_mdc_sect021htm

For such models one can also use PROC PHREG which is designed for the Coxproportional hazards model for survival analysis because the partial likelihood forthat analysis has the same form as the likelihood for the multinomial model (Allison1999 Chap 7 Chen and Kuo 2001)

Chapters 9ndash10 Loglinear Models

For details on the use of SAS (mainly with PROC GENMOD) for loglinear modelingof contingency tables and discrete response variables see Advanced Log-Linear ModelsUsing SAS by D Zelterman (published by SAS 2002)

Table 17 uses PROC GENMOD to fit loglinear model (AC AM CM ) to Table 93from the survey of high school students about using alcohol cigarettes and marijuana

Table 18 uses PROC GENMOD for table raking of Table 915 from the textbookNote the artificial pseudo counts used for the response to ensure the smoothed mar-gins

Table 19 uses PROC GENMOD to fit the linear-by-linear association model (105)and the row effects model (107) to Table 103 in the textbook (with column scores 12 4 5) The defined variable ldquoassocrdquo represents the cross-product of row and columnscores which has β parameter as coefficient in model (105)

Correspondence analysis is available in SAS with PROC CORRESP

17

Table 15 SAS Code for Cumulative Logit and Probit Models withHappiness Data in Table 85

------------------------------------------------------------------------data gss

input race $ trauma happy race is 0=white 1=blackdatalines

0 0 10 0 10 0 10 0 10 0 10 0 10 0 10 0 2

1 3 3

ods graphics onods htmlproc genmod class racemodel happy = trauma race dist=multinomial link=clogit lrci type3effectplot slicefitrunproc logistic data=gss plots=effect(at(race=all)) class racemodel happy = trauma race output out=predict p=hi_hat lower=LCL upper=UCLproc print data=predict

runproc logistic data=gss plots=all class racemodel happy = trauma racerunods html closeods graphics offrun------------------------------------------------------------------------

15

Table 16 SAS Code for Adjacent-Categories Logit Model Fitted toTable 85 on Happiness Traumatic Events and Race

-------------------------------------------------------------------data gssinput race trauma happy countcount2 = count + 000000001datalines0 0 1 70 0 2 150 1 1 80 1 2 120 1 3 10 2 1 50 2 2 160 2 3 10 3 1 10 3 2 90 3 3 10 4 1 10 4 2 40 4 3 00 5 1 00 5 2 00 5 3 21 0 1 01 0 2 11 0 3 11 1 1 01 1 2 31 1 3 11 2 1 01 2 2 31 2 3 11 3 1 01 3 2 21 3 3 11 4 1 01 4 2 01 4 3 01 5 1 01 5 2 01 5 3 0

proc catmod order=data weight count2population race traumamodel happy = (1 0 0 0 0 1 0 0

1 0 2 0 0 1 1 01 0 4 0 0 1 2 01 0 6 0 0 1 3 01 0 8 0 0 1 4 01 0 10 0 0 1 5 01 0 0 2 0 1 0 11 0 2 2 0 1 1 11 0 4 2 0 1 2 11 0 6 2 0 1 3 11 0 8 2 0 1 4 11 0 10 2 0 1 5 1) ML NOGLS

run-------------------------------------------------------------------

16

Table 17 SAS Code for Fitting Loglinear Models to High School DrugSurvey Data in Table 93

-------------------------------------------------------------------------data drugsinput a c m count datalines1 1 1 911 1 1 2 538 1 2 1 44 1 2 2 4562 1 1 3 2 1 2 43 2 2 1 2 2 2 2 279proc genmod class a c mmodel count = a c m am ac cm dist=poi link=log lrci type3 obstats

run--------------------------------------------------------------------------

the ldquononzero correlationrdquo alternative treats X and Y as ordinal [statistic (819)] SeeStokes et al (2012 Sec 62) for a detailed discussion of various generalized CMHanalyses

PROC MDC fits multinomial discrete choice models with logit and probit links(including multinomial probit models) See

supportsascomdocumentationcdlenetsug60372HTMLdefaultviewerhtm

etsug_mdc_sect021htm

For such models one can also use PROC PHREG which is designed for the Coxproportional hazards model for survival analysis because the partial likelihood forthat analysis has the same form as the likelihood for the multinomial model (Allison1999 Chap 7 Chen and Kuo 2001)

Chapters 9ndash10 Loglinear Models

For details on the use of SAS (mainly with PROC GENMOD) for loglinear modelingof contingency tables and discrete response variables see Advanced Log-Linear ModelsUsing SAS by D Zelterman (published by SAS 2002)

Table 17 uses PROC GENMOD to fit loglinear model (AC AM CM ) to Table 93from the survey of high school students about using alcohol cigarettes and marijuana

Table 18 uses PROC GENMOD for table raking of Table 915 from the textbookNote the artificial pseudo counts used for the response to ensure the smoothed mar-gins

Table 19 uses PROC GENMOD to fit the linear-by-linear association model (105)and the row effects model (107) to Table 103 in the textbook (with column scores 12 4 5) The defined variable ldquoassocrdquo represents the cross-product of row and columnscores which has β parameter as coefficient in model (105)

Correspondence analysis is available in SAS with PROC CORRESP

17

Table 16 SAS Code for Adjacent-Categories Logit Model Fitted toTable 85 on Happiness Traumatic Events and Race

-------------------------------------------------------------------data gssinput race trauma happy countcount2 = count + 000000001datalines0 0 1 70 0 2 150 1 1 80 1 2 120 1 3 10 2 1 50 2 2 160 2 3 10 3 1 10 3 2 90 3 3 10 4 1 10 4 2 40 4 3 00 5 1 00 5 2 00 5 3 21 0 1 01 0 2 11 0 3 11 1 1 01 1 2 31 1 3 11 2 1 01 2 2 31 2 3 11 3 1 01 3 2 21 3 3 11 4 1 01 4 2 01 4 3 01 5 1 01 5 2 01 5 3 0

proc catmod order=data weight count2population race traumamodel happy = (1 0 0 0 0 1 0 0

1 0 2 0 0 1 1 01 0 4 0 0 1 2 01 0 6 0 0 1 3 01 0 8 0 0 1 4 01 0 10 0 0 1 5 01 0 0 2 0 1 0 11 0 2 2 0 1 1 11 0 4 2 0 1 2 11 0 6 2 0 1 3 11 0 8 2 0 1 4 11 0 10 2 0 1 5 1) ML NOGLS

run-------------------------------------------------------------------

16

Table 17 SAS Code for Fitting Loglinear Models to High School DrugSurvey Data in Table 93

-------------------------------------------------------------------------data drugsinput a c m count datalines1 1 1 911 1 1 2 538 1 2 1 44 1 2 2 4562 1 1 3 2 1 2 43 2 2 1 2 2 2 2 279proc genmod class a c mmodel count = a c m am ac cm dist=poi link=log lrci type3 obstats

run--------------------------------------------------------------------------

the ldquononzero correlationrdquo alternative treats X and Y as ordinal [statistic (819)] SeeStokes et al (2012 Sec 62) for a detailed discussion of various generalized CMHanalyses

PROC MDC fits multinomial discrete choice models with logit and probit links(including multinomial probit models) See

supportsascomdocumentationcdlenetsug60372HTMLdefaultviewerhtm

etsug_mdc_sect021htm

For such models one can also use PROC PHREG which is designed for the Coxproportional hazards model for survival analysis because the partial likelihood forthat analysis has the same form as the likelihood for the multinomial model (Allison1999 Chap 7 Chen and Kuo 2001)

Chapters 9ndash10 Loglinear Models

For details on the use of SAS (mainly with PROC GENMOD) for loglinear modelingof contingency tables and discrete response variables see Advanced Log-Linear ModelsUsing SAS by D Zelterman (published by SAS 2002)

Table 17 uses PROC GENMOD to fit loglinear model (AC AM CM ) to Table 93from the survey of high school students about using alcohol cigarettes and marijuana

Table 18 uses PROC GENMOD for table raking of Table 915 from the textbookNote the artificial pseudo counts used for the response to ensure the smoothed mar-gins

Table 19 uses PROC GENMOD to fit the linear-by-linear association model (105)and the row effects model (107) to Table 103 in the textbook (with column scores 12 4 5) The defined variable ldquoassocrdquo represents the cross-product of row and columnscores which has β parameter as coefficient in model (105)

Correspondence analysis is available in SAS with PROC CORRESP

17

Table 17 SAS Code for Fitting Loglinear Models to High School DrugSurvey Data in Table 93

-------------------------------------------------------------------------data drugsinput a c m count datalines1 1 1 911 1 1 2 538 1 2 1 44 1 2 2 4562 1 1 3 2 1 2 43 2 2 1 2 2 2 2 279proc genmod class a c mmodel count = a c m am ac cm dist=poi link=log lrci type3 obstats

run--------------------------------------------------------------------------

the ldquononzero correlationrdquo alternative treats X and Y as ordinal [statistic (819)] SeeStokes et al (2012 Sec 62) for a detailed discussion of various generalized CMHanalyses

PROC MDC fits multinomial discrete choice models with logit and probit links(including multinomial probit models) See

supportsascomdocumentationcdlenetsug60372HTMLdefaultviewerhtm

etsug_mdc_sect021htm

For such models one can also use PROC PHREG which is designed for the Coxproportional hazards model for survival analysis because the partial likelihood forthat analysis has the same form as the likelihood for the multinomial model (Allison1999 Chap 7 Chen and Kuo 2001)

Chapters 9ndash10 Loglinear Models

For details on the use of SAS (mainly with PROC GENMOD) for loglinear modelingof contingency tables and discrete response variables see Advanced Log-Linear ModelsUsing SAS by D Zelterman (published by SAS 2002)

Table 17 uses PROC GENMOD to fit loglinear model (AC AM CM ) to Table 93from the survey of high school students about using alcohol cigarettes and marijuana

Table 18 uses PROC GENMOD for table raking of Table 915 from the textbookNote the artificial pseudo counts used for the response to ensure the smoothed mar-gins

Table 19 uses PROC GENMOD to fit the linear-by-linear association model (105)and the row effects model (107) to Table 103 in the textbook (with column scores 12 4 5) The defined variable ldquoassocrdquo represents the cross-product of row and columnscores which has β parameter as coefficient in model (105)

Correspondence analysis is available in SAS with PROC CORRESP

17

Table 18 SAS Code for Raking Table 915

-----------------------------------------------------------------------data rakeinput partyid polviews countlog_ct = log(count) pseudo = 1003datalines1 1 3061 2 2791 3 1162 1 1852 2 3122 3 1943 1 263 2 1343 3 338proc genmod class partyid polviewsmodel pseudo = partyid polviews dist=poi link=log offset=log_ct

obstatsrun-----------------------------------------------------------------------

Table 19 SAS Code for Fitting Linear-by-Linear Association and RowEffects Models to GSS Data in Table 103

-------------------------------------------------------------------data sexinput premar birth u v count assoc = uv datalines1 1 1 1 81 1 2 1 2 68 1 3 1 4 60 1 4 1 5 38proc genmod class premar birthmodel count = premar birth assoc dist=poi link=log

proc genmod class premar birthmodel count = premar birth premarv dist=poi link=log

run-------------------------------------------------------------------

18

Table 20 SAS Code for McNemarrsquos Test and Comparing Proportionsfor Matched Samples in Table 111

--------------------------------------------------------------data matchedinput first second count datalines1 1 175 1 2 16 2 1 54 2 2 188

proc freq weight count

tables firstsecond agree exact mcnemproc catmod weight count

response marginalsmodel firstsecond = (1 0

1 1 ) run--------------------------------------------------------------

Chapter 11 Models for Matched Pairs

Table 20 analyzes Table 111 on presidential voting in two elections For square tablesthe AGREE option in PROC FREQ provides the McNemar chi-squared statistic forbinary matched pairs the X2 test of fit of the symmetry model (also called Bowkerrsquostest) and Cohenrsquos kappa and weighted kappa with SE values The MCNEM keywordin the EXACT statement provides a small-sample binomial version of McNemarrsquostest PROC CATMOD can provide the Wald confidence interval for the difference ofproportions The code forms a model for the marginal proportions in the first rowand the first column specifying a model matrix in the model statement that has anintercept parameter (the first column) that applies to both proportions and a slopeparameter that applies only to the second hence the second parameter is the differencebetween the second and first marginal proportions

PROC LOGISTIC can conduct conditional logistic regression

Table 21 shows ways of testing marginal homogeneity for the migration data inTable 115 of the textbook The PROC GENMOD code shows the Lipsitz et al(1990) approach expressing the I2 expected frequencies in terms of parameters forthe (I minus 1)2 cells in the first I minus 1 rows and I minus 1 columns the cell in the last row andlast column and I minus 1 marginal totals (which are the same for rows and columns)Here m11 denotes expected frequency micro11 m1 denotes micro1+ = micro+1 and so on Thisparameterization uses formulas such as micro14 = micro1+ minus micro11 minus micro12 minus micro13 for terms in thelast column or last row CATMOD provides the Bhapkar test (1115) of marginalhomogeneity as shown

Table 22 shows various square-table analyses of Table 116 of the textbook onpremarital and extramarital sex The ldquosymmrdquo factor indexes the pairs of cells thathave the same association terms in the symmetry and quasi-symmetry models Forinstance ldquosymmrdquo takes the same value for cells (1 2) and (2 1) Including this term

19

Table 21 SAS Code for Testing Marginal Homogeneity with MigrationData in Table 115

----------------------------------------------------------------------------data migrateinput then $ now $ count m11 m12 m13 m21 m22 m23 m31 m32 m33 m44 m1 m2 m3datalinesne ne 266 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1ne mw 15 0 1 0 0 0 0 0 0 0 0 0 0 0 2 5ne s 61 0 0 1 0 0 0 0 0 0 0 0 0 0 3 5ne w 28 -1 -1 -1 0 0 0 0 0 0 0 1 0 0 4 5mw ne 10 0 0 0 1 0 0 0 0 0 0 0 0 0 2 5mw mw 414 0 0 0 0 1 0 0 0 0 0 0 0 0 5 2mw s 50 0 0 0 0 0 1 0 0 0 0 0 0 0 6 5mw w 40 0 0 0 -1 -1 -1 0 0 0 0 0 1 0 7 5s ne 8 0 0 0 0 0 0 1 0 0 0 0 0 0 3 5s mw 22 0 0 0 0 0 0 0 1 0 0 0 0 0 6 5s s 578 0 0 0 0 0 0 0 0 1 0 0 0 0 8 3s w 22 0 0 0 0 0 0 -1 -1 -1 0 0 0 1 9 5w ne 7 -1 0 0 -1 0 0 -1 0 0 0 1 0 0 4 5w mw 6 0 -1 0 0 -1 0 0 -1 0 0 0 1 0 7 5w s 27 0 0 -1 0 0 -1 0 0 -1 0 0 0 1 9 5w w 301 0 0 0 0 0 0 0 0 0 1 0 0 0 10 4

proc genmodmodel count = m11 m12 m13 m21 m22 m23 m31 m32 m33 m44 m1 m2 m3 dist=poi link=identity

proc catmod weight count response marginalsmodel thennow = _response_ freqrepeated time 2

run---------------------------------------------------------------------------

20

as a factor in a model invokes a parameter λij satisfying λij = λji The first modelfits this factor alone providing the symmetry model The second model looks like thethird except that it identifies ldquopremarrdquo and ldquoextramarrdquo as class variables (for quasi-symmetry) whereas the third model statement does not (for ordinal quasi-symmetry)The fourth model fits quasi-independence The ldquoqirdquo factor invokes the δi parametersIt takes a separate level for each cell on the main diagonal and a common value forall other cells The fifth model fits a quasi-uniform association model that takes theuniform association version of the linear-by-linear association model and imposes aperfect fit on the main diagonal

The bottom of Table 22 fits square-table models as logit models The pairs of cellcounts (nij nji) labeled as ldquoaboverdquo and ldquobelowrdquo with reference to the main diagonalare six sets of binomial counts The variable defined as ldquoscorerdquo is the distance (uj minusui) = j minus i The first two cases are symmetry and ordinal quasi-symmetry Neithermodel contains an intercept (NOINT) and the ordinal model uses ldquoscorerdquo as thepredictor The third model allows an intercept and is the conditional symmetry modelmentioned in Note 112

Table 23 uses PROC GENMOD for logit fitting of the BradleyndashTerry model (1130)to the baseball data of Table 1110 forming an artificial explanatory variable foreach team For a given observation the variable for team i is 1 if it wins minus1 ifit loses and 0 if it is not one of the teams for that match Each observation liststhe number of wins (ldquowinsrdquo) for the team with variate-level equal to 1 out of thenumber of games (ldquogamesrdquo) against the team with variate-level equal to minus1 Themodel has these artificial variates one of which is redundant as explanatory variableswith no intercept term The COVB option provides the estimated covariance matrixof parameter estimators

Table 24 uses PROC GENMOD for fitting the complete symmetry and quasi-symmetry models to Table 1113 on attitudes toward legalized abortion

Table 25 shows the likelihood-ratio test of marginal homogeneity for the attitudestoward abortion data of Table 1113 where for instance m11p denotes micro11+ Themarginal homogeneity model expresses the eight cell expected frequencies in termsof micro111 micro11+ micro1+1 micro+11 micro1++ and micro222 (since micro+1+ = micro++1 = micro1++) Note forinstance that micro112 = micro11+ minus micro111 and micro122 = micro111 + micro1++ minus micro11+ minus micro1+1 CATMODprovides the generalized Bhapkar test (1137) of marginal homogeneity

Chapter 12 Clustered Categorical Responses MarginalModels

Table 26 uses PROC GENMOD to analyze Table 121 from the textbook on depres-sion using GEE Possible working correlation structures are TYPE = EXCH for ex-changeable TYPE = AR for autoregressive TYPE = INDEP for independence andTYPE = UNSTR for unstructured Output shows estimates and standard errors un-der the naive working correlation and based on the sandwich matrix incorporating theempirical dependence Alternatively the working association structure in the binarycase can use the log odds ratio (eg using LOGOR = EXCH for exchangeability) Thetype 3 option in GEE provides score-type tests about effects See Stokes et al (2012)for the use of GEE with missing data PROC GENMOD also provides deletion anddiagnostics statistics for its GEE analyses and provides graphics for these statistics

21

Table 22 SAS Code Showing Square-Table Analysis of Table 116 onPremarital and Extramarital Sex

--------------------------------------------------------------------------data sexinput premar extramar symm qi count unif = premarextramardatalines1 1 1 1 144 1 2 2 5 2 1 3 3 5 0 1 4 4 5 02 1 2 5 33 2 2 5 2 4 2 3 6 5 2 2 4 7 5 03 1 3 5 84 3 2 6 5 14 3 3 8 3 6 3 4 9 5 14 1 4 5 126 4 2 7 5 29 4 3 9 5 25 4 4 10 4 5proc genmod class symmmodel count = symm dist=poi link=log symmetry

proc genmod class extramar premar symmmodel count = symm extramar premar dist=poi link=log QS

proc genmod class symmmodel count = symm extramar premar dist=poi link=log ordinal QS

proc genmod class extramar premar qimodel count = extramar premar qi dist=poi link=log quasi indep

proc genmod class extramar premarmodel count = extramar premar qi unif dist=poi link=log

quasi unif assoc data sex2input score below above trials = below + abovedatalines1 33 2 1 14 2 1 25 1 2 84 0 2 29 0 3 126 0proc genmod data=sex2model abovetrials = dist=bin link=logit noint symmetry

proc genmod data=sex2model abovetrials = score dist=bin link=logit noint ordinal QS

proc genmod data=sex2model abovetrials = dist=bin link=logit conditional symm

run--------------------------------------------------------------------------

22

Table 23 SAS Code for Fitting BradleyndashTerry Model to Baseball Dataof Table 1110

-------------------------------------------------------------------------data baseballinput wins games boston newyork tampabay toronto baltimore datalines12 18 1 -1 0 0 06 18 1 0 -1 0 010 18 1 0 0 -1 010 18 1 0 0 0 -19 18 0 1 -1 0 011 18 0 1 0 -1 013 18 0 1 0 0 -112 18 0 0 1 -1 09 18 0 0 1 0 -112 18 0 0 0 1 -1

proc genmod

model winsgames = boston newyork tampabay toronto baltimore dist=bin link=logit noint covb obstats

run-------------------------------------------------------------------------

23

Table 24 SAS Code for Fitting Complete Symmetry and Quasi-Symmetry Models to Attitudes toward Legalized Abortion Data ofTable 1113

-------------------------------------------------------------------------data gss

input absingle abhlth abpoor countsum = absingle + abhlth + abpoor

datalines1 1 1 4661 1 0 391 0 1 31 0 0 10 1 1 710 1 0 4230 0 1 30 0 0 147

proc genmod data=gss class sum

model count = sum dist=poisson link=log symmetryrunproc genmod data=gss class sum

model count = sum absingle abhlth abpoor dist=poisson link=logobstats quasi-symmetry

run-------------------------------------------------------------------------

24

Table 25 SAS Code for Testing Marginal Homogeneity with Attitudestoward Legalized Abortion Data of Table 1113

---------------------------------------------------------------------------data abortioninput a b c count m111 m11p m1p1 mp11 m1pp m222 datalines1 1 1 466 1 0 0 0 0 0 1 1 2 3 -1 1 0 0 0 01 2 1 71 -1 0 1 0 0 0 1 2 2 3 1 -1 -1 0 1 02 1 1 39 -1 0 0 1 0 0 2 1 2 1 1 -1 0 -1 1 02 2 1 423 1 0 -1 -1 1 0 2 2 2 147 0 0 0 0 0 1

proc genmod

model count = m111 m11p m1p1 mp11 m1pp m222 dist=poi link=identityproc catmod weight count response marginals

model abc = _response_ freqrepeated item 3

run---------------------------------------------------------------------------

Table 26 SAS Code for Marginal (GEE) and Random Effects Modelingof Depression Data in Table 121

-------------------------------------------------------------------------------data depressinput case diagnose drug time outcome outcome=1 is normaldatalines1 0 0 0 1 1 0 0 1 1 1 0 0 2 1

340 1 1 0 0 340 1 1 1 0 340 1 1 2 0proc genmod descending class casemodel outcome = diagnose drug time drugtime dist=bin link=logit type3repeated subject=case type=exch corrw

proc nlmixed qpoints=200parms alpha=-03 beta1=-13 beta2=-06 beta3=48 beta4=102 sigma=066eta = alpha + beta1diagnose + beta2drug + beta3time + beta4drugtime + up = exp(eta)(1 + exp(eta))model outcome ~ binary(p)random u ~ normal(0 sigmasigma) subject = case

run-------------------------------------------------------------------------------

25

Table 27 uses PROC GENMOD to implement GEE for a cumulative logit model forthe insomnia data of Table 123 For multinomial responses independence is currentlythe only working correlation structure

Chapter 13 Clustered Categorical Responses Random Ef-fects Models

PROC NLMIXED extends GLMs to GLMMs by including random effects Table 28analyzes the matched pairs model (133) for the change in presidential voting data inTable 131

Table 29 analyzes the Presidential voting data in Table 132 of the text using aone-way random effects model

Table 26 above uses PROC NLMIXED for fitting the random effects model to Ta-ble 121 on depression with initial estimates specified Table 27 uses PROC NLMIXEDfor ordinal random effects modeling of Table 123 on insomnia defining a general multi-nomial log likelihood

Agresti et al (2000) showed PROC NLMIXED examples for clustered data Table31 shows code for multicenter trials such as Table 137 Table 32 shows code formulticenter trials with an ordinal response for data analyzed in Hartzel et al (2001a)on comparing two drugs for a three-category response at eight centers

Table 33 shows a correlated bivariate random effect analysis of Table 138 onattitudes toward the leading crowd

Table 34 shows PROC NLMIXED code for an adjacent-categories logit model anda nominal model for the movie rater data analyzed in Hartzel et al (2001b)

Chen and Kuo (2001) discussed fitting multinomial logit models including discrete-choice models with random effects

PROC NLMIXED allows only one RANDOM statement which makes it difficultto incorporate random effects at different levels PROC GLIMMIX has more flexibilityIt also fits random effects models and provides built-in distributions and associatedvariance functions as well as link functions for categorical responses See

supportsascomrndappdastatproceduresglimmixhtml

PROC GLIMMIX is easier to use than PROC NLMIXED for GLMMs Table 35 showsGEE and random effects modeling (using GLIMMIX) of the opinions about legalizedabortion in Table 133 Table 36 shows them for the ordiinal insomnia study data ofTable 123

Chapter 14 Other Mixture Models for Categorical Data

PROC LOGISTIC provides two overdispersion approaches for binary data TheSCALE = WILLIAMS option uses variance function of the beta-binomial form (1410)and SCALE = PEARSON uses the scaled binomial variance (1411) Table 37 illus-trates for Table 47 from a teratology study That table also uses PROC NLMIXEDfor adding litter random intercepts

For Table 146 on homicides Table 38 uses PROC GENMOD to fit a negativebinomial model and a quasi-likelihood model with scaled Poisson variance using the

26

Table 20 SAS Code for McNemarrsquos Test and Comparing Proportionsfor Matched Samples in Table 111

--------------------------------------------------------------data matchedinput first second count datalines1 1 175 1 2 16 2 1 54 2 2 188

proc freq weight count

tables firstsecond agree exact mcnemproc catmod weight count

response marginalsmodel firstsecond = (1 0

1 1 ) run--------------------------------------------------------------

Chapter 11 Models for Matched Pairs

Table 20 analyzes Table 111 on presidential voting in two elections For square tablesthe AGREE option in PROC FREQ provides the McNemar chi-squared statistic forbinary matched pairs the X2 test of fit of the symmetry model (also called Bowkerrsquostest) and Cohenrsquos kappa and weighted kappa with SE values The MCNEM keywordin the EXACT statement provides a small-sample binomial version of McNemarrsquostest PROC CATMOD can provide the Wald confidence interval for the difference ofproportions The code forms a model for the marginal proportions in the first rowand the first column specifying a model matrix in the model statement that has anintercept parameter (the first column) that applies to both proportions and a slopeparameter that applies only to the second hence the second parameter is the differencebetween the second and first marginal proportions

PROC LOGISTIC can conduct conditional logistic regression

Table 21 shows ways of testing marginal homogeneity for the migration data inTable 115 of the textbook The PROC GENMOD code shows the Lipsitz et al(1990) approach expressing the I2 expected frequencies in terms of parameters forthe (I minus 1)2 cells in the first I minus 1 rows and I minus 1 columns the cell in the last row andlast column and I minus 1 marginal totals (which are the same for rows and columns)Here m11 denotes expected frequency micro11 m1 denotes micro1+ = micro+1 and so on Thisparameterization uses formulas such as micro14 = micro1+ minus micro11 minus micro12 minus micro13 for terms in thelast column or last row CATMOD provides the Bhapkar test (1115) of marginalhomogeneity as shown

Table 22 shows various square-table analyses of Table 116 of the textbook onpremarital and extramarital sex The ldquosymmrdquo factor indexes the pairs of cells thathave the same association terms in the symmetry and quasi-symmetry models Forinstance ldquosymmrdquo takes the same value for cells (1 2) and (2 1) Including this term

19

Table 21 SAS Code for Testing Marginal Homogeneity with MigrationData in Table 115

----------------------------------------------------------------------------data migrateinput then $ now $ count m11 m12 m13 m21 m22 m23 m31 m32 m33 m44 m1 m2 m3datalinesne ne 266 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1ne mw 15 0 1 0 0 0 0 0 0 0 0 0 0 0 2 5ne s 61 0 0 1 0 0 0 0 0 0 0 0 0 0 3 5ne w 28 -1 -1 -1 0 0 0 0 0 0 0 1 0 0 4 5mw ne 10 0 0 0 1 0 0 0 0 0 0 0 0 0 2 5mw mw 414 0 0 0 0 1 0 0 0 0 0 0 0 0 5 2mw s 50 0 0 0 0 0 1 0 0 0 0 0 0 0 6 5mw w 40 0 0 0 -1 -1 -1 0 0 0 0 0 1 0 7 5s ne 8 0 0 0 0 0 0 1 0 0 0 0 0 0 3 5s mw 22 0 0 0 0 0 0 0 1 0 0 0 0 0 6 5s s 578 0 0 0 0 0 0 0 0 1 0 0 0 0 8 3s w 22 0 0 0 0 0 0 -1 -1 -1 0 0 0 1 9 5w ne 7 -1 0 0 -1 0 0 -1 0 0 0 1 0 0 4 5w mw 6 0 -1 0 0 -1 0 0 -1 0 0 0 1 0 7 5w s 27 0 0 -1 0 0 -1 0 0 -1 0 0 0 1 9 5w w 301 0 0 0 0 0 0 0 0 0 1 0 0 0 10 4

proc genmodmodel count = m11 m12 m13 m21 m22 m23 m31 m32 m33 m44 m1 m2 m3 dist=poi link=identity

proc catmod weight count response marginalsmodel thennow = _response_ freqrepeated time 2

run---------------------------------------------------------------------------

20

as a factor in a model invokes a parameter λij satisfying λij = λji The first modelfits this factor alone providing the symmetry model The second model looks like thethird except that it identifies ldquopremarrdquo and ldquoextramarrdquo as class variables (for quasi-symmetry) whereas the third model statement does not (for ordinal quasi-symmetry)The fourth model fits quasi-independence The ldquoqirdquo factor invokes the δi parametersIt takes a separate level for each cell on the main diagonal and a common value forall other cells The fifth model fits a quasi-uniform association model that takes theuniform association version of the linear-by-linear association model and imposes aperfect fit on the main diagonal

The bottom of Table 22 fits square-table models as logit models The pairs of cellcounts (nij nji) labeled as ldquoaboverdquo and ldquobelowrdquo with reference to the main diagonalare six sets of binomial counts The variable defined as ldquoscorerdquo is the distance (uj minusui) = j minus i The first two cases are symmetry and ordinal quasi-symmetry Neithermodel contains an intercept (NOINT) and the ordinal model uses ldquoscorerdquo as thepredictor The third model allows an intercept and is the conditional symmetry modelmentioned in Note 112

Table 23 uses PROC GENMOD for logit fitting of the BradleyndashTerry model (1130)to the baseball data of Table 1110 forming an artificial explanatory variable foreach team For a given observation the variable for team i is 1 if it wins minus1 ifit loses and 0 if it is not one of the teams for that match Each observation liststhe number of wins (ldquowinsrdquo) for the team with variate-level equal to 1 out of thenumber of games (ldquogamesrdquo) against the team with variate-level equal to minus1 Themodel has these artificial variates one of which is redundant as explanatory variableswith no intercept term The COVB option provides the estimated covariance matrixof parameter estimators

Table 24 uses PROC GENMOD for fitting the complete symmetry and quasi-symmetry models to Table 1113 on attitudes toward legalized abortion

Table 25 shows the likelihood-ratio test of marginal homogeneity for the attitudestoward abortion data of Table 1113 where for instance m11p denotes micro11+ Themarginal homogeneity model expresses the eight cell expected frequencies in termsof micro111 micro11+ micro1+1 micro+11 micro1++ and micro222 (since micro+1+ = micro++1 = micro1++) Note forinstance that micro112 = micro11+ minus micro111 and micro122 = micro111 + micro1++ minus micro11+ minus micro1+1 CATMODprovides the generalized Bhapkar test (1137) of marginal homogeneity

Chapter 12 Clustered Categorical Responses MarginalModels

Table 26 uses PROC GENMOD to analyze Table 121 from the textbook on depres-sion using GEE Possible working correlation structures are TYPE = EXCH for ex-changeable TYPE = AR for autoregressive TYPE = INDEP for independence andTYPE = UNSTR for unstructured Output shows estimates and standard errors un-der the naive working correlation and based on the sandwich matrix incorporating theempirical dependence Alternatively the working association structure in the binarycase can use the log odds ratio (eg using LOGOR = EXCH for exchangeability) Thetype 3 option in GEE provides score-type tests about effects See Stokes et al (2012)for the use of GEE with missing data PROC GENMOD also provides deletion anddiagnostics statistics for its GEE analyses and provides graphics for these statistics

21

Table 22 SAS Code Showing Square-Table Analysis of Table 116 onPremarital and Extramarital Sex

--------------------------------------------------------------------------data sexinput premar extramar symm qi count unif = premarextramardatalines1 1 1 1 144 1 2 2 5 2 1 3 3 5 0 1 4 4 5 02 1 2 5 33 2 2 5 2 4 2 3 6 5 2 2 4 7 5 03 1 3 5 84 3 2 6 5 14 3 3 8 3 6 3 4 9 5 14 1 4 5 126 4 2 7 5 29 4 3 9 5 25 4 4 10 4 5proc genmod class symmmodel count = symm dist=poi link=log symmetry

proc genmod class extramar premar symmmodel count = symm extramar premar dist=poi link=log QS

proc genmod class symmmodel count = symm extramar premar dist=poi link=log ordinal QS

proc genmod class extramar premar qimodel count = extramar premar qi dist=poi link=log quasi indep

proc genmod class extramar premarmodel count = extramar premar qi unif dist=poi link=log

quasi unif assoc data sex2input score below above trials = below + abovedatalines1 33 2 1 14 2 1 25 1 2 84 0 2 29 0 3 126 0proc genmod data=sex2model abovetrials = dist=bin link=logit noint symmetry

proc genmod data=sex2model abovetrials = score dist=bin link=logit noint ordinal QS

proc genmod data=sex2model abovetrials = dist=bin link=logit conditional symm

run--------------------------------------------------------------------------

22

Table 23 SAS Code for Fitting BradleyndashTerry Model to Baseball Dataof Table 1110

-------------------------------------------------------------------------data baseballinput wins games boston newyork tampabay toronto baltimore datalines12 18 1 -1 0 0 06 18 1 0 -1 0 010 18 1 0 0 -1 010 18 1 0 0 0 -19 18 0 1 -1 0 011 18 0 1 0 -1 013 18 0 1 0 0 -112 18 0 0 1 -1 09 18 0 0 1 0 -112 18 0 0 0 1 -1

proc genmod

model winsgames = boston newyork tampabay toronto baltimore dist=bin link=logit noint covb obstats

run-------------------------------------------------------------------------

23

Table 24 SAS Code for Fitting Complete Symmetry and Quasi-Symmetry Models to Attitudes toward Legalized Abortion Data ofTable 1113

-------------------------------------------------------------------------data gss

input absingle abhlth abpoor countsum = absingle + abhlth + abpoor

datalines1 1 1 4661 1 0 391 0 1 31 0 0 10 1 1 710 1 0 4230 0 1 30 0 0 147

proc genmod data=gss class sum

model count = sum dist=poisson link=log symmetryrunproc genmod data=gss class sum

model count = sum absingle abhlth abpoor dist=poisson link=logobstats quasi-symmetry

run-------------------------------------------------------------------------

24

Table 25 SAS Code for Testing Marginal Homogeneity with Attitudestoward Legalized Abortion Data of Table 1113

---------------------------------------------------------------------------data abortioninput a b c count m111 m11p m1p1 mp11 m1pp m222 datalines1 1 1 466 1 0 0 0 0 0 1 1 2 3 -1 1 0 0 0 01 2 1 71 -1 0 1 0 0 0 1 2 2 3 1 -1 -1 0 1 02 1 1 39 -1 0 0 1 0 0 2 1 2 1 1 -1 0 -1 1 02 2 1 423 1 0 -1 -1 1 0 2 2 2 147 0 0 0 0 0 1

proc genmod

model count = m111 m11p m1p1 mp11 m1pp m222 dist=poi link=identityproc catmod weight count response marginals

model abc = _response_ freqrepeated item 3

run---------------------------------------------------------------------------

Table 26 SAS Code for Marginal (GEE) and Random Effects Modelingof Depression Data in Table 121

-------------------------------------------------------------------------------data depressinput case diagnose drug time outcome outcome=1 is normaldatalines1 0 0 0 1 1 0 0 1 1 1 0 0 2 1

340 1 1 0 0 340 1 1 1 0 340 1 1 2 0proc genmod descending class casemodel outcome = diagnose drug time drugtime dist=bin link=logit type3repeated subject=case type=exch corrw

proc nlmixed qpoints=200parms alpha=-03 beta1=-13 beta2=-06 beta3=48 beta4=102 sigma=066eta = alpha + beta1diagnose + beta2drug + beta3time + beta4drugtime + up = exp(eta)(1 + exp(eta))model outcome ~ binary(p)random u ~ normal(0 sigmasigma) subject = case

run-------------------------------------------------------------------------------

25

Table 27 uses PROC GENMOD to implement GEE for a cumulative logit model forthe insomnia data of Table 123 For multinomial responses independence is currentlythe only working correlation structure

Chapter 13 Clustered Categorical Responses Random Ef-fects Models

PROC NLMIXED extends GLMs to GLMMs by including random effects Table 28analyzes the matched pairs model (133) for the change in presidential voting data inTable 131

Table 29 analyzes the Presidential voting data in Table 132 of the text using aone-way random effects model

Table 26 above uses PROC NLMIXED for fitting the random effects model to Ta-ble 121 on depression with initial estimates specified Table 27 uses PROC NLMIXEDfor ordinal random effects modeling of Table 123 on insomnia defining a general multi-nomial log likelihood

Agresti et al (2000) showed PROC NLMIXED examples for clustered data Table31 shows code for multicenter trials such as Table 137 Table 32 shows code formulticenter trials with an ordinal response for data analyzed in Hartzel et al (2001a)on comparing two drugs for a three-category response at eight centers

Table 33 shows a correlated bivariate random effect analysis of Table 138 onattitudes toward the leading crowd

Table 34 shows PROC NLMIXED code for an adjacent-categories logit model anda nominal model for the movie rater data analyzed in Hartzel et al (2001b)

Chen and Kuo (2001) discussed fitting multinomial logit models including discrete-choice models with random effects

PROC NLMIXED allows only one RANDOM statement which makes it difficultto incorporate random effects at different levels PROC GLIMMIX has more flexibilityIt also fits random effects models and provides built-in distributions and associatedvariance functions as well as link functions for categorical responses See

supportsascomrndappdastatproceduresglimmixhtml

PROC GLIMMIX is easier to use than PROC NLMIXED for GLMMs Table 35 showsGEE and random effects modeling (using GLIMMIX) of the opinions about legalizedabortion in Table 133 Table 36 shows them for the ordiinal insomnia study data ofTable 123

Chapter 14 Other Mixture Models for Categorical Data

PROC LOGISTIC provides two overdispersion approaches for binary data TheSCALE = WILLIAMS option uses variance function of the beta-binomial form (1410)and SCALE = PEARSON uses the scaled binomial variance (1411) Table 37 illus-trates for Table 47 from a teratology study That table also uses PROC NLMIXEDfor adding litter random intercepts

For Table 146 on homicides Table 38 uses PROC GENMOD to fit a negativebinomial model and a quasi-likelihood model with scaled Poisson variance using the

26

Table 21 SAS Code for Testing Marginal Homogeneity with MigrationData in Table 115

----------------------------------------------------------------------------data migrateinput then $ now $ count m11 m12 m13 m21 m22 m23 m31 m32 m33 m44 m1 m2 m3datalinesne ne 266 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1ne mw 15 0 1 0 0 0 0 0 0 0 0 0 0 0 2 5ne s 61 0 0 1 0 0 0 0 0 0 0 0 0 0 3 5ne w 28 -1 -1 -1 0 0 0 0 0 0 0 1 0 0 4 5mw ne 10 0 0 0 1 0 0 0 0 0 0 0 0 0 2 5mw mw 414 0 0 0 0 1 0 0 0 0 0 0 0 0 5 2mw s 50 0 0 0 0 0 1 0 0 0 0 0 0 0 6 5mw w 40 0 0 0 -1 -1 -1 0 0 0 0 0 1 0 7 5s ne 8 0 0 0 0 0 0 1 0 0 0 0 0 0 3 5s mw 22 0 0 0 0 0 0 0 1 0 0 0 0 0 6 5s s 578 0 0 0 0 0 0 0 0 1 0 0 0 0 8 3s w 22 0 0 0 0 0 0 -1 -1 -1 0 0 0 1 9 5w ne 7 -1 0 0 -1 0 0 -1 0 0 0 1 0 0 4 5w mw 6 0 -1 0 0 -1 0 0 -1 0 0 0 1 0 7 5w s 27 0 0 -1 0 0 -1 0 0 -1 0 0 0 1 9 5w w 301 0 0 0 0 0 0 0 0 0 1 0 0 0 10 4

proc genmodmodel count = m11 m12 m13 m21 m22 m23 m31 m32 m33 m44 m1 m2 m3 dist=poi link=identity

proc catmod weight count response marginalsmodel thennow = _response_ freqrepeated time 2

run---------------------------------------------------------------------------

20

as a factor in a model invokes a parameter λij satisfying λij = λji The first modelfits this factor alone providing the symmetry model The second model looks like thethird except that it identifies ldquopremarrdquo and ldquoextramarrdquo as class variables (for quasi-symmetry) whereas the third model statement does not (for ordinal quasi-symmetry)The fourth model fits quasi-independence The ldquoqirdquo factor invokes the δi parametersIt takes a separate level for each cell on the main diagonal and a common value forall other cells The fifth model fits a quasi-uniform association model that takes theuniform association version of the linear-by-linear association model and imposes aperfect fit on the main diagonal

The bottom of Table 22 fits square-table models as logit models The pairs of cellcounts (nij nji) labeled as ldquoaboverdquo and ldquobelowrdquo with reference to the main diagonalare six sets of binomial counts The variable defined as ldquoscorerdquo is the distance (uj minusui) = j minus i The first two cases are symmetry and ordinal quasi-symmetry Neithermodel contains an intercept (NOINT) and the ordinal model uses ldquoscorerdquo as thepredictor The third model allows an intercept and is the conditional symmetry modelmentioned in Note 112

Table 23 uses PROC GENMOD for logit fitting of the BradleyndashTerry model (1130)to the baseball data of Table 1110 forming an artificial explanatory variable foreach team For a given observation the variable for team i is 1 if it wins minus1 ifit loses and 0 if it is not one of the teams for that match Each observation liststhe number of wins (ldquowinsrdquo) for the team with variate-level equal to 1 out of thenumber of games (ldquogamesrdquo) against the team with variate-level equal to minus1 Themodel has these artificial variates one of which is redundant as explanatory variableswith no intercept term The COVB option provides the estimated covariance matrixof parameter estimators

Table 24 uses PROC GENMOD for fitting the complete symmetry and quasi-symmetry models to Table 1113 on attitudes toward legalized abortion

Table 25 shows the likelihood-ratio test of marginal homogeneity for the attitudestoward abortion data of Table 1113 where for instance m11p denotes micro11+ Themarginal homogeneity model expresses the eight cell expected frequencies in termsof micro111 micro11+ micro1+1 micro+11 micro1++ and micro222 (since micro+1+ = micro++1 = micro1++) Note forinstance that micro112 = micro11+ minus micro111 and micro122 = micro111 + micro1++ minus micro11+ minus micro1+1 CATMODprovides the generalized Bhapkar test (1137) of marginal homogeneity

Chapter 12 Clustered Categorical Responses MarginalModels

Table 26 uses PROC GENMOD to analyze Table 121 from the textbook on depres-sion using GEE Possible working correlation structures are TYPE = EXCH for ex-changeable TYPE = AR for autoregressive TYPE = INDEP for independence andTYPE = UNSTR for unstructured Output shows estimates and standard errors un-der the naive working correlation and based on the sandwich matrix incorporating theempirical dependence Alternatively the working association structure in the binarycase can use the log odds ratio (eg using LOGOR = EXCH for exchangeability) Thetype 3 option in GEE provides score-type tests about effects See Stokes et al (2012)for the use of GEE with missing data PROC GENMOD also provides deletion anddiagnostics statistics for its GEE analyses and provides graphics for these statistics

21

Table 22 SAS Code Showing Square-Table Analysis of Table 116 onPremarital and Extramarital Sex

--------------------------------------------------------------------------data sexinput premar extramar symm qi count unif = premarextramardatalines1 1 1 1 144 1 2 2 5 2 1 3 3 5 0 1 4 4 5 02 1 2 5 33 2 2 5 2 4 2 3 6 5 2 2 4 7 5 03 1 3 5 84 3 2 6 5 14 3 3 8 3 6 3 4 9 5 14 1 4 5 126 4 2 7 5 29 4 3 9 5 25 4 4 10 4 5proc genmod class symmmodel count = symm dist=poi link=log symmetry

proc genmod class extramar premar symmmodel count = symm extramar premar dist=poi link=log QS

proc genmod class symmmodel count = symm extramar premar dist=poi link=log ordinal QS

proc genmod class extramar premar qimodel count = extramar premar qi dist=poi link=log quasi indep

proc genmod class extramar premarmodel count = extramar premar qi unif dist=poi link=log

quasi unif assoc data sex2input score below above trials = below + abovedatalines1 33 2 1 14 2 1 25 1 2 84 0 2 29 0 3 126 0proc genmod data=sex2model abovetrials = dist=bin link=logit noint symmetry

proc genmod data=sex2model abovetrials = score dist=bin link=logit noint ordinal QS

proc genmod data=sex2model abovetrials = dist=bin link=logit conditional symm

run--------------------------------------------------------------------------

22

Table 23 SAS Code for Fitting BradleyndashTerry Model to Baseball Dataof Table 1110

-------------------------------------------------------------------------data baseballinput wins games boston newyork tampabay toronto baltimore datalines12 18 1 -1 0 0 06 18 1 0 -1 0 010 18 1 0 0 -1 010 18 1 0 0 0 -19 18 0 1 -1 0 011 18 0 1 0 -1 013 18 0 1 0 0 -112 18 0 0 1 -1 09 18 0 0 1 0 -112 18 0 0 0 1 -1

proc genmod

model winsgames = boston newyork tampabay toronto baltimore dist=bin link=logit noint covb obstats

run-------------------------------------------------------------------------

23

Table 24 SAS Code for Fitting Complete Symmetry and Quasi-Symmetry Models to Attitudes toward Legalized Abortion Data ofTable 1113

-------------------------------------------------------------------------data gss

input absingle abhlth abpoor countsum = absingle + abhlth + abpoor

datalines1 1 1 4661 1 0 391 0 1 31 0 0 10 1 1 710 1 0 4230 0 1 30 0 0 147

proc genmod data=gss class sum

model count = sum dist=poisson link=log symmetryrunproc genmod data=gss class sum

model count = sum absingle abhlth abpoor dist=poisson link=logobstats quasi-symmetry

run-------------------------------------------------------------------------

24

Table 25 SAS Code for Testing Marginal Homogeneity with Attitudestoward Legalized Abortion Data of Table 1113

---------------------------------------------------------------------------data abortioninput a b c count m111 m11p m1p1 mp11 m1pp m222 datalines1 1 1 466 1 0 0 0 0 0 1 1 2 3 -1 1 0 0 0 01 2 1 71 -1 0 1 0 0 0 1 2 2 3 1 -1 -1 0 1 02 1 1 39 -1 0 0 1 0 0 2 1 2 1 1 -1 0 -1 1 02 2 1 423 1 0 -1 -1 1 0 2 2 2 147 0 0 0 0 0 1

proc genmod

model count = m111 m11p m1p1 mp11 m1pp m222 dist=poi link=identityproc catmod weight count response marginals

model abc = _response_ freqrepeated item 3

run---------------------------------------------------------------------------

Table 26 SAS Code for Marginal (GEE) and Random Effects Modelingof Depression Data in Table 121

-------------------------------------------------------------------------------data depressinput case diagnose drug time outcome outcome=1 is normaldatalines1 0 0 0 1 1 0 0 1 1 1 0 0 2 1

340 1 1 0 0 340 1 1 1 0 340 1 1 2 0proc genmod descending class casemodel outcome = diagnose drug time drugtime dist=bin link=logit type3repeated subject=case type=exch corrw

proc nlmixed qpoints=200parms alpha=-03 beta1=-13 beta2=-06 beta3=48 beta4=102 sigma=066eta = alpha + beta1diagnose + beta2drug + beta3time + beta4drugtime + up = exp(eta)(1 + exp(eta))model outcome ~ binary(p)random u ~ normal(0 sigmasigma) subject = case

run-------------------------------------------------------------------------------

25

Table 27 uses PROC GENMOD to implement GEE for a cumulative logit model forthe insomnia data of Table 123 For multinomial responses independence is currentlythe only working correlation structure

Chapter 13 Clustered Categorical Responses Random Ef-fects Models

PROC NLMIXED extends GLMs to GLMMs by including random effects Table 28analyzes the matched pairs model (133) for the change in presidential voting data inTable 131

Table 29 analyzes the Presidential voting data in Table 132 of the text using aone-way random effects model

Table 26 above uses PROC NLMIXED for fitting the random effects model to Ta-ble 121 on depression with initial estimates specified Table 27 uses PROC NLMIXEDfor ordinal random effects modeling of Table 123 on insomnia defining a general multi-nomial log likelihood

Agresti et al (2000) showed PROC NLMIXED examples for clustered data Table31 shows code for multicenter trials such as Table 137 Table 32 shows code formulticenter trials with an ordinal response for data analyzed in Hartzel et al (2001a)on comparing two drugs for a three-category response at eight centers

Table 33 shows a correlated bivariate random effect analysis of Table 138 onattitudes toward the leading crowd

Table 34 shows PROC NLMIXED code for an adjacent-categories logit model anda nominal model for the movie rater data analyzed in Hartzel et al (2001b)

Chen and Kuo (2001) discussed fitting multinomial logit models including discrete-choice models with random effects

PROC NLMIXED allows only one RANDOM statement which makes it difficultto incorporate random effects at different levels PROC GLIMMIX has more flexibilityIt also fits random effects models and provides built-in distributions and associatedvariance functions as well as link functions for categorical responses See

supportsascomrndappdastatproceduresglimmixhtml

PROC GLIMMIX is easier to use than PROC NLMIXED for GLMMs Table 35 showsGEE and random effects modeling (using GLIMMIX) of the opinions about legalizedabortion in Table 133 Table 36 shows them for the ordiinal insomnia study data ofTable 123

Chapter 14 Other Mixture Models for Categorical Data

PROC LOGISTIC provides two overdispersion approaches for binary data TheSCALE = WILLIAMS option uses variance function of the beta-binomial form (1410)and SCALE = PEARSON uses the scaled binomial variance (1411) Table 37 illus-trates for Table 47 from a teratology study That table also uses PROC NLMIXEDfor adding litter random intercepts

For Table 146 on homicides Table 38 uses PROC GENMOD to fit a negativebinomial model and a quasi-likelihood model with scaled Poisson variance using the

26

as a factor in a model invokes a parameter λij satisfying λij = λji The first modelfits this factor alone providing the symmetry model The second model looks like thethird except that it identifies ldquopremarrdquo and ldquoextramarrdquo as class variables (for quasi-symmetry) whereas the third model statement does not (for ordinal quasi-symmetry)The fourth model fits quasi-independence The ldquoqirdquo factor invokes the δi parametersIt takes a separate level for each cell on the main diagonal and a common value forall other cells The fifth model fits a quasi-uniform association model that takes theuniform association version of the linear-by-linear association model and imposes aperfect fit on the main diagonal

The bottom of Table 22 fits square-table models as logit models The pairs of cellcounts (nij nji) labeled as ldquoaboverdquo and ldquobelowrdquo with reference to the main diagonalare six sets of binomial counts The variable defined as ldquoscorerdquo is the distance (uj minusui) = j minus i The first two cases are symmetry and ordinal quasi-symmetry Neithermodel contains an intercept (NOINT) and the ordinal model uses ldquoscorerdquo as thepredictor The third model allows an intercept and is the conditional symmetry modelmentioned in Note 112

Table 23 uses PROC GENMOD for logit fitting of the BradleyndashTerry model (1130)to the baseball data of Table 1110 forming an artificial explanatory variable foreach team For a given observation the variable for team i is 1 if it wins minus1 ifit loses and 0 if it is not one of the teams for that match Each observation liststhe number of wins (ldquowinsrdquo) for the team with variate-level equal to 1 out of thenumber of games (ldquogamesrdquo) against the team with variate-level equal to minus1 Themodel has these artificial variates one of which is redundant as explanatory variableswith no intercept term The COVB option provides the estimated covariance matrixof parameter estimators

Table 24 uses PROC GENMOD for fitting the complete symmetry and quasi-symmetry models to Table 1113 on attitudes toward legalized abortion

Table 25 shows the likelihood-ratio test of marginal homogeneity for the attitudestoward abortion data of Table 1113 where for instance m11p denotes micro11+ Themarginal homogeneity model expresses the eight cell expected frequencies in termsof micro111 micro11+ micro1+1 micro+11 micro1++ and micro222 (since micro+1+ = micro++1 = micro1++) Note forinstance that micro112 = micro11+ minus micro111 and micro122 = micro111 + micro1++ minus micro11+ minus micro1+1 CATMODprovides the generalized Bhapkar test (1137) of marginal homogeneity

Chapter 12 Clustered Categorical Responses MarginalModels

Table 26 uses PROC GENMOD to analyze Table 121 from the textbook on depres-sion using GEE Possible working correlation structures are TYPE = EXCH for ex-changeable TYPE = AR for autoregressive TYPE = INDEP for independence andTYPE = UNSTR for unstructured Output shows estimates and standard errors un-der the naive working correlation and based on the sandwich matrix incorporating theempirical dependence Alternatively the working association structure in the binarycase can use the log odds ratio (eg using LOGOR = EXCH for exchangeability) Thetype 3 option in GEE provides score-type tests about effects See Stokes et al (2012)for the use of GEE with missing data PROC GENMOD also provides deletion anddiagnostics statistics for its GEE analyses and provides graphics for these statistics

21

Table 22 SAS Code Showing Square-Table Analysis of Table 116 onPremarital and Extramarital Sex

--------------------------------------------------------------------------data sexinput premar extramar symm qi count unif = premarextramardatalines1 1 1 1 144 1 2 2 5 2 1 3 3 5 0 1 4 4 5 02 1 2 5 33 2 2 5 2 4 2 3 6 5 2 2 4 7 5 03 1 3 5 84 3 2 6 5 14 3 3 8 3 6 3 4 9 5 14 1 4 5 126 4 2 7 5 29 4 3 9 5 25 4 4 10 4 5proc genmod class symmmodel count = symm dist=poi link=log symmetry

proc genmod class extramar premar symmmodel count = symm extramar premar dist=poi link=log QS

proc genmod class symmmodel count = symm extramar premar dist=poi link=log ordinal QS

proc genmod class extramar premar qimodel count = extramar premar qi dist=poi link=log quasi indep

proc genmod class extramar premarmodel count = extramar premar qi unif dist=poi link=log

quasi unif assoc data sex2input score below above trials = below + abovedatalines1 33 2 1 14 2 1 25 1 2 84 0 2 29 0 3 126 0proc genmod data=sex2model abovetrials = dist=bin link=logit noint symmetry

proc genmod data=sex2model abovetrials = score dist=bin link=logit noint ordinal QS

proc genmod data=sex2model abovetrials = dist=bin link=logit conditional symm

run--------------------------------------------------------------------------

22

Table 23 SAS Code for Fitting BradleyndashTerry Model to Baseball Dataof Table 1110

-------------------------------------------------------------------------data baseballinput wins games boston newyork tampabay toronto baltimore datalines12 18 1 -1 0 0 06 18 1 0 -1 0 010 18 1 0 0 -1 010 18 1 0 0 0 -19 18 0 1 -1 0 011 18 0 1 0 -1 013 18 0 1 0 0 -112 18 0 0 1 -1 09 18 0 0 1 0 -112 18 0 0 0 1 -1

proc genmod

model winsgames = boston newyork tampabay toronto baltimore dist=bin link=logit noint covb obstats

run-------------------------------------------------------------------------

23

Table 24 SAS Code for Fitting Complete Symmetry and Quasi-Symmetry Models to Attitudes toward Legalized Abortion Data ofTable 1113

-------------------------------------------------------------------------data gss

input absingle abhlth abpoor countsum = absingle + abhlth + abpoor

datalines1 1 1 4661 1 0 391 0 1 31 0 0 10 1 1 710 1 0 4230 0 1 30 0 0 147

proc genmod data=gss class sum

model count = sum dist=poisson link=log symmetryrunproc genmod data=gss class sum

model count = sum absingle abhlth abpoor dist=poisson link=logobstats quasi-symmetry

run-------------------------------------------------------------------------

24

Table 25 SAS Code for Testing Marginal Homogeneity with Attitudestoward Legalized Abortion Data of Table 1113

---------------------------------------------------------------------------data abortioninput a b c count m111 m11p m1p1 mp11 m1pp m222 datalines1 1 1 466 1 0 0 0 0 0 1 1 2 3 -1 1 0 0 0 01 2 1 71 -1 0 1 0 0 0 1 2 2 3 1 -1 -1 0 1 02 1 1 39 -1 0 0 1 0 0 2 1 2 1 1 -1 0 -1 1 02 2 1 423 1 0 -1 -1 1 0 2 2 2 147 0 0 0 0 0 1

proc genmod

model count = m111 m11p m1p1 mp11 m1pp m222 dist=poi link=identityproc catmod weight count response marginals

model abc = _response_ freqrepeated item 3

run---------------------------------------------------------------------------

Table 26 SAS Code for Marginal (GEE) and Random Effects Modelingof Depression Data in Table 121

-------------------------------------------------------------------------------data depressinput case diagnose drug time outcome outcome=1 is normaldatalines1 0 0 0 1 1 0 0 1 1 1 0 0 2 1

340 1 1 0 0 340 1 1 1 0 340 1 1 2 0proc genmod descending class casemodel outcome = diagnose drug time drugtime dist=bin link=logit type3repeated subject=case type=exch corrw

proc nlmixed qpoints=200parms alpha=-03 beta1=-13 beta2=-06 beta3=48 beta4=102 sigma=066eta = alpha + beta1diagnose + beta2drug + beta3time + beta4drugtime + up = exp(eta)(1 + exp(eta))model outcome ~ binary(p)random u ~ normal(0 sigmasigma) subject = case

run-------------------------------------------------------------------------------

25

Table 27 uses PROC GENMOD to implement GEE for a cumulative logit model forthe insomnia data of Table 123 For multinomial responses independence is currentlythe only working correlation structure

Chapter 13 Clustered Categorical Responses Random Ef-fects Models

PROC NLMIXED extends GLMs to GLMMs by including random effects Table 28analyzes the matched pairs model (133) for the change in presidential voting data inTable 131

Table 29 analyzes the Presidential voting data in Table 132 of the text using aone-way random effects model

Table 26 above uses PROC NLMIXED for fitting the random effects model to Ta-ble 121 on depression with initial estimates specified Table 27 uses PROC NLMIXEDfor ordinal random effects modeling of Table 123 on insomnia defining a general multi-nomial log likelihood

Agresti et al (2000) showed PROC NLMIXED examples for clustered data Table31 shows code for multicenter trials such as Table 137 Table 32 shows code formulticenter trials with an ordinal response for data analyzed in Hartzel et al (2001a)on comparing two drugs for a three-category response at eight centers

Table 33 shows a correlated bivariate random effect analysis of Table 138 onattitudes toward the leading crowd

Table 34 shows PROC NLMIXED code for an adjacent-categories logit model anda nominal model for the movie rater data analyzed in Hartzel et al (2001b)

Chen and Kuo (2001) discussed fitting multinomial logit models including discrete-choice models with random effects

PROC NLMIXED allows only one RANDOM statement which makes it difficultto incorporate random effects at different levels PROC GLIMMIX has more flexibilityIt also fits random effects models and provides built-in distributions and associatedvariance functions as well as link functions for categorical responses See

supportsascomrndappdastatproceduresglimmixhtml

PROC GLIMMIX is easier to use than PROC NLMIXED for GLMMs Table 35 showsGEE and random effects modeling (using GLIMMIX) of the opinions about legalizedabortion in Table 133 Table 36 shows them for the ordiinal insomnia study data ofTable 123

Chapter 14 Other Mixture Models for Categorical Data

PROC LOGISTIC provides two overdispersion approaches for binary data TheSCALE = WILLIAMS option uses variance function of the beta-binomial form (1410)and SCALE = PEARSON uses the scaled binomial variance (1411) Table 37 illus-trates for Table 47 from a teratology study That table also uses PROC NLMIXEDfor adding litter random intercepts

For Table 146 on homicides Table 38 uses PROC GENMOD to fit a negativebinomial model and a quasi-likelihood model with scaled Poisson variance using the

26

Table 22 SAS Code Showing Square-Table Analysis of Table 116 onPremarital and Extramarital Sex

--------------------------------------------------------------------------data sexinput premar extramar symm qi count unif = premarextramardatalines1 1 1 1 144 1 2 2 5 2 1 3 3 5 0 1 4 4 5 02 1 2 5 33 2 2 5 2 4 2 3 6 5 2 2 4 7 5 03 1 3 5 84 3 2 6 5 14 3 3 8 3 6 3 4 9 5 14 1 4 5 126 4 2 7 5 29 4 3 9 5 25 4 4 10 4 5proc genmod class symmmodel count = symm dist=poi link=log symmetry

proc genmod class extramar premar symmmodel count = symm extramar premar dist=poi link=log QS

proc genmod class symmmodel count = symm extramar premar dist=poi link=log ordinal QS

proc genmod class extramar premar qimodel count = extramar premar qi dist=poi link=log quasi indep

proc genmod class extramar premarmodel count = extramar premar qi unif dist=poi link=log

quasi unif assoc data sex2input score below above trials = below + abovedatalines1 33 2 1 14 2 1 25 1 2 84 0 2 29 0 3 126 0proc genmod data=sex2model abovetrials = dist=bin link=logit noint symmetry

proc genmod data=sex2model abovetrials = score dist=bin link=logit noint ordinal QS

proc genmod data=sex2model abovetrials = dist=bin link=logit conditional symm

run--------------------------------------------------------------------------

22

Table 23 SAS Code for Fitting BradleyndashTerry Model to Baseball Dataof Table 1110

-------------------------------------------------------------------------data baseballinput wins games boston newyork tampabay toronto baltimore datalines12 18 1 -1 0 0 06 18 1 0 -1 0 010 18 1 0 0 -1 010 18 1 0 0 0 -19 18 0 1 -1 0 011 18 0 1 0 -1 013 18 0 1 0 0 -112 18 0 0 1 -1 09 18 0 0 1 0 -112 18 0 0 0 1 -1

proc genmod

model winsgames = boston newyork tampabay toronto baltimore dist=bin link=logit noint covb obstats

run-------------------------------------------------------------------------

23

Table 24 SAS Code for Fitting Complete Symmetry and Quasi-Symmetry Models to Attitudes toward Legalized Abortion Data ofTable 1113

-------------------------------------------------------------------------data gss

input absingle abhlth abpoor countsum = absingle + abhlth + abpoor

datalines1 1 1 4661 1 0 391 0 1 31 0 0 10 1 1 710 1 0 4230 0 1 30 0 0 147

proc genmod data=gss class sum

model count = sum dist=poisson link=log symmetryrunproc genmod data=gss class sum

model count = sum absingle abhlth abpoor dist=poisson link=logobstats quasi-symmetry

run-------------------------------------------------------------------------

24

Table 25 SAS Code for Testing Marginal Homogeneity with Attitudestoward Legalized Abortion Data of Table 1113

---------------------------------------------------------------------------data abortioninput a b c count m111 m11p m1p1 mp11 m1pp m222 datalines1 1 1 466 1 0 0 0 0 0 1 1 2 3 -1 1 0 0 0 01 2 1 71 -1 0 1 0 0 0 1 2 2 3 1 -1 -1 0 1 02 1 1 39 -1 0 0 1 0 0 2 1 2 1 1 -1 0 -1 1 02 2 1 423 1 0 -1 -1 1 0 2 2 2 147 0 0 0 0 0 1

proc genmod

model count = m111 m11p m1p1 mp11 m1pp m222 dist=poi link=identityproc catmod weight count response marginals

model abc = _response_ freqrepeated item 3

run---------------------------------------------------------------------------

Table 26 SAS Code for Marginal (GEE) and Random Effects Modelingof Depression Data in Table 121

-------------------------------------------------------------------------------data depressinput case diagnose drug time outcome outcome=1 is normaldatalines1 0 0 0 1 1 0 0 1 1 1 0 0 2 1

340 1 1 0 0 340 1 1 1 0 340 1 1 2 0proc genmod descending class casemodel outcome = diagnose drug time drugtime dist=bin link=logit type3repeated subject=case type=exch corrw

proc nlmixed qpoints=200parms alpha=-03 beta1=-13 beta2=-06 beta3=48 beta4=102 sigma=066eta = alpha + beta1diagnose + beta2drug + beta3time + beta4drugtime + up = exp(eta)(1 + exp(eta))model outcome ~ binary(p)random u ~ normal(0 sigmasigma) subject = case

run-------------------------------------------------------------------------------

25

Table 27 uses PROC GENMOD to implement GEE for a cumulative logit model forthe insomnia data of Table 123 For multinomial responses independence is currentlythe only working correlation structure

Chapter 13 Clustered Categorical Responses Random Ef-fects Models

PROC NLMIXED extends GLMs to GLMMs by including random effects Table 28analyzes the matched pairs model (133) for the change in presidential voting data inTable 131

Table 29 analyzes the Presidential voting data in Table 132 of the text using aone-way random effects model

Table 26 above uses PROC NLMIXED for fitting the random effects model to Ta-ble 121 on depression with initial estimates specified Table 27 uses PROC NLMIXEDfor ordinal random effects modeling of Table 123 on insomnia defining a general multi-nomial log likelihood

Agresti et al (2000) showed PROC NLMIXED examples for clustered data Table31 shows code for multicenter trials such as Table 137 Table 32 shows code formulticenter trials with an ordinal response for data analyzed in Hartzel et al (2001a)on comparing two drugs for a three-category response at eight centers

Table 33 shows a correlated bivariate random effect analysis of Table 138 onattitudes toward the leading crowd

Table 34 shows PROC NLMIXED code for an adjacent-categories logit model anda nominal model for the movie rater data analyzed in Hartzel et al (2001b)

Chen and Kuo (2001) discussed fitting multinomial logit models including discrete-choice models with random effects

PROC NLMIXED allows only one RANDOM statement which makes it difficultto incorporate random effects at different levels PROC GLIMMIX has more flexibilityIt also fits random effects models and provides built-in distributions and associatedvariance functions as well as link functions for categorical responses See

supportsascomrndappdastatproceduresglimmixhtml

PROC GLIMMIX is easier to use than PROC NLMIXED for GLMMs Table 35 showsGEE and random effects modeling (using GLIMMIX) of the opinions about legalizedabortion in Table 133 Table 36 shows them for the ordiinal insomnia study data ofTable 123

Chapter 14 Other Mixture Models for Categorical Data

PROC LOGISTIC provides two overdispersion approaches for binary data TheSCALE = WILLIAMS option uses variance function of the beta-binomial form (1410)and SCALE = PEARSON uses the scaled binomial variance (1411) Table 37 illus-trates for Table 47 from a teratology study That table also uses PROC NLMIXEDfor adding litter random intercepts

For Table 146 on homicides Table 38 uses PROC GENMOD to fit a negativebinomial model and a quasi-likelihood model with scaled Poisson variance using the

26

Table 23 SAS Code for Fitting BradleyndashTerry Model to Baseball Dataof Table 1110

-------------------------------------------------------------------------data baseballinput wins games boston newyork tampabay toronto baltimore datalines12 18 1 -1 0 0 06 18 1 0 -1 0 010 18 1 0 0 -1 010 18 1 0 0 0 -19 18 0 1 -1 0 011 18 0 1 0 -1 013 18 0 1 0 0 -112 18 0 0 1 -1 09 18 0 0 1 0 -112 18 0 0 0 1 -1

proc genmod

model winsgames = boston newyork tampabay toronto baltimore dist=bin link=logit noint covb obstats

run-------------------------------------------------------------------------

23

Table 24 SAS Code for Fitting Complete Symmetry and Quasi-Symmetry Models to Attitudes toward Legalized Abortion Data ofTable 1113

-------------------------------------------------------------------------data gss

input absingle abhlth abpoor countsum = absingle + abhlth + abpoor

datalines1 1 1 4661 1 0 391 0 1 31 0 0 10 1 1 710 1 0 4230 0 1 30 0 0 147

proc genmod data=gss class sum

model count = sum dist=poisson link=log symmetryrunproc genmod data=gss class sum

model count = sum absingle abhlth abpoor dist=poisson link=logobstats quasi-symmetry

run-------------------------------------------------------------------------

24

Table 25 SAS Code for Testing Marginal Homogeneity with Attitudestoward Legalized Abortion Data of Table 1113

---------------------------------------------------------------------------data abortioninput a b c count m111 m11p m1p1 mp11 m1pp m222 datalines1 1 1 466 1 0 0 0 0 0 1 1 2 3 -1 1 0 0 0 01 2 1 71 -1 0 1 0 0 0 1 2 2 3 1 -1 -1 0 1 02 1 1 39 -1 0 0 1 0 0 2 1 2 1 1 -1 0 -1 1 02 2 1 423 1 0 -1 -1 1 0 2 2 2 147 0 0 0 0 0 1

proc genmod

model count = m111 m11p m1p1 mp11 m1pp m222 dist=poi link=identityproc catmod weight count response marginals

model abc = _response_ freqrepeated item 3

run---------------------------------------------------------------------------

Table 26 SAS Code for Marginal (GEE) and Random Effects Modelingof Depression Data in Table 121

-------------------------------------------------------------------------------data depressinput case diagnose drug time outcome outcome=1 is normaldatalines1 0 0 0 1 1 0 0 1 1 1 0 0 2 1

340 1 1 0 0 340 1 1 1 0 340 1 1 2 0proc genmod descending class casemodel outcome = diagnose drug time drugtime dist=bin link=logit type3repeated subject=case type=exch corrw

proc nlmixed qpoints=200parms alpha=-03 beta1=-13 beta2=-06 beta3=48 beta4=102 sigma=066eta = alpha + beta1diagnose + beta2drug + beta3time + beta4drugtime + up = exp(eta)(1 + exp(eta))model outcome ~ binary(p)random u ~ normal(0 sigmasigma) subject = case

run-------------------------------------------------------------------------------

25

Table 27 uses PROC GENMOD to implement GEE for a cumulative logit model forthe insomnia data of Table 123 For multinomial responses independence is currentlythe only working correlation structure

Chapter 13 Clustered Categorical Responses Random Ef-fects Models

PROC NLMIXED extends GLMs to GLMMs by including random effects Table 28analyzes the matched pairs model (133) for the change in presidential voting data inTable 131

Table 29 analyzes the Presidential voting data in Table 132 of the text using aone-way random effects model

Table 26 above uses PROC NLMIXED for fitting the random effects model to Ta-ble 121 on depression with initial estimates specified Table 27 uses PROC NLMIXEDfor ordinal random effects modeling of Table 123 on insomnia defining a general multi-nomial log likelihood

Agresti et al (2000) showed PROC NLMIXED examples for clustered data Table31 shows code for multicenter trials such as Table 137 Table 32 shows code formulticenter trials with an ordinal response for data analyzed in Hartzel et al (2001a)on comparing two drugs for a three-category response at eight centers

Table 33 shows a correlated bivariate random effect analysis of Table 138 onattitudes toward the leading crowd

Table 34 shows PROC NLMIXED code for an adjacent-categories logit model anda nominal model for the movie rater data analyzed in Hartzel et al (2001b)

Chen and Kuo (2001) discussed fitting multinomial logit models including discrete-choice models with random effects

PROC NLMIXED allows only one RANDOM statement which makes it difficultto incorporate random effects at different levels PROC GLIMMIX has more flexibilityIt also fits random effects models and provides built-in distributions and associatedvariance functions as well as link functions for categorical responses See

supportsascomrndappdastatproceduresglimmixhtml

PROC GLIMMIX is easier to use than PROC NLMIXED for GLMMs Table 35 showsGEE and random effects modeling (using GLIMMIX) of the opinions about legalizedabortion in Table 133 Table 36 shows them for the ordiinal insomnia study data ofTable 123

Chapter 14 Other Mixture Models for Categorical Data

PROC LOGISTIC provides two overdispersion approaches for binary data TheSCALE = WILLIAMS option uses variance function of the beta-binomial form (1410)and SCALE = PEARSON uses the scaled binomial variance (1411) Table 37 illus-trates for Table 47 from a teratology study That table also uses PROC NLMIXEDfor adding litter random intercepts

For Table 146 on homicides Table 38 uses PROC GENMOD to fit a negativebinomial model and a quasi-likelihood model with scaled Poisson variance using the

26

Table 24 SAS Code for Fitting Complete Symmetry and Quasi-Symmetry Models to Attitudes toward Legalized Abortion Data ofTable 1113

-------------------------------------------------------------------------data gss

input absingle abhlth abpoor countsum = absingle + abhlth + abpoor

datalines1 1 1 4661 1 0 391 0 1 31 0 0 10 1 1 710 1 0 4230 0 1 30 0 0 147

proc genmod data=gss class sum

model count = sum dist=poisson link=log symmetryrunproc genmod data=gss class sum

model count = sum absingle abhlth abpoor dist=poisson link=logobstats quasi-symmetry

run-------------------------------------------------------------------------

24

Table 25 SAS Code for Testing Marginal Homogeneity with Attitudestoward Legalized Abortion Data of Table 1113

---------------------------------------------------------------------------data abortioninput a b c count m111 m11p m1p1 mp11 m1pp m222 datalines1 1 1 466 1 0 0 0 0 0 1 1 2 3 -1 1 0 0 0 01 2 1 71 -1 0 1 0 0 0 1 2 2 3 1 -1 -1 0 1 02 1 1 39 -1 0 0 1 0 0 2 1 2 1 1 -1 0 -1 1 02 2 1 423 1 0 -1 -1 1 0 2 2 2 147 0 0 0 0 0 1

proc genmod

model count = m111 m11p m1p1 mp11 m1pp m222 dist=poi link=identityproc catmod weight count response marginals

model abc = _response_ freqrepeated item 3

run---------------------------------------------------------------------------

Table 26 SAS Code for Marginal (GEE) and Random Effects Modelingof Depression Data in Table 121

-------------------------------------------------------------------------------data depressinput case diagnose drug time outcome outcome=1 is normaldatalines1 0 0 0 1 1 0 0 1 1 1 0 0 2 1

340 1 1 0 0 340 1 1 1 0 340 1 1 2 0proc genmod descending class casemodel outcome = diagnose drug time drugtime dist=bin link=logit type3repeated subject=case type=exch corrw

proc nlmixed qpoints=200parms alpha=-03 beta1=-13 beta2=-06 beta3=48 beta4=102 sigma=066eta = alpha + beta1diagnose + beta2drug + beta3time + beta4drugtime + up = exp(eta)(1 + exp(eta))model outcome ~ binary(p)random u ~ normal(0 sigmasigma) subject = case

run-------------------------------------------------------------------------------

25

Table 27 uses PROC GENMOD to implement GEE for a cumulative logit model forthe insomnia data of Table 123 For multinomial responses independence is currentlythe only working correlation structure

Chapter 13 Clustered Categorical Responses Random Ef-fects Models

PROC NLMIXED extends GLMs to GLMMs by including random effects Table 28analyzes the matched pairs model (133) for the change in presidential voting data inTable 131

Table 29 analyzes the Presidential voting data in Table 132 of the text using aone-way random effects model

Table 26 above uses PROC NLMIXED for fitting the random effects model to Ta-ble 121 on depression with initial estimates specified Table 27 uses PROC NLMIXEDfor ordinal random effects modeling of Table 123 on insomnia defining a general multi-nomial log likelihood

Agresti et al (2000) showed PROC NLMIXED examples for clustered data Table31 shows code for multicenter trials such as Table 137 Table 32 shows code formulticenter trials with an ordinal response for data analyzed in Hartzel et al (2001a)on comparing two drugs for a three-category response at eight centers

Table 33 shows a correlated bivariate random effect analysis of Table 138 onattitudes toward the leading crowd

Table 34 shows PROC NLMIXED code for an adjacent-categories logit model anda nominal model for the movie rater data analyzed in Hartzel et al (2001b)

Chen and Kuo (2001) discussed fitting multinomial logit models including discrete-choice models with random effects

PROC NLMIXED allows only one RANDOM statement which makes it difficultto incorporate random effects at different levels PROC GLIMMIX has more flexibilityIt also fits random effects models and provides built-in distributions and associatedvariance functions as well as link functions for categorical responses See

supportsascomrndappdastatproceduresglimmixhtml

PROC GLIMMIX is easier to use than PROC NLMIXED for GLMMs Table 35 showsGEE and random effects modeling (using GLIMMIX) of the opinions about legalizedabortion in Table 133 Table 36 shows them for the ordiinal insomnia study data ofTable 123

Chapter 14 Other Mixture Models for Categorical Data

PROC LOGISTIC provides two overdispersion approaches for binary data TheSCALE = WILLIAMS option uses variance function of the beta-binomial form (1410)and SCALE = PEARSON uses the scaled binomial variance (1411) Table 37 illus-trates for Table 47 from a teratology study That table also uses PROC NLMIXEDfor adding litter random intercepts

For Table 146 on homicides Table 38 uses PROC GENMOD to fit a negativebinomial model and a quasi-likelihood model with scaled Poisson variance using the

26

Table 25 SAS Code for Testing Marginal Homogeneity with Attitudestoward Legalized Abortion Data of Table 1113

---------------------------------------------------------------------------data abortioninput a b c count m111 m11p m1p1 mp11 m1pp m222 datalines1 1 1 466 1 0 0 0 0 0 1 1 2 3 -1 1 0 0 0 01 2 1 71 -1 0 1 0 0 0 1 2 2 3 1 -1 -1 0 1 02 1 1 39 -1 0 0 1 0 0 2 1 2 1 1 -1 0 -1 1 02 2 1 423 1 0 -1 -1 1 0 2 2 2 147 0 0 0 0 0 1

proc genmod

model count = m111 m11p m1p1 mp11 m1pp m222 dist=poi link=identityproc catmod weight count response marginals

model abc = _response_ freqrepeated item 3

run---------------------------------------------------------------------------

Table 26 SAS Code for Marginal (GEE) and Random Effects Modelingof Depression Data in Table 121

-------------------------------------------------------------------------------data depressinput case diagnose drug time outcome outcome=1 is normaldatalines1 0 0 0 1 1 0 0 1 1 1 0 0 2 1

340 1 1 0 0 340 1 1 1 0 340 1 1 2 0proc genmod descending class casemodel outcome = diagnose drug time drugtime dist=bin link=logit type3repeated subject=case type=exch corrw

proc nlmixed qpoints=200parms alpha=-03 beta1=-13 beta2=-06 beta3=48 beta4=102 sigma=066eta = alpha + beta1diagnose + beta2drug + beta3time + beta4drugtime + up = exp(eta)(1 + exp(eta))model outcome ~ binary(p)random u ~ normal(0 sigmasigma) subject = case

run-------------------------------------------------------------------------------

25

Table 27 uses PROC GENMOD to implement GEE for a cumulative logit model forthe insomnia data of Table 123 For multinomial responses independence is currentlythe only working correlation structure

Chapter 13 Clustered Categorical Responses Random Ef-fects Models

PROC NLMIXED extends GLMs to GLMMs by including random effects Table 28analyzes the matched pairs model (133) for the change in presidential voting data inTable 131

Table 29 analyzes the Presidential voting data in Table 132 of the text using aone-way random effects model

Table 26 above uses PROC NLMIXED for fitting the random effects model to Ta-ble 121 on depression with initial estimates specified Table 27 uses PROC NLMIXEDfor ordinal random effects modeling of Table 123 on insomnia defining a general multi-nomial log likelihood

Agresti et al (2000) showed PROC NLMIXED examples for clustered data Table31 shows code for multicenter trials such as Table 137 Table 32 shows code formulticenter trials with an ordinal response for data analyzed in Hartzel et al (2001a)on comparing two drugs for a three-category response at eight centers

Table 33 shows a correlated bivariate random effect analysis of Table 138 onattitudes toward the leading crowd

Table 34 shows PROC NLMIXED code for an adjacent-categories logit model anda nominal model for the movie rater data analyzed in Hartzel et al (2001b)

Chen and Kuo (2001) discussed fitting multinomial logit models including discrete-choice models with random effects

PROC NLMIXED allows only one RANDOM statement which makes it difficultto incorporate random effects at different levels PROC GLIMMIX has more flexibilityIt also fits random effects models and provides built-in distributions and associatedvariance functions as well as link functions for categorical responses See

supportsascomrndappdastatproceduresglimmixhtml

PROC GLIMMIX is easier to use than PROC NLMIXED for GLMMs Table 35 showsGEE and random effects modeling (using GLIMMIX) of the opinions about legalizedabortion in Table 133 Table 36 shows them for the ordiinal insomnia study data ofTable 123

Chapter 14 Other Mixture Models for Categorical Data

PROC LOGISTIC provides two overdispersion approaches for binary data TheSCALE = WILLIAMS option uses variance function of the beta-binomial form (1410)and SCALE = PEARSON uses the scaled binomial variance (1411) Table 37 illus-trates for Table 47 from a teratology study That table also uses PROC NLMIXEDfor adding litter random intercepts

For Table 146 on homicides Table 38 uses PROC GENMOD to fit a negativebinomial model and a quasi-likelihood model with scaled Poisson variance using the

26

Table 27 uses PROC GENMOD to implement GEE for a cumulative logit model forthe insomnia data of Table 123 For multinomial responses independence is currentlythe only working correlation structure

Chapter 13 Clustered Categorical Responses Random Ef-fects Models

PROC NLMIXED extends GLMs to GLMMs by including random effects Table 28analyzes the matched pairs model (133) for the change in presidential voting data inTable 131

Table 29 analyzes the Presidential voting data in Table 132 of the text using aone-way random effects model

Table 26 above uses PROC NLMIXED for fitting the random effects model to Ta-ble 121 on depression with initial estimates specified Table 27 uses PROC NLMIXEDfor ordinal random effects modeling of Table 123 on insomnia defining a general multi-nomial log likelihood

Agresti et al (2000) showed PROC NLMIXED examples for clustered data Table31 shows code for multicenter trials such as Table 137 Table 32 shows code formulticenter trials with an ordinal response for data analyzed in Hartzel et al (2001a)on comparing two drugs for a three-category response at eight centers

Table 33 shows a correlated bivariate random effect analysis of Table 138 onattitudes toward the leading crowd

Table 34 shows PROC NLMIXED code for an adjacent-categories logit model anda nominal model for the movie rater data analyzed in Hartzel et al (2001b)

Chen and Kuo (2001) discussed fitting multinomial logit models including discrete-choice models with random effects

PROC NLMIXED allows only one RANDOM statement which makes it difficultto incorporate random effects at different levels PROC GLIMMIX has more flexibilityIt also fits random effects models and provides built-in distributions and associatedvariance functions as well as link functions for categorical responses See

supportsascomrndappdastatproceduresglimmixhtml

PROC GLIMMIX is easier to use than PROC NLMIXED for GLMMs Table 35 showsGEE and random effects modeling (using GLIMMIX) of the opinions about legalizedabortion in Table 133 Table 36 shows them for the ordiinal insomnia study data ofTable 123

Chapter 14 Other Mixture Models for Categorical Data

PROC LOGISTIC provides two overdispersion approaches for binary data TheSCALE = WILLIAMS option uses variance function of the beta-binomial form (1410)and SCALE = PEARSON uses the scaled binomial variance (1411) Table 37 illus-trates for Table 47 from a teratology study That table also uses PROC NLMIXEDfor adding litter random intercepts

For Table 146 on homicides Table 38 uses PROC GENMOD to fit a negativebinomial model and a quasi-likelihood model with scaled Poisson variance using the

26

Table 27 SAS Code for Marginal (GEE) and Random Intercept Cu-mulative Logit Analysis of Insomnia Data in Table 123

--------------------------------------------------------------------------data francominput case treat time outcome y1=0y2=0y3=0y4=0if outcome=1 then y1=1if outcome=2 then y2=1if outcome=3 then y3=1if outcome=4 then y4=1

datalines1 1 0 1 1 1 1 1

239 0 0 4 239 0 1 4

proc genmod class casemodel outcome = treat time treattime dist=multinomial link=clogitrepeated subject=case type=indep corrw

proc nlmixed qpoints=40bounds i2 gt 0 bounds i3 gt 0eta1 = i1 + treatbeta1 + timebeta2 + treattimebeta3 + ueta2 = i1 + i2 + treatbeta1 + timebeta2 + treattimebeta3 + ueta3 = i1 + i2 + i3 + treatbeta1 + timebeta2 + treattimebeta3 + up1 = exp(eta1)(1 + exp(eta1))p2 = exp(eta2)(1 + exp(eta2)) - exp(eta1)(1 + exp(eta1))p3 = exp(eta3)(1 + exp(eta3)) - exp(eta2)(1 + exp(eta2))p4 = 1 - exp(eta3)(1 + exp(eta3))ll = y1log(p1) + y2log(p2) + y3log(p3) + y4log(p4)model y1 ~ general(ll)estimate rsquointerc2rsquo i1+i2 this is alpha_2 in model and i1 is alpha_1estimate rsquointerc3rsquo i1+i2+i3 this is alpha_3 in modelrandom u ~ normal(0 sigmasigma) subject=case

run--------------------------------------------------------------------------

27

Table 28 SAS Code for Fitting Model (133) for Matched Pairs toTable 131

------------------------------------------------------------------------data kerryobamainput case occasion response count

datalines1 0 1 1751 1 1 1752 0 1 162 1 0 163 0 0 543 1 1 544 0 0 1884 1 0 188

proc nlmixed data=kerryobama qpoints=1000eta = alpha + betaoccasion + up = exp(eta)(1 + exp(eta))model response ~ binary(p)random u ~ normal(0 sigmasigma) subject = casereplicate count

run------------------------------------------------------------------------

28

Table 29 SAS Code for GLMM Analysis of Election Data in Table 132

---------------------------------------------------------------------data voteinput y ncase = _n_datalines1 516 321 4proc nlmixed

eta = alpha + u p = exp(eta) (1 + exp(eta))model y ~ binomial(np)random u ~ normal(0sigmasigma) subject=casepredict p out=new

proc print data=newrun---------------------------------------------------------------------

Pearson statistic and PROC NLMIXED to fit a Poisson GLMM PROC NLMIXEDcan also fit negative binomial models

The PROC GENMOD procedure fits zero-inflated Poisson regression models

Chapter 15 Non-Model-Based Classification and Cluster-ing

PROC DISCRIM in SAS can perform discriminant analysis For example for theungrouped horseshoe crab as analyzed above in Table 9 you can add code such asshown in Table 39 The statement ldquopriors proprdquo sets the prior probabilities equal to thesample proportions Alternatively ldquopriors equalrdquo would have equal prior proportionsin the two categories

PROC DISCRIM can also be used to perform smoothing using kernel methodsand using k-nearest neighbor methods See

supportsascomdocumentationcdlenstatug63347HTMLdefaultviewer

htmstatug_discrim_sect017htm

You can construct and assess classification trees using PROC HPSPLIT See

httpssupportsascomdocumentationonlinedocstat141hpsplitpdf

PROC DISTANCE can form distances such as simple matching and the Jaccardindex between pairs of variables Then PROC CLUSTER can perform a clusteranalysis Table 40 illustrates for Table 156 of the textbook on statewide groundsfor divorce using the average linkage method for pairs of clusters with the Jaccard

29

Table 28 SAS Code for Fitting Model (133) for Matched Pairs toTable 131

------------------------------------------------------------------------data kerryobamainput case occasion response count

datalines1 0 1 1751 1 1 1752 0 1 162 1 0 163 0 0 543 1 1 544 0 0 1884 1 0 188

proc nlmixed data=kerryobama qpoints=1000eta = alpha + betaoccasion + up = exp(eta)(1 + exp(eta))model response ~ binary(p)random u ~ normal(0 sigmasigma) subject = casereplicate count

run------------------------------------------------------------------------

28

Table 29 SAS Code for GLMM Analysis of Election Data in Table 132

---------------------------------------------------------------------data voteinput y ncase = _n_datalines1 516 321 4proc nlmixed

eta = alpha + u p = exp(eta) (1 + exp(eta))model y ~ binomial(np)random u ~ normal(0sigmasigma) subject=casepredict p out=new

proc print data=newrun---------------------------------------------------------------------

Pearson statistic and PROC NLMIXED to fit a Poisson GLMM PROC NLMIXEDcan also fit negative binomial models

The PROC GENMOD procedure fits zero-inflated Poisson regression models

Chapter 15 Non-Model-Based Classification and Cluster-ing

PROC DISCRIM in SAS can perform discriminant analysis For example for theungrouped horseshoe crab as analyzed above in Table 9 you can add code such asshown in Table 39 The statement ldquopriors proprdquo sets the prior probabilities equal to thesample proportions Alternatively ldquopriors equalrdquo would have equal prior proportionsin the two categories

PROC DISCRIM can also be used to perform smoothing using kernel methodsand using k-nearest neighbor methods See

supportsascomdocumentationcdlenstatug63347HTMLdefaultviewer

htmstatug_discrim_sect017htm

You can construct and assess classification trees using PROC HPSPLIT See

httpssupportsascomdocumentationonlinedocstat141hpsplitpdf

PROC DISTANCE can form distances such as simple matching and the Jaccardindex between pairs of variables Then PROC CLUSTER can perform a clusteranalysis Table 40 illustrates for Table 156 of the textbook on statewide groundsfor divorce using the average linkage method for pairs of clusters with the Jaccard

29

Table 29 SAS Code for GLMM Analysis of Election Data in Table 132

---------------------------------------------------------------------data voteinput y ncase = _n_datalines1 516 321 4proc nlmixed

eta = alpha + u p = exp(eta) (1 + exp(eta))model y ~ binomial(np)random u ~ normal(0sigmasigma) subject=casepredict p out=new

proc print data=newrun---------------------------------------------------------------------

Pearson statistic and PROC NLMIXED to fit a Poisson GLMM PROC NLMIXEDcan also fit negative binomial models

The PROC GENMOD procedure fits zero-inflated Poisson regression models

Chapter 15 Non-Model-Based Classification and Cluster-ing

PROC DISCRIM in SAS can perform discriminant analysis For example for theungrouped horseshoe crab as analyzed above in Table 9 you can add code such asshown in Table 39 The statement ldquopriors proprdquo sets the prior probabilities equal to thesample proportions Alternatively ldquopriors equalrdquo would have equal prior proportionsin the two categories

PROC DISCRIM can also be used to perform smoothing using kernel methodsand using k-nearest neighbor methods See

supportsascomdocumentationcdlenstatug63347HTMLdefaultviewer

htmstatug_discrim_sect017htm

You can construct and assess classification trees using PROC HPSPLIT See

httpssupportsascomdocumentationonlinedocstat141hpsplitpdf

PROC DISTANCE can form distances such as simple matching and the Jaccardindex between pairs of variables Then PROC CLUSTER can perform a clusteranalysis Table 40 illustrates for Table 156 of the textbook on statewide groundsfor divorce using the average linkage method for pairs of clusters with the Jaccard

29

Table 30 SAS Code for GLMM for Multicenter Trial Data in Ta-ble 137

----------------------------------------------------------------------------------data binomialinput center treat y n y successes out of n trialsif treat = 1 then treat = 5 else treat = -5cards1 1 11 36 1 0 10 37 2 1 16 20 2 0 22 323 1 14 19 3 0 7 19 4 1 2 16 4 0 1 175 1 6 17 5 0 0 12 6 1 1 11 6 0 0 107 1 1 5 7 0 1 9 8 1 4 6 8 0 6 7run

proc genmod data=binomial fixed effects no interaction modelclass centermodel yn = treat center dist=bin link=logit noint

run

proc nlmixed data=binomial qpoints=15 random effects no interactionparms alpha=-1 beta=1 sig=1 initial values for parameter estimatespi = exp(a + betatreat)(1+exp(a + betatreat)) logistic formula for probmodel y ~ binomial(n pi)random a ~ normal(alpha sigsig) subject=centerpredict a + betatreat out=OUT1

run

proc nlmixed data=binomial qpoints=15 random effects interactionparms alpha=-1 beta=1 sig_a=1 sig_b=1 initial valuespi = exp(a + btreat)(1+exp(a + btreat))model y ~ binomial(n pi)random a b ~ normal([alphabeta] [sig_asig_a0sig_bsig_b]) subject=center

run----------------------------------------------------------------------------------

30

Table 31 SAS Code for GLMM for cumulative logit random effectsheterogeneity model fitted to data in Hartzel et al (2001a)

----------------------------------------------------------------------------------data ordinal

do center = 1 to 8do trt = 1 to 0 by -1

do resp = 3 to 1 by -1input count output

endend

enddatalines13 7 6 1 1 10 2 5 10 2 2 111 23 7 2 8 2 7 11 8 0 3 215 3 5 1 1 5 13 5 5 4 0 17 4 13 1 1 11 15 9 2 3 2 2

run

proc nlmixed data=ordinal qpoints=15 To maintain the threshold ordering define thresholds such that threshold 1 = 0 and threshold 2 = i2 where i2 gt 0 Use starting value of 0 for sig_cb bounds i2gt0 parms sig_cb=0eta1 = c - btrteta2 = i2 - c - btrtif (resp=1) then z = 1(1+exp(-eta1))

else if (resp=2) then z = 1(1+exp(-eta2)) - 1(1+exp(-eta1))else z = 1 - 1(1+exp(-eta2))

if (z gt 1e-8) then ll = countlog(z) Check for small values of z else ll=-1e100

model resp ~ general(ll) Define general log-likelihood random c b ~ normal([gammabeta][sig_csig_c sig_cb sig_bsig_b])

subject = center out = out1 OUT1 contains predicted center- specific cumulative log odds ratios

run----------------------------------------------------------------------------------

31

Table 32 SAS Code for GLMM for Leading Crowd Data in Table 138

------------------------------------------------------------------------------data crowdinput mem1 att1 mem2 att2 countdatalines1 1 1 1 4580 0 0 0 554

data new set crowdcase=_n_x1m=1 x1a=0 x2m=0 x2a=0 var=1 resp=mem1 outputx1m=0 x1a=1 x2m=0 x2a=0 var=0 resp=att1 outputx1m=0 x1a=0 x2m=1 x2a=0 var=1 resp=mem2 outputx1m=0 x1a=0 x2m=0 x2a=1 var=0 resp=att2 outputdrop mem1 att1 mem2 att2

proc nlmixed data=neweta=beta1mx1m + beta1ax1a + beta2mx2m + beta2ax2a + umvar + ua(1-var)p=exp(eta)(1+exp(eta))model resp ~ binary(p)random um ua ~ normal([00][s1s1 cov12 s2s2]) subject=casereplicate countestimate rsquomem changersquo beta2m-beta1m estimate rsquoatt changersquo beta2a-beta1a

run------------------------------------------------------------------------------

32

Table 33 SAS Code for GLMM for Movie Rater Data in Hartzel etal (2001b)

----------------------------------------------------------------------------------Input Datadata moviedo lyons = 1 to 3do siskel = 1 to 3do medved = 1 to 3do ebert = 1 to 3

input count if (count ne 0) then output

endendend

enddatalines15 2 0 2 0 0 8 0 20 3 0 2 3 0 3 2 11 0 1 1 2 1 1 1 2

2 0 0 2 1 0 0 1 01 1 0 0 1 0 0 1 01 0 0 0 0 0 0 1 1

3 1 1 0 0 0 4 1 00 0 1 0 0 0 1 3 00 0 1 2 1 0 2 2 4

Create subject variable movie and code dummy variablesdata movieset moviemovie = _n_x1 = 1 x2 = 0 x3 = 0 resp = siskel outputx1 = 0 x2 = 1 x3 = 0 resp = ebert outputx1 = 0 x2 = 0 x3 = 1 resp = lyons outputx1 = 0 x2 = 0 x3 = 0 resp = medved outputdrop lyons siskel medved ebert

Recode responses as multivariate Bernoullidata movieset movieif resp = 1 then do

y1=1 y2=0 y3=0endelse if resp = 2 then do

y1=0 y2=1 y3=0endelse if resp = 3 then do

y1=0 y2=0 y3=1endelse deleteoutputdrop resp

run

Run NLMIXED add qpoints= to change number of quadrature pointsAdjacent Category Logit Modelproc nlmixed data=movie

Code linear predictorseta1 = alpha1 + x1 beta1 + x2 beta2 + x3 beta3 + u1eta2 = alpha2 + x1 beta1 + x2 beta2 + x3 beta3 + u2

Adjacent category modelpi1 = exp(eta1+eta2) (exp(eta1+eta2) + exp(eta2) + 1)pi2 = exp(eta2) (exp(eta1+eta2) + exp(eta2) + 1)pi3 = 1 (exp(eta1+eta2) + exp(eta2) + 1)

Define likelihoodz = (pi1y1)(pi2y2)(pi3y3)if (z gt 1e-8) then ll = log(z)else ll=-1e100model y1 ~ general(ll)

Specify random effect distributionrandom u1 u2 ~ normal([00] [s1s1 cov12 s2s2]) subject = movie

Replicate data points with identical likelihood valuesreplicate count

run

Adjacent Category Logit Model with Uncorrelated Random Effectsproc nlmixed data=movie qpoints = 50

Code linear predictorseta1 = alpha1 + x1 beta1 + x2 beta2 + x3 beta3 + u1eta2 = alpha2 + x1 beta1 + x2 beta2 + x3 beta3 + u2

Adjacent categorypi1 = exp(eta1+eta2) (exp(eta1+eta2) + exp(eta2) + 1)pi2 = exp(eta2) (exp(eta1+eta2) + exp(eta2) + 1)pi3 = 1 (exp(eta1+eta2) + exp(eta2) + 1)

z = (pi1y1)(pi2y2)(pi3y3)if (z gt 1e-8) then ll = log(z)else ll=-1e100model y1 ~ general(ll)random u1 u2 ~ normal([00] [s1s1 0 s2s2]) subject = moviereplicate count

run

Baseline Category Logit Modelproc nlmixed data=movie

Code linear predictorseta1 = alpha1 + x1 beta11 + x2 beta21 + x3 beta31 + u1eta2 = alpha2 + x1 beta12 + x2 beta22 + x3 beta32 + u2

Baseline category modelpi1 = exp(eta1) (exp(eta1) + exp(eta2) + 1)pi2 = exp(eta2) (exp(eta1) + exp(eta2) + 1)pi3 = 1 (exp(eta1) + exp(eta2) + 1)

Define likelihoodz = (pi1y1)(pi2y2)(pi3y3)if (z gt 1e-8) then ll = log(z)else ll = -1e100model y1 ~ general(ll)

Specify random effect distributionrandom u1 u2 ~ normal([00] [s1s1 cov12 s2s2]) subject = movie

Replicate data points with identical likelihood valuesreplicate count

run------------------------------------------------------------------------------

33

Table 34 SAS Code for Marginal and Random Effects Modeling of Opinionsabout Legalized Abortion Data in Table 133

-----------------------------------------------------------------------------

data abortion

input case gender question response response=1 is yes

datalines

1 1 1 1

1 1 2 1

1 1 3 1

1850 0 1 0

1850 0 2 0

1850 0 3 0

proc genmod descending class case question

model response = gender question dist=bin link=logit type3

repeated subject=case type=exch corrw

proc glimmix method=quad(qpoints=1000) data=abortion

class question case

model response = question gender dist=binomial link=logit solution

random intercept subject=case

run

-----------------------------------------------------------------------------

Table 35 SAS Code for Marginal and Random Effects Modeling of InsomniaData in Table 123

----------------------------------------------------------------------------------------

data sleep

input case treat occasion outcome

datalines

1 1 0 1

1 1 1 1

2 1 0 1

2 1 1 1

239 0 0 4

239 0 1 4

proc genmod data=sleep class case

model outcome = treat occasion treatoccasion dist=multinomial link=cumlogit

repeated subject=case type=indep corrw

run

proc glimmix method=quad(qpoints=50) data=sleep class case

model outcome = treat occasion treatoccasion link=cumlogit dist=multinomial solution

random int subject=case

run

----------------------------------------------------------------------------------------

34

Table 36 SAS Code for Overdispersion Analysis of Teratology StudyData of Table 47

--------------------------------------------------------------------------data mooreinput litter group n y z2=0 z3=0 z4=0if group=2 then z2=1 if group=3 then z3=1 if group=4 then z4=1

datalines1 1 10 1 2 1 11 4 3 1 12 9 4 1 4 455 4 14 1 56 4 8 0 57 4 6 0 58 4 17 0proc logistic

model yn = z2 z3 z4 scale=williamsproc logistic

model yn = z2 z3 z4 scale=pearsonproc nlmixed qpoints=200eta = alpha + beta2z2 + beta3z3 + beta4z4 + u p = exp(eta)(1 + exp(eta))model y ~ binomial(np) random u ~ normal(0 sigmasigma) subject=litter

run--------------------------------------------------------------------------

35

Table 37 SAS Code for Fitting Models to Murder Data in Table 146

--------------------------------------------------------------------------data newinput white black other responsedatalines1070 119 55 060 16 5 1

1 0 0 6

data new set new count = white race = 0 output

count = black race = 1 output drop white black otherdata new2 set new do i = 1 to count output end drop iproc genmod data=new2

model response = race dist=negbin link=logproc genmod data=new2

model response = race dist=poi link=log scale=pearsondata new set new case = _n_proc nlmixed data = new qpoints=400

parms alpha=-37 beta=190 sigma=16eta = alpha + betarace + u mu = exp(eta)model response ~ poisson(mu)random u ~ normal(0 sigmasigma) subject=casereplicate count

run--------------------------------------------------------------------------

Table 38 SAS Code for Discriminant Analysis for Horseshoe CrabData Analyzed in Section 1512

--------------------------------------proc discrim data=crab crossvalidate

priors propclass yvar width color

run--------------------------------------

36

dissimilarity index

Chapter 16 Large- and Small-Sample Theory for Multino-mial Models

Exact conditional logistic regression is available in PROC LOGISTIC with the EXACTstatement It provides ordinary and mid-P -values as well as confidence limits for eachmodel parameter and the corresponding odds ratio with the ESTIMATE = BOTHoption Or you can pick the type of confidence interval you want by specifyingCLTYPE=EXACT or CLTYPE=MIDP In particular this enables you to get theCornfield exact interval for an odds ratio (also available with PROC FREQ as shownabove in Table 4) or its mid-P adaptation You can also conduct the exact condi-tional version of the CochranndashArmitage test using the TREND option in the EXACTstatement with PROC FREQ One can also conduct an asymptotic conditional logisticregression using a STRATA statement to indicate the stratification parameters to beconditioned out PROC PHREG can also do this (Stokes et al 2012) For a 2times2timesKtable using the EQOR option in an EXACT statement in PROC FREQ provides anexact test for equal odds ratios proposed by Zelen (1971) as shown above in Table 12

37

Table 39 SAS Code for Cluster Analysis for Table 156 on StatewideGrounds for Divorce

-----------------------------------------------------------------------data divorceinput state $ incompat cruelty desertn non_supp alcohol

felony impotenc insanity separate datalinesCalifornia 1 0 0 0 0 0 0 1 0Florida 1 0 0 0 0 0 0 1 0Illinois 0 1 1 0 1 1 1 0 0Massachusetts 1 1 1 1 1 1 1 0 1Michigan 1 0 0 0 0 0 0 0 0NewYork 0 1 1 0 0 1 0 0 1Texas 1 1 1 0 0 1 0 1 1Washington 1 0 0 0 0 0 0 0 1

title Grounds for Divorceproc distance data=divorce method=djaccard absent=0 out=distjaccvar anominal(incompat--separate)id state

runproc print data=distjacc(obs=8)id statetitle2 Only 8 states

runproc cluster data=distjacc method=averagepseudo outtree=treeid state

runproc tree data=tree horizontal n=7 out=out id staterun-----------------------------------------------------------------------

38

Table 31 SAS Code for GLMM for cumulative logit random effectsheterogeneity model fitted to data in Hartzel et al (2001a)

----------------------------------------------------------------------------------data ordinal

do center = 1 to 8do trt = 1 to 0 by -1

do resp = 3 to 1 by -1input count output

endend

enddatalines13 7 6 1 1 10 2 5 10 2 2 111 23 7 2 8 2 7 11 8 0 3 215 3 5 1 1 5 13 5 5 4 0 17 4 13 1 1 11 15 9 2 3 2 2

run

proc nlmixed data=ordinal qpoints=15 To maintain the threshold ordering define thresholds such that threshold 1 = 0 and threshold 2 = i2 where i2 gt 0 Use starting value of 0 for sig_cb bounds i2gt0 parms sig_cb=0eta1 = c - btrteta2 = i2 - c - btrtif (resp=1) then z = 1(1+exp(-eta1))

else if (resp=2) then z = 1(1+exp(-eta2)) - 1(1+exp(-eta1))else z = 1 - 1(1+exp(-eta2))

if (z gt 1e-8) then ll = countlog(z) Check for small values of z else ll=-1e100

model resp ~ general(ll) Define general log-likelihood random c b ~ normal([gammabeta][sig_csig_c sig_cb sig_bsig_b])

subject = center out = out1 OUT1 contains predicted center- specific cumulative log odds ratios

run----------------------------------------------------------------------------------

31

Table 32 SAS Code for GLMM for Leading Crowd Data in Table 138

------------------------------------------------------------------------------data crowdinput mem1 att1 mem2 att2 countdatalines1 1 1 1 4580 0 0 0 554

data new set crowdcase=_n_x1m=1 x1a=0 x2m=0 x2a=0 var=1 resp=mem1 outputx1m=0 x1a=1 x2m=0 x2a=0 var=0 resp=att1 outputx1m=0 x1a=0 x2m=1 x2a=0 var=1 resp=mem2 outputx1m=0 x1a=0 x2m=0 x2a=1 var=0 resp=att2 outputdrop mem1 att1 mem2 att2

proc nlmixed data=neweta=beta1mx1m + beta1ax1a + beta2mx2m + beta2ax2a + umvar + ua(1-var)p=exp(eta)(1+exp(eta))model resp ~ binary(p)random um ua ~ normal([00][s1s1 cov12 s2s2]) subject=casereplicate countestimate rsquomem changersquo beta2m-beta1m estimate rsquoatt changersquo beta2a-beta1a

run------------------------------------------------------------------------------

32

Table 33 SAS Code for GLMM for Movie Rater Data in Hartzel etal (2001b)

----------------------------------------------------------------------------------Input Datadata moviedo lyons = 1 to 3do siskel = 1 to 3do medved = 1 to 3do ebert = 1 to 3

input count if (count ne 0) then output

endendend

enddatalines15 2 0 2 0 0 8 0 20 3 0 2 3 0 3 2 11 0 1 1 2 1 1 1 2

2 0 0 2 1 0 0 1 01 1 0 0 1 0 0 1 01 0 0 0 0 0 0 1 1

3 1 1 0 0 0 4 1 00 0 1 0 0 0 1 3 00 0 1 2 1 0 2 2 4

Create subject variable movie and code dummy variablesdata movieset moviemovie = _n_x1 = 1 x2 = 0 x3 = 0 resp = siskel outputx1 = 0 x2 = 1 x3 = 0 resp = ebert outputx1 = 0 x2 = 0 x3 = 1 resp = lyons outputx1 = 0 x2 = 0 x3 = 0 resp = medved outputdrop lyons siskel medved ebert

Recode responses as multivariate Bernoullidata movieset movieif resp = 1 then do

y1=1 y2=0 y3=0endelse if resp = 2 then do

y1=0 y2=1 y3=0endelse if resp = 3 then do

y1=0 y2=0 y3=1endelse deleteoutputdrop resp

run

Run NLMIXED add qpoints= to change number of quadrature pointsAdjacent Category Logit Modelproc nlmixed data=movie

Code linear predictorseta1 = alpha1 + x1 beta1 + x2 beta2 + x3 beta3 + u1eta2 = alpha2 + x1 beta1 + x2 beta2 + x3 beta3 + u2

Adjacent category modelpi1 = exp(eta1+eta2) (exp(eta1+eta2) + exp(eta2) + 1)pi2 = exp(eta2) (exp(eta1+eta2) + exp(eta2) + 1)pi3 = 1 (exp(eta1+eta2) + exp(eta2) + 1)

Define likelihoodz = (pi1y1)(pi2y2)(pi3y3)if (z gt 1e-8) then ll = log(z)else ll=-1e100model y1 ~ general(ll)

Specify random effect distributionrandom u1 u2 ~ normal([00] [s1s1 cov12 s2s2]) subject = movie

Replicate data points with identical likelihood valuesreplicate count

run

Adjacent Category Logit Model with Uncorrelated Random Effectsproc nlmixed data=movie qpoints = 50

Code linear predictorseta1 = alpha1 + x1 beta1 + x2 beta2 + x3 beta3 + u1eta2 = alpha2 + x1 beta1 + x2 beta2 + x3 beta3 + u2

Adjacent categorypi1 = exp(eta1+eta2) (exp(eta1+eta2) + exp(eta2) + 1)pi2 = exp(eta2) (exp(eta1+eta2) + exp(eta2) + 1)pi3 = 1 (exp(eta1+eta2) + exp(eta2) + 1)

z = (pi1y1)(pi2y2)(pi3y3)if (z gt 1e-8) then ll = log(z)else ll=-1e100model y1 ~ general(ll)random u1 u2 ~ normal([00] [s1s1 0 s2s2]) subject = moviereplicate count

run

Baseline Category Logit Modelproc nlmixed data=movie

Code linear predictorseta1 = alpha1 + x1 beta11 + x2 beta21 + x3 beta31 + u1eta2 = alpha2 + x1 beta12 + x2 beta22 + x3 beta32 + u2

Baseline category modelpi1 = exp(eta1) (exp(eta1) + exp(eta2) + 1)pi2 = exp(eta2) (exp(eta1) + exp(eta2) + 1)pi3 = 1 (exp(eta1) + exp(eta2) + 1)

Define likelihoodz = (pi1y1)(pi2y2)(pi3y3)if (z gt 1e-8) then ll = log(z)else ll = -1e100model y1 ~ general(ll)

Specify random effect distributionrandom u1 u2 ~ normal([00] [s1s1 cov12 s2s2]) subject = movie

Replicate data points with identical likelihood valuesreplicate count

run------------------------------------------------------------------------------

33

Table 34 SAS Code for Marginal and Random Effects Modeling of Opinionsabout Legalized Abortion Data in Table 133

-----------------------------------------------------------------------------

data abortion

input case gender question response response=1 is yes

datalines

1 1 1 1

1 1 2 1

1 1 3 1

1850 0 1 0

1850 0 2 0

1850 0 3 0

proc genmod descending class case question

model response = gender question dist=bin link=logit type3

repeated subject=case type=exch corrw

proc glimmix method=quad(qpoints=1000) data=abortion

class question case

model response = question gender dist=binomial link=logit solution

random intercept subject=case

run

-----------------------------------------------------------------------------

Table 35 SAS Code for Marginal and Random Effects Modeling of InsomniaData in Table 123

----------------------------------------------------------------------------------------

data sleep

input case treat occasion outcome

datalines

1 1 0 1

1 1 1 1

2 1 0 1

2 1 1 1

239 0 0 4

239 0 1 4

proc genmod data=sleep class case

model outcome = treat occasion treatoccasion dist=multinomial link=cumlogit

repeated subject=case type=indep corrw

run

proc glimmix method=quad(qpoints=50) data=sleep class case

model outcome = treat occasion treatoccasion link=cumlogit dist=multinomial solution

random int subject=case

run

----------------------------------------------------------------------------------------

34

Table 36 SAS Code for Overdispersion Analysis of Teratology StudyData of Table 47

--------------------------------------------------------------------------data mooreinput litter group n y z2=0 z3=0 z4=0if group=2 then z2=1 if group=3 then z3=1 if group=4 then z4=1

datalines1 1 10 1 2 1 11 4 3 1 12 9 4 1 4 455 4 14 1 56 4 8 0 57 4 6 0 58 4 17 0proc logistic

model yn = z2 z3 z4 scale=williamsproc logistic

model yn = z2 z3 z4 scale=pearsonproc nlmixed qpoints=200eta = alpha + beta2z2 + beta3z3 + beta4z4 + u p = exp(eta)(1 + exp(eta))model y ~ binomial(np) random u ~ normal(0 sigmasigma) subject=litter

run--------------------------------------------------------------------------

35

Table 37 SAS Code for Fitting Models to Murder Data in Table 146

--------------------------------------------------------------------------data newinput white black other responsedatalines1070 119 55 060 16 5 1

1 0 0 6

data new set new count = white race = 0 output

count = black race = 1 output drop white black otherdata new2 set new do i = 1 to count output end drop iproc genmod data=new2

model response = race dist=negbin link=logproc genmod data=new2

model response = race dist=poi link=log scale=pearsondata new set new case = _n_proc nlmixed data = new qpoints=400

parms alpha=-37 beta=190 sigma=16eta = alpha + betarace + u mu = exp(eta)model response ~ poisson(mu)random u ~ normal(0 sigmasigma) subject=casereplicate count

run--------------------------------------------------------------------------

Table 38 SAS Code for Discriminant Analysis for Horseshoe CrabData Analyzed in Section 1512

--------------------------------------proc discrim data=crab crossvalidate

priors propclass yvar width color

run--------------------------------------

36

dissimilarity index

Chapter 16 Large- and Small-Sample Theory for Multino-mial Models

Exact conditional logistic regression is available in PROC LOGISTIC with the EXACTstatement It provides ordinary and mid-P -values as well as confidence limits for eachmodel parameter and the corresponding odds ratio with the ESTIMATE = BOTHoption Or you can pick the type of confidence interval you want by specifyingCLTYPE=EXACT or CLTYPE=MIDP In particular this enables you to get theCornfield exact interval for an odds ratio (also available with PROC FREQ as shownabove in Table 4) or its mid-P adaptation You can also conduct the exact condi-tional version of the CochranndashArmitage test using the TREND option in the EXACTstatement with PROC FREQ One can also conduct an asymptotic conditional logisticregression using a STRATA statement to indicate the stratification parameters to beconditioned out PROC PHREG can also do this (Stokes et al 2012) For a 2times2timesKtable using the EQOR option in an EXACT statement in PROC FREQ provides anexact test for equal odds ratios proposed by Zelen (1971) as shown above in Table 12

37

Table 39 SAS Code for Cluster Analysis for Table 156 on StatewideGrounds for Divorce

-----------------------------------------------------------------------data divorceinput state $ incompat cruelty desertn non_supp alcohol

felony impotenc insanity separate datalinesCalifornia 1 0 0 0 0 0 0 1 0Florida 1 0 0 0 0 0 0 1 0Illinois 0 1 1 0 1 1 1 0 0Massachusetts 1 1 1 1 1 1 1 0 1Michigan 1 0 0 0 0 0 0 0 0NewYork 0 1 1 0 0 1 0 0 1Texas 1 1 1 0 0 1 0 1 1Washington 1 0 0 0 0 0 0 0 1

title Grounds for Divorceproc distance data=divorce method=djaccard absent=0 out=distjaccvar anominal(incompat--separate)id state

runproc print data=distjacc(obs=8)id statetitle2 Only 8 states

runproc cluster data=distjacc method=averagepseudo outtree=treeid state

runproc tree data=tree horizontal n=7 out=out id staterun-----------------------------------------------------------------------

38

Table 32 SAS Code for GLMM for Leading Crowd Data in Table 138

------------------------------------------------------------------------------data crowdinput mem1 att1 mem2 att2 countdatalines1 1 1 1 4580 0 0 0 554

data new set crowdcase=_n_x1m=1 x1a=0 x2m=0 x2a=0 var=1 resp=mem1 outputx1m=0 x1a=1 x2m=0 x2a=0 var=0 resp=att1 outputx1m=0 x1a=0 x2m=1 x2a=0 var=1 resp=mem2 outputx1m=0 x1a=0 x2m=0 x2a=1 var=0 resp=att2 outputdrop mem1 att1 mem2 att2

proc nlmixed data=neweta=beta1mx1m + beta1ax1a + beta2mx2m + beta2ax2a + umvar + ua(1-var)p=exp(eta)(1+exp(eta))model resp ~ binary(p)random um ua ~ normal([00][s1s1 cov12 s2s2]) subject=casereplicate countestimate rsquomem changersquo beta2m-beta1m estimate rsquoatt changersquo beta2a-beta1a

run------------------------------------------------------------------------------

32

Table 33 SAS Code for GLMM for Movie Rater Data in Hartzel etal (2001b)

----------------------------------------------------------------------------------Input Datadata moviedo lyons = 1 to 3do siskel = 1 to 3do medved = 1 to 3do ebert = 1 to 3

input count if (count ne 0) then output

endendend

enddatalines15 2 0 2 0 0 8 0 20 3 0 2 3 0 3 2 11 0 1 1 2 1 1 1 2

2 0 0 2 1 0 0 1 01 1 0 0 1 0 0 1 01 0 0 0 0 0 0 1 1

3 1 1 0 0 0 4 1 00 0 1 0 0 0 1 3 00 0 1 2 1 0 2 2 4

Create subject variable movie and code dummy variablesdata movieset moviemovie = _n_x1 = 1 x2 = 0 x3 = 0 resp = siskel outputx1 = 0 x2 = 1 x3 = 0 resp = ebert outputx1 = 0 x2 = 0 x3 = 1 resp = lyons outputx1 = 0 x2 = 0 x3 = 0 resp = medved outputdrop lyons siskel medved ebert

Recode responses as multivariate Bernoullidata movieset movieif resp = 1 then do

y1=1 y2=0 y3=0endelse if resp = 2 then do

y1=0 y2=1 y3=0endelse if resp = 3 then do

y1=0 y2=0 y3=1endelse deleteoutputdrop resp

run

Run NLMIXED add qpoints= to change number of quadrature pointsAdjacent Category Logit Modelproc nlmixed data=movie

Code linear predictorseta1 = alpha1 + x1 beta1 + x2 beta2 + x3 beta3 + u1eta2 = alpha2 + x1 beta1 + x2 beta2 + x3 beta3 + u2

Adjacent category modelpi1 = exp(eta1+eta2) (exp(eta1+eta2) + exp(eta2) + 1)pi2 = exp(eta2) (exp(eta1+eta2) + exp(eta2) + 1)pi3 = 1 (exp(eta1+eta2) + exp(eta2) + 1)

Define likelihoodz = (pi1y1)(pi2y2)(pi3y3)if (z gt 1e-8) then ll = log(z)else ll=-1e100model y1 ~ general(ll)

Specify random effect distributionrandom u1 u2 ~ normal([00] [s1s1 cov12 s2s2]) subject = movie

Replicate data points with identical likelihood valuesreplicate count

run

Adjacent Category Logit Model with Uncorrelated Random Effectsproc nlmixed data=movie qpoints = 50

Code linear predictorseta1 = alpha1 + x1 beta1 + x2 beta2 + x3 beta3 + u1eta2 = alpha2 + x1 beta1 + x2 beta2 + x3 beta3 + u2

Adjacent categorypi1 = exp(eta1+eta2) (exp(eta1+eta2) + exp(eta2) + 1)pi2 = exp(eta2) (exp(eta1+eta2) + exp(eta2) + 1)pi3 = 1 (exp(eta1+eta2) + exp(eta2) + 1)

z = (pi1y1)(pi2y2)(pi3y3)if (z gt 1e-8) then ll = log(z)else ll=-1e100model y1 ~ general(ll)random u1 u2 ~ normal([00] [s1s1 0 s2s2]) subject = moviereplicate count

run

Baseline Category Logit Modelproc nlmixed data=movie

Code linear predictorseta1 = alpha1 + x1 beta11 + x2 beta21 + x3 beta31 + u1eta2 = alpha2 + x1 beta12 + x2 beta22 + x3 beta32 + u2

Baseline category modelpi1 = exp(eta1) (exp(eta1) + exp(eta2) + 1)pi2 = exp(eta2) (exp(eta1) + exp(eta2) + 1)pi3 = 1 (exp(eta1) + exp(eta2) + 1)

Define likelihoodz = (pi1y1)(pi2y2)(pi3y3)if (z gt 1e-8) then ll = log(z)else ll = -1e100model y1 ~ general(ll)

Specify random effect distributionrandom u1 u2 ~ normal([00] [s1s1 cov12 s2s2]) subject = movie

Replicate data points with identical likelihood valuesreplicate count

run------------------------------------------------------------------------------

33

Table 34 SAS Code for Marginal and Random Effects Modeling of Opinionsabout Legalized Abortion Data in Table 133

-----------------------------------------------------------------------------

data abortion

input case gender question response response=1 is yes

datalines

1 1 1 1

1 1 2 1

1 1 3 1

1850 0 1 0

1850 0 2 0

1850 0 3 0

proc genmod descending class case question

model response = gender question dist=bin link=logit type3

repeated subject=case type=exch corrw

proc glimmix method=quad(qpoints=1000) data=abortion

class question case

model response = question gender dist=binomial link=logit solution

random intercept subject=case

run

-----------------------------------------------------------------------------

Table 35 SAS Code for Marginal and Random Effects Modeling of InsomniaData in Table 123

----------------------------------------------------------------------------------------

data sleep

input case treat occasion outcome

datalines

1 1 0 1

1 1 1 1

2 1 0 1

2 1 1 1

239 0 0 4

239 0 1 4

proc genmod data=sleep class case

model outcome = treat occasion treatoccasion dist=multinomial link=cumlogit

repeated subject=case type=indep corrw

run

proc glimmix method=quad(qpoints=50) data=sleep class case

model outcome = treat occasion treatoccasion link=cumlogit dist=multinomial solution

random int subject=case

run

----------------------------------------------------------------------------------------

34

Table 36 SAS Code for Overdispersion Analysis of Teratology StudyData of Table 47

--------------------------------------------------------------------------data mooreinput litter group n y z2=0 z3=0 z4=0if group=2 then z2=1 if group=3 then z3=1 if group=4 then z4=1

datalines1 1 10 1 2 1 11 4 3 1 12 9 4 1 4 455 4 14 1 56 4 8 0 57 4 6 0 58 4 17 0proc logistic

model yn = z2 z3 z4 scale=williamsproc logistic

model yn = z2 z3 z4 scale=pearsonproc nlmixed qpoints=200eta = alpha + beta2z2 + beta3z3 + beta4z4 + u p = exp(eta)(1 + exp(eta))model y ~ binomial(np) random u ~ normal(0 sigmasigma) subject=litter

run--------------------------------------------------------------------------

35

Table 37 SAS Code for Fitting Models to Murder Data in Table 146

--------------------------------------------------------------------------data newinput white black other responsedatalines1070 119 55 060 16 5 1

1 0 0 6

data new set new count = white race = 0 output

count = black race = 1 output drop white black otherdata new2 set new do i = 1 to count output end drop iproc genmod data=new2

model response = race dist=negbin link=logproc genmod data=new2

model response = race dist=poi link=log scale=pearsondata new set new case = _n_proc nlmixed data = new qpoints=400

parms alpha=-37 beta=190 sigma=16eta = alpha + betarace + u mu = exp(eta)model response ~ poisson(mu)random u ~ normal(0 sigmasigma) subject=casereplicate count

run--------------------------------------------------------------------------

Table 38 SAS Code for Discriminant Analysis for Horseshoe CrabData Analyzed in Section 1512

--------------------------------------proc discrim data=crab crossvalidate

priors propclass yvar width color

run--------------------------------------

36

dissimilarity index

Chapter 16 Large- and Small-Sample Theory for Multino-mial Models

Exact conditional logistic regression is available in PROC LOGISTIC with the EXACTstatement It provides ordinary and mid-P -values as well as confidence limits for eachmodel parameter and the corresponding odds ratio with the ESTIMATE = BOTHoption Or you can pick the type of confidence interval you want by specifyingCLTYPE=EXACT or CLTYPE=MIDP In particular this enables you to get theCornfield exact interval for an odds ratio (also available with PROC FREQ as shownabove in Table 4) or its mid-P adaptation You can also conduct the exact condi-tional version of the CochranndashArmitage test using the TREND option in the EXACTstatement with PROC FREQ One can also conduct an asymptotic conditional logisticregression using a STRATA statement to indicate the stratification parameters to beconditioned out PROC PHREG can also do this (Stokes et al 2012) For a 2times2timesKtable using the EQOR option in an EXACT statement in PROC FREQ provides anexact test for equal odds ratios proposed by Zelen (1971) as shown above in Table 12

37

Table 39 SAS Code for Cluster Analysis for Table 156 on StatewideGrounds for Divorce

-----------------------------------------------------------------------data divorceinput state $ incompat cruelty desertn non_supp alcohol

felony impotenc insanity separate datalinesCalifornia 1 0 0 0 0 0 0 1 0Florida 1 0 0 0 0 0 0 1 0Illinois 0 1 1 0 1 1 1 0 0Massachusetts 1 1 1 1 1 1 1 0 1Michigan 1 0 0 0 0 0 0 0 0NewYork 0 1 1 0 0 1 0 0 1Texas 1 1 1 0 0 1 0 1 1Washington 1 0 0 0 0 0 0 0 1

title Grounds for Divorceproc distance data=divorce method=djaccard absent=0 out=distjaccvar anominal(incompat--separate)id state

runproc print data=distjacc(obs=8)id statetitle2 Only 8 states

runproc cluster data=distjacc method=averagepseudo outtree=treeid state

runproc tree data=tree horizontal n=7 out=out id staterun-----------------------------------------------------------------------

38

Table 33 SAS Code for GLMM for Movie Rater Data in Hartzel etal (2001b)

----------------------------------------------------------------------------------Input Datadata moviedo lyons = 1 to 3do siskel = 1 to 3do medved = 1 to 3do ebert = 1 to 3

input count if (count ne 0) then output

endendend

enddatalines15 2 0 2 0 0 8 0 20 3 0 2 3 0 3 2 11 0 1 1 2 1 1 1 2

2 0 0 2 1 0 0 1 01 1 0 0 1 0 0 1 01 0 0 0 0 0 0 1 1

3 1 1 0 0 0 4 1 00 0 1 0 0 0 1 3 00 0 1 2 1 0 2 2 4

Create subject variable movie and code dummy variablesdata movieset moviemovie = _n_x1 = 1 x2 = 0 x3 = 0 resp = siskel outputx1 = 0 x2 = 1 x3 = 0 resp = ebert outputx1 = 0 x2 = 0 x3 = 1 resp = lyons outputx1 = 0 x2 = 0 x3 = 0 resp = medved outputdrop lyons siskel medved ebert

Recode responses as multivariate Bernoullidata movieset movieif resp = 1 then do

y1=1 y2=0 y3=0endelse if resp = 2 then do

y1=0 y2=1 y3=0endelse if resp = 3 then do

y1=0 y2=0 y3=1endelse deleteoutputdrop resp

run

Run NLMIXED add qpoints= to change number of quadrature pointsAdjacent Category Logit Modelproc nlmixed data=movie

Code linear predictorseta1 = alpha1 + x1 beta1 + x2 beta2 + x3 beta3 + u1eta2 = alpha2 + x1 beta1 + x2 beta2 + x3 beta3 + u2

Adjacent category modelpi1 = exp(eta1+eta2) (exp(eta1+eta2) + exp(eta2) + 1)pi2 = exp(eta2) (exp(eta1+eta2) + exp(eta2) + 1)pi3 = 1 (exp(eta1+eta2) + exp(eta2) + 1)

Define likelihoodz = (pi1y1)(pi2y2)(pi3y3)if (z gt 1e-8) then ll = log(z)else ll=-1e100model y1 ~ general(ll)

Specify random effect distributionrandom u1 u2 ~ normal([00] [s1s1 cov12 s2s2]) subject = movie

Replicate data points with identical likelihood valuesreplicate count

run

Adjacent Category Logit Model with Uncorrelated Random Effectsproc nlmixed data=movie qpoints = 50

Code linear predictorseta1 = alpha1 + x1 beta1 + x2 beta2 + x3 beta3 + u1eta2 = alpha2 + x1 beta1 + x2 beta2 + x3 beta3 + u2

Adjacent categorypi1 = exp(eta1+eta2) (exp(eta1+eta2) + exp(eta2) + 1)pi2 = exp(eta2) (exp(eta1+eta2) + exp(eta2) + 1)pi3 = 1 (exp(eta1+eta2) + exp(eta2) + 1)

z = (pi1y1)(pi2y2)(pi3y3)if (z gt 1e-8) then ll = log(z)else ll=-1e100model y1 ~ general(ll)random u1 u2 ~ normal([00] [s1s1 0 s2s2]) subject = moviereplicate count

run

Baseline Category Logit Modelproc nlmixed data=movie

Code linear predictorseta1 = alpha1 + x1 beta11 + x2 beta21 + x3 beta31 + u1eta2 = alpha2 + x1 beta12 + x2 beta22 + x3 beta32 + u2

Baseline category modelpi1 = exp(eta1) (exp(eta1) + exp(eta2) + 1)pi2 = exp(eta2) (exp(eta1) + exp(eta2) + 1)pi3 = 1 (exp(eta1) + exp(eta2) + 1)

Define likelihoodz = (pi1y1)(pi2y2)(pi3y3)if (z gt 1e-8) then ll = log(z)else ll = -1e100model y1 ~ general(ll)

Specify random effect distributionrandom u1 u2 ~ normal([00] [s1s1 cov12 s2s2]) subject = movie

Replicate data points with identical likelihood valuesreplicate count

run------------------------------------------------------------------------------

33

Table 34 SAS Code for Marginal and Random Effects Modeling of Opinionsabout Legalized Abortion Data in Table 133

-----------------------------------------------------------------------------

data abortion

input case gender question response response=1 is yes

datalines

1 1 1 1

1 1 2 1

1 1 3 1

1850 0 1 0

1850 0 2 0

1850 0 3 0

proc genmod descending class case question

model response = gender question dist=bin link=logit type3

repeated subject=case type=exch corrw

proc glimmix method=quad(qpoints=1000) data=abortion

class question case

model response = question gender dist=binomial link=logit solution

random intercept subject=case

run

-----------------------------------------------------------------------------

Table 35 SAS Code for Marginal and Random Effects Modeling of InsomniaData in Table 123

----------------------------------------------------------------------------------------

data sleep

input case treat occasion outcome

datalines

1 1 0 1

1 1 1 1

2 1 0 1

2 1 1 1

239 0 0 4

239 0 1 4

proc genmod data=sleep class case

model outcome = treat occasion treatoccasion dist=multinomial link=cumlogit

repeated subject=case type=indep corrw

run

proc glimmix method=quad(qpoints=50) data=sleep class case

model outcome = treat occasion treatoccasion link=cumlogit dist=multinomial solution

random int subject=case

run

----------------------------------------------------------------------------------------

34

Table 36 SAS Code for Overdispersion Analysis of Teratology StudyData of Table 47

--------------------------------------------------------------------------data mooreinput litter group n y z2=0 z3=0 z4=0if group=2 then z2=1 if group=3 then z3=1 if group=4 then z4=1

datalines1 1 10 1 2 1 11 4 3 1 12 9 4 1 4 455 4 14 1 56 4 8 0 57 4 6 0 58 4 17 0proc logistic

model yn = z2 z3 z4 scale=williamsproc logistic

model yn = z2 z3 z4 scale=pearsonproc nlmixed qpoints=200eta = alpha + beta2z2 + beta3z3 + beta4z4 + u p = exp(eta)(1 + exp(eta))model y ~ binomial(np) random u ~ normal(0 sigmasigma) subject=litter

run--------------------------------------------------------------------------

35

Table 37 SAS Code for Fitting Models to Murder Data in Table 146

--------------------------------------------------------------------------data newinput white black other responsedatalines1070 119 55 060 16 5 1

1 0 0 6

data new set new count = white race = 0 output

count = black race = 1 output drop white black otherdata new2 set new do i = 1 to count output end drop iproc genmod data=new2

model response = race dist=negbin link=logproc genmod data=new2

model response = race dist=poi link=log scale=pearsondata new set new case = _n_proc nlmixed data = new qpoints=400

parms alpha=-37 beta=190 sigma=16eta = alpha + betarace + u mu = exp(eta)model response ~ poisson(mu)random u ~ normal(0 sigmasigma) subject=casereplicate count

run--------------------------------------------------------------------------

Table 38 SAS Code for Discriminant Analysis for Horseshoe CrabData Analyzed in Section 1512

--------------------------------------proc discrim data=crab crossvalidate

priors propclass yvar width color

run--------------------------------------

36

dissimilarity index

Chapter 16 Large- and Small-Sample Theory for Multino-mial Models

Exact conditional logistic regression is available in PROC LOGISTIC with the EXACTstatement It provides ordinary and mid-P -values as well as confidence limits for eachmodel parameter and the corresponding odds ratio with the ESTIMATE = BOTHoption Or you can pick the type of confidence interval you want by specifyingCLTYPE=EXACT or CLTYPE=MIDP In particular this enables you to get theCornfield exact interval for an odds ratio (also available with PROC FREQ as shownabove in Table 4) or its mid-P adaptation You can also conduct the exact condi-tional version of the CochranndashArmitage test using the TREND option in the EXACTstatement with PROC FREQ One can also conduct an asymptotic conditional logisticregression using a STRATA statement to indicate the stratification parameters to beconditioned out PROC PHREG can also do this (Stokes et al 2012) For a 2times2timesKtable using the EQOR option in an EXACT statement in PROC FREQ provides anexact test for equal odds ratios proposed by Zelen (1971) as shown above in Table 12

37

Table 39 SAS Code for Cluster Analysis for Table 156 on StatewideGrounds for Divorce

-----------------------------------------------------------------------data divorceinput state $ incompat cruelty desertn non_supp alcohol

felony impotenc insanity separate datalinesCalifornia 1 0 0 0 0 0 0 1 0Florida 1 0 0 0 0 0 0 1 0Illinois 0 1 1 0 1 1 1 0 0Massachusetts 1 1 1 1 1 1 1 0 1Michigan 1 0 0 0 0 0 0 0 0NewYork 0 1 1 0 0 1 0 0 1Texas 1 1 1 0 0 1 0 1 1Washington 1 0 0 0 0 0 0 0 1

title Grounds for Divorceproc distance data=divorce method=djaccard absent=0 out=distjaccvar anominal(incompat--separate)id state

runproc print data=distjacc(obs=8)id statetitle2 Only 8 states

runproc cluster data=distjacc method=averagepseudo outtree=treeid state

runproc tree data=tree horizontal n=7 out=out id staterun-----------------------------------------------------------------------

38

Table 34 SAS Code for Marginal and Random Effects Modeling of Opinionsabout Legalized Abortion Data in Table 133

-----------------------------------------------------------------------------

data abortion

input case gender question response response=1 is yes

datalines

1 1 1 1

1 1 2 1

1 1 3 1

1850 0 1 0

1850 0 2 0

1850 0 3 0

proc genmod descending class case question

model response = gender question dist=bin link=logit type3

repeated subject=case type=exch corrw

proc glimmix method=quad(qpoints=1000) data=abortion

class question case

model response = question gender dist=binomial link=logit solution

random intercept subject=case

run

-----------------------------------------------------------------------------

Table 35 SAS Code for Marginal and Random Effects Modeling of InsomniaData in Table 123

----------------------------------------------------------------------------------------

data sleep

input case treat occasion outcome

datalines

1 1 0 1

1 1 1 1

2 1 0 1

2 1 1 1

239 0 0 4

239 0 1 4

proc genmod data=sleep class case

model outcome = treat occasion treatoccasion dist=multinomial link=cumlogit

repeated subject=case type=indep corrw

run

proc glimmix method=quad(qpoints=50) data=sleep class case

model outcome = treat occasion treatoccasion link=cumlogit dist=multinomial solution

random int subject=case

run

----------------------------------------------------------------------------------------

34

Table 36 SAS Code for Overdispersion Analysis of Teratology StudyData of Table 47

--------------------------------------------------------------------------data mooreinput litter group n y z2=0 z3=0 z4=0if group=2 then z2=1 if group=3 then z3=1 if group=4 then z4=1

datalines1 1 10 1 2 1 11 4 3 1 12 9 4 1 4 455 4 14 1 56 4 8 0 57 4 6 0 58 4 17 0proc logistic

model yn = z2 z3 z4 scale=williamsproc logistic

model yn = z2 z3 z4 scale=pearsonproc nlmixed qpoints=200eta = alpha + beta2z2 + beta3z3 + beta4z4 + u p = exp(eta)(1 + exp(eta))model y ~ binomial(np) random u ~ normal(0 sigmasigma) subject=litter

run--------------------------------------------------------------------------

35

Table 37 SAS Code for Fitting Models to Murder Data in Table 146

--------------------------------------------------------------------------data newinput white black other responsedatalines1070 119 55 060 16 5 1

1 0 0 6

data new set new count = white race = 0 output

count = black race = 1 output drop white black otherdata new2 set new do i = 1 to count output end drop iproc genmod data=new2

model response = race dist=negbin link=logproc genmod data=new2

model response = race dist=poi link=log scale=pearsondata new set new case = _n_proc nlmixed data = new qpoints=400

parms alpha=-37 beta=190 sigma=16eta = alpha + betarace + u mu = exp(eta)model response ~ poisson(mu)random u ~ normal(0 sigmasigma) subject=casereplicate count

run--------------------------------------------------------------------------

Table 38 SAS Code for Discriminant Analysis for Horseshoe CrabData Analyzed in Section 1512

--------------------------------------proc discrim data=crab crossvalidate

priors propclass yvar width color

run--------------------------------------

36

dissimilarity index

Chapter 16 Large- and Small-Sample Theory for Multino-mial Models

Exact conditional logistic regression is available in PROC LOGISTIC with the EXACTstatement It provides ordinary and mid-P -values as well as confidence limits for eachmodel parameter and the corresponding odds ratio with the ESTIMATE = BOTHoption Or you can pick the type of confidence interval you want by specifyingCLTYPE=EXACT or CLTYPE=MIDP In particular this enables you to get theCornfield exact interval for an odds ratio (also available with PROC FREQ as shownabove in Table 4) or its mid-P adaptation You can also conduct the exact condi-tional version of the CochranndashArmitage test using the TREND option in the EXACTstatement with PROC FREQ One can also conduct an asymptotic conditional logisticregression using a STRATA statement to indicate the stratification parameters to beconditioned out PROC PHREG can also do this (Stokes et al 2012) For a 2times2timesKtable using the EQOR option in an EXACT statement in PROC FREQ provides anexact test for equal odds ratios proposed by Zelen (1971) as shown above in Table 12

37

Table 39 SAS Code for Cluster Analysis for Table 156 on StatewideGrounds for Divorce

-----------------------------------------------------------------------data divorceinput state $ incompat cruelty desertn non_supp alcohol

felony impotenc insanity separate datalinesCalifornia 1 0 0 0 0 0 0 1 0Florida 1 0 0 0 0 0 0 1 0Illinois 0 1 1 0 1 1 1 0 0Massachusetts 1 1 1 1 1 1 1 0 1Michigan 1 0 0 0 0 0 0 0 0NewYork 0 1 1 0 0 1 0 0 1Texas 1 1 1 0 0 1 0 1 1Washington 1 0 0 0 0 0 0 0 1

title Grounds for Divorceproc distance data=divorce method=djaccard absent=0 out=distjaccvar anominal(incompat--separate)id state

runproc print data=distjacc(obs=8)id statetitle2 Only 8 states

runproc cluster data=distjacc method=averagepseudo outtree=treeid state

runproc tree data=tree horizontal n=7 out=out id staterun-----------------------------------------------------------------------

38

Table 36 SAS Code for Overdispersion Analysis of Teratology StudyData of Table 47

--------------------------------------------------------------------------data mooreinput litter group n y z2=0 z3=0 z4=0if group=2 then z2=1 if group=3 then z3=1 if group=4 then z4=1

datalines1 1 10 1 2 1 11 4 3 1 12 9 4 1 4 455 4 14 1 56 4 8 0 57 4 6 0 58 4 17 0proc logistic

model yn = z2 z3 z4 scale=williamsproc logistic

model yn = z2 z3 z4 scale=pearsonproc nlmixed qpoints=200eta = alpha + beta2z2 + beta3z3 + beta4z4 + u p = exp(eta)(1 + exp(eta))model y ~ binomial(np) random u ~ normal(0 sigmasigma) subject=litter

run--------------------------------------------------------------------------

35

Table 37 SAS Code for Fitting Models to Murder Data in Table 146

--------------------------------------------------------------------------data newinput white black other responsedatalines1070 119 55 060 16 5 1

1 0 0 6

data new set new count = white race = 0 output

count = black race = 1 output drop white black otherdata new2 set new do i = 1 to count output end drop iproc genmod data=new2

model response = race dist=negbin link=logproc genmod data=new2

model response = race dist=poi link=log scale=pearsondata new set new case = _n_proc nlmixed data = new qpoints=400

parms alpha=-37 beta=190 sigma=16eta = alpha + betarace + u mu = exp(eta)model response ~ poisson(mu)random u ~ normal(0 sigmasigma) subject=casereplicate count

run--------------------------------------------------------------------------

Table 38 SAS Code for Discriminant Analysis for Horseshoe CrabData Analyzed in Section 1512

--------------------------------------proc discrim data=crab crossvalidate

priors propclass yvar width color

run--------------------------------------

36

dissimilarity index

Chapter 16 Large- and Small-Sample Theory for Multino-mial Models

Exact conditional logistic regression is available in PROC LOGISTIC with the EXACTstatement It provides ordinary and mid-P -values as well as confidence limits for eachmodel parameter and the corresponding odds ratio with the ESTIMATE = BOTHoption Or you can pick the type of confidence interval you want by specifyingCLTYPE=EXACT or CLTYPE=MIDP In particular this enables you to get theCornfield exact interval for an odds ratio (also available with PROC FREQ as shownabove in Table 4) or its mid-P adaptation You can also conduct the exact condi-tional version of the CochranndashArmitage test using the TREND option in the EXACTstatement with PROC FREQ One can also conduct an asymptotic conditional logisticregression using a STRATA statement to indicate the stratification parameters to beconditioned out PROC PHREG can also do this (Stokes et al 2012) For a 2times2timesKtable using the EQOR option in an EXACT statement in PROC FREQ provides anexact test for equal odds ratios proposed by Zelen (1971) as shown above in Table 12

37

Table 39 SAS Code for Cluster Analysis for Table 156 on StatewideGrounds for Divorce

-----------------------------------------------------------------------data divorceinput state $ incompat cruelty desertn non_supp alcohol

felony impotenc insanity separate datalinesCalifornia 1 0 0 0 0 0 0 1 0Florida 1 0 0 0 0 0 0 1 0Illinois 0 1 1 0 1 1 1 0 0Massachusetts 1 1 1 1 1 1 1 0 1Michigan 1 0 0 0 0 0 0 0 0NewYork 0 1 1 0 0 1 0 0 1Texas 1 1 1 0 0 1 0 1 1Washington 1 0 0 0 0 0 0 0 1

title Grounds for Divorceproc distance data=divorce method=djaccard absent=0 out=distjaccvar anominal(incompat--separate)id state

runproc print data=distjacc(obs=8)id statetitle2 Only 8 states

runproc cluster data=distjacc method=averagepseudo outtree=treeid state

runproc tree data=tree horizontal n=7 out=out id staterun-----------------------------------------------------------------------

38

Table 37 SAS Code for Fitting Models to Murder Data in Table 146

--------------------------------------------------------------------------data newinput white black other responsedatalines1070 119 55 060 16 5 1

1 0 0 6

data new set new count = white race = 0 output

count = black race = 1 output drop white black otherdata new2 set new do i = 1 to count output end drop iproc genmod data=new2

model response = race dist=negbin link=logproc genmod data=new2

model response = race dist=poi link=log scale=pearsondata new set new case = _n_proc nlmixed data = new qpoints=400

parms alpha=-37 beta=190 sigma=16eta = alpha + betarace + u mu = exp(eta)model response ~ poisson(mu)random u ~ normal(0 sigmasigma) subject=casereplicate count

run--------------------------------------------------------------------------

Table 38 SAS Code for Discriminant Analysis for Horseshoe CrabData Analyzed in Section 1512

--------------------------------------proc discrim data=crab crossvalidate

priors propclass yvar width color

run--------------------------------------

36

dissimilarity index

Chapter 16 Large- and Small-Sample Theory for Multino-mial Models

Exact conditional logistic regression is available in PROC LOGISTIC with the EXACTstatement It provides ordinary and mid-P -values as well as confidence limits for eachmodel parameter and the corresponding odds ratio with the ESTIMATE = BOTHoption Or you can pick the type of confidence interval you want by specifyingCLTYPE=EXACT or CLTYPE=MIDP In particular this enables you to get theCornfield exact interval for an odds ratio (also available with PROC FREQ as shownabove in Table 4) or its mid-P adaptation You can also conduct the exact condi-tional version of the CochranndashArmitage test using the TREND option in the EXACTstatement with PROC FREQ One can also conduct an asymptotic conditional logisticregression using a STRATA statement to indicate the stratification parameters to beconditioned out PROC PHREG can also do this (Stokes et al 2012) For a 2times2timesKtable using the EQOR option in an EXACT statement in PROC FREQ provides anexact test for equal odds ratios proposed by Zelen (1971) as shown above in Table 12

37

Table 39 SAS Code for Cluster Analysis for Table 156 on StatewideGrounds for Divorce

-----------------------------------------------------------------------data divorceinput state $ incompat cruelty desertn non_supp alcohol

felony impotenc insanity separate datalinesCalifornia 1 0 0 0 0 0 0 1 0Florida 1 0 0 0 0 0 0 1 0Illinois 0 1 1 0 1 1 1 0 0Massachusetts 1 1 1 1 1 1 1 0 1Michigan 1 0 0 0 0 0 0 0 0NewYork 0 1 1 0 0 1 0 0 1Texas 1 1 1 0 0 1 0 1 1Washington 1 0 0 0 0 0 0 0 1

title Grounds for Divorceproc distance data=divorce method=djaccard absent=0 out=distjaccvar anominal(incompat--separate)id state

runproc print data=distjacc(obs=8)id statetitle2 Only 8 states

runproc cluster data=distjacc method=averagepseudo outtree=treeid state

runproc tree data=tree horizontal n=7 out=out id staterun-----------------------------------------------------------------------

38

dissimilarity index

Chapter 16 Large- and Small-Sample Theory for Multino-mial Models

Exact conditional logistic regression is available in PROC LOGISTIC with the EXACTstatement It provides ordinary and mid-P -values as well as confidence limits for eachmodel parameter and the corresponding odds ratio with the ESTIMATE = BOTHoption Or you can pick the type of confidence interval you want by specifyingCLTYPE=EXACT or CLTYPE=MIDP In particular this enables you to get theCornfield exact interval for an odds ratio (also available with PROC FREQ as shownabove in Table 4) or its mid-P adaptation You can also conduct the exact condi-tional version of the CochranndashArmitage test using the TREND option in the EXACTstatement with PROC FREQ One can also conduct an asymptotic conditional logisticregression using a STRATA statement to indicate the stratification parameters to beconditioned out PROC PHREG can also do this (Stokes et al 2012) For a 2times2timesKtable using the EQOR option in an EXACT statement in PROC FREQ provides anexact test for equal odds ratios proposed by Zelen (1971) as shown above in Table 12

37

Table 39 SAS Code for Cluster Analysis for Table 156 on StatewideGrounds for Divorce

-----------------------------------------------------------------------data divorceinput state $ incompat cruelty desertn non_supp alcohol

felony impotenc insanity separate datalinesCalifornia 1 0 0 0 0 0 0 1 0Florida 1 0 0 0 0 0 0 1 0Illinois 0 1 1 0 1 1 1 0 0Massachusetts 1 1 1 1 1 1 1 0 1Michigan 1 0 0 0 0 0 0 0 0NewYork 0 1 1 0 0 1 0 0 1Texas 1 1 1 0 0 1 0 1 1Washington 1 0 0 0 0 0 0 0 1

title Grounds for Divorceproc distance data=divorce method=djaccard absent=0 out=distjaccvar anominal(incompat--separate)id state

runproc print data=distjacc(obs=8)id statetitle2 Only 8 states

runproc cluster data=distjacc method=averagepseudo outtree=treeid state

runproc tree data=tree horizontal n=7 out=out id staterun-----------------------------------------------------------------------

38

Table 39 SAS Code for Cluster Analysis for Table 156 on StatewideGrounds for Divorce

-----------------------------------------------------------------------data divorceinput state $ incompat cruelty desertn non_supp alcohol

felony impotenc insanity separate datalinesCalifornia 1 0 0 0 0 0 0 1 0Florida 1 0 0 0 0 0 0 1 0Illinois 0 1 1 0 1 1 1 0 0Massachusetts 1 1 1 1 1 1 1 0 1Michigan 1 0 0 0 0 0 0 0 0NewYork 0 1 1 0 0 1 0 0 1Texas 1 1 1 0 0 1 0 1 1Washington 1 0 0 0 0 0 0 0 1

title Grounds for Divorceproc distance data=divorce method=djaccard absent=0 out=distjaccvar anominal(incompat--separate)id state

runproc print data=distjacc(obs=8)id statetitle2 Only 8 states

runproc cluster data=distjacc method=averagepseudo outtree=treeid state

runproc tree data=tree horizontal n=7 out=out id staterun-----------------------------------------------------------------------

38