S052/ § I.1(a): Applied Data Analysis Roadmap of the Course – What Is Today’s Topic?

© Willett, Harvard University Graduate School of Education S052/I.1(a) – Slide 1

S052/§I.1(a): Applied Data AnalysisRoadmap of the Course – What Is Today’s Topic?

More details can be found in the “Course Objectives and Content” handout on the course webpage.

Multiple RegressionAnalysis (MRA) iiii XXY 22110

Do your residuals meet the required assumptions?

Test for residual

normality

Use influence statistics to

detect atypical datapoints

If your residuals are not independent,

replace OLS by GLS regression analysis

Use Individual

growth modeling

Specify a Multi-level

Model

If your sole predictor is continuous, MRA is

identical to correlational analysis

If your sole predictor is dichotomous, MRA is identical to a t-test

If your several predictors are

categorical, MRA is identical to ANOVA

If time is a predictor, you need discrete-

time survival analysis…

If your outcome is categorical, you need to

use…

Binomial logistic

regression analysis

(dichotomous outcome)

Multinomial logistic

regression analysis

(polytomous outcome)

If you have more predictors than you

can deal with,

Create taxonomies of fitted models and compare

them.Form composites of the indicators of any common

construct.

Conduct a Principal Components Analysis

Use Cluster Analysis

Use non-linear regression analysis.

Transform the outcome or predictor

If your outcome vs. predictor relationship

is non-linear,

How do you deal with missing

data?

Today’s Topic Area

© Willett, Harvard University Graduate School of Education S052/I.1(a) – Slide 2

S052/§I.1(a): Applied Data AnalysisWhere Does Today’s Topic Appear in the Printed Syllabus?

In the future, I ask you to keep automatically tabs on the inter-connections among the Roadmap, the Daily Topic Area, the Printed Syllabus, and the content of the day’s class, when you first download and pre-read the required day’s class materials.

Today’s topic, Deciding Which Regression Models to Fit, is from Syllabus Section I.1(a) and includes: Slides 3-4: Introducing the ILLCAUSE data-example. Slides 5-6: Imagining a “Universe Of All Possible

Models”. Slides 7-8: Two Strategies For Choosing Sensible

Subsets of Regression Models To Fit. Slides 9-14: Exploratory Univariate & Bivariate

Analyses in the ILLCAUSE Dataset. Slide 15: Establishing Priorities Among the Predictors. Slides 16-18: Fitting a Taxonomy of Regression Models

in the ILLCAUSE Dataset. Slide 19: Decoding Standard Regression Output. Slide 20: APA-Style Table Displaying a Taxonomy Of

Fitted Regression Models. Slides 21-22: Appendix 1. Slide 23: Appendix 2.

© Willett, Harvard University Graduate School of Education S052/I.1(a) – Slide 3

S052/§I.1(a): Deciding Which Multiple Regression Models To Fit What Question, and Dataset, Will Drive Our Presentation Today?

As an introductory example of this part of the syllabus -- “Deciding Which Multiple Regression Models To Fit” -- here’s a multiple regression analysis to address the following broad question:Do children suffering from different types of chronic

illness understand the causes of illness differently from each other, and from healthy children?

Dataset on the website: ILLCAUSE.txtCodebook on the website: ILLCAUSE_info

Dataset ILLCAUSE.txt

Overview Data for investigating differences in children’s understanding of the causes of illness, by their health status.

Source Perrin E.C., Sayer A.G., and Willett J.B. (1991). Sticks And Stones May Break My Bones: Reasoning About Illness Causality And Body Functioning In Children Who Have A Chronic Illness, Pediatrics, 88(3), 608-19.

Sample size 301 children, including a sub-sample of 205 who were described as asthmatic, diabetic, or healthy. After further reductions due to the list-wise deletion of cases with missing data on one or more variables, the analytic sub-sample used in class ends up containing 33 diabetic children, 68 asthmatic children and 93 healthy children.

More info Chronically-ill children were recruited into the study through their pediatricians; healthy children were a matched random sample drawn from the same schools as the ill children.

Updated September 16, 2005

© Willett, Harvard University Graduate School of Education S052/I.1(a) – Slide 4

S052/§I.1(a): Deciding Which Multiple Regression Models To Fit What Variables Will We Focus On In Our Analyses?

Col Variable Variable Description Variable Metric/Labels 1 ID Child identification code Integers

2 ILLCAUSE Child’s score on a measure of the understanding of illness causality.

Ordinal score obtained by averaging child responses to 7 interview questions on the causes of illness, with responses rated on a “developmental” scale: 1 = No response. 2 = Phenomenistic or circular response. 3 = External agency cited as sole cause. 4 = Internalization in understanding illness, once agent

internalized, illness is inevitable. 5 = Interaction of host and agent described. 6 = Mechanisms of illness causation described, including notions

of treatment and bodily response.

3 SES

Family socio-economic status, rated using the education and employment levels of the primary bread-winner with Hollingshead Two-Factor Index of Social Position. (Hollingshead & Frederick. Social Class and Mental Illness. NY: Wiley, 1958)

Ordinal rating of social class: 1 = upper 2 = upper middle 3 = middle 4 = lower middle 5 = low (Notice the ordering of the numerical values is counterintuitive).

4 PPVT Child’s normed score on the Peabody Picture Vocabulary Test.

Continuous score, mean of 100 & standard deviation of 15 in population.

5 AGE Child age Continuous variable, months since birth.

6 GENREAS Child’s score on a measure of general reasoning.

Ordinal score, from 1 through 6. Similar to ILLCAUSE, but requires general reasoning, rather than reasoning about illness.

7 HEALTH Child Health Status Indicator

Categorical variable with multiple categories, of which we are interested in:

3 = Diabetic 5 = Asthmatic 6 = Healthy

© Willett, Harvard University Graduate School of Education S052/I.1(a) – Slide 5

S052/§I.1(a): Deciding Which Multiple Regression Models To Fit Even With A Few Predictors, There Are So Many Models You Can Possibly Fit!

To address RQ about children’s Understanding of Illness Causality: You must identify ILLCAUSE as the outcome. You must identify “HEALTH,” AGE, and SES as predictors.

And proceed with a multiple regression analysis …

… and then, what about non-linear expressions of the continuous predictors, or categorical versions, or what if you add another predictor like gender or race, or … How Many Potential Models Would There Be, Then?

The task seems reasonable until you begin to enumerate how many possible models you can actually specify using just these few predictors …

Three models with a 1 main effect

ii

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SESILLCAUSEAGEILLCAUSEHEALTHILLCAUSE

10

10

10 ""

Three models with 2 main effects and 1 two-way interaction

ii

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SESAGESESAGEILLCAUSESESHEALTHSESHEALTHILLCAUSEAGEHEALTHAGEHEALTHILLCAUSE

3210

3210

3210

""""""""

One model with 3 main effects

ii SESAGEHEALTHILLCAUSE 3210 ""

Three models with 2 main effects

ii

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SESAGEILLCAUSESESHEALTHILLCAUSEAGEHEALTHILLCAUSE

210

210

210

""""

Three models with 3 main effects and 1 two-way interaction

ii

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SESAGESESAGEHEALTHILLCAUSESESHEALTHSESAGEHEALTHILLCAUSEAGEHEALTHSESAGEHEALTHILLCAUSE

43210

43210

43210

""""""""""

Three models with 3 main effects and 2 two-way interactions

One model with 3 main effects and 3 two-way interactions

and so on ...

© Willett, Harvard University Graduate School of Education S052/I.1(a) – Slide 6

S052/§I.1(a): Deciding Which Multiple Regression Models To Fit How Big Is the Universe of All Possible Models, and How Can You Map it?

Number of feasible regression models that can be specified increases exponentially as the number of potential predictors rises:

predictors of 5.141 numbere

“Initial” model contains main effect of the question

predictor HEALTH?

*

Second model adds the main effect of control

predictor AGE?

*Third model adds two-

way interaction of HEALTH and AGE?

*

Fourth model adds the main effect of control

predictor, SES? *

next?

You are here!

So, with a single outcome and thirteen predictors …

The “Universe of All Possible Models” contains 73,566,892 potential specifications …

It seems plausible to ask, then …In this Universe, What Strategy Can Lead You To The “Best” Subset Of Models?

© Willett, Harvard University Graduate School of Education S052/I.1(a) – Slide 7

S052/§I.1(a): Deciding Which Multiple Regression Models To Fit Two Broad Strategies For Deciding Which Models To Specify And Fit

Two Broad Classes of Model-Specification Strategy

More Thoughtful Methods: Use your knowledge of the research

question, the supporting substantive theory and logic to specify and fit a systematic taxonomy of regression models.

Make sure your decisions are driven by your need to answer specific research questions, to test reasonable hypotheses, and to tell a good story.

Example Follows … Using ILLCAUSE Data:

• We’ll use the same example later to refine our ability to conduct regression analyses by adding new tools:General Linear Hypothesis (GLH) Testing,Influence Statistics,Innovations in Residual Analysis,Strategies For Improved Interpretation Of

Fitted Models.

Well-known “Automated” Methods:• Forward selection …• Backward elimination …• Stepwise regression …• All-possible-subsets regression … erk!

I Don’t Recommend These Methods At All:• Choice of models is abdicated to a computer.• Choice of each subsequent model can be strongly

impacted by relationships between the predictors already in the model and any potential subsequent predictors.

Please Don’t Ever Use These Methods …but do read about them in the post-class readings so

that you can recognize what to avoid!

© Willett, Harvard University Graduate School of Education S052/I.1(a) – Slide 8

S052/§I.1(a): Deciding Which Multiple Regression Models To Fit Example of Specifying a Sensible Taxonomy of Regression Models to Data

Do children suffering from different types of chronic illness understand the causes of illness differently from

each other, and from healthy children?Here starts the data-example – my illustrative data-analyses are

contained in ...

Data-Analytic Handout I.1(a).1 Available on class website. Features exploratory univariate and

bivariate analyses of the ILLCAUSE data.

Data-Analytic Handout I.1(a).2 Available on class website. Features the fitting of one sensible taxonomy of

regression models to the ILLCAUSE data.

A “Do it Yourself” PC-SAS Activity is available on the course website: Full_PCSAS_Manual. Trimmed_PCSAS_Manual. Additional Support Materials.

All S052 Data-Analytic Handoutscontain “model” PC-SAS Code and Statistical Output

I asked you to print these handouts out, and include them in your package of materials for today’s class.

Serve as “models” for your own future data-analyses, such as our regular Data-Analytic Memos (DAMs), and your future research.

Consult them carefully as you work on the DAM assignments.A few programming comments follow …

© Willett, Harvard University Graduate School of Education S052/I.1(a) – Slide 9

S052/§I.1(a): Deciding Which Multiple Regression Models To FitAnnotated PC-SAS Code, From The Exploratory Analysis

OPTIONS Nodate Pageno=1; TITLE1 'S-052: APPLIED DATA ANALYSIS';TITLE2 'I: Conducting Sensible Multiple Regression Analyses';TITLE3 'I.1(a): Fitting Taxonomies of Multiple Regression Models';TITLE4 'Handout #1: Introducing the Data';TITLE5 'Understanding of Illness Causality and Chronic Illness in Children'; *-------------------------------------------------------------------------*Input data, name and label variables in dataset*-------------------------------------------------------------------------*; DATA ILLCAUSE; INFILE 'C:\DATA\S052\ILLCAUSE.txt'; INPUT ID ILLCAUSE SES PPVT AGE GENREAS HEALTH; IF HEALTH=3 OR HEALTH=5 OR HEALTH=6; LABEL ID = 'Subject Identification Code' ILLCAUSE = 'Understanding of Illness Causality' SES = 'Hollingshead SES' PPVT = 'Score on Peabody Picture Vocabulary Test' AGE = 'Chronological Age (Months)' GENREAS = 'General Reasoning Ability' HEALTH = 'Health Status';

PROC FORMAT; VALUE HFMT 3 = 'Diabetic‘ 5 = 'Asthmatic‘ 6 = 'Healthy‘;

Data-Analytic Handout I.1(a).1 starts like this…

As many TITLEs as you like – the title text is in single quotes,

and ends in a semicolon

In the DATA step, raw data is read from an external file into a

temporary SAS dataset, here called ILLCAUSE

In this PROCedure, labels are created for the values of the

categorical HEALTH variable.

In the INPUT command,

variables are listed in the order they

appear in the dataset.

This IF statement selects out the

diabetic, asthmatic and healthy children.

In the INFILE command, you specify the location of the raw data.

Comments can run over several lines – they start with an asterisk

and end with a semicolon

© Willett, Harvard University Graduate School of Education S052/I.1(a) – Slide 10

S052/§I.1(a): Deciding Which Multiple Regression Models To Fit Annotated PC-SAS Code For Conducting Exploratory Analysis

*-------------------------------------------------------------------------* List data for subsample of 30 cases *-------------------------------------------------------------------------*; PROC PRINT LABEL DATA=ILLCAUSE(obs=35); TITLE6 'Outcome and Predictor Values For the First 35 Children'; FORMAT HEALTH HFMT.; VAR ID ILLCAUSE AGE HEALTH SES;

*-------------------------------------------------------------------------* Display selected univariate and bivariate summary information *-------------------------------------------------------------------------*; PROC TABULATE DATA=ILLCAUSE; TITLE6 'Selected Univariate Summaries of ILLCAUSE, AGE and SES'; CLASS HEALTH; FORMAT HEALTH HFMT.; VAR ILLCAUSE AGE SES; TABLE (ILLCAUSE AGE SES)*(HEALTH ALL), N MEAN STD MIN MAX; PROC PLOT DATA=ILLCAUSE; TITLE6 'Bivariate Scatterplots for ILLCAUSE by AGE and SES'; PLOT ILLCAUSE*(AGE SES); PROC CORR NOSIMPLE DATA=ILLCAUSE; TITLE6 'Estimated Bivariate Correlations ...'; VAR ILLCAUSE AGE SES; RUN;

Print the first 35 cases for inspection and review

Add a sixth title to the output for this procedure

Format the values of the HEALTH variable, just for this procedure

List the variables in the order you want them printing

Inspect the bivariate relationships between outcome, ILLCAUSE,

and the two continuous controls, AGE & SES, by plotting the

values of the outcome against each predictor, in scatter-plots

Create a tabulation of summary statistics (mean, standard deviation, minimum and maximum) for the

three continuous variables in the analysis -- ILLCAUSE (outcome), AGE and SES (control

predictors) – separately by the values of categorical (“CLASSification”) variable HEALTH.

Estimate bivariate correlation coefficients to summarize the bivariate relationships between the outcome, ILLCAUSE, and the continuous

controls, AGE & SES

Don’t forget to “RUN”

and continues like this …

© Willett, Harvard University Graduate School of Education S052/I.1(a) – Slide 11

S052/§I.1(a): Deciding Which Multiple Regression Models To Fit Listing a Few Cases from the ILLCAUSE Datasest

Subject Understanding Identification of Illness Chronological Health Hollingshead Obs Code Causality Age (Months) Status SES 1 301 . 128 Diabetic 2 2 302 2.857 79 Diabetic 2 3 303 3.429 151 Diabetic 3 4 304 4.286 178 Diabetic 3 5 305 4.286 113 Diabetic 4 6 306 3.286 81 Diabetic 2 7 307 2.857 194 Diabetic 4 ...

Notice that a period (.) is used in PC-SAS as the default

missing value code.

Notice that the children's values of SES are

heterogeneous and remember that higher

values mean lower SES!!!

Notice the heterogeneous ages of the sampled children

(in months)

“Obs” represents the values of a SAS “system” variable that counts and identifies the

observations in the order in which they appear in the temporary SAS dataset

Selected output from Data-Analytic Handout I.1(a).1 … data on a few early cases

The values of the outcome, ILLCAUSE, are listed here for each

child

Notice that the health status of the children has been reformatted by my PC-SAS program from a

numerical to an alphabetic label

© Willett, Harvard University Graduate School of Education S052/I.1(a) – Slide 12

S052/§I.1(a): Deciding Which Multiple Regression Models To Fit Exploratory Bivariate Tabulations of the ILLCAUSE Data

What Four Things Do You Notice That May Usefully

Inform Subsequent Regression Analysis?

Tabular descriptive output on the principal variables, by children’s health status … from Data-Analytic Handout I.1(a).1

© Willett, Harvard University Graduate School of Education S052/I.1(a) – Slide 13

S052/§I.1(a): Deciding Which Multiple Regression Models To Fit Exploratory Bivariate Scatterplot of the ILLCAUSE/AGE Relationship

What Four Things Do You Notice In This Plot That May Usefully Inform Subsequent Regression

Analysis?

Scatterplot of the bivariate relationship between outcome ILLCAUSE and control predictor AGE …

© Willett, Harvard University Graduate School of Education S052/I.1(a) – Slide 14

S052/§I.1(a): Deciding Which Multiple Regression Models To Fit Exploratory Bivariate Scatterplot of the ILLCAUSE/SES Relationship

What Four Things Do You Notice In This Plot That May Usefully Inform Subsequent Regression

Analysis?

Scatterplot of the bivariate relationship between outcome ILLCAUSE and control predictor SES …

© Willett, Harvard University Graduate School of Education S052/I.1(a) – Slide 15

S052/§I.1(a): Deciding Which Multiple Regression Models To Fit How Do You Come Up With A “Thoughtful Taxonomy Of Fitted Regression Models”?

There is no single “right way.” You needto fit and examine a sensible subset of fitted models that respond directly to your research question, without exploring the entire universe of possible models! It’s a good idea to:First, identify your outcome variable -- here, ILLCAUSE (Duh!).Second, establish important classes of predictors, based on substance (i.e., your research questions and theoretical

framework), your research design, … etc, and establish the priorities among them.Third, choose a sensible order in which to enter the classes into the regression model, again based on substance --

an example follows, but see the appendix. Enter the predictors systematically in their classes, exhausting one class before proceeding to the next.

At each step, once the main effects have been exhausted, consider the interactions.

Priority Predictor Comment

High HEALTH is the key question predictor. Without health status in the final model, we cannot address the research questions!

MediumAGE is a key control predictor because it represents the multi-cohort nature of the research design:

Sample contains multiple sub-samples of children of different ages. By controlling for AGE, we can include all the children in the same analysis, regardless of their age, rather than doing an “age-slice by age-slice” analysis, as was suggested by one ill-informed reviewer!!!

LowSES is a subsidiary control predictor. It is often worth including because some twit will always ask you if it had an effect.

In these data, descriptive analyses suggest that ill children have lower SES. So, if understanding illness also depends on home resources, the effect of SES could masquerade as an effect of HEALTH.

*-------------------------------------------------------------------------* Input the data, name and label the variables in the dataset*-------------------------------------------------------------------------*; DATA ILLCAUSE; INFILE 'C:\DATA\S052\ILLCAUSE.txt'; INPUT ID ILLCAUSE SES PPVT AGE GENREAS HEALTH; IF HEALTH=3 OR HEALTH=5 OR HEALTH=6;*-------------------------------------------------------------------------* Construction of the required dummy predictors and interesting interactions *-------------------------------------------------------------------------*; DATA ILLCAUSE; SET ILLCAUSE; * Creating illness-group dummies to serve as principal question predictors; IF HEALTH=3 THEN D=1; ELSE D=0; IF HEALTH=5 THEN A=1; ELSE A=0; IF HEALTH=6 THEN H=1; ELSE H=0; * Creating two-way health status by age interactions;

DxAGE = D*AGE; AxAGE = A*AGE; HxAGE = H*AGE;

* Creating two-way health status by SES interactions; DxSES = D*SES; AxSES = A*SES; HxSES = H*SES;

* Creating two-way AGE by SES interaction; AGExSES = AGE*SES;

* Creating three-way health status by age by SES interactions; DxAGExSES = D*AGE*SES; AxAGExSES = A*AGE*SES; HxAGExSES = H*AGE*SES;

© Willett, Harvard University Graduate School of Education S052/I.1(a) – Slide 16

S052/§I.1(a): Deciding Which Multiple Regression Models To Fit Annotated PC-SAS Code For Conducting the Multiple Regression Analyses

Read the data into a temporary SAS dataset called ILLCAUSE

Access the temporary SAS dataset, ILLCAUSE, a second

time to manipulate the data prior to regression analysis

Convert the categorical HEALTH status variable into a

system of dummy variables

Here’s the example from Data-Analytic Handout I.1(a).2, starting with the Data Input steps in which I first read and recode/create the variables that will be needed in the regression analysis …

Create a set of two-way HEALTH by AGE interactions

Create a set of two-way HEALTH by SES interactions

Create an AGE by SES interactions

Create a set of three-way HEALTH by AGE by SES

interactions

*-------------------------------------------------------------------------* Fitting one possible sensible taxonomy of nested regression models*-------------------------------------------------------------------------*; PROC REG DATA=ILLCAUSE; VAR ILLCAUSE D A H AGE SES;

* Estimating the total main effect of health status; M1: MODEL ILLCAUSE = D A;

* Accounting for important issues of research design; * Controlling for the presence of multiple age-cohorts of children; * Checking the main effect of AGE; M2: MODEL ILLCAUSE = D A AGE; * Checking the two-way interaction of health status and AGE; M3: MODEL ILLCAUSE = D A AGE DxAGE AxAGE;

* Controlling for additional substantive effects; * Checking the main effect of socioeconomic status; M4: MODEL ILLCAUSE = D A AGE DxAGE AxAGE SES; * Checking that all interactions with SES, simultaneously, are not needed; M5: MODEL ILLCAUSE = D A AGE DxAGE AxAGE SES DxSES AxSES AGExSES DxAGExSES AxAGExSES;

© Willett, Harvard University Graduate School of Education S052/I.1(a) – Slide 17

S052/§I.1(a): Deciding Which Multiple Regression Models To Fit Annotated PC-SAS Code For Conducting the Multiple Regression Analyses

And here’s the PROC REG step …

Specify the predictors you want included in each hypothesized model

Nominate the variable that is the outcome

Indicate that you are about to specify a regression model

Optionally label each model so that you can

refer to it later

You can specify many regression models in one PROC REG “paragraph.”

PROC REG is the PC-SAS procedure for conducting

multiple regression analysis

Begin by listing all variables that will be used in any regression model in order to ensure listwise deletion of cases with missing values

The categorical question predictor, HEALTH, is

represented by only two of the health status dummies: D (for diabetics) and A (for

asthmatics). For a discussion of this, see

Appendix 1.

© Willett, Harvard University Graduate School of Education S052/I.1(a) – Slide 18

S052/§I.1(a): Deciding Which Multiple Regression Models To Fit Annotated PC-SAS Code For Conducting the Multiple Regression Analyses

What Logic Drove The Specification Of

This Sequence Of Regression Models In

My Taxonomy?

*-------------------------------------------------------------------------* Fitting one possible sensible taxonomy of nested regression models*-------------------------------------------------------------------------*; PROC REG DATA=ILLCAUSE; VAR ILLCAUSE D A H AGE SES;

* Estimating the total main effect of health status; M1: MODEL ILLCAUSE = D A;

* Accounting for important issues of research design; * Controlling for the presence of multiple age-cohorts of children; * Checking the main effect of AGE; M2: MODEL ILLCAUSE = D A AGE; * Checking the two-way interaction of health status and AGE; M3: MODEL ILLCAUSE = D A AGE DxAGE AxAGE;

* Controlling for additional substantive effects; * Checking the main effect of socioeconomic status; M4: MODEL ILLCAUSE = D A AGE DxAGE AxAGE SES; * Checking that all interactions with SES, simultaneously, are not needed; M5: MODEL ILLCAUSE = D A AGE DxAGE AxAGE SES DxSES AxSES AGExSES DxAGExSES AxAGExSES;

*-------------------------------------------------------------------------* Rendering M4 more parsimonious by reconceptualizing health status *-------------------------------------------------------------------------*; DATA ILLCAUSE; SET ILLCAUSE; * Creating a new question predictor to identify ill children; IF D=1 OR A=1 THEN ILL=1; ELSE ILL=0; * Creating a new two-way interaction of ILL and AGE; ILLxAGE = ILL*AGE;

* Simplifying M4 by replacing former health status dummies by predictor ILL; PROC REG DATA=ILLCAUSE; VAR ILLCAUSE D A H AGE SES; M6: MODEL ILLCAUSE = ILL AGE ILLxAGE SES;

The REG Procedure Model: M1 Dependent Variable: ILLCAUSE

Number of Observations Read 205 Number of Observations Used 194 Number of Observations with Missing Values 11

Analysis of Variance

Sum of MeanSource DF Squares Square F Value Pr > F

Model 2 39.73731 19.86866 23.45 <.0001Error 191 161.80982 0.84717Corrected Total 193 201.54714

Root MSE 0.92042 R-Square 0.1972 Dependent Mean 4.13328 Adj R-Sq 0.1888 Coeff Var 22.26848

Parameter Estimates

Parameter Standard Variable DF Estimate Error t Value Pr > |t|

Intercept 1 4.60366 0.09544 48.23 <.0001 D 1 -0.83732 0.18650 -4.49 <.0001 A 1 -0.93560 0.14686 -6.37 <.0001

© Willett, Harvard University Graduate School of Education S052/I.1(a) – Slide 19

S052/§I.1(a): Deciding Which Multiple Regression Models To Fit What Does Regular Multiple Regression Output Look Like?

Can You Navigate & Interpret Typical Regression Output?

What hypothesis do these statistics test?

Can you interpret the estimated intercept?

What hypothesis does each of these pairs of statistics test?

Conceptually, what is standard error?

Can you interpret the R2 statistic?

Can you interpret the estimated coefficient associated with predictors D and A?

Can you interpret the “Root MSE” statistic?

Can you interpret the “Sum of Squares Error” -- or SSError -- statistic?

Can you interpret the “Sum of Squares Model” -- or SSModel -- statistic?

Can you interpret the “Sum of Squares Total” -- or SSTotal -- statistic?

© Willett, Harvard University Graduate School of Education S052/I.1(a) – Slide 20

S052/§I.1(a): Deciding Which Multiple Regression Models To Fit Finally, Assemble The Taxonomy of Fitted Models In An APA-Style Table

What are the critical features of APA formatting for tables?

Consult the style manuals and exemplars on the course website

We’ll dissect this taxonomy, and interpret its substantive story in great detail, in subsequent classes ...

© Willett, Harvard University Graduate School of Education S052/I.1(a) – Slide 21

S052/§I.1(a): Deciding Which Multiple Regression Models To Fit Appendix 1: Why Can Two Dummy Predictors Distinguish Among Three Groups?

The first fitted regression model (M1) from Data-Analytic Handout I.1(a).1 is:

From it, you can estimate the predicted value of ILLCAUSE in each health status group by substituting numerical values of the health status predictors that represent prototypical individuals in the dataset:

Notice that the predicted outcome values corresponding to one of the groups – the reference, omitted or comparison group (here, healthy children) – are obtained when the two dichotomous predictors that distinguish the chronically-ill children are both set to zero. This means that, if you have an intercept in the model, you need one less dummy predictor in the model than there are groups compared, as the fitted value for the “reference (or omitted) group” is provided by the estimated intercept.Another way of thinking about this is to understand that, although there are three distinct health status groups present, only two independent pieces of information are needed to indicate the health status of a child because if a child is neither diabetic nor asthmatic then s/he must be healthy, by default.Of course, you get to choose which of the health status groups serves as the reference, because you are the one who picks which dummy predictor is omitted from the regression model. Typically, you make this choice for substantive, not statistical, reasons.

iii ADUSEAILLC 94.084.060.4ˆ

76.394.060.4194.0084.060.4ˆ:

66.384.060.4094.0184.060.4ˆ:60.4094.0084.060.4ˆ:

i

i

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USEAILLC1A0;D AsthmaticUSEAILLC0A1;D DiabeticUSEAILLC0A0;DHealthy

© Willett, Harvard University Graduate School of Education S052/I.1(a) – Slide 22

S052/§I.1(a): Deciding Which Multiple Regression Models To Fit Appendix 1: Why Can Two Dummy Predictors Distinguish Among Three Groups?

Inspection of the fitted values computed on the previous slide indicate that the fitted regression parameters that we obtained in the analysis – that is, the estimated intercept parameter and the two estimated slope parameters associated with the dummy predictors representing health status, can be interpreted as follows:

iii ADUSEAILLC 210ˆˆˆˆ

The fitted slope parameter associated with dummy

predictor A represents the difference in the predicted

value of ILLCAUSE between the asthmatic and

“reference” healthy children – it is our best estimate of

the difference between asthmatic and healthy

children, on average, in the population (-0.94).

The fitted slope parameter associated with dummy

predictor D represents the difference in the predicted

value of ILLCAUSE between diabetic and “reference” healthy

children – it is our best estimate of the difference

between diabetic and healthy children, on

average, in the population (-0.84).

The fitted intercept represents the predicted

value of ILLCAUSE (4.60) for those in the reference

(or omitted) category –it is our best estimate of the

understanding of healthy children, on average, in the

population.

© Willett, Harvard University Graduate School of Education S052/I.1(a) – Slide 23

S052/§I.1(a): Deciding Which Multiple Regression Models To Fit Appendix 2: Other Strategies For Specifying Taxonomies Of Regression Models

“Baseline Control Model” Approach:• Form a baseline control model, by sequentially adding

control predictors, highest priority first, and testing for appropriate interactions as you go along.

• Then, add the main effects of the question predictors to the new baseline control model.

• Then, add interactions between the question predictors and the control predictors in the baseline control model, sequentially.

• Finally, add interactions between the question predictors.

Here, the objective is to obtain a parsimonious model that controls away all extraneous variation first, and then focus attention on the impact of the question predictors. While this approach refines your view of the impact of the question predictors, removing that part of their effect that may depend on the inter-relationships with the controls, it never reveals the “total” impact of the question predictors on the outcome for a person who has been randomly selected from the population without regard to any of their other characteristics.

“Work Back From The End” Approach:• Include all possible predictors in the model, both their

main effects and interactions.• The, remove statistically unimportant predictors

sequentially to achieve a more parsimonious model, starting with those of lowest declared priority that do not appear to have statistically significant effects (i.e., remove question predictors last).

• Make sure that you remove any statistically unimportant ahead of any of the main effects from which they are constituted.

Here, the objective is to obtain a final parsimonious model by sequentially removing predictors that appear unimportant. The idea is that you get to see the impact of “everything” to start with, and then you can “slim down” the fitted model to a final model. However, the impact of main effects is always masked when interactions are present in the model, and you still may remove an important predictor whose correlation with another predictor makes it look unimportant.

Devise Your Own Strategy?• It’s acceptable to devise your own strategy, in fact it’s

probably the best approach as you know the field the best!.

It’s perfectly acceptable to devise your own strategy providing that it is systematic, sensible and that you explain it explicitly to your reader, describing the logic that underpins its construction.