Experimental design and statistical analyses of data Lesson 4: Analysis of variance II A posteriori tests Model control How to choose the best model

Experimental design and statistical analyses of data

Lesson 4:

Analysis of variance II

A posteriori tests

Model control

How to choose the best model

Growth of bean plants in four different media

Zn Cu Mn Control Overall

Biomass (y)

61.7

59.4

60.5

59.2

57.6

57.0

58.4

57.3

57.8

59.9

62.3

66.2

65.2

63.7

64.1

58.1

56.3

58.9

57.4

56.1

ni 5

59.7

2.35

5

58.1

1.33

5

64.3

2.20

5

57.4

1.43

20

59.86

9.207

Completely randomized design (one-way anova)

iy2is

3322110 xxxy

How to do it with SAS

DATA medium;

/* 20 bean plants exposed to 4 different treatments

(5 plants per treatment)

Mn = extra mangan added to the soil

Zn = ekstra zink added to the soil

Cu = ekstra cupper added to the soil

K = control soil

The dependent variable (Mass) is the biomass of the plants at harvest */

INPUT treat $ mass ;

/* treat = treatment */

/* mass = biomass of a plant */

CARDS;

zn 61.7

zn 59.4

zn 60.5

zn 59.2

zn 57.6

cu 57.0

cu 58.4

cu 57.3

cu 57.8

cu 59.9

mn 62.3

mn 66.2

mn 65.2

mn 63.7

mn 64.1

k 58.1

k 56.3

k 58.9

k 57.4

k 56.1

;

PROC SORT; /* sort the observations according

to treatment */

BY treat;

RUN;

/* compute average and 95% confidence limits for each treatment */

PROC MEANS N MEAN CLM;

BY treat;

RUN;

1 14:09 Wednesday, November 7, 2001 Analysis Variable : MASS ------------------------------------- TREAT=cu --------------------------------- N Mean Lower 95.0% CLM Upper 95.0% CLM -------------------------------------------------- 5 58.0800000 56.6550587 59.5049413 -------------------------------------------------- -------------------------------------- TREAT=k --------------------------------- N Mean Lower 95.0% CLM Upper 95.0% CLM -------------------------------------------------- 5 57.3600000 55.8866517 58.8333483 -------------------------------------------------- ------------------------------------- TREAT=mn --------------------------------- N Mean Lower 95.0% CLM Upper 95.0% CLM -------------------------------------------------- 5 64.3000000 62.4562230 66.1437770 -------------------------------------------------- ------------------------------------- TREAT=zn --------------------------------- N Mean Lower 95.0% CLM Upper 95.0% CLM -------------------------------------------------- 5 59.6800000 57.7777805 61.5822195 --------------------------------------------------

PROC GLM;

CLASS treat;

MODEL mass = treat /SOLUTION;

/* SOLUTION gives the estimated parameter values */

RUN;

Class Levels Values TREAT 4 cu k mn zn Number of observations in data set = 20 General Linear Models Procedure Dependent Variable: MASS Sum of MeanSource DF Squares Square F Value Pr > FModel 3 145.82150 48.60717 26.72 0.0001Error 16 29.10800 1.81925 Corrected Total 19 174.92950 R-Square C.V. Root MSE MASS Mean 0.833602 2.253439 1.3488 59.855 Source DF Type I SS Mean Square F Value Pr > FTREAT 3 145.82150 48.60717 26.72 0.0001 Source DF Type III SS Mean Square F Value Pr > FTREAT 3 145.82150 48.60717 26.72 0.0001

T for H0: Pr > |T| Std Error ofParameter Estimate Parameter=0 Estimate INTERCEPT 59.68000000 B 98.94 0.0001 0.60319980TREAT cu -1.60000000 B -1.88 0.0791 0.85305334 k -2.32000000 B -2.72 0.0151 0.85305334 mn 4.62000000 B 5.42 0.0001 0.85305334 zn 0.00000000 B . . . NOTE: The X'X matrix has been found to be singular and a generalized inverse was used to solve the normal equations. Estimates followed by the letter 'B' are biased, and are not unique estimators of the parameters.

PROC GLM;

CLASS treat;



/*Test for pairwise differences between treatments by

linear contrasts */

CONTRAST 'Cu vs K' Treat 1 -1 0 0;

CONTRAST 'Cu vs Mn' Treat 1 0 -1 0;

CONTRAST 'Cu vs Zn' Treat 1 0 0 -1;

CONTRAST 'K vs Mn' Treat 0 1 -1 0;

CONTRAST 'K vs Zn' Treat 0 1 0 -1;

CONTRAST 'Mn vs Zn' Treat 0 0 1 -1;

/* test for whether the 3 treatments with added minerals are different from the control */

CONTRAST 'K vs Cu, Mn Zn' Treat 1 -3 1 1;

RUN;

Contrast DF Contrast SS Mean Square F Value Pr > F Cu vs K 1 1.29600 1.29600 0.71 0.4111Cu vs Mn 1 96.72100 96.72100 53.17 0.0001Cu vs Zn 1 6.40000 6.40000 3.52 0.0791K vs Mn 1 120.40900 120.40900 66.19 0.0001K vs Zn 1 13.45600 13.45600 7.40 0.0151Mn vs Zn 1 53.36100 53.36100 29.33 0.0001K vs Cu, Mn Zn 1 41.50017 41.50017 22.81 0.0002

PROC GLM;

CLASS treat;



/* Test for differences between levels of treatment */

MEANS treat / BON DUNCAN SCHEFFE TUKEY DUNNETT('k');

RUN;

Tukey's Studentized Range (HSD) Test for variable: MASS NOTE: This test controls the type I experimentwise error rate. Alpha= 0.05 Confidence= 0.95 df= 16 MSE= 1.81925 Critical Value of Studentized Range= 4.046 Minimum Significant Difference= 2.4406 Comparisons significant at the 0.05 level are indicated by '***'. Simultaneous Simultaneous Lower Difference Upper TREAT Confidence Between Confidence Comparison Limit Means Limit mn - zn 2.1794 4.6200 7.0606 *** mn - cu 3.7794 6.2200 8.6606 *** mn - k 4.4994 6.9400 9.3806 *** zn - mn -7.0606 -4.6200 -2.1794 *** zn - cu -0.8406 1.6000 4.0406 zn - k -0.1206 2.3200 4.7606 cu - mn -8.6606 -6.2200 -3.7794 *** cu - zn -4.0406 -1.6000 0.8406 cu - k -1.7206 0.7200 3.1606 k - mn -9.3806 -6.9400 -4.4994 *** k - zn -4.7606 -2.3200 0.1206 k - cu -3.1606 -0.7200 1.7206

Bonferroni (Dunn) T tests for variable: MASS NOTE: This test controls the type I experimentwise error rate but generally has a higher type II error rate than Tukey's for all pairwise comparisons. Alpha= 0.05 Confidence= 0.95 df= 16 MSE= 1.81925 Critical Value of T= 3.00833 Minimum Significant Difference= 2.5663 Comparisons significant at the 0.05 level are indicated by '***'. Simultaneous Simultaneous Lower Difference Upper TREAT Confidence Between Confidence Comparison Limit Means Limit mn - zn 2.0537 4.6200 7.1863 *** mn - cu 3.6537 6.2200 8.7863 *** mn - k 4.3737 6.9400 9.5063 *** zn - mn -7.1863 -4.6200 -2.0537 *** zn - cu -0.9663 1.6000 4.1663 zn - k -0.2463 2.3200 4.8863 cu - mn -8.7863 -6.2200 -3.6537 *** cu - zn -4.1663 -1.6000 0.9663 cu - k -1.8463 0.7200 3.2863 k - mn -9.5063 -6.9400 -4.3737 *** k - zn -4.8863 -2.3200 0.2463 k - cu -3.2863 -0.7200 1.8463

Scheffe's test for variable: MASS NOTE: This test controls the type I experimentwise error rate but generally has a higher type II error rate than Tukey's for all pairwise comparisons. Alpha= 0.05 Confidence= 0.95 df= 16 MSE= 1.81925 Critical Value of F= 3.23887 Minimum Significant Difference= 2.6591 Comparisons significant at the 0.05 level are indicated by '***'. Simultaneous Simultaneous Lower Difference Upper TREAT Confidence Between Confidence Comparison Limit Means Limit mn - zn 1.9609 4.6200 7.2791 *** mn - cu 3.5609 6.2200 8.8791 *** mn - k 4.2809 6.9400 9.5991 *** zn - mn -7.2791 -4.6200 -1.9609 *** zn - cu -1.0591 1.6000 4.2591 zn - k -0.3391 2.3200 4.9791 cu - mn -8.8791 -6.2200 -3.5609 *** cu - zn -4.2591 -1.6000 1.0591 cu - k -1.9391 0.7200 3.3791 k - mn -9.5991 -6.9400 -4.2809 *** k - zn -4.9791 -2.3200 0.3391 k - cu -3.3791 -0.7200 1.9391

Dunnett's T tests for variable: MASS NOTE: This tests controls the type I experimentwise error for comparisons of all treatments against a control. Alpha= 0.05 Confidence= 0.95 df= 16 MSE= 1.81925 Critical Value of Dunnett's T= 2.592 Minimum Significant Difference= 2.2115 Comparisons significant at the 0.05 level are indicated by '***'. Simultaneous Simultaneous Lower Difference Upper TREAT Confidence Between Confidence Comparison Limit Means Limit mn - k 4.7285 6.9400 9.1515 *** zn - k 0.1085 2.3200 4.5315 *** cu - k -1.4915 0.7200 2.9315

Duncan’s test exaggarates the risk of Type I errors

Comparison between multiple tests

Test Minimum significant difference

Duncan

Dunnett

Tukey

Bonferroni

Scheffe

1.951

2.2115

2.4406

2.5663

2.6591

Type I

Scheffe’s test exaggarates the risk of Type II errrors

Type II

Tukey’s test is recommended as the best!

PROC GLM;

CLASS treat;



/* Test for differences between different levels of treatment */

MEANS treat / BON DUNCAN SCHEFFE TUKEY lines;

RUN;

General Linear Models Procedure Duncan's Multiple Range Test for variable: MASS NOTE: This test controls the type I comparisonwise error rate, not the experimentwise error rate Alpha= 0.05 df= 16 MSE= 1.81925 Number of Means 2 3 4 Critical Range 1.808 1.896 1.951 Means with the same letter are not significantly different. Duncan Grouping Mean N TREAT A 64.3000 5 mn B 59.6800 5 zn B C B 58.0800 5 cu C C 57.3600 5 k

General Linear Models Procedure Tukey's Studentized Range (HSD) Test for variable: MASS NOTE: This test controls the type I experimentwise error rate, but generally has a higher type II error rate than REGWQ. Alpha= 0.05 df= 16 MSE= 1.81925 Critical Value of Studentized Range= 4.046 Minimum Significant Difference= 2.4406 Means with the same letter are not significantly different. Tukey Grouping Mean N TREAT A 64.3000 5 mn B 59.6800 5 zn B B 58.0800 5 cu B B 57.3600 5 k

General Linear Models Procedure Bonferroni (Dunn) T tests for variable: MASS NOTE: This test controls the type I experimentwise error rate, but generally has a higher type II error rate than REGWQ. Alpha= 0.05 df= 16 MSE= 1.81925 Critical Value of T= 3.01 Minimum Significant Difference= 2.5663 Means with the same letter are not significantly different. Bon Grouping Mean N TREAT A 64.3000 5 mn B 59.6800 5 zn B B 58.0800 5 cu B B 57.3600 5 k

General Linear Models Procedure Scheffe's test for variable: MASS NOTE: This test controls the type I experimentwise error rate but generally has a higher type II error rate than REGWF for all pairwise comparisons Alpha= 0.05 df= 16 MSE= 1.81925 Critical Value of F= 3.23887 Minimum Significant Difference= 2.6591 Means with the same letter are not significantly different. Scheffe Grouping Mean N TREAT A 64.3000 5 mn B 59.6800 5 zn B B 58.0800 5 cu B B 57.3600 5 k

PROC GLM;

CLASS treat;



/* In unbalanced (and balanced) designs LSMEANS can be used: */

LSMEANS treat /TDIF PDIFF;

RUN;

The GLM Procedure Least Squares Means LSMEAN treat mass LSMEAN Number cu 58.0800000 1 k 57.3600000 2 mn 64.3000000 3 zn 59.6800000 4 Least Squares Means for Effect treat t for H0: LSMean(i)=LSMean(j) / Pr > |t| Dependent Variable: mass i/j 1 2 3 4 1 0.844027 -7.29145 -1.87562 0.4111 <.0001 0.0791 2 -0.84403 -8.13548 -2.71964 0.4111 <.0001 0.0151 3 7.291455 8.135482 5.41584 <.0001 <.0001 <.0001 4 1.875615 2.719642 -5.41584 0.0791 0.0151 <.0001 NOTE: To ensure overall protection level, only probabilities associated with pre-planned

comparisons should be used.

Er denne P-værdi signifikant?

Den sekventielle Bonferroni-testDen sekventielle Bonferroni-test er mindre konservativ end den ordinære Bonferroni-test. Procedure:Først ordnes de k P-værdier i voksende rækkefølge. Lad P(i) betegne den i’te P-værdi efter at værdierne er

blevet ordnet i voksende rækkefølge. Herefter beregnes

1)(

iki

hvor α er det signifikansniveau, der benyttes, hvis der kun var en enkelt P-værdi (sædvanligvis 0.05). Hvis P(i) < α(i) er den i’te P-værdi signifikant.

i P(i) α(i) P(i)-α(i)

123456

0.00010.00010.00010.01510.07910.4111

0.00830.01

0.01250.01670.0250.05

-0.0082-0.0099-0.0124

-0.001570.05410.3611

Signifikante P-værdier

Model assumptions and model control

• All GLM’s are based on the assumption that

(1) ε is independently distributed

(2) ε is normally distributed with the mean = 0

(3) The variance of ε (denoted σ2) is the same for all values of the independent variable(s) (variance homogeneity)

(4) Mathematically this is written as ε is iid ND(0; σ2)

iid = independently and identically distributed

Transformation of data

• Transformation of data serves two purposes

(1) To remove variance heteroscedasticity

(2) To make data more normal

Usually a transformation meets both purposes, but if this is not possible, variance homoscedasticity is regarded as the most important, especially if sample sizes are large

How to choose the appropriate transformation?

y* = yp

We have to find a value of p, so that the transformedvalues of y (denoted y*)

meet the condition of being normally distributedand with a variance that is independent of y*.

A useful method to find p is to fit Taylor’s power law to data

Taylor’s power lawbyas 2

It can be shown that

p = 1- b/2

is the appropriate transformation we search for

ybas logloglog 2

yyypb 2

1

*2

1

2

111

yypb log*02

212

yypb *10

If y is a proportion, i.e. 0 <= y <= 1, an appropriate transformation is often

yy arcsin*

10-02

10-01

1000

1001

1002

1003

1004

Mean number of T. urticae per plant

10-02

10-01

1000

1001

1002

1003

1004

1005

1006

1007

1008

Sp

ati

al

va

ria

nc

e

10-02

10-01

1000

1001

1002

Mean number of P. persimilis per plant

10-02

10-01

1000

1001

1002

1003

1004

Sp

ati

al

va

ria

nc

e

(b)

(a)

T. urticae:log s2 = 1.303 + 1.943 log xr2 = 0.994y* = log(y+1)

P. persimilis:log s2 = 1.193 + 1.900 log xr2 = 0.992y* = log(y+1)

Exponential growth

Exponent.exe

Deterministic model: rNNdb

dt

dN

rtt eNN 0

Stochastic model:

tdNtbNtDBN )()()(

b = birth rate/capitad = death rate/capita

Instantaneous growth rate

N = population size at time t

r = net growth rate/capita

ΔN = change in N during Δt

B = birth rate D = death rate

ε = noise associated with birthsδ = noise associated with deaths

The number of births during a time interval follows a Poisson distribution with mean BΔt

The number of deaths during a time interval is binomially distributed with parameters (θ, N)

The probability that an individual dies during Δt is θ = DΔt/N

H:\LIN-MOD\Powerpoint presentations\Exponent.exe

Type I, II, III and IV SS

Example: Mites in stored grain influenced by temperature (T) and humidity (H)

THHHTTy 52

432

210

DATA mites;

INFILE'h:\lin-mod\besvar\opg1-1.prn' FIRSTOBS=2;

INPUT pos $ depth T H Mites;

/* pos = position in store */

/* depth = depth in m */

/* T = Temperature of grain */

/* H = Humidity of grain */

/* Mites = number of mites in sampling unit */

logMites = log10(Mites+1); /* log transformation of Mites */

T2 = T**2; /* square temperature */

H2 = H**2; /* square humidity */

TH = T*H; /* product of temperature and humidity */

PROC GLM;

CLASS pos;

MODEL logMites = T T2 H H2 TH /SOLUTION SS1 SS3;

RUN;

General Linear Models Procedure

Dependent Variable: LOGMITES Source DF Sum of Squares Mean Square F Value Pr > F Model 5 2.72839285 0.54567857 2.94 0.0265 Error 33 6.12429305 0.18558464 Corrected Total 38 8.85268590 R-Square C.V. Root MSE LOGMITES Mean 0.308199 85.66578 0.43079535 0.50287914 T for H0: Pr > |T| Std Error ofParameter Estimate Parameter=0 Estimate INTERCEPT 28.03994955 0.54 0.5902 51.56270293T -0.86682324 -1.27 0.2147 0.68517409T2 0.02333784 2.19 0.0358 0.01066368H -3.52741058 -0.50 0.6235 7.11853025H2 0.12548846 0.51 0.6161 0.24789107TH 0.02315214 0.43 0.6702 0.05388643


Dependent Variable: LOGMITES Source DF Sum of Squares Mean Square F Value Pr > F Model 5 2.72839285 0.54567857 2.94 0.0265 Error 33 6.12429305 0.18558464 Corrected Total 38 8.85268590 R-Square C.V. Root MSE LOGMITES Mean 0.308199 85.66578 0.43079535 0.50287914 Source DF Type I SS Mean Square F Value Pr > F T 1 0.22115656 0.22115656 1.19 0.2829T2 1 1.38171889 1.38171889 7.45 0.0101H 1 1.03546840 1.03546840 5.58 0.0242H2 1 0.05579073 0.05579073 0.30 0.5872TH 1 0.03425827 0.03425827 0.18 0.6702 Source DF Type III SS Mean Square F Value Pr > F T 1 0.29703065 0.29703065 1.60 0.2147T2 1 0.88889243 0.88889243 4.79 0.0358H 1 0.04556941 0.04556941 0.25 0.6235H2 1 0.04755847 0.04755847 0.26 0.6161TH 1 0.03425827 0.03425827 0.18 0.6702

Example: β3

SS I is used to compare the model:

HTTy 32

210

with

2210 TTy

SS III is used to compare the model

HTHHTTy 352

42

210

with

THHTTy 52

42

210


Dependent Variable: LOGMITES Source DF Sum of Squares Mean Square F Value Pr > F Model 5 2.72839285 0.54567857 2.94 0.0265 Error 33 6.12429305 0.18558464 Corrected Total 38 8.85268590 R-Square C.V. Root MSE LOGMITES Mean 0.308199 85.66578 0.43079535 0.50287914 Source DF Type I SS Mean Square F Value Pr > F T 1 0.22115656 0.22115656 1.19 0.2829T2 1 1.38171889 1.38171889 7.45 0.0101H 1 1.03546840 1.03546840 5.58 0.0242H2 1 0.05579073 0.05579073 0.30 0.5872TH 1 0.03425827 0.03425827 0.18 0.6702 Source DF Type III SS Mean Square F Value Pr > F T 1 0.29703065 0.29703065 1.60 0.2147T2 1 0.88889243 0.88889243 4.79 0.0358H 1 0.04556941 0.04556941 0.25 0.6235H2 1 0.04755847 0.04755847 0.26 0.6161TH 1 0.03425827 0.03425827 0.18 0.6702

H is significant if it is added after T and T2

H is not significant if it is added after T, T2, H2, and TH

How do we choose the best model?

DATA mites;



/* pos = position in store */

/* depth = depth in m */

/* T = Temperature of grain */

/* H = Humidity of grain */

/* Mites = number of mites in sampling unit */

logMites = log10(Mites+1); /* log transformation of Mites */



TH = T*H; /* product of temperature and humidity */

PROC STEPWISE;

MODEL logMites = T T2 H H2 TH /MAXR;

RUN;

Maximum R-square Improvement for Dependent Variable LOGMITES

Step 1 Variable H2 Entered R-square = 0.11939020 C(p) = 7.00650467

DF Sum of Squares Mean Square F Prob>F

Regression 1 1.05692394 1.05692394 5.02 0.0312

Error 37 7.79576196 0.21069627

Total 38 8.85268590

Parameter Standard Type II

Variable Estimate Error Sum of Squares F Prob>F

INTERCEP -2.00838948 1.12364950 0.67311767 3.19 0.0821

H2 0.01218833 0.00544190 1.05692394 5.02 0.0312

Bounds on condition number: 1, 1

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

The above model is the best 1-variable model found.

Step 2 Variable T Entered R-square = 0.14111324 C(p) = 7.97028096


Regression 2 1.24923115 0.62461557 2.96 0.0647

Error 36 7.60345475 0.21120708

Total 38 8.85268590



INTERCEP -1.75010488 1.15711557 0.48315129 2.29 0.1391

T -0.02071757 0.02171178 0.19230720 0.91 0.3463

H2 0.01202664 0.00545113 1.02807459 4.87 0.0338

Bounds on condition number: 1.000967, 4.003869

Step 3 Variable H2 Removed R-square = 0.18352305 C(p) = 5.94726448

Variable TH Entered


Regression 2 1.62467193 0.81233596 4.05 0.0260

Error 36 7.22801397 0.20077817

Total 38 8.85268590



INTERCEP 0.72634839 0.24079746 1.82684898 9.10 0.0047

T -0.52367627 0.19084468 1.51175757 7.53 0.0094

TH 0.03507940 0.01326789 1.40351537 6.99 0.0121



Step 4 Variable T2 Entered R-square = 0.30260874 C(p) = 2.26668560


Regression 3 2.67890010 0.89296670 5.06 0.0051

Error 35 6.17378580 0.17639388

Total 38 8.85268590



INTERCEP 3.12310821 1.00603496 1.69993153 9.64 0.0038

T -0.94651154 0.24882419 2.55240831 14.47 0.0005

T2 0.02187819 0.00894923 1.05422818 5.98 0.0197

TH 0.03099168 0.01254804 1.07602465 6.10 0.0185


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


Step 5 Variable H2 Entered R-square = 0.30305192 C(p) = 4.24554515


Regression 4 2.68282344 0.67070586 3.70 0.0133

Error 34 6.16986246 0.18146654

Total 38 8.85268590



INTERCEP 2.56528922 3.92853717 0.07737622 0.43 0.5182

T -0.85413336 0.67705608 0.28880097 1.59 0.2157

T2 0.02264025 0.01045241 0.85138509 4.69 0.0374

H2 0.00311049 0.02115432 0.00392334 0.02 0.8840

TH 0.02338380 0.05328321 0.03494992 0.19 0.6635


Step 6 Variable TH Removed R-square = 0.30432962 C(p) = 4.18459648 Variable H Entered DF Sum of Squares Mean Square F Prob>F Regression 4 2.69413458 0.67353364 3.72 0.0129 Error 34 6.15855132 0.18113386 Total 38 8.85268590

Parameter Standard Type II Variable Estimate Error Sum of Squares F Prob>F INTERCEP 26.64541542 50.83962556 0.04975537 0.27 0.6036 T -0.58573429 0.20112070 1.53634027 8.48 0.0063 T2 0.02565523 0.00908804 1.44347920 7.97 0.0079 H -3.55394542 7.03238763 0.04626106 0.26 0.6166 H2 0.13533410 0.24385185 0.05579073 0.31 0.5825 Bounds on condition number: 2335.775, 19486.21----------------------------------------------------------------------------------- The above model is the best 4-variable model found.

Step 7 Variable TH Entered R-square = 0.30819944 C(p) = 6.00000000 DF Sum of Squares Mean Square F Prob>F Regression 5 2.72839285 0.54567857 2.94 0.0265 Error 33 6.12429305 0.18558464 Total 38 8.85268590 Parameter Standard Type II Variable Estimate Error Sum of Squares F Prob>F INTERCEP 28.03994954 51.56270293 0.05488139 0.30 0.5902 T -0.86682324 0.68517409 0.29703065 1.60 0.2147 T2 0.02333784 0.01066368 0.88889243 4.79 0.0358 H -3.52741058 7.11853025 0.04556941 0.25 0.6235 H2 0.12548846 0.24789107 0.04755847 0.26 0.6161 TH 0.02315214 0.05388643 0.03425827 0.18 0.6702 Bounds on condition number: 2355.783, 37061.88----------------------------------------------------------------------------------- The above model is the best 5-variable model found. No further improvement in R-square is possible.

Models with 1 variable

Model R2 F P

T 0.0250 0.948 0.3365

T2 0.0138 0.516 0.4770

H 0.1193 5.012 0.0313

H2 0.1194 5.016 0.0312

T*H 0.0128 0.478 0.4936

Models with 2 variables

Model R2 F P

T T2 0.1811 3.980 0.0274

T H 0.1409 2.952 0.0650

T H2 0.1411 2.957 0.0647

T T*H 0.1835 4.046 0.0260

T2 H 0.1305 2.703 0.0806

T2 H2 0.1307 2.707 0.0803

T2 T*H 0.0143 0.261 0.7718

H H2 0.1194 2.441 0.1014

H T*H 0.1386 2.896 0.0632

H2 T*H 0.1388 2.902 0.4936


Model R2 F P

T T2 H 0.2980 4.953 0.0057

T T2 H2 0.2991 4.979 0.0056

T T2 T*H 0.3026 5.062 0.0051

T H H2 0.1413 1.919 0.1444

T H T*H 0.2074 3.054 0.0411

T H2 T*H 0.2069 3.043 0.0416

T2 H H2 0.1308 1.755 0.1737

T2 H T*H 0.2681 4.273 0.0113

T2 H2 T*H 0.2704 4.324 0.0108

H H2 T*H 0.1390 1.883 0.1504


Model R2 F P

T T2 H H2 0.3043 3.718 0.0129

T T2 H T*H 0.3028 3.692 0.0134

T T2 H2 T*H 0.3031 3.696 0.0133

T H H2 T*H 0.2078 2.229 0.0864

T2 H H2 T*H 0.2746 3.218 0.0241


Model R2 F P

T T2 H H2 TH 0.3082 2.940 0.0265

Best models

Model R2 F P C(p)

H2 0.1194 5.016 0.0312 7.007

T T*H 0.1835 4.046 0.0260 5.947

T T2 T*H 0.3026 5.062 0.0051 2.267

T T2 H H2 0.3043 3.718 0.0129 4.185

T T2 H H2 T*H 0.3082 2.940 0.0265 6.000

Overall, this may considered the best model

Mallow’s C(p)

Model control

DATA mites;



LogMites = log10(Mites+1); /* transform dependent variable */



TH = T*H; /* interaction between temperature and humidity */

PROC REG; /* Multiple regression analysis */

MODEL logMites = T T2 H H2 TH;

OUTPUT out = new P = pred R = res;

RUN;

/*Model control */

PROC GPLOT;

PLOT LogMites*pred pred*pred /OVERLAY;

/*plot observed values against predicted values together with line of equality */

SYMBOL1 COLOR=blue VALUE=circle HEIGHT=1;

SYMBOL2 COLOR=red INTERPOL=line WIDTH = 1;

PLOT res*pred; /* plot residuals against the predicted values */


RUN;

Observed values of LogMites against predicted values

DATA mites;



LogMites = log10(Mites+1); /* transform dependent variable */



TH = T*H; /* interaction between temperature and humidity */

PROC REG; /* Multiple regression analysis */

MODEL logMites = T T2 H H2 TH;

OUTPUT out = new P = pred R = res;

RUN;

/*Model control */

PROC GPLOT;

PLOT LogMites*pred pred*pred /OVERLAY;

/*plot observed values against predicted values together with line of equality */


SYMBOL2 COLOR=red INTERPOL=line WIDTH = 1;

PLOT res*pred; /* plot residuals against the predicted values */


RUN;

Residuals plotted against predicted values of LogMites

PROC UNIVARIATE FREQ PLOT NORMAL data= Newdata;

/* PROC UNIVARIATE gives information about the variables defined by VAR */

/* FREQ, PLOT, NORMAL etc are options

FREQ = number of observations of a given value

PLOT = plot of observations

NORMAL = test for the variable is normally distributed */

VAR res; /* information about the residuals */

RUN;

Univariate Procedure

Variable=RES

Moments Quantiles(Def=5)

N 20 Sum Wgts 20 100% Max 2.02 99% 2.02

Mean 0 Sum 0 75% Q3 0.86 95% 1.96

Std Dev 1.23774 Variance 1.532 50% Med -0.24 90% 1.86

Skewness 0.129454 Kurtosis -0.82681 25% Q1 -0.92 10% -1.63

USS 29.108 CSS 29.108 0% Min -2.08 5% -2.04

CV . Std Mean 0.276767 1% -2.08

T:Mean=0 0 Pr>|T| 1.0000 Range 4.1

Num ^= 0 20 Num > 0 9 Q3-Q1 1.78

M(Sign) -1 Pr>=|M| 0.8238 Mode -2.08

Sgn Rank -3 Pr>=|S| 0.9273

W:Normal 0.956524 Pr<W 0.4851

H0: The residuals are normally distributed

This is the probability of getting a deviation from the normaldistribution equal to or greater than the observed one by chancegiven H0 is true

Extremes

Lowest Obs Highest Obs

-2.08( 20) 0.9( 13)

-2( 11) 1.54( 8)

-1.26( 10) 1.82( 5)

-1.08( 1) 1.9( 12)

-1.06( 7) 2.02( 16)

Stem Leaf # Boxplot

2 0 1 |

1 589 3 |

1 |

0 789 3 +-----+

0 03 2 | + |

-0 332 3 *-----*

-0 865 3 +-----+

-1 311 3 |

-1 |

-2 10 2 |

----+----+----+----+

2.0

1.51.81.9

-2.0-2.1

Normal Probability Plot

2.25+ ++*+

| * * +*++

| ++++

0.75+ *+**+

| ++**+

| +*** *

-0.75+ ++*+*

| +*+* *

| ++*+

-2.25+ +*++

+----+----+----+----+----+----+----+----+----+----+

-2 -1 0 +1 +2

Points should follow a straight line if data are normally distributed

Frequency Table Percents Percents Value Count Cell Cum Value Count Cell Cum -2.08 1 5.0 5.0 -0.2 1 5.0 55.0 -2 1 5.0 10.0 0.04 1 5.0 60.0 -1.26 1 5.0 15.0 0.32 1 5.0 65.0 -1.08 1 5.0 20.0 0.74 1 5.0 70.0 -1.06 1 5.0 25.0 0.82 1 5.0 75.0 -0.78 1 5.0 30.0 0.9 1 5.0 80.0 -0.6 1 5.0 35.0 1.54 1 5.0 85.0 -0.48 1 5.0 40.0 1.82 1 5.0 90.0 -0.28 1 5.0 45.0 1.9 1 5.0 95.0 -0.28 1 5.0 50.0 2.02 1 5.0 100.0

Documents

Experimental design and statistical analyses of data Lesson 4: Analysis of variance II A posteriori tests Model control How to choose the best model