Effect on R 2 of fitting means and calculation of lack of fit (LOF) Page 1

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1 Effect on R of fitting means and calculation of lack of fit (LOF) Page dm'log;clear;output;clear'; OPTIONS PS=5 LS=0 NOCENTER NODTE NONUMBER FORMCHR=" =-/\<>*"; ODS HTML style=minimal body='ppendix09 LOF & Rsquare Suppliment.html'; ODS listing; TITLE 'EXST705: Marathon Footrace from Pennsylvania 00'; filename input 'ppendix09 MReg-0K Polynomial.DT'; ***********************************************; *** Finish times in a race ***; *** Data taken from various sites on the ***; *** internet reporting race results ***; ***********************************************; data P00; length hometown $ 3 gender $ 3; infile input missover; input Marathon $ ge gender $ TIME HomeTown $ 35-57; if age eq 99 then age =.; *(apparently 99 represents missing for 5 P race participants); * ; cards; run; ; data P00; set P00; if marathon = 'P0500'; proc sort data=p00; by gender age; run; TITLE 'Scatter plot of raw data'; proc plot data=p00; by gender; plot time*age; options ls=3 ps=65; run; options ps=56 ls=80; proc glm data=p00; BY gender; TITLE 'Quadratic model fitted to raw data - separate by gender'; model time= age age*age; run; proc means data=p00 noprint; by gender age; var time; output out=meansbygender n=n mean=timemean std=std; run; TITLE 'Scatter plot of means'; proc plot data=meansbygender; by gender; plot timemean*age; options ls=3 ps=65; run; options ps=56 ls=80; proc glm data=meansbygender; BY gender; TITLE 'Quadratic model fitted on UNweighted means - separate by gender'; model timemean = age age*age; run; proc glm data=meansbygender; BY gender; weight n; TITLE 'Quadratic model fitted on weighted means - separate by gender'; model timemean = age age*age; run; proc print data=meansbygender; var gender age n timemean std; run; proc glm data=p00; BY gender; class age; TITLE 'nalysis of Variance of GE - separate by gender'; model time = age; run; data P00; set P00; age_again = age; proc glm data=p00; BY gender; class age_again; TITLE 'nalysis of Variance of GE - separate by gender'; model time = age age*age age_again; run; ODS HTML close; run; quit;

2 Effect on R of fitting means and calculation of lack of fit (LOF) Page EXST705: Marathon Footrace from Pennsylvania 00 Scatter plot of raw data Plot of TIME*ge. Legend: = obs, B = obs, etc. TIME B C B B B B B B B B B B B C B B C B D C B B B B B B B B B C C B B C B B B E B C D B E B B B C D B B C D B D C C B B C B C B B B E D E B D B B B C B C B B C B E D B B C D B B D C B B B C B B B C B B B C C D B D C D B B B C 50 + B D B D C C C B C C B C B B B D B B B D B B B B F B B F B B B B B B B B C C D B D B B C B B C B B B B B C B C B B B B C B B B C B B B B B B B 00 + B ge NOTE: obs had missing values. Plot of TIME*ge. Legend: = obs, B = obs, etc. TIME B B B B B C B B B B CB B B B C B B B C BB B B B B C CB B CB E C B B B CB C B CB BB EC B BD B B B C B B BC D B B CB BC CB EB B B B CC B BC BB B B B BC BC BC FB D C B B E B D C B BB E B GC BC BC BC BB DB B D C C C D B C C FE BE B E EC EC CC ID B D C B B B D C BC D CC DC BC CB B HG C HE G CF D B B D 50 + D BB BE BB B C E CE D D B FJ BE B E CE B G CE BB HD BG GE BF GC BD EF CE D B BB BE B D FD CD CK BC FC CF EI EB LH JC CB D DB B E BE C BD ED BD CE D DI DD G FB CD DE JF EC B BB B C BB BC DE BB CF EC CH GC C FE E F EB D B B B BB B C DC EB B FC BC BE DC DB CB DC B EB C B B B C C BD BC D CG C CB DE EF GD C D BC 00 + B C B BC BD BC CB C C CB BB CE D D B D B C B B E B C CE BB BB BD C B B B B C BB B B CB BB BC B BD E D B B BB B B B B B ge NOTE: 3 obs had missing values.

3 Effect on R of fitting means and calculation of lack of fit (LOF) Page 3 EXST705: Marathon Footrace from Pennsylvania 00 Quadratic model fitted to raw data - separate by gender The GLM Procedure Read Used Model Error Corrected Total Squares Meann Square R-Square Coeff Var Root MSE TIME Mean ge ge* *ge Type I SS Meann Square ge ge* *ge Type III SS Meann Square Parameter Intercept ge ge* *ge Standardd Estimate Errorr t Value Pr > t < <. 000 Read Used Model Error Corrected Total Squares Meann Square R-Square Coeff Var Root MSE TIME Mean ge ge* *ge Type I SS Meann Square ge ge* *ge Type III SS Meann Square Parameter Intercept ge ge* *ge Standardd Estimate Errorr t Value Pr > t <. 000 <. 000 <. 000

4 Effect on R of fitting means and calculation of lack of fit (LOF) Page 4 EXST705: Marathon Footrace from Pennsylvania 00 Scatter plot of means Plot of timemean*ge. Legend: = obs, B = obs, etc. timemean ge NOTE: obs had missing values. Plot of timemean*ge. Legend: = obs, B = obs, etc. timemean NOTE: obs had missing values. ge

5 Effect on R of fitting means and calculation of lack of fit (LOF) Page 5 EXST705: Marathon Footrace from Pennsylvania 00 Quadratic model fitted on UNweighted means - separate by gender The GLM Procedure Read 46 Used 45 Dependent Variable: timemean Squares Mean Square Model Error Corrected Total R-Square Coeff Var Root MSE timemean Mean Type I SS Mean Square ge ge*ge Type III SS Mean Square ge ge*ge Standard Parameter Estimate Error t Value Pr > t Intercept ge ge*ge Read 60 Used 59 Dependent Variable: timemean Squares Mean Square Model Error Corrected Total R-Square Coeff Var Root MSE timemean Mean Type I SS Mean Square ge ge*ge Type III SS Mean Square ge ge*ge Standard Parameter Estimate Error t Value Pr > t Intercept ge ge*ge

6 Effect on R of fitting means and calculation of lack of fit (LOF) Page 6 EXST705: Marathon Footrace from Pennsylvania 00 Quadratic model fitted on weighted means - separate by gender The GLM Procedure Read 46 Used 45 Dependent Variable: timemean Weight: n Squares Mean Square Model Error Corrected Total R-Square Coeff Var Root MSE timemean Mean Type I SS Mean Square ge ge*ge Type III SS Mean Square ge ge*ge Standard Parameter Estimate Error t Value Pr > t Intercept ge ge*ge Read 60 Used 59 Dependent Variable: timemean Weight: n Squares Mean Square Model Error Corrected Total R-Square Coeff Var Root MSE timemean Mean Type I SS Mean Square ge ge*ge Type III SS Mean Square ge ge*ge Standard Parameter Estimate Error t Value Pr > t Intercept ge ge*ge

7 Effect on R of fitting means and calculation of lack of fit (LOF) Page 7 EXST705: Marathon Footrace from Pennsylvania 00 Quadratic model fitted on weighted means - separate by gender Obs gender ge n timemean std F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M Comparison of fits to Raw data and fits to means (weighted and unweighted) Females Males Statistic Raw data Means Weighted means Raw data Means Weighted means b b b R MSE

8 Effect on R of fitting means and calculation of lack of fit (LOF) EXST705: Marathon Footrace from Pennsylvania 00 nalysis of Variance of GE - separate by gender Page 8 The GLM Procedure Class Level Informationn Class Levels Values ge Read 753 Used Model Error Corrected Total Squares Meann Square R-Square Coeff Var Root MSE TIME Mean ge 44 Type I SS Meann Square ge 44 Type III SS Meann Square Class Level Informationn Class Levels Values ge Read 808 Used Model Error Corrected Total Squares Meann Square R-Square Coeff Var Root MSE TIME Mean ge 58 Type I SS Meann Square ge 58 Type III SS Meann Square

9 Effect on R of fitting means and calculation of lack of fit (LOF) Page 9 Test of Lack of Fit a measure of adequacy of the model Regression on means for Females Squares Mean Square Model Error Corrected Total R = Regression on raw data for Females Squares Mean Square Model Error Corrected Total R = nalysis of variance on GE using raw data for Females Squares Mean Square Model Error Corrected Total R = Test of lack of fit for Females d.f. SSE MSE F P>F Reduced model Full model Difference Full model Regression on means for Males Squares Mean Square Model Error Corrected Total R = Regression on raw data for Males Squares Mean Square Model Error Corrected Total R = nalysis of variance on GE using raw data for Males Squares Mean Square Model Error Corrected Total R = Test of lack of ft for Males d.f. SSE MSE F P>F Reduced model Full model Difference Full model

10 Effect on R of fitting means and calculation of lack of fit (LOF) Page 0 Model to test LOF for Quadratic model fitted to raw data - separate by gender The GLM Procedure It is possible to test LOF in a single model by including the two sets of the independent variable and putting one in the class statement. The regression is fitted first with the quantitative X variable (not in the class statement) and the categorical version (the one in the class statement) is Class Level Information Class Levels Values fitted last to account for any remaining variation which is LOF. age_again Read 753 Used 75 The usual Deviations from regression are = ( Y ˆ ij Yi.) Squares Mean Square Model Error PE = ( Yij Yi.) Corrected Total CSS = ( Y Y ) R-Square Coeff Var Root MSE TIME Mean ij.. Type I SS Mean Square ge ge*ge age_again LOF = ( Y Yˆ ) i. i. Type III SS Mean Square ge ge*ge age_again Class Level Information Class Levels Values age_again Read 808 Used 805 Squares Mean Square Model Error Corrected Total R-Square Coeff Var Root MSE TIME Mean Type I SS Mean Square ge ge*ge age_again Type III SS Mean Square ge ge*ge age_again Compare the tests of age_again to the tests of Lack of Fit on the preceding page.