The FREQ Procedure. Table of Sex by Gym Sex(Sex) Gym(Gym) No Yes Total Male Female Total

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1 Jenn Selensky gathered data from students in an introduction to psychology course. The data are weights, sex/gender, and whether or not the student worked-out in the gym. Here is the output from a 2 x 2 Sex x ANOVA and related other analyses. Proc Freq; Tables Sex* / CHISQ nopercent nocol; run; Frequency Row Pct The FREQ Procedure Table of Sex by Sex(Sex) () No Yes Total Male Female Total Statistics for Table of Sex by Statistic DF Value Prob Chi Phi Coefficient Chi-square and phi do not equal zero. The ANOVA will be nonorthogonal. Men attended gym at a higher rate than did women. Sample Size = 200 Omnibus Analysis and Simple Main Effects Using Pooled Error PROC GLM data=gym; CLASS Sex ; MODEL Weight=Sex / SS3 EFFECTSIZE alpha=0.1; means Sex ; LSMEANS Sex* / SLICE=Sex; LSMEANS ; LSMEANS Sex; title 'Omnibus Analysis and Simple Main Effects Using Pooled Error'; run; quit; The GLM Procedure Class Level Information Class Levels Values Sex 2 Female Male 2 No Yes Number of Observations Read 200 Number of Observations Used 200

2 Dependent Variable: Weight Source The GLM Procedure DF Sum of s F Value Pr > F Model <.0001 Error Corrected Total R- Coeff Var Root MSE Weight Proportion of Variation Accounted for Eta % Confidence (0.13,0.29) Statistics associated with the total model (combined effect of sex and gym) are generally ignored in factorial ANOVA. Source DF Type III SS F Value Pr > F Sex < Sex* Total Variation Accounted For Eta- Conservative 90% Confidence Partial Variation Accounted For Partial Eta- Partial 90% Confidence Sex Sex x All three effects are significant. Converting Cohen s benchmarks for r into proportions of variance (r 2 ).01 = small.09 = medium.25 = large

3 The interaction is monotonic. The effect of gym is larger in men than in women The schematic plot above calls into doubt the normality assumption. Observation 133 has weight = 265 pounds. Warning: SAS observation numbers will not match SPSS case numbers. Furthermore, the SAS observation numbers may vary from one importation of the SPSS data to another importation. Level of Sex N Weight Female Male The men weigh significantly more (29.7 pound difference) than the women, d = 1.10.

4 Level of N Weight No Yes Those who attend gym weigh significantly more than those who do not, d =.36. Level of Sex Level of N Weight Female No Female Yes Male No Male Yes

5 Sex The GLM Procedure Least s s DF Sum of s F Value Pr > F Female Male The simple main effect of gym attendance is significant for the men but not for the women. Eta- Total Variation Accounted For Conservative 90% Confidence Partial Variation Accounted For Partial Eta- Partial 90% Confidence Female Male The proportions of variance here have as the denominator all of the variance in weights, for both men and for women. I think it makes more sense to use a denominator that is the within-sex total variance, which I compute below. Least s s Weight LSMEAN Observed No Yes Notice that the adjusted means differ by more (10.2) than do the observed means (9.3). Least s s Sex Weight LSMEAN Observed Female Male Notice that the adjusted means differ by less (28.6) than do the observed means (29.8). The tests for the main effects here test the null that the adjusted mean are equal, not the null that the unadjusted means are equal. A simple t test would test the null that the unadjusted means are equal. Proc Sort; By Sex; run; Proc GLM; Class ; Model Weight= / SS1 EFFECTSIZE alpha=0.1; By Sex; title 'simple main effects, individual error'; run; quit; simple main effects, individual error The GLM Procedure Sex=Male Class Level Information Class Levels Values 2 No Yes

6 Number of Observations Read 51 Number of Observations Used 51 Dependent Variable: Weight Sex=Male Source DF Sum of s F Value Pr > F Model Error Corrected Total R- Coeff Var Root MSE Weight Source DF Type I SS F Value Pr > F Total Variation Accounted For Eta- Conservative 90% Confidence In the men, 11% of the variance in weights is associated with gym usage. Sex=Female Class Level Information Class Levels Values 2 No Yes Number of Observations Read 149 Number of Observations Used 149 Dependent Variable: Weight Sex=Female Source DF Sum of s F Value Pr > F Model Error Corrected Total R- Coeff Var Root MSE Weight

7 Source DF Type I SS F Value Pr > F Total Variation Accounted For Eta- Conservative 90% Confidence As noted earlier, the boxplots called into question the normality assumption. Let s see how badly skewed the within-cell distributions are. proc sort; by sex gym; proc means skewness kurtosis; var weight; by sex gym; run;

8 Sex=Male =No Analysis Variable : Weight Skewness Kurtosis Sex=Male =Yes Analysis Variable : Weight Skewness Kurtosis Sex=Female =No Analysis Variable : Weight Skewness Kurtosis Sex=Female =Yes Analysis Variable : Weight Skewness Kurtosis We really ought to try a nonparametric analysis here. proc npar1way wilcoxon; class gym; by sex; run; The NPAR1WAY Procedure Sex=Male Wilcoxon Scores (Rank Sums) for Variable Weight Classified by Variable N Sum of Scores Expected Under H0 Under H0 Score No Yes Average scores were used for ties.

9 Wilcoxon Two-Sample Test Statistic Normal Approximation Z One-Sided Pr < Z Two-Sided Pr > Z attendance is significantly associated with weight in the men. Sex=Female Wilcoxon Scores (Rank Sums) for Variable Age Classified by Variable N Sum of Scores Expected Under H0 Under H0 Score No Yes Average scores were used for ties. Wilcoxon Two-Sample Test Statistic Normal Approximation Z One-Sided Pr > Z Two-Sided Pr > Z attendance is not significantly related to weight in the women.