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library(Keng)library(effectsize)library(car)#> Loading required package: carDatadata("depress")

PRE is called partial R-squared in regression, and partialEta-squared in ANOVA. This vignette will examine their equivalence usingthe internal datadepress.

depress collecteddepression,gender, andclass at Time 1. Traditionally, weexamine the effect ofgender andclass usinganova. We firstly let R knowgender andclassare factors (i.e., categorical variables). Then we conduct anova usingcar::Anova() and compute partial Eta-squared usingeffectsize::eta_squared().

# factor gender and classdepress_factor<- depressdepress_factor$class<-factor(depress_factor$class,labels =c(3,5,9,12))depress_factor$gender<-factor(depress_factor$gender,labels =c(0,1))anova.fit<-lm(dm1~ gender+ class, depress_factor)Anova(anova.fit,type =3)#> Anova Table (Type III tests)#>#> Response: dm1#>             Sum Sq Df  F value Pr(>F)#> (Intercept) 67.166  1 464.5565 <2e-16 ***#> gender       0.025  1   0.1758 0.6760#> class        0.729  3   1.6808 0.1768#> Residuals   12.868 89#> ---#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1cat("\n\n")print(eta_squared(Anova(anova.fit,type =3),partial =TRUE),digits =6)#> # Effect Size for ANOVA (Type III)#>#> Parameter | Eta2 (partial) |               95% CI#> -------------------------------------------------#> gender    |       0.001972 | [0.000000, 1.000000]#> class     |       0.053619 | [0.000000, 1.000000]#>#> - One-sided CIs: upper bound fixed at [1.000000].

Then we conduct regression analysis and computePRE. Forclass with four levels: 3, 5, 9, and 12, we dummy-code itusingifelse() with the class12 as the reference group.

# class3 indicates whether the class is class3depress$class3<-ifelse(depress$class==3,1,0)# class5 indicates whether the class is class5depress$class5<-ifelse(depress$class==5,1,0)# class9 indicates whether the class is class9depress$class9<-ifelse(depress$class==9,1,0)

We compute thePRE ofgender though comparingModel A withgender against Model C withoutgender.

fitC<-lm(dm1~ class3+ class5+ class9, depress)fitA<-lm(dm1~ class3+ class5+ class9+ gender, depress)print(compare_lm(fitC, fitA),digits =3)#>                      Baseline       C       A  A vs. C#> SSE                      13.6 12.8932 12.8678  0.02542#> n                        94.0 94.0000 94.0000 94.00000#> Number of parameters      1.0  4.0000  5.0000  1.00000#> df                       93.0 90.0000 89.0000  1.00000#> R_squared                  NA  0.0530  0.0549  0.00187#> f_squared                  NA  0.0560  0.0581  0.00198#> R_squared_adj              NA  0.0214  0.0124       NA#> PRE                        NA  0.0530  0.0549  0.00197#> F(PA-PC,n-PA)              NA  1.6791  1.2917  0.17583#> p                          NA  0.1771  0.2794  0.67599#> PRE_adj                    NA  0.0214  0.0124 -0.00924#> power_post                 NA  0.4263  0.3887  0.06993

We compute thePRE ofclass. Note that inregression, thePRE ofclass is thePREof allclass’s dummy codes:class3,class5, andclass9.

fitC<-lm(dm1~ gender, depress)fitA<-lm(dm1~ class3+ class5+ class9+ gender, depress)print(compare_lm(fitC, fitA),digits =3)#>                      Baseline        C       A A vs. C#> SSE                      13.6 13.59681 12.8678  0.7291#> n                        94.0 94.00000 94.0000 94.0000#> Number of parameters      1.0  2.00000  5.0000  3.0000#> df                       93.0 92.00000 89.0000  3.0000#> R_squared                  NA  0.00132  0.0549  0.0535#> f_squared                  NA  0.00132  0.0581  0.0567#> R_squared_adj              NA -0.00953  0.0124      NA#> PRE                        NA  0.00132  0.0549  0.0536#> F(PA-PC,n-PA)              NA  0.12165  1.2917  1.6808#> p                          NA  0.72805  0.2794  0.1768#> PRE_adj                    NA -0.00953  0.0124  0.0217#> power_post                 NA  0.06376  0.3887  0.4266

We compute thePRE of the full model(Model A). ThePRE (partial R-squared or partial Eta-squared) of the fullmodel is commonly known as the R-squared or Eta-squared of the fullmodel.

fitC<-lm(dm1~1, depress)fitA<-lm(dm1~ class3+ class5+ class9+ gender, depress)print(compare_lm(fitC, fitA),digits =3)#>                      Baseline        C       A A vs. C#> SSE                      13.6 1.36e+01 12.8678  0.7470#> n                        94.0 9.40e+01 94.0000 94.0000#> Number of parameters      1.0 1.00e+00  5.0000  4.0000#> df                       93.0 9.30e+01 89.0000  4.0000#> R_squared                  NA 6.52e-16  0.0549  0.0549#> f_squared                  NA 6.66e-16  0.0581  0.0581#> R_squared_adj              NA 7.77e-16  0.0124      NA#> PRE                        NA 6.66e-16  0.0549  0.0549#> F(PA-PC,n-PA)              NA       NA  1.2917  1.2917#> p                          NA       NA  0.2794  0.2794#> PRE_adj                    NA 6.66e-16  0.0124  0.0124#> power_post                 NA       NA  0.3887  0.3887

As shown, thePRE of Model A against Model C is equal toModel A’s R_squared. Taken the loss of precision into consideration,Model C’s R_squared is zero.