PSYC 6140 November 16, 2005 ANOVA output in R

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1 PSYC 6140 November 16, 2005 ANOVA output in R Type I, Type II and Type III Sums of Squares are displayed in ANOVA tables in a mumber of packages. The car library in R makes these available in R. This handout discusses how to use and interpret anova output for regression models with interaction. The following script shows output for a model with interaction using all three types of sum of squares. In summary: Note: 1. Type I and Type II SS compare models that obey the principle of marginality and, thus, these tables remain the same if the categorical variables is recoded or if the continuous variable is subject to an affine transformation. 2. Type III SS depend on the coding the library(car) source(" coursefun.r Last update: Oct. 27, 2005 An R script containing functions and some datasets for the courses PSYC6140 and MATH6630 in Help on the following functions is available by typing the name of the function. This text is available by typing coursefun A current version of this file can be sourced or downloaded from A copy is kept at: Please make corrections, changes and additions to the version on the wiki. They will be periodically transferred to the downloadable version. Functions: Tables: atotal: border an array with sums

2 ANOVA output in R 2 abind : glue two arrays together on a selected dimension Graphics: td : easy front end to trellis.device and related functions xqplot: extended quantile plots 3D graphics by John Fox: scatter3d identify3d ellipsoid Inference cell - a modified version of car::confidence.ellipse.lm that can add to a plot Graphics for linear algebra vplot - plots column vectors adding to current plot vell - ellipse as a 2 x n matrix vbox - box around unit circle orthog - 2 x 2 rotation orthog.prog 2 x 2 matrix of orthog projection To add functions, modify data( Prestige ) scatterplot.matrix(prestige) ## Let's just use those with non.missing type dd <- na.omit(prestige) dim(dd) [1] 98 6 dim(prestige) [1] attach(dd)

3 ANOVA output in R 3 The following object(s) are masked from package:datasets : women table(type) type bc prof wc income.log <- log(income ) scatterplot( income.log, prestige, groups = type)

4 ANOVA output in R 4 prestige bc prof wc income.log

5 ANOVA output in R 5 Model: E( Y ) = R Syntax P. of Marg. Comments 1 α + β X + γ 1D1+ γ 2D2 + δ1dx 1 + δ2dx Y ~ X*G 2 Yes Full model 2N α + γ 1D1+ γ 2D2 + δ1dx 1 + δ2dx Y ~ G + X:G 2 No Y ~ X*G X 3N α + β X + δ1dx 1 + δ2dx Y ~ X + X:G 2 No Y ~ X*G G 4N α + δ1dx 1 + δ2dx Y ~ X:G 2 No 5 α β X + γ 1D1 γ 2D2 + + Y ~ X + G Yes Additive model 6 α + γ 1D1+ γ 2D2 Y ~ G Yes 7 α + β X Y ~ X Yes 8 α Y ~ 1 Yes Intercept only model fit.add <- lm( prestige ~ income.log + type ) Additive model fit.int <- lm( prestige ~ income.log * type ) Full interaction model summary(fit.int) Call: lm(formula = prestige ~ income.log * type) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr( t ) (Intercept) e-09 *** income.log e-10 *** typeprof e-05 *** typewc * income.log:typeprof *** income.log:typewc * Exercise: Draw a sketch of the fitted model and label it with each of the estimated values in this table

6 ANOVA output in R 6 Residual standard error: on 92 degrees of freedom Multiple R-Squared: , Adjusted R-squared: F-statistic: on 5 and 92 DF, p-value: < 2.2e-16 anova(fit.int) ANOVA for full model using anova Analysis of Variance Table Type I SS (sequential) Df Sum Sq Mean Sq F value Pr(F) income.log < 2.2e-16 *** (8) (7) / (1) M:ok type < 2.2e-16 *** (7) (5) / (1) M:ok income.log:type *** Residuals (5) (1) / (1) M:ok EXERCISE: What would you get if you specified the model as prestige ~ income.log * type anova( fit.add, fit.int) Analysis of Variance Table Model 1: prestige ~ income.log + type Model 2: prestige ~ income.log * type Res.Df RSS Df Sum of Sq F Pr(F) *** (5) (1) / (1) ## ## Interpreting coefficients ## contrasts(type) # bc is reference level prof wc bc 0 0 bc is reference level prof 1 0 wc 0 1

7 ANOVA output in R 7 fit.int$contrasts $type [1] "contr.treatment" anova( fit.int) Analysis of Variance Table Df Sum Sq Mean Sq F value Pr(F) income.log < 2.2e-16 *** type < 2.2e-16 *** income.log:type *** Residuals Anova( fit.int, type = "II") ANOVA using Anova Type II Anova Table (Type II tests) in library(car) NOTE: All satisfy PoM Sum Sq Df F value Pr(F) income.log e-11 *** (6) (5) / (1) [Different] type < 2.2e-16 *** (7) (5) / (1) [Same as Type I] income.log:type *** (5) (1) / (1) [Same as Type I] Residuals Anova( fit.int, type = "III") ANOVA using Anova Type III Anova Table (Type III tests) in library(car) Sum Sq Df F value Pr(F) (Intercept) e-09 *** income.log e-10 *** (2N) (1) / (1) [Depends on coding of G] type *** (3N) (1) / (1) [Depends on 0 of X] income.log:type *** (5) (1) / (1) [Same as I and II] Residuals

8 ANOVA output in R 8 fit.int.s <- lm( prestige ~ income.log * type, contrasts = list( type = contr.sum ) ) Sum to 0 coding fit.int.s$contrasts $type [,1] [,2] bc 1 0 Note: No type is reference level. prof 0 1 Reference level is at the mean wc -1-1 of the 3 types. summary(fit.int.s) Call: lm(formula = prestige ~ income.log * type, contrasts = list(type = contr.sum)) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr( t ) (Intercept) e-08 *** income.log e-10 *** type *** type ** income.log:type ** income.log:type * EXERCISE: Sketch the fitted model and label with values of fitted coefficients Residual standard error: on 92 degrees of freedom Multiple R-Squared: , Adjusted R-squared: F-statistic: on 5 and 92 DF, p-value: < 2.2e-16

9 ANOVA output in R 9 anova( fit.int.s ) Analysis of Variance Table Df Sum Sq Mean Sq F value Pr(F) income.log < 2.2e-16 *** type < 2.2e-16 *** income.log:type *** Residuals anova( fit.add, fit.int.s ) Analysis of Variance Table EXERCISE: Fill in models Model 1: prestige ~ income.log + type Model 2: prestige ~ income.log * type Res.Df RSS Df Sum of Sq F Pr(F) EXERCISE: Fill in models *** EXERCISE: What happens if we use fit.add.s instead? anova( fit.int.s) Analysis of Variance Table Df Sum Sq Mean Sq F value Pr(F) income.log < 2.2e-16 *** type < 2.2e-16 *** income.log:type *** Residuals

10 ANOVA output in R 10 Anova Type II SS Anova( fit.int.s, type = "II") EXERCISE: Fill in models? Anova Table (Type II tests) Do you get same output as before? Sum Sq Df F value Pr(F) income.log e-11 *** type < 2.2e-16 *** income.log:type *** Residuals Anova Type III SS Anova( fit.int.s, type = "III") EXERCISE: Draw a sketch showing Anova Table (Type III tests) what is tested in this table. Sum Sq Df F value Pr(F) (Intercept) e-08 *** income.log e-10 *** type *** income.log:type *** Residuals Note: Type III SS using sum to 0 are quite popular in Psychology and Sociology. They are readily produced in SAS output. Two cautions: 1. Is the hypothesis meaningful in the presence of interaction? i.e. is the hypothesis that the average slope, averaging over levels of the factor, equals 0 of real interest? 2. If the number of observations in each category of the factor is quite variable, then the Type III SS will have low power in comparison with a hypothesis that uses a weighted average of slopes -- as the Type II SS.

11 ANOVA output in R 11 ## ## 3 way interaction ## summary( lm( prestige ~ income.log * type * women )) 3 way interaction EXERCISE: Draw 2 sketches, one for Call: women = 0 and another for lm(formula = prestige ~ income.log * type * women) women = 100. Label sketches appropriately Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr( t ) (Intercept) e-09 *** income.log e-10 *** typeprof e-05 *** typewc women income.log:typeprof e-05 *** income.log:typewc income.log:women typeprof:women typewc:women income.log:typeprof:women income.log:typewc:women Residual standard error: on 86 degrees of freedom Multiple R-Squared: , Adjusted R-squared: F-statistic: on 11 and 86 DF, p-value: < 2.2e-16 fit3.s <- lm( prestige ~ income.log * type * women, contrasts = list( type = contr.sum)) Sum to 0 coding

12 ANOVA output in R 12 ANOVA tables with treatment coding and with sum to 0 coding anova(fit3) Analysis of Variance Table Df Sum Sq Mean Sq F value Pr(F) income.log < 2.2e-16 *** type < 2.2e-16 *** women * income.log:type e-05 *** income.log:women type:women ** income.log:type:women Residuals Anova(fit3, type = "II") Anova Table (Type II tests) Sum Sq Df F value Pr(F) income.log e-11 *** type e-12 *** women *** income.log:type e-06 *** income.log:women type:women ** income.log:type:women Residuals Anova(fit3, type = "III") Anova Table (Type III tests) Sum Sq Df F value Pr(F) (Intercept) e-09 *** income.log e-10 *** type e-05 *** women

13 ANOVA output in R 13 income.log:type e-05 *** income.log:women type:women income.log:type:women Residuals anova(fit3.s) Analysis of Variance Table Df Sum Sq Mean Sq F value Pr(F) income.log < 2.2e-16 *** type < 2.2e-16 *** women * income.log:type e-05 *** income.log:women type:women ** income.log:type:women Residuals Anova(fit3.s, type = "II") Anova Table (Type II tests) Sum Sq Df F value Pr(F) income.log e-11 *** type e-12 *** women *** income.log:type e-06 *** income.log:women type:women ** income.log:type:women Residuals

14 ANOVA output in R 14 Anova(fit3.s, type = "III") Anova Table (Type III tests) Sum Sq Df F value Pr(F) (Intercept) e-07 *** income.log e-08 *** type e-05 *** women income.log:type e-05 *** income.log:women type:women income.log:type:women Residuals ## Refit without 3-way interaction: fit3.2 <- lm( prestige ~ (income.log + type + women)^2 ) summary(fit3.2) Call: lm(formula = prestige ~ (income.log + type + women)^2) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr( t ) (Intercept) e-10 *** income.log e-11 *** typeprof e-06 *** typewc women income.log:typeprof e-06 *** income.log:typewc income.log:women

15 ANOVA output in R 15 typeprof:women * typewc:women Residual standard error: on 88 degrees of freedom Multiple R-Squared: , Adjusted R-squared: F-statistic: on 9 and 88 DF, p-value: < 2.2e-16 Anova( fit3.2, type = "II") Why would you prefer Type II here Anova Table (Type II tests) Sum Sq Df F value Pr(F) income.log e-12 *** type e-12 *** women *** income.log:type e-06 *** income.log:women type:women ** Residuals Drop income.log:women interaction fit3.22 <- lm( prestige ~ (income.log + women) * type ) summary(fit3.22) Call: lm(formula = prestige ~ (income.log + women) * type) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr( t ) (Intercept) e-11 ***

16 ANOVA output in R 16 income.log e-12 *** women ** typeprof e-06 *** typewc income.log:typeprof e-06 *** income.log:typewc women:typeprof * women:typewc Residual standard error: on 89 degrees of freedom Multiple R-Squared: , Adjusted R-squared: F-statistic: 71.2 on 8 and 89 DF, p-value: < 2.2e-16 Anova( fit3.22, type = "II") Anova Table (Type II tests) Sum Sq Df F value Pr(F) income.log e-12 *** women *** type e-15 *** income.log:type e-07 *** women:type ** Residuals

17 ANOVA output in R 17 Graphical presentation of models with higher-order interactions summary(income.log) Min. 1st Qu. Median Mean 3rd Qu. Max pred <- expand.grid( income.log = seq(9,10.5,.1), + type = levels(type), + women = c(0,100)) pred$prestige <- predict( fit3.22, newdata = pred) library(lattice) td(new=t) xyplot( prestige ~ income.log women, pred, groups = type, type = 'l', + auto.key = T)

18 ANOVA output in R 18 bc prof wc women women 100 prestige income.log

19 ANOVA output in R 19 td( col = c('red','blue','black'), lwd = 1.5) xyplot( prestige ~ income.log + factor(paste("women =",women,"%")), pred, + groups = type, type = 'l', + auto.key = list(columns= 3, lines=t, points = F))

20 ANOVA output in R 20 bc prof wc women = 0 % women = 100 % prestige income.log

21 ANOVA output in R 21 xyplot( prestige ~ income.log type, + pred, + groups = factor(paste("women =",women,"%")), type = 'l', + auto.key = list(columns= 2, lines=t, points = F))

22 ANOVA output in R 22 women = 0 % women = 100 % wc prestige 150 bc prof \ income.log