Statistics 333 Random Effects Models and Nested Factors Spring 2003In class, I described the ideas of random effects and fixed effects. It is appropriate to model a factor as arandom effect if you wish to think of the levels as having been randomly selected from a larger population whereasfixed effects are factors where the levels in the problem are the only ones of interest. We also talked about thedifference between factors that are crossed or nested. Two factors are crossed if there is at least one observationfor each pair of levels. Two factors are nested if the levels of one factor are completely contained in the levels ofanother factor. For the chimp example in Chapter 14, the factors sign and chimp are crossed because each chimpis measured at each sign. If we created a factor sex, then chimp would be nested within sex as the levels Brunoand Booee are nested within male and Cindy and Thelma are female.For this problem, it makes sense to consider treating chimp as a random effect. However, R is quite limitedin the types of mixed effects models that can be fit. Fitting nested factors works quite well, but fitting crossedfactors when one or more are random is far from straight-forward. So, I’m not going to show you how to do it.But here is an example of fitting a model with nested factors adding the factor sex. In the formula, we indicatethe nesting of chimp within sex by sex/CHIMP. We will have to create the sex factor within R. (Alternatively, wecould edit the file in which we read in the data). The strategy will be to create a variable that is female everywhereand then change the proper observations to male.> case1401 <- read.table("sleuth/case1401.csv", header = T, sep = ",")> attach(case1401)> sex = rep("female", nrow(case1401))> sex[CHIMP == "BOOEE"] <- "male"> sex[CHIMP == "BRUNO"] <- "male"> sex <- factor(sex)> new1401 <- data.frame(case1401, sex)> detach()> attach(new1401)> fit1 <- lm(log(MINUTES) ~ SIGN + CHIMP)> fit2 <- lm(log(MINUTES) ~ SIGN + sex/CHIMP)> anova(fit1)Analysis of Variance TableResponse: log(MINUTES)Df Sum Sq Mean Sq F value Pr(>F)SIGN 9 45.690 5.077 7.7649 1.498e-05 ***CHIMP 3 5.333 1.778 2.7189 0.06421 .Residuals 27 17.653 0.654---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1> anova(fit2)Analysis of Variance TableResponse: log(MINUTES)Df Sum Sq Mean Sq F value Pr(>F)SIGN 9 45.690 5.077 7.7649 1.498e-05 ***sex 1 0.167 0.167 0.2558 0.61710sex:CHIMP 2 5.166 2.583 3.9505 0.03127 *Residuals 27 17.653 0.654Bret Larget April 18, 2003Statistics 333 Random Effects Models and Nested Factors Spring 2003---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1Notice that the second model just breaks up the sum of squares from chimp into two pieces. There is little evidenceof a sex effect to explain the differences between the chimps. The estimated regression coefficients for sign areidentical, and the estimates for chimp are simply parameterized in a different way.> summary(fit1)Call:lm(formula = log(MINUTES) ~ SIGN + CHIMP)Residuals:Min 1Q Median 3Q Max-1.70580 -0.38195 0.03512 0.41762 1.73890Coefficients:Estimate Std. Error t value Pr(>|t|)(Intercept) 2.69487 0.46096 5.846 3.18e-06 ***SIGNfood 0.96866 0.57175 1.694 0.101736SIGNfruit 1.35652 0.57175 2.373 0.025047 *SIGNhat 1.85663 0.57175 3.247 0.003108 **SIGNkey 0.12460 0.57175 0.218 0.829124SIGNlisten -1.05303 0.57175 -1.842 0.076520 .SIGNlook 2.05422 0.57175 3.593 0.001286 **SIGNmore 0.30099 0.57175 0.526 0.602881SIGNshoe 0.05467 0.57175 0.096 0.924527SIGNstring 2.55070 0.57175 4.461 0.000129 ***CHIMPBRUNO 0.68133 0.36161 1.884 0.070351 .CHIMPCINDY 0.09287 0.36161 0.257 0.799270CHIMPTHELMA 0.84713 0.36161 2.343 0.026766 *---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1Residual standard error: 0.8086 on 27 degrees of freedomMultiple R-Squared: 0.743, Adjusted R-squared: 0.6287F-statistic: 6.503 on 12 and 27 DF, p-value: 2.776e-05> summary(fit2)Call:lm(formula = log(MINUTES) ~ SIGN + sex/CHIMP)Residuals:Min 1Q Median 3Q Max-1.70580 -0.38195 0.03512 0.41762 1.73890Coefficients: (4 not defined because of singularities)Estimate Std. Error t value Pr(>|t|)Bret Larget April 18, 2003Statistics 333 Random Effects Models and Nested Factors Spring 2003(Intercept) 3.54200 0.46096 7.684 2.9e-08 ***SIGNfood 0.96866 0.57175 1.694 0.101736SIGNfruit 1.35652 0.57175 2.373 0.025047 *SIGNhat 1.85663 0.57175 3.247 0.003108 **SIGNkey 0.12460 0.57175 0.218 0.829124SIGNlisten -1.05303 0.57175 -1.842 0.076520 .SIGNlook 2.05422 0.57175 3.593 0.001286 **SIGNmore 0.30099 0.57175 0.526 0.602881SIGNshoe 0.05467 0.57175 0.096 0.924527SIGNstring 2.55070 0.57175 4.461 0.000129 ***sexmale -0.84713 0.36161 -2.343 0.026766 *sexmale:CHIMPBRUNO 0.68133 0.36161 1.884 0.070351 .sexfemale:CHIMPCINDY -0.75426 0.36161 -2.086 0.046571 *---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1Residual standard error: 0.8086 on 27 degrees of freedomMultiple R-Squared: 0.743, Adjusted R-squared: 0.6287F-statistic: 6.503 on 12 and 27 DF, p-value: 2.776e-05Bret Larget April 18,