22s:152 Applied Linear Regression DeCook Fall 2011 Lab 3 Monday October 3

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1 s:5 Applied Linear Regression DeCook all 0 Lab onday October The data Set In 004, a study was done to examine if gender, after controlling for other variables, was a significant predictor of salary for science, technology, engineering, and math (STE) faculty at Iowa State University All the information is publicly available, but the names have been removed, and this is a subset of the full variables set The subsetted data can be found in salary ISU datacsv VARIABLES Department : one of 8 different departments Rank Code : rank of faculty ull professor Associate professor Assistant professor Gender : male or female Salary 9 mo : 9 month salary for faculty member for 004 ## We wish to exclude faculty members without any contracts or ## grants for this analysis ## ) Use a boolean statement to pull-out certain rows: > salary=salaryoriginal[salaryoriginal[,5]!=0,] or ## ) Use a boolean statement within the subset function: > salary=subset(salaryoriginal, salaryoriginal[,5]!=0) > head(salary) Department Gender Salary_9_mo Avg_Cont_Grants CCE E EEOBS AN S SHN AGRON CO S > attach(salary) Avg Cont Grants : average contracts and grants for fiscal years 00, 00, 00 Subsetting the data The data set salary ISU datacsv is available from our class website Either download the file and then read it into R, or use the following commands > salaryoriginal=readcsv(" datasets/salary_isu_datacsv") > head(salaryoriginal) Department Gender Salary_9_mo Avg_Cont_Grants CCE E EEOBS EEOBS ISE CO S AN S Exploring the data Let s look at the salary variable, the grants variable, and some of their transformations Income or money variables are often right-skewed irst, let s rename them for ease of scripting: > ACG=Avg_Cont_Grants > Sal=Salary_9_mo > par(mfrow=c(,)) > hist(sal) > hist(log(sal)) > hist(acg) > hist(log(acg)) It turns out that using the transformed variables will help meet our assumptions

2 Let s look at the rank variable, which is a categorical variable How many faculty members are in each category? > table() ## Recall: ull Professor== What percentage is in each category? > table()/length() Why do you think there s so many more full professors? it the simple linear regression: > SLRout=lm(log(Sal) ~ log(acg)) > abline(slrout) > summary(slrout) (Intercept) < e-6 *** log(acg) *** Residual standard error: 0565 on 4 degrees of freedom ultiple R-Squared: 007,Adjusted R-squared: statistic: 48 on and 4 D, p-value: The relationship between log(salary) and log(grants) is significant 4 Relationship between log(salary) and log(grants) Write the model: > plot(log(acg),log(sal)) There doesn t seem to be a strong relationship, but on top of that, there are three individuals with very large grant amounts Let s look at these observations: > subset(salary,log(acg)>5) Turns out these individuals are different from the others in that they are administrators in university centers In this case, after discussion with those involved, we felt it was justifiable to remove these observations before further analysis We ll remove these three observations and proceed: > salary=subset(salary,log(acg)<5) > detach(salary) ## The old data set > attach(salary) ## After removal of the > ACG=Avg_Cont_Grants ## ACG after removal > Sal=Salary_9_mo ## Sal after removal 5 Inclusion of Rank It was known that rank would have an impact on salary If we re interested in how gender affects salary, we should also include any other variables known to affect salary Rank is a categorical variable Let s create dummy variables Because there are three categories, we ll need dummy variables: rankdummy=rep(0,nrow(salary)) rankdummy[==]= rankdummy=rep(0,nrow(salary)) rankdummy[==]= ## All zeroes at first ## Place s appropriately > plot(log(acg),log(sal)) 4

3 What is the coding we used? (Recall: ull Professor== in dataset) it an additive model (ie no interaction) with grants and rank: What is the baseline group? Let s check our coding: Assistant Associate ull dummy dummy > lmbothout=lm(log(sal) ~ log(acg) + rankdummy + rankdummy) > summary(lmbothout) (Intercept) < e-6 *** log(acg) ** rankdummy < e-6 *** rankdummy < e-6 *** > dataframe(,rankdummy,rankdummy) rankdummy rankdummy Rank Code was already numeric, why can t we just use that variable in our model? What model are you fitting if you regress log(sal) on Rank Code here? > isnumeric() [] TRUE Residual standard error: 004 on 40 degrees of freedom ultiple R-Squared: 0906,Adjusted R-squared: 086 -statistic: 897 on and 40 D, p-value: < e-6 Write the model: Y i = β 0 + β ACG x i + β D D i + β D D i + i What does the hypothesis of H 0 : β D = 0 test? (In the context of data) How do I test if Rank is useful in the model at all? H 0 : β D = β D =0 a Partial -test: > anova(slrout,lmbothout) Analysis of Variance Table odel : log(sal) ~ log(acg) odel : log(sal) ~ log(acg) + rankdummy + rankdummy ResDf RSS Df Sum of Sq Pr(>) < e-6 *** 5 6

4 If we wanted to include an interaction between rank and grants, what other variables would be needed in the model? What test would be used to test for interaction? 6 Inclusion of gender > sexdummy=rep(0,nrow(salary)) > sexdummy[gender==""]= > lmout=lm(log(sal)~log(acg)+rankdummy+rankdummy+sexdummy) > summary(lmout) (Intercept) < e-6 *** log(acg) ** rankdummy < e-6 *** rankdummy < e-6 *** sexdummy ** Residual standard error: 00 on 49 degrees of freedom ultiple R-Squared: 0404,Adjusted R-squared: statistic: 705 on 4 and 49 D, p-value: < e-6 In this model, which sex is estimated to have a slight advantage? What if we didn t include rank (only grants and gender)? > lmoutnorank=lm(log(sal)~log(acg)+sexdummy) > summary(lmoutnorank) (Intercept) < e-6 *** log(acg) e-05 *** sexdummy e-08 *** Residual standard error: 0479 on 4 degrees of freedom ultiple R-Squared: ,Adjusted R-squared: statistic: on and 4 D, p-value: 08e-0 Why is gender so much stronger without rank included? (Recall the fundamentals of multiple regression) > table() > table(gender,) Gender rank and gender are not independent of each other Knowing the rank of a randomly chosen individual gives you some information on the likelihood of their gender A large proportion of the women are in lower ranks If you don t account for rank, it will look like women are paid less (but that s not a good analysis) It turns out that if we also include department (which is also associated with salary), the significant sex effect disappears 7 8

5 7 Lattice Plot Lattice plots can be useful when considering a quantitative response and categorical predictors, or factors R can actually make dummy variables on its own using the asfactor() command (more on this later) Here, we can see how log(sal) is related to log(acg) for each of the six combinations of Sex/Rank: ## lattice is an attachable package > library(lattice) > xyplot(log(sal)~log(acg) asfactor(gender)+asfactor()) log(sal) log(acg) 9