Introduction

The purpose of this lab is to help you gain more experience working with multilevel linear regression models. The lab is based on Exercise 6 of Section 12.11 of Data Analysis Using Regression and Multilevel/Hierarchical Models by Gelman A., and Hill, J. The data is from the following article:
Hamermesh, D. S. and Parker, A. (2005), “Beauty in the classroom: instructors’ pulchritude and putative pedagogical productivity”, Economics of Education Review, v. 24 (4), pp. 369-376.

The data contains information about student evaluations of instructor’s beauty and teaching quality for several courses at the University of Texas from 2000 to 2002. Evaluations were carried out during the last 3 weeks of the 15-week semester. Students administer the evaluation instrument while the instructor is absent from the classroom. The beauty judgements were made later using each instructor’s pictures by six undergraduate students (3 women and 3 men) who had not attended the classes and were not aware of the course evaluations. The sample contains a total of 94 professors across 463 classes, with the number of classes taught by each professor ranging from 1 to 13. Underlying the 463 observations are 16,957 completed evaluations from 25,547 registered students. The data you will use for this lab excludes some variables in the original dataset.

Read the article (available via Duke library) for more information about the problem.

In this lab you will once again work in pairs just like you did for lab two. You should work with the same student you worked with for lab 2. The names of the two students on each team MUST be on the team’s lab report. Each team should submit only one report for this lab. Gradescope will let you select your team mate when submitting, so make sure to do so. Only one person needs to submit the report.

Housekeeping

The Data

Download the data (named Beauty.txt) from Sakai and save it locally to the same directory as your R markdown file. To find the data file on Sakai, go to Resources \(\rightarrow\) Datasets \(\rightarrow\) Lab Datasets \(\rightarrow\) Lab 4. Once you have downloaded the data file into the SAME folder as your R markdown file, load the data and packages needed for multilevel models by using the following R code.

⊕Remember to double-check the dimensions and first few rows of the data to confirm you have the right file.

library(lme4)
library(lmerTest)
# Read in the data
Beauty <- read.table ("Beauty.txt", header=T, sep=" ")

Exercises

Treat the variable eval as your response variable and the other variables as potential predictors.

1. Is the distribution of eval normal? If not, try the log transformation. Does that look more “normal”?
   If no, use eval as your response variable for the remaining questions. If yes, use log(eval) as your response variable instead for the remaining questions.
   If you think the log transformation does not help, what other transformation(s) do you think might work? Examine and describe the distribution(s) for the transformation(s).
   Either way, for ease of interpreting the results for this lab, answer the remaining questions using either eval or `log(eval).

2. Describe the overall relationship between eval and beauty. Also examine the same relationship by CourseID. Are there any courses for which the relationship looks potentially different than others?
   There are 31 levels of CourseID in all, which might be tough to explore graphically, so you should probably take a look at a random sample of say 9 classes plus class CourseID == 0, making a total of 10 classes. Note that level CourseID == 0 actually includes so many other classes which were not identified in the data. For the purposes of this lab, we will treat that class as one single huge class.

3. Is it meaningful to fit a model that includes random slopes for beauty by profnumber? Why or why not?

4. Now, explore the relationship between eval and the other potential predictors, excluding profnumber, profevaluation, and CourseID. Don’t include any of the plots, just briefly describe the most interesting relationships. We should not include profevaluation as a predictor for eval. Why?

5. Fit a varying-intercept model for these data by profnumber with beauty as the only predictor. Interpret the results in the context of the question.

6. Now identify three instructor-level predictors excluding beauty (with profnumber being the instructor-level identifier) that you think we should control for based on your EDA. Fit a varying-intercept model for these data by profnumber and include beauty plus the other three variables you have identified as predictors. Interpret the results in the context of the question.

7. Using the model from question 5, how does the variation in average ratings across professors compare to the variation in ratings across the different course evaluations for the same instructor?

8. Extend the model from question 6 by also allowing the intercept and coefficient for beauty to vary by CourseID. Fit the model in R and interpret the results: the coefficient estimates and the estimated standard deviations. Did any of the results for the fixed effects change? If yes, why do you think that is?

9. Using two or three (at most!) plots, tell a visual story about your most interesting or surprising findings. If you need to present results by profnumber or CourseID, you should consider either looking at a random sample of them or focusing on the groups that are the most different, since there are so many of them.

10. Identify one class-level predictor (with CourseID being the class-level identifier) that you think we should control for in the model. Include it as a fixed predictor in the model from Question 8. Is the variable significant? If yes, interpret the coefficient.

Grading

10 points: 1 point for each question