Instructions

  1. Start with your repo for this assignment
    • located at organization for this class
    • name contains HW 2 and your name

Clone this repo in your local directory on gort. (Remember, the address is http://gort.stat.duke.edu:8787/.)

  1. Edit the README.md to include some relevant information about the repository, commit, and push. (This is just to check everything is working fine, and you know what you’re doing.)

  2. Open a new R Markdown file, name it the same name as your repository, and save it.

  3. Include answers to all exercises in your R Markdown file. Your answers should always include any summary and/or plot you use to answer that particular question.

Grading the professor

Many college courses conclude by giving students the opportunity to evaluate the course and the instructor anonymously. However, the use of these student evaluations as an indicator of course quality and teaching effectiveness is often criticized because these measures may reflect the influence of non-teaching related characteristics, such as the physical appearance of the instructor. The article titled, “Beauty in the classroom: instructors’ pulchritude and putative pedagogical productivity” (Hamermesh and Parker 2005) found that instructors who are viewed to be better looking receive higher instructional ratings.

In this lab we will analyze the data from this study in order to learn what goes into a positive professor evaluation.

The data

The data were gathered from end of semester student evaluations for a large sample of professors from the University of Texas at Austin. In addition, six students rated the professors’ physical appearance. (This is a slightly modified version of the original data set that was released as part of the replication data for Data Analysis Using Regression and Multilevel/Hierarchical Models (Gelman and Hill 2007).) The result is a data frame where each row contains a different course and columns represent variables about the courses and professors.

Let’s first load the packages we’ll use for this exercise. Note that you might not have some of them installed. If that’s the case, use install.packages to install them first.

library(dplyr)
library(ggplot2)
library(stringr)
library(GGally)

And let’s also load the data:

load(url("http://www.openintro.org/stat/data/evals.RData"))
variable description
score average professor evaluation score: (1) very unsatisfactory - (5) excellent.
rank rank of professor: teaching, tenure track, tenured.
ethnicity ethnicity of professor: not minority, minority.
gender gender of professor: female, male.
language language of school where professor received education: english or non-english.
age age of professor.
cls_perc_eval percent of students in class who completed evaluation.
cls_did_eval number of students in class who completed evaluation.
cls_students total number of students in class.
cls_level class level: lower, upper.
cls_profs number of professors teaching sections in course in sample: single, multiple.
cls_credits number of credits of class: one credit (lab, PE, etc.), multi credit.
bty_f1lower beauty rating of professor from lower level female: (1) lowest - (10) highest.
bty_f1upper beauty rating of professor from upper level female: (1) lowest - (10) highest.
bty_f2upper beauty rating of professor from second upper level female: (1) lowest - (10) highest.
bty_m1lower beauty rating of professor from lower level male: (1) lowest - (10) highest.
bty_m1upper beauty rating of professor from upper level male: (1) lowest - (10) highest.
bty_m2upper beauty rating of professor from second upper level male: (1) lowest - (10) highest.
bty_avg average beauty rating of professor.
pic_outfit outfit of professor in picture: not formal, formal.
pic_color color of professor’s picture: color, black & white.

Exploring the data

  1. Is this an observational study or an experiment? The original research question posed in the paper is whether beauty leads directly to the differences in course evaluations. Given the study design, is it possible to answer this question as it is phrased? If not, rephrase the question.

  2. Describe the distribution of score. Is the distribution skewed? What does that tell you about how students rate courses? Is this what you expected to see? Why, or why not? Include any summary statistics and visualizations you use in your response.

  3. Excluding score, select two other variables and describe their relationship using an appropriate visualizations.

Simple linear regression

The fundamental phenomenon suggested by the study is that better looking teachers are evaluated more favorably. Let’s create a scatterplot to see if this appears to be the case:

ggplot(data = evals, aes(x = bty_avg, y = score)) +
  geom_point()

Before we draw conclusions about the trend, compare the number of observations in the data frame with the approximate number of points on the scatterplot. Is anything awry?

  1. Replot the scatterplot, but this time use geom_point(position = "jitter"). What does “jitter” mean? What was misleading about the initial scatterplot?

  2. Let’s see if the apparent trend in the plot is something more than natural variation. Fit a linear model called m_bty to predict average professor score by average beauty rating and add the regression line to your plot. Write out the equation for the linear model and interpret the slope. Does there appear to be a practically significant relationship between professor score and average beauty rating?

Multiple linear regression

The data set contains several variables on the beauty score of the professor: individual ratings from each of the six students who were asked to score the physical appearance of the professors and the average of these six scores. Let’s take a look at the relationship between one of these scores and the average beauty score.

ggplot(data = evals, aes(x = bty_avg, y = bty_f1lower)) +
  geom_point(position = "jitter")
evals %>% 
  summarise(cor(bty_avg, bty_f1lower))

As expected the relationship is quite strong. Note that the correlation coefficient measures the linear association between two variables, and ranges between -1 and 1, -1 indicating a perfect negative relationship and 1 indicating a perfect positive relationship. It is nor surprising that the correlation between these two variables is quite strong - after all, the average score is calculated using the individual scores. We can actually take a look at the relationships between all beauty variables (variables that have the character string bty in them) using the following command:

bty_cols <- str_detect(names(evals), "bty_")
ggpairs(evals, columns = which(bty_cols == TRUE))
  1. Describe how the str_detect function works and how we use information resulting from that function to make the pairwise plot.

These variables are collinear (correlated), and adding more than one of these variables to the model would not add much value to the model. In this application and with these highly-correlated predictors, it is reasonable to use the average beauty score as the single representative of these variables.

In order to see if beauty is still a a good predictor of professor score after we’ve accounted for the gender of the professor, we can add the gender term into the model.

m_bty_gen <- lm(score ~ bty_avg + gender, data = evals)
m_bty_gen
  1. How do the AIC and the adjusted \(R^2\) of this model and the previous one compare? Has the addition of gender to the model changed the parameter estimate (slope) for bty_avg?

Note that the estimate for gender is now called gendermale. You’ll see this name change whenever you introduce a categorical variable. The reason is that R recodes gender from having the values of female and male to being an indicator variable called gendermale that takes a value of \(0\) for females and a value of \(1\) for males. (Such variables are often referred to as “dummy” variables.)

As a result, for females, the parameter estimate is multiplied by zero, leaving the intercept and slope form familiar from simple regression.

\[ \begin{aligned} \widehat{score} &= \hat{\beta}_0 + \hat{\beta}_1 \times bty\_avg + \hat{\beta}_2 \times (0) \\ &= \hat{\beta}_0 + \hat{\beta}_1 \times bty\_avg\end{aligned} \]

  1. What is the equation of the line corresponding to males? (Hint: For males, the parameter estimate is multiplied by 1.) For two professors who received the same beauty rating, which gender tends to have the higher course evaluation score?

The decision to call the indicator variable gendermale instead ofgenderfemale has no deeper meaning. R simply codes the category that comes first alphabetically as a \(0\). (You can change the reference level of a categorical variable, which is the level that is coded as a 0, using therelevel function. Use ?relevel to learn more.)

  1. Create a new model called m_bty_rank with gender removed and rank added in. How does R appear to handle categorical variables that have more than two levels? Note that the rank variable has three levels: teaching, tenure track, tenured.

The interpretation of the coefficients in multiple regression is slightly different from that of simple regression. The estimate for bty_avg reflects how much higher a group of professors is expected to score if they have a beauty rating that is one point higher while holding all other variables constant. In this case, that translates into considering only professors of the same rank with bty_avg scores that are one point apart.

We can go a step further and include an interaction variable between bty_avg and gender:

m_bty_gen_int <- lm(score ~ bty_avg * gender, data = evals)
m_bty_gen_int

We can also visualize this model with the following:

ggplot(data = evals, aes(x = bty_avg, y = score, color = gender)) +
  geom_point() +
  stat_smooth(method = "lm")
  1. How does the relationship between beauty and evaluation score vary between male and female professors?

The search for the best model

We will start with a full model that predicts professor score based on rank, ethnicity, gender, language of the university where they got their degree, age, proportion of students that filled out evaluations, class size, course level, number of professors, number of credits, average beauty rating, outfit, and picture color.

  1. Which variable would you expect to be the worst predictor of evaluation scores? Why? Hint: Think about which variable would you expect to not have any association with the professor score.

Let’s run the model…

m_full <- lm(score ~ rank + ethnicity + gender + language + age + cls_perc_eval 
             + cls_students + cls_level + cls_profs + cls_credits + bty_avg 
             + pic_outfit + pic_color, data = evals)
summary(m_full)

For simplicity we’ll work with a model that does not have interaction effects.

  1. Check your suspicions from the previous exercise. Include the model output in your response.

  2. Interpret the coefficient associated with the ethnicity variable.

  3. Drop the variable that results in the highest gain of adjusted R-squared. Did the coefficients and significance of the other explanatory variables change? (One of the things that makes multiple regression interesting is that coefficient estimates depend on the other variables that are included in the model.) If not, what does this say about whether or not the dropped variable was collinear with the other explanatory variables?

  4. Using backward-selection and either AIC or adjusted R-squared as the selection criterion, determine the best model. You do not need to show all steps in your answer, just the output for the final model. Also, write out the linear model for predicting score based on the final model you settle on.

  5. Based on your final model, describe the characteristics of a professor and course at University of Texas at Austin that would be associated with a high evaluation score.

  6. Would you be comfortable generalizing your conclusions to apply to professors generally (at any university)? Why or why not?

License

This is a product of OpenIntro that is released under a Creative Commons Attribution-ShareAlike 3.0 Unported. This lab was written by Mine Çetinkaya-Rundel and Andrew Bray.

References

Gelman, Andrew, and Jennifer Hill. 2007. Data Analysis Using Regression and Multilevel/Hierarchical Models. 1st ed. Cambridge University Press.

Hamermesh, Daniel S., and Amy Parker. 2005. “Beauty in the Classroom - Instructors’ Pulchritude and Putative Pedagogical Productivity” 24 (4). Economics of Education Review: 369–76. doi:10.1016/j.econedurev.2004.07.013.