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. (Daniel S. Hamermesh, Amy Parker, Beauty in the classroom: instructors pulchritude and putative pedagogical productivity, Economics of Education Review, Volume 24, Issue 4, August 2005, Pages 369-376, ISSN 0272-7757, 10.1016/j.econedurev.2004.07.013. http://www.sciencedirect.com/science/article/pii/S0272775704001165.)

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.

load("more/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.

    This is an observational study; they are retrieving data that was collected by questionnaire without regard to control and test groups, blinding and other experimental controls.

  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?

hist(evals$score, main = "Histogram of Professor Evals", xlab = "Scores  (1-5)")

These data have a right skew and is unimodal. The mode is at 4.5 which is a very high score. This means that most professors are very satisfactory, with a relatively smaller number of professors whose teaching is less satisfactory. For a major research university, this seems reasonable, since such institutions are able to hire top academics in their fields, but some professors are better researchers than teachers.

  1. Excluding score, select two other variables and describe their relationship using an appropriate visualization (scatterplot, side-by-side boxplots, or mosaic plot).
plot(evals$cls_perc_eval, evals$bty_avg, xlab = "% eval completion", ylab = "Average Beuaty Rating of Professor")

boxplot(evals$bty_m1lower,evals$bty_f1lower, names = c("Male Professor Beauty Lower Level","Female Professor Beauty Lower Level"))

plot(evals$age,evals$bty_avg, xlab = "Age (years)", ylab = "Average Beuaty Score")

It appears that there might a weak correlation between Average Beauty and % Student Eval completion. There might be a weak negative correlation between average beauty score and age. Male professors are rated as less beautiful than female professors on average.

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:

plot(evals$score ~ evals$bty_avg)

It looks like a weak correlation at best.

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 the function jitter() on the \(y\)- or the \(x\)-coordinate. (Use ?jitter to learn more.) What was misleading about the initial scatterplot?
plot(evals$score ~ jitter(evals$bty_avg))

“jitter(x, …) returns a numeric of the same length as x, but with an amount of noise added in order to break ties.”

It appears as if there were several tied scores in the data.

  1. 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 line to your plot using abline(m_bty). Write out the equation for the linear model and interpret the slope. Is average beauty score a statistically significant predictor? Does it appear to be a practically significant predictor?
m_bty <- lm(evals$score ~ evals$bty_avg)
summary(m_bty)
plot(evals$score ~ evals$bty_avg)
abline(m_bty)

\[ score = 3.88034 + 0.06664*bty_{avg} \]

The p-value of the slope is much smaller than 0.05, which means the slope is significant at a 95% confidence interval. However, the R\(^2\) value is tiny, 0.03502 which means 3.502% of the score’s variation about the mean is explained by this model. Further more the slope is also small, for each point of beauty, the score increases by 0.06664. On a scale of 1-10 this is insignificant. There is no practical significance here.

  1. Use residual plots to evaluate whether the conditions of least squares regression are reasonable. Provide plots and comments for each one (see the Simple Regression Lab for a reminder of how to make these).
# The document fails on this line: 
#plot(evals$bty_avg, m_bty$residuals)
#hist(m_bty$residuals)

There is a strong right skew to the data, so it does not meet the condition of normally distributed residuals. Furthermore, the residuals are not uniformly scattered about 0; the range is approximately (-2,1). This model only marginally meets the residual conditions for linear regression.

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.

plot(evals$bty_avg ~ evals$bty_f1lower)
cor(evals$bty_avg, evals$bty_f1lower)

As expected the relationship 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 (columns 13 through 19) using the following command:

plot(evals[,13:19])

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 significant 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)
summary(m_bty_gen)
  1. P-values and parameter estimates should only be trusted if the conditions for the regression are reasonable. Verify that the conditions for this model are reasonable using diagnostic plots.
# The document will also not knit on this line:
#plot(fitted(m_bty_gen), resid(m_bty_gen))
#hist(resid(m_bty_gen))

Normal Distribution and Heteroskedasticity conditions are not fully met with this model. The validity of the model is dubious.

  1. Is bty_avg still a significant predictor of score? Has the addition of gender to the model changed the parameter estimate for bty_avg?

The p-value is still <0.05 and actually improved by an order of magnitude, but if the residual conditions are not met that is pointless.

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 re-codes 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.

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

We can plot this line and the line corresponding to males with the following custom function.

multiLines(m_bty_gen)
  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?

Males have the higher score on average for a given beauty score. The formula for males is:

\[ score = 3.74734 + 0.17239*gendermale \]

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.
m_bty_rank <- lm(score ~ bty_avg + rank, data = evals)
summary(m_bty_rank)

The first category alphabetically, teaching is set to ‘0’ and the next two tenure track and tenured are treated as separate variables.

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.

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 have the highest p-value in this model? Why? Hint: Think about which variable would you expect to not have any association with the professor score.

I would think that the color of the faculty photo would have the least significant effect. I cannot think of a reason why the faculty photo would have any outcome on evaluation 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)
  1. Check your suspicions from the previous exercise. Include the model output in your response.
summary(m_full)

If a class is team taught or not seems to have the least significant effect. Surprisingly a color photo will knock 0.22 \(\pm\) 0.07 off your score at better than 99% level of confidence.

  1. Interpret the coefficient associated with the ethnicity variable.

Appearing as not a minority adds 0.12 \(\pm\) 0.08 to a professors evaluation, but this effect is not significant at a 90% level of confidence.

  1. Drop the variable with the highest p-value and re-fit the model. 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?
m_full2 <- lm(score ~ rank + ethnicity + gender + language + age + cls_perc_eval 
             + cls_students + cls_level  + cls_credits + bty_avg 
             + pic_outfit + pic_color, data = evals)
summary(m_full2)

The p-values of all the other variables changed slightly some went down, e.g., pic_color \(0.00252 \rightarrow 0.002205\) and some went up, e.g., bty_avg \(0.02267 \rightarrow 0.023032\). This suggests that the removed variable was collinear with the rest, to at least a minor degree.

  1. Using backward-selection and p-value 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.
m_full3 <- lm(score ~ rank + ethnicity + gender + language + age + cls_perc_eval 
             + cls_credits + bty_avg 
             + pic_outfit + pic_color, data = evals)
summary(m_full3)

After I removed ‘rank’, the adjusted R\(^2\) went from 0.1632 to 0.161. Removing more variables such as pic outfit, made adjusted R\(^2\) fall further, meaning less of the variability of the score about it’s mean was explained by the model.

\[ score = 4.152893 + -0.142239*tenure track + -0.083092*tenured + 0.143509*ethnicity + 0.208080*gender + -0.222515 *language + -0.009074*age + 0.004841*cls_perc_eval + 0.472669*cls_credit + 0.043578 + -0.136594*pic_outfit + -0.189905*pic_color \]

  1. Verify that the conditions for this model are reasonable using diagnostic plots.
#plot(fitted(m_full3), resid(m_full3))
#hist(resid(m_full3))

The residuals only marginally meet the requirements of residual distribution and constant variance.

  1. The original paper describes how these data were gathered by taking a sample of professors from the University of Texas at Austin and including all courses that they have taught. Considering that each row represents a course, could this new information have an impact on any of the conditions of linear regression?

Professors teach multiple courses, so depending on the sample size, the data might not be linearly independent.

  1. 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.

A non-tenured, male, fluent english-speaking, white, young, handsome professor who gets his students to fill out the evals, dresses on the formal side, teaches single credit courses and has a black and white faculty photo.

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

Different universities have different campus cultures (serious vs party school) and demographics (socioeconomic status of private vs. public schools). This could easily invalidate the study for any other university.

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.