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 aslightly 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.

Answer

This is an observational study, not an experiment. Data was collected via surveys from students. The question is too vague. A better question would use a hypothesis test:

Null Hypothesis: A professor’s appearance is unrelated to course evaluations. Alt Hypothesis: A professor’s appearance is related to course evaluations.

  1. 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?

Answer

The distribution of the score variable is not normal. It’s skewed left which means that most of the students rated courses very high. I did expect to see this. Students who complete evaluations often will score the course very high. Those unsatisfied with a course probably won’t evaluate it or would have dropped out earlier.

hist(evals$score)

  1. Excluding score, select two other variables and describe their relationship using an appropriate visualization (scatterplot, side-by-side boxplots, or mosaic plot).

Answer

There does not seem to be any correlation between age and the average beauty rating of professor. Also, it appears as though male professors over the age of 55 have a higher beauty average.

library(ggplot2)
## Warning: package 'ggplot2' was built under R version 3.5.1
ggplot(evals, aes(age,bty_avg, colour = gender)) + 
  geom_point()

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)

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?

Answer

There are 463 observations in the dataframe which appears to be less in the scatterplot above.

  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?

Answer

Some data points on the above scatter plot over-laid each other.

plot(jitter(evals$score) ~ evals$bty_avg)

  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?

Answer

The linear model predicting the average score is

y = 3.88 + 0.06664*bty_avg

The slope in this context means that for every additional increase in bty_avg, the average score for the professor increases by 0.06664.

The p-value is < 0.05 which means that this model is statistically significant and we can reject the null hypothesis that score is independent of bty_avg.

m_bty <- lm(score ~ bty_avg, data = evals)
summary(m_bty)
## 
## Call:
## lm(formula = score ~ bty_avg, data = evals)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.9246 -0.3690  0.1420  0.3977  0.9309 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  3.88034    0.07614   50.96  < 2e-16 ***
## bty_avg      0.06664    0.01629    4.09 5.08e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.5348 on 461 degrees of freedom
## Multiple R-squared:  0.03502,    Adjusted R-squared:  0.03293 
## F-statistic: 16.73 on 1 and 461 DF,  p-value: 5.083e-05
plot(jitter(evals$score) ~ evals$bty_avg)
abline(m_bty)

  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 conditions of least squared residuals are not reasonably met. The distribution of the residuals are not normally distributed. There seems to be a left skew in the distribution.

plot(m_bty$residuals ~ evals$bty_avg, main="A. Constant Variability of Residuals")
abline(h = 0, lty = 3)

A. There is constant variability of the residuals, no distinct pattern.

hist(m_bty$residuals, main="B. Nearly normal residuals")

B. The histogram is left skewed.

qqnorm(m_bty$residuals, main="C. Normal Q-Q Plot")
qqline(m_bty$residuals)

C. The Q-Q plot shows a deviation from the normal distribution.

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.
plot(abs(m_bty_gen$residuals) ~ m_bty_gen$fitted.values, main="D. Variability of Residuals for m_bty_gen")

D. Variability is not nearly as diverse.

hist(m_bty_gen$residuals, main="E. Left Skewed residuals for m_bty_gen")

E. Left skewed histogram.

qqnorm(m_bty_gen$residuals, main="F. Normal Q-Q Plot for m_bty_gen ")
qqline(m_bty_gen$residuals)

F. Significant deviation from normality.

  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?

Answer

bty_avg is still more a significant predictor of score. The addition of the gender variable does show a higher correlation, but the residual plots do not fully support this finding.

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.

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

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?

Answer

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

Given the model, the male gendered professor would have a higher score if he receives the same beauty rating as the female gendered professor.

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.

Answer

R reduced the “rank” variable with three categories into just two, “rankenture track” and “ranktenured”.

m_bty_rank <- lm(score ~ bty_avg + rank, data = evals)
summary(m_bty_rank)
## 
## Call:
## lm(formula = score ~ bty_avg + rank, data = evals)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.8713 -0.3642  0.1489  0.4103  0.9525 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)       3.98155    0.09078  43.860  < 2e-16 ***
## bty_avg           0.06783    0.01655   4.098 4.92e-05 ***
## ranktenure track -0.16070    0.07395  -2.173   0.0303 *  
## ranktenured      -0.12623    0.06266  -2.014   0.0445 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.5328 on 459 degrees of freedom
## Multiple R-squared:  0.04652,    Adjusted R-squared:  0.04029 
## F-statistic: 7.465 on 3 and 459 DF,  p-value: 6.88e-05

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.

Answer

My expectation is that the variable cls_credits (number of credits) will not have any association with the score of the professor.

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.

Answer

My suspicion was incorrect. It seems as though the number of classes that a professor teaches cls_profssingle has the highest p-value.

cls_profssingle -0.0146619 0.0519885 -0.282 0.77806

  1. Interpret the coefficient associated with the ethnicity variable.

Answer

When the ethnicity is not a minority, the score improves by .124 if all other coefficients are zero.

  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?

Answer

When cls_profs is dropped, the coefficients and significance of the other explanatory variables changed.

m_full <- 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_full)
  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.

Answer

m_best_model <- lm(score ~ ethnicity + gender + language + age + cls_perc_eval 
             +  cls_credits + bty_avg 
             + pic_color, data = evals)
summary(m_best_model)

\[ \begin{aligned} \widehat{score} &= 3.77 + 0.17 \times ethnicity + 0.21 \times gender + -0.21 \times language + -0.01 \times age + 0.005\times cls\_perc\_eval + 0.5\times cls\_credits + 0.05\times bty\_avg + -0.19 \times pic\_color \end{aligned} \]

  1. Verify that the conditions for this model are reasonable using diagnostic plots.

Answer

Plots of the residuals do not show normality which leads to doubts about the model’s ability to predict accurately.

qqnorm(m_best_model$residuals, main="Normal Q-Q Plot for m_best_model ")
qqline(m_best_model$residuals)
plot(abs(m_best_model$residuals) ~ m_best_model$fitted.values, main="Variability of Residuals for m_best_model")
  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?

Answer

This new information calls into question the Indenpendence of the observations which may affect the model. Some the evaluations may be from students who took multiple courses from the same professor, and may have chosen that professor because of a positive prior experience.

  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.

Answer

A. Be of a non-ethnic minority (ethnicitynot minority 0.167872)
B. Be male (gendermale 0.207112) c. Educated in an English speaking school (languagenon-english -0.206178)
D. Be young (age -0.006046 ) E. Have a high percentage of students complete the evaluations (cls_perc_eval 0.004656)
F. Teach only one course for credit (cls_creditsone credit 0.505306)
G. Have a high beauty rating (bty_avg 0.051069)
H. Have a B&W photo rather than a color one (pic_colorcolor -0.190579)

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

No. I would not. There are several questions as to how the data was collected which may indicate bias. Also, given how education is distributed more online, the question itself may be irrelevant.

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.