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” by Hamermesh and Parker found that instructors who are viewed to be better looking receive higher instructional ratings.

Here, you will analyze the data from this study in order to learn what goes into a positive professor evaluation.

Getting Started

Load packages

In this lab, you will explore and visualize the data using the tidyverse suite of packages. The data can be found in the companion package for OpenIntro resources, openintro.

Let’s load the packages.

library(tidyverse)
library(openintro)
library(GGally)

This is the first time we’re using the GGally package. You will be using the ggpairs function from this package later in the lab.

Creating a reproducible lab report

To create your new lab report, in RStudio, go to New File -> R Markdown… Then, choose From Template and then choose Lab Report for OpenIntro Statistics Labs from the list of templates.

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. The result is a data frame where each row contains a different course and columns represent variables about the courses and professors. It’s called evals.

We have observations on 21 different variables, some categorical and some numerical. The meaning of each variable can be found by bringing up the help file:

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

It appears to be an observational study, though it would be better written as: “Whether perceived beauty leads directly to the differences in observed 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

Per the histogram below, the data is unimodal. 4.3 is the median. Given the low number of possibilities (1 through 5), it is difficult to say whether it is skewed or not, although per this visualization it does seem to be left-skewed. The data seems to indicate that students tend to rate favorably, with very few low ratings and the majority of ratings of at least 4. I would have expected to see the mode closer to 3 with a more even distribution.

ggplot(evals, aes(x=score)) + geom_histogram(binwidth=.3)

median(evals$score)
## [1] 4.3
  1. Excluding score, select two other variables and describe their relationship with each other using an appropriate visualization.

ANSWER

Below is a scatter plot showing the relationship between age and beauty, with regression lines showing the correlation of the two variables separated by gender. There appears to be a negative correlation between perceived beauty and age, with a slight difference between gender which increases as age increases. The perceived beauty of women is higher than man as age increases.

evals %>%
  ggplot(aes(x=age, 
             y=bty_avg, 
             color=gender))+
  geom_point()+
  geom_smooth(method="lm")

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 you 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

From the glimpse() function used above we can see there are 463 observations, and on the scatterplot there are seems to be quite a few less for some reason.

  1. Replot the scatterplot, but this time use geom_jitter as your layer. What was misleading about the initial scatterplot?

ANSWER

It seems that many of the data points were plotted on top of each other, which presents a misleading picture of the data.

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

  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. 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

As seen below via the summary() function, the slope for this model is 0.06, with an adjusted r-squared value of 0.03293, meaning that only 3% of the variability is explained by the model. Taking the square root of the r squared, we get an R value of 0.1815 which indicates a very weak relationship.

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
sqrt(.03293)
## [1] 0.1814663

Add the line of the best fit model to your plot using the following:

ggplot(data = evals, aes(x = bty_avg, y = score)) +
  geom_jitter() +
  geom_smooth(method = "lm")

The blue line is the model. The shaded gray area around the line tells you about the variability you might expect in your predictions. To turn that off, use se = FALSE.

ggplot(data = evals, aes(x = bty_avg, y = score)) +
  geom_jitter() +
  geom_smooth(method = "lm", se = FALSE)

  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).

ANSWER

  • Linearity: The regression line in the plot above shows what appears to be a linear relationship, though very slight (possibly due to random chance). The QQ plot below of the residuals however shows that beyond the theoretical value of 0 the slope of the relationship changes significantly.
  • Nearly normal residuals: The histogram below indicates that the residuals are NOT nearly normal, and are very left skewed.
  • Constant variability: The scatter plot below shows fairly constant variability.
  • Independent data: Assumed yes.
# Constant variability / Residuals scatter plot
ggplot(data = m_bty, aes(x = .fitted, y = .resid)) +
  geom_point() +
  geom_hline(yintercept = 0, linetype = "dashed") +
  xlab("Fitted values") +
  ylab("Residuals")

# Nearly normal residuals
ggplot(data = m_bty, aes(x = .resid)) +
  geom_histogram(binwidth = .25) +
  xlab("Residuals")

# QQ / Normality plot
ggplot(data = m_bty, aes(sample = .resid)) +
  stat_qq() +
  stat_qq_line()

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_f1lower, y = bty_avg)) +
  geom_point()

evals %>% 
  summarise(cor(bty_avg, bty_f1lower))
## # A tibble: 1 x 1
##   `cor(bty_avg, bty_f1lower)`
##                         <dbl>
## 1                       0.844

As expected, the relationship is quite strong—after all, the average score is calculated using the individual scores. You can actually look at the relationships between all beauty variables (columns 13 through 19) using the following command:

evals %>%
  select(contains("bty")) %>%
  ggpairs()

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 you’ve accounted for the professor’s gender, you can add the gender term into the model.

m_bty_gen <- lm(score ~ bty_avg + gender, data = evals)
summary(m_bty_gen)
## 
## Call:
## lm(formula = score ~ bty_avg + gender, data = evals)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.8305 -0.3625  0.1055  0.4213  0.9314 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  3.74734    0.08466  44.266  < 2e-16 ***
## bty_avg      0.07416    0.01625   4.563 6.48e-06 ***
## gendermale   0.17239    0.05022   3.433 0.000652 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.5287 on 460 degrees of freedom
## Multiple R-squared:  0.05912,    Adjusted R-squared:  0.05503 
## F-statistic: 14.45 on 2 and 460 DF,  p-value: 8.177e-07
  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.

ANSWER

Per the plots below very similar conditions and challenges exist as with the original model, although the residuals variability does seem to be less.

# Constant variability / Residuals scatter plot
ggplot(data = m_bty_gen, aes(x = .fitted, y = .resid)) +
  geom_point() +
  geom_hline(yintercept = 0, linetype = "dashed") +
  xlab("Fitted values") +
  ylab("Residuals")

# Nearly normal residuals
ggplot(data = m_bty_gen, aes(x = .resid)) +
  geom_histogram(binwidth = .25) +
  xlab("Residuals")

# QQ / Normality plot
ggplot(data = m_bty_gen, aes(sample = .resid)) +
  stat_qq()

  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

The adjusted R squared value has increased from approximately 3% to approximately 6%, meaning that 6% of the variability can be explained by the model. Though still low, this is twice as much as before so a good improvement. Likewise, the R value has increased from 0.18 to 0.23, indicating a slightly stronger correlation.

summary(m_bty_gen)
## 
## Call:
## lm(formula = score ~ bty_avg + gender, data = evals)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.8305 -0.3625  0.1055  0.4213  0.9314 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  3.74734    0.08466  44.266  < 2e-16 ***
## bty_avg      0.07416    0.01625   4.563 6.48e-06 ***
## gendermale   0.17239    0.05022   3.433 0.000652 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.5287 on 460 degrees of freedom
## Multiple R-squared:  0.05912,    Adjusted R-squared:  0.05503 
## F-statistic: 14.45 on 2 and 460 DF,  p-value: 8.177e-07
# Find the R value
sqrt(.05503)
## [1] 0.2345847

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 male and female to being an indicator variable called gendermale that takes a value of \(0\) for female professors and a value of \(1\) for male professors. (Such variables are often referred to as “dummy” variables.)

As a result, for female professors, 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 those with color pictures? (Hint: For those with color pictures, the parameter estimate is multiplied by 1.) For two professors who received the same beauty rating, which color picture tends to have the higher course evaluation score?

ANSWER

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

ggplot(data = evals, aes(x = bty_avg, y = score, color = pic_color)) +
  geom_smooth(method = "lm", formula = y ~ x, se = FALSE)

The decision to call the indicator variable gendermale instead of genderfemale 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

Per the summary of the model below, it seems that the summary includes all separate variables, minus 1. This is likely the same as above with gender where the first one per alphabetical order is multiplied by zero and thus excluded.

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, gender, ethnicity, 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

I would expect the cls_credits variable to have a high p-value, as it (I assume!) would have little if any bearing on the perceived beauty of the professor.

Let’s run the model…

m_full <- lm(score ~ rank + gender + ethnicity + language + age + cls_perc_eval 
             + cls_students + cls_level + cls_profs + cls_credits + bty_avg 
             + pic_outfit + pic_color, data = evals)
summary(m_full)
## 
## Call:
## lm(formula = score ~ rank + gender + ethnicity + language + age + 
##     cls_perc_eval + cls_students + cls_level + cls_profs + cls_credits + 
##     bty_avg + pic_outfit + pic_color, data = evals)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.77397 -0.32432  0.09067  0.35183  0.95036 
## 
## Coefficients:
##                         Estimate Std. Error t value Pr(>|t|)    
## (Intercept)            4.0952141  0.2905277  14.096  < 2e-16 ***
## ranktenure track      -0.1475932  0.0820671  -1.798  0.07278 .  
## ranktenured           -0.0973378  0.0663296  -1.467  0.14295    
## gendermale             0.2109481  0.0518230   4.071 5.54e-05 ***
## ethnicitynot minority  0.1234929  0.0786273   1.571  0.11698    
## languagenon-english   -0.2298112  0.1113754  -2.063  0.03965 *  
## age                   -0.0090072  0.0031359  -2.872  0.00427 ** 
## cls_perc_eval          0.0053272  0.0015393   3.461  0.00059 ***
## cls_students           0.0004546  0.0003774   1.205  0.22896    
## cls_levelupper         0.0605140  0.0575617   1.051  0.29369    
## cls_profssingle       -0.0146619  0.0519885  -0.282  0.77806    
## cls_creditsone credit  0.5020432  0.1159388   4.330 1.84e-05 ***
## bty_avg                0.0400333  0.0175064   2.287  0.02267 *  
## pic_outfitnot formal  -0.1126817  0.0738800  -1.525  0.12792    
## pic_colorcolor        -0.2172630  0.0715021  -3.039  0.00252 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.498 on 448 degrees of freedom
## Multiple R-squared:  0.1871, Adjusted R-squared:  0.1617 
## F-statistic: 7.366 on 14 and 448 DF,  p-value: 6.552e-14
  1. Check your suspicions from the previous exercise. Include the model output in your response.

ANSWER

It seems that my assumption was incorrect. In fact, ‘cls_creditsone credit’ is marked with 3 asterisks which indicates it has a high effect on the response variable. The output is summarized as follows:

Variable: cls_creditsone credit
Estimate: 0.5020432
Std. Error: 0.1159388
t value: 4.330 Pr(>|t): 1.84e-05 ***

  1. Interpret the coefficient associated with the ethnicity variable.

ANSWER

Ethnicity, which here is indicates as ‘ethnicitynot minority’ has an estimate of 0.1234, which means non-minorities would be expected to score that much higher than minorities. However with P(>|t) value of 0.11698 which indicates that overall it does not have much affect on the response variable.

  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

With a p-value of almost 0.78, cls_prof is the least significant variable and has been dropped. The result is that the coefficients and significance level of the other variables changed, though very slightly in most cases. This indicates a lack of significant collinearity with the cls_prof variable.

m_less_full <- lm(score ~ rank + gender + ethnicity + language + age + cls_perc_eval 
             + cls_students + cls_level + cls_credits + bty_avg 
             + pic_outfit + pic_color, data = evals)
summary(m_less_full)
## 
## Call:
## lm(formula = score ~ rank + gender + ethnicity + language + age + 
##     cls_perc_eval + cls_students + cls_level + cls_credits + 
##     bty_avg + pic_outfit + pic_color, data = evals)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.7836 -0.3257  0.0859  0.3513  0.9551 
## 
## Coefficients:
##                         Estimate Std. Error t value Pr(>|t|)    
## (Intercept)            4.0872523  0.2888562  14.150  < 2e-16 ***
## ranktenure track      -0.1476746  0.0819824  -1.801 0.072327 .  
## ranktenured           -0.0973829  0.0662614  -1.470 0.142349    
## gendermale             0.2101231  0.0516873   4.065 5.66e-05 ***
## ethnicitynot minority  0.1274458  0.0772887   1.649 0.099856 .  
## languagenon-english   -0.2282894  0.1111305  -2.054 0.040530 *  
## age                   -0.0089992  0.0031326  -2.873 0.004262 ** 
## cls_perc_eval          0.0052888  0.0015317   3.453 0.000607 ***
## cls_students           0.0004687  0.0003737   1.254 0.210384    
## cls_levelupper         0.0606374  0.0575010   1.055 0.292200    
## cls_creditsone credit  0.5061196  0.1149163   4.404 1.33e-05 ***
## bty_avg                0.0398629  0.0174780   2.281 0.023032 *  
## pic_outfitnot formal  -0.1083227  0.0721711  -1.501 0.134080    
## pic_colorcolor        -0.2190527  0.0711469  -3.079 0.002205 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4974 on 449 degrees of freedom
## Multiple R-squared:  0.187,  Adjusted R-squared:  0.1634 
## F-statistic: 7.943 on 13 and 449 DF,  p-value: 2.336e-14
  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

As indicated in the summary below, the best linear model is as follows:

\[ \begin{aligned} \widehat{score} &= 3.907 + .2026 \times gendermale + .1638 \times ethnicitynot\_minority - \\ &\qquad .2467 \times languagenonenglish - 0.0069 \times age + .0049 \times cls\_perc\_eval + \\ &\qquad .5172 \times cls\_creditsone credit + .0467 \times bty\_avg - .1139 \times pic\_outfitnot formal - \\ &\qquad .1809 \times pic\_colorcolor \end{aligned} \]

# Load MASS library and use for backward selection
library(MASS)
m_best <-  stepAIC(m_full, direction = "backward", trace = FALSE)
summary(m_best)
## 
## Call:
## lm(formula = score ~ gender + ethnicity + language + age + cls_perc_eval + 
##     cls_credits + bty_avg + pic_outfit + pic_color, data = evals)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.8455 -0.3221  0.1013  0.3745  0.9051 
## 
## Coefficients:
##                        Estimate Std. Error t value Pr(>|t|)    
## (Intercept)            3.907030   0.244889  15.954  < 2e-16 ***
## gendermale             0.202597   0.050102   4.044 6.18e-05 ***
## ethnicitynot minority  0.163818   0.075158   2.180 0.029798 *  
## languagenon-english   -0.246683   0.106146  -2.324 0.020567 *  
## age                   -0.006925   0.002658  -2.606 0.009475 ** 
## cls_perc_eval          0.004942   0.001442   3.427 0.000666 ***
## cls_creditsone credit  0.517205   0.104141   4.966 9.68e-07 ***
## bty_avg                0.046732   0.017091   2.734 0.006497 ** 
## pic_outfitnot formal  -0.113939   0.067168  -1.696 0.090510 .  
## pic_colorcolor        -0.180870   0.067456  -2.681 0.007601 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4982 on 453 degrees of freedom
## Multiple R-squared:  0.1774, Adjusted R-squared:  0.161 
## F-statistic: 10.85 on 9 and 453 DF,  p-value: 2.441e-15
  1. Verify that the conditions for this model are reasonable using diagnostic plots.

ANSWER

See plots below. The variability of the residuals is a bit high but overall the conditions seem to be met.

# Constant variability / Residuals scatter plot
ggplot(data = m_best, aes(x = .fitted, y = .resid)) +
  geom_point() +
  geom_hline(yintercept = 0, linetype = "dashed") +
  xlab("Fitted values") +
  ylab("Residuals")

# Nearly normal residuals
ggplot(data = m_best, aes(x = .resid)) +
  geom_histogram(binwidth = .25) +
  xlab("Residuals")

# QQ / Normality plot
ggplot(data = m_best, aes(sample = .resid)) +
  stat_qq() +
  stat_qq_line()

  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

Yes, it raises concerns regarding independence of the residuals.

  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

Based on the variables with the lowest p-values, the highest evaluation would be for a ale professor with a high percentage of students who provided reviews and taught a one credit class.

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

ANSWER

No, in large part due to the independence concerns mentioned above as well as the fact that the conditions for regression have been arguably met, at best.

References

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