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.

    This is an observational study. Therefore, causation cannot be concluded. I’d rephrase the question to, “Is there an association between beauty and differences in course evaluations?”

  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?

    Below is a histogram of the variable score. It shows a strong left-skewed distribution, with a center of 4.5. This shows that most students give high marks and very few give low marks. I was expecting this to be more evenly distributed because students tend to rate lower in classes they are forced to take and do not like.

    hist(evals$score)

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

    I decided to find the relationship between professor’s age and rank. As expected, the average age for tenured professors is much higher than that of professors who are on a tenure track. I also expected the age of teaching professors to be high as well since a lot of them are doing this as a second job and are not in academia.

    plot(evals$age ~ evals$rank)

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?

  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?

    The jitter function allows ust to see that there are multiple points that are stacked on top each other. In the previous plot, we could not see that, and it appeared that there were significantly less points.

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

  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 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?

    From R-code below, we can see that the linear regression line is:

    \(\hat{y} = 3.88034 + 0.06664\times\text{avg. beauty rating}\)

    In this case, the slope means that for every unit increase of average beauty score, the professor’s evaluation will increase by an average of 0.06664. The \(P\)-value is \(5.08\times10^{-5}\), which is significant at very high confidence levels. However, it can be argued that this is not practically significant since the slope is very small and only increases by very little.

    m_bty <- lm(evals$score ~ evals$bty_avg)
plot(jitter(evals$score) ~ evals$bty_avg)
abline(m_bty)

    summary(m_bty)
    
Call:
lm(formula = evals$score ~ evals$bty_avg)

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 ***
evals$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

  3. 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).

    From the residual plot, it shows that there is no apparent pattern, so this relationship appears to be linear. However, the residuals are left-skewed, and the data deviates from the normal line. Therefore, the residuals are not normally distributed, and a linear model may not be the best fit.

    plot(jitter(m_bty$residuals) ~ evals$bty_avg)
abline(h = 0, lty = 3)

    hist(m_bty$residuals)

    qqnorm(m_bty$residuals)
qqline(m_bty$residuals)

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)
## 
## 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.

    I created a histogram for the residuals and have confirmed a left-skew distibution. While I would be hesitant to assume normality based on this, in order to coninute with this lab, I will say this is approximately normal.

    From the residual plot, while I see that there are some values that are greater in magnitude on the negative side, I can see a fairly even distribution between positive and negative. Therefore, I can assume constant variability.

    hist(m_bty_gen$residuals)

    plot(m_bty_gen$residuals ~ m_bty_gen$fitted)
abline(h = 0, lty = 3)

    I will also assume that the observations were independent.

    While the results are not perfect, I would be ok with reporting this data.

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

    Yes. While the addition of gender has changed the parameter estimate for bty_avg, the \(P\)-value is still significant. Therefore, bty_avg is still a significant predictor of score.

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?

    In the females, we remove \(\hat{\beta_{2}}\) because the value was 0. Now, we will keep it:

    \(\widehat{score} = \hat{\beta}_{0} + \hat{\beta}_{1}\times bty\_avg + \hat{\beta}_{2}\)

    Since \(\hat{\beta}_2\) is a positive value, the male gender will have a higher rating between two opposite-sex professors that have the same bty_avg.

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.

    R split up the variable into two variables, tenure track and tenured. I am not sure how teaching is accounted for in this.

    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.

    I expected the number of credits to not have any effect on professor’s score. I can’t think of any reason for someone to base their rating on a professor due to number of credits of a course.

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)
## 
## Call:
## lm(formula = 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)
## 
## 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    
## ethnicitynot minority  0.1234929  0.0786273   1.571  0.11698    
## gendermale             0.2109481  0.0518230   4.071 5.54e-05 ***
## 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.

    It looks like credits produced a \(P\)-value that was very close to 0, and is therefore significant. The highest \(P\)-value actually belonged to cls_prof.

    summary(m_full)
    ## 
## Call:
## lm(formula = 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)
## 
## 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    
## ethnicitynot minority  0.1234929  0.0786273   1.571  0.11698    
## gendermale             0.2109481  0.0518230   4.071 5.54e-05 ***
## 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

  2. Interpret the coefficient associated with the ethnicity variable.

    When every other variable in the model is held constant, the score will increase by 0.1234929, when ethnicity is not a minority.

  3. 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?

    All variables changed slightly, but each variables \(P\)-value was close to what it was before. In addition, all of the slopes remained of the same sign, even though they were altered slightly. This shows that the dropped variable had minimal colinearity with the other variables.

    m_drop_cls_prof <- 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_drop_cls_prof)
    ## 
## Call:
## lm(formula = score ~ rank + ethnicity + gender + 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    
## ethnicitynot minority  0.1274458  0.0772887   1.649 0.099856 .  
## gendermale             0.2101231  0.0516873   4.065 5.66e-05 ***
## 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

  4. 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.

    Below is a function from the MASS package that does step-wise calculation and grabs the best \(R^{2}_{adj}\) value. The equation that this function came up with is:

    \[ \begin{aligned} \widehat{score} &= 3.907 + 0.164\times ethnicity\_not\_minority + 0.203\times gendermale \\ &- 0.247\times language\_non\_english - 0.007 \times age + 0.005 \times cls\_perc\_eval \\ &+ 0.517\times cls\_credits\_one\_credit +0.047\times bty\_avg - 0.114\times pic\_outfit\_not\_formal \\ &- 0.181\times pic\_color\_color \end{aligned} \]

    library(MASS)
b_selec <- stepAIC(m_full, direction="backward", trace=FALSE)
summary(b_selec)
    
Call:
lm(formula = score ~ ethnicity + gender + 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 ***
ethnicitynot minority  0.163818   0.075158   2.180 0.029798 *  
gendermale             0.202597   0.050102   4.044 6.18e-05 ***
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

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

    It appears that there is an equal amount of residual data in the positive and negative (maybe slightly more negative). It shows that there is no apparent pattern, so it appears to be linear. This also tells us that there is equal variability.

    The histogram and 11-plot show that there is a left-skew. However, the shape is unimodal. I would say that this is a normal distribution.

    plot(b_selec$residuals ~ b_selec$fitted)
abline(h = 0, lty = 3)

    hist(b_selec$residuals)

    qqnorm(b_selec$residuals)
qqline(b_selec$residuals)

  6. 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?

    This will increase the sample size since professors will now be represented multiple times. This may effect the independence of observations since a student may have a predisposition to like/hate the professor because of a different course they had with them.

  7. 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.

    Based on my final model, a higher score will correlate with:
    • not a member of a minority group
    • gender is male
    • english speaking school
    • younger is preferred
    • higher percentage of students filling out evaluations
    • the class being one credit
    • higher beauty ranking
    • dresses formally in picture
    • black and white picture

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

    I would be comfortable, but hesitant. The education system is pretty consistent throughout the US. However, I would be more comfortable associating these results with colleges of the same type. For example, if we had results of Ivy League schools, I would feel comfortable generalizing the results with other Ivy League schools.

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.