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. We can rephrase the question as: Is there an association between beauty and course evaluation score?

  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?
summary(evals$score)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   2.300   3.800   4.300   4.175   4.600   5.000
hist(evals$score)

Left skewed. The distribution of score is between 2.3 and 5 with a median score of 4.3.

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

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?

Initial scatterplot did not reveal any relationship between two variables score and bty_avg. It suggested relationship may not be linear. After adding jitter() function, scatterplots show some pattern with jitter() function added to score. However, it still does not reveal form and direction.

par(mfrow = c(1, 2))

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

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?
options("scipen"=100, "digits"=4)
m_bty <- lm(evals$score ~ evals$bty_avg)
plot(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 
## -1.925 -0.369  0.142  0.398  0.931 
## 
## Coefficients:
##               Estimate Std. Error t value             Pr(>|t|)    
## (Intercept)     3.8803     0.0761   50.96 < 0.0000000000000002 ***
## evals$bty_avg   0.0666     0.0163    4.09             0.000051 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.535 on 461 degrees of freedom
## Multiple R-squared:  0.035,  Adjusted R-squared:  0.0329 
## F-statistic: 16.7 on 1 and 461 DF,  p-value: 0.0000508

Linear equation to predict average professor score(\(\hat{y}_{professor\ score}\)) by average beauty rating(\(x_{beauty\ rating}\)), with \(\beta_0 = 3.8803, \beta_1 = 0.0666\)

\(\hat{y}_{professor\ score} = \beta_0 + \beta_1{x}_{beauty\ rating}\)

\(\hat{y}_{professor\ score} = 3.8803 + 0.0666{x}_{beauty\ rating}\)

As \(\beta_0 = 3.88034\ and\ \beta_1 = 0.06664\) are positive, linear equation suggests there is positive relation between average professor score and average beauty rating. The equation also suggests as beauty rating increases, professor score will be increased by 0.06664. Also, as beauty rating approaches zero, professor score will be 3.88034.

As the p-value is 0.0000508, which is much less than 0.05, the model suggests there is a significant relationship between the variables in the linear regression model. Using \(R^2 = 0.035\), only 3.5 percent data can be explained by the model. Since it is low value, model is considered to be statistically significant predictor and not a practically significant predictor.

  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).
#Scatter residual plot 
m_bty <- lm(evals$score ~ evals$bty_avg)
plot(m_bty$residuals ~ evals$bty_avg)
abline(h = 0, lty = 3)

#Residuals Histogram
hist(m_bty$residuals)

# Nearly normal residuals using Normal probability plot 
qqnorm(m_bty$residuals)
qqline(m_bty$residuals)  # adds diagonal line to the normal prob plot

Looking at scatterplot, residuals are randomly scattered around zero on the horizontal axis. This indicates Linearity condition is met.

Histogram suggests data are left skewed. This indicates there are some outliers. Normal probability plot shows that most data points are close to the line. Both graphs show enough evidence that data meets Nearly Normal condition.

Looking at scatterplot, the points have constant variance, with the residuals scattered randomly around zero on the horizontal axis. Since residuals do not show increasing or decreasing pattern, we can assume Constant Variance exists.

Using residual plots we can conclude conditions for least squares regression are reasonable.

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.

If data meets following conditions it is reasonable to apply regression model.

**1_: The residuals of the model are nearly normal._**

**2_: The variability of the residuals is nearly constant._**

**3_: The residuals are independent._**

**4_: Each variable is linearly related to the outcome._**

#Scatter residual plot 
m_bty_gen <- lm(score ~ bty_avg + gender, data = evals)

#Regression diagnostic plots
plot(m_bty_gen)

hist(m_bty_gen$res)

plot(evals$bty_avg, residuals(m_bty_gen), xlab="Beauty", ylab="Residuals", main = "Beauty Vs. Residuals")
abline(h=0, col="red")

plot(evals$gender, residuals(m_bty_gen), xlab="Gender", ylab="Residuals", main = "Gender Vs. Residuals")