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")
plot_ss <- function(x, y, showSquares = FALSE, leastSquares = FALSE){
  plot(y~x, asp = 1)# xlab = paste(substitute(x)), ylab = paste(substitute(y)))
  
  if(leastSquares){
    m1 <- lm(y~x)
    y.hat <- m1$fit
  } else{
    cat("Click two points to make a line.")
    pt1 <- locator(1)
    points(pt1$x, pt1$y, pch = 4)
    pt2 <- locator(1)
    points(pt2$x, pt2$y, pch = 4)
    pts <- data.frame("x" = c(pt1$x, pt2$x),"y" = c(pt1$y, pt2$y))
    m1 <- lm(y ~ x, data = pts)
    y.hat <- predict(m1, newdata = data.frame(x))
  }
  r <- y - y.hat
  abline(m1)

  oSide <- x - r
  LLim <- par()$usr[1]
  RLim <- par()$usr[2]
  oSide[oSide < LLim | oSide > RLim] <- c(x + r)[oSide < LLim | oSide > RLim] # move boxes to avoid margins

  n <- length(y.hat)
  for(i in 1:n){
    lines(rep(x[i], 2), c(y[i], y.hat[i]), lty = 2, col = "blue")
    if(showSquares){
    lines(rep(oSide[i], 2), c(y[i], y.hat[i]), lty = 3, col = "orange")
    lines(c(oSide[i], x[i]), rep(y.hat[i],2), lty = 3, col = "orange")
    lines(c(oSide[i], x[i]), rep(y[i],2), lty = 3, col = "orange")
    }
  }

  SS <- round(sum(r^2), 3)
  cat("\r                                ")
  print(m1)
  cat("Sum of Squares: ", SS)
}
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: Yes, this is an observation study. Are course evaluations impacted by beauty.

  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 score is left skewed. Majority of the students rate their courses higher or we can say they are satisfatory with the results.

  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$rank ~ evals$language)
  plot(evals$rank ~ evals$age)

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

**Answer: score = 3.88 +.066*bty_avg**

No, the bty_avg is quite small to make a significant impact on the overall score.

  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).
plot_ss(evals$score, evals$bty_avg)

## Click two points to make a line.
                                
## Call:
## lm(formula = y ~ x, data = pts)
## 
## Coefficients:
## (Intercept)            x  
##      2.2237       0.5256  
## 
## Sum of Squares:  1040.048

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.

Answer: Below are the conditions

a. Residuals should be nearly normal
 b. Variance in the residuals should be nearly constant
 c. Residuals should be independent
 d. Each variable should be linearly related to the outcome

From the below plots we can say that above conditions are satisfied

m_bty_gen <- lm(score ~ bty_avg + gender, data = evals)
hist(m_bty_gen$residuals)

plot(evals$score ~ evals$gender)

plot(m_bty_gen)

  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 model when included with gender would be below. It shows that bty_avg is a more significant predictor than gendermale, since the gendermale is always .17 where as bty_avg with an avg value of 4.4 in the evals dataset is higher than .17.

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: score = 3.747 + .07 * bty_avg + .17 * 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.

Answer:
The below model provides data only for two levels i.e., tenure track and tenured. When there are more than two levels then R seems to be reducing 1 level from the total number of levels.

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

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: I expect the rank to have highest p-value and age to be least

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.
m_rank_score <- lm(score ~ rank+age, data = evals)
summary(m_rank_score)
## 
## Call:
## lm(formula = score ~ rank + age, data = evals)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.83092 -0.34739  0.09644  0.41076  0.90727 
## 
## Coefficients:
##                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)       4.739994   0.164159  28.874  < 2e-16 ***
## ranktenure track -0.236802   0.082698  -2.863  0.00438 ** 
## ranktenured      -0.129720   0.063255  -2.051  0.04086 *  
## age              -0.009080   0.003094  -2.934  0.00351 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.5374 on 459 degrees of freedom
## Multiple R-squared:  0.02983,    Adjusted R-squared:  0.02349 
## F-statistic: 4.704 on 3 and 459 DF,  p-value: 0.003024
  1. Interpret the coefficient associated with the ethnicity variable.

**Answer: score = 4.07 + .119*ethnicity_not_minority**

Here .119 is a positive number and professors who fall under this category have a positive trend towards higher score.

m_eth_score <- lm(score ~ ethnicity, data = evals)
summary(m_eth_score)
## 
## Call:
## lm(formula = score ~ ethnicity, data = evals)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.8912 -0.3816  0.1088  0.4088  0.9281 
## 
## Coefficients:
##                       Estimate Std. Error t value Pr(>|t|)    
## (Intercept)            4.07188    0.06786  60.003   <2e-16 ***
## ethnicitynot minority  0.11935    0.07310   1.633    0.103    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.5429 on 461 degrees of freedom
## Multiple R-squared:  0.005749,   Adjusted R-squared:  0.003593 
## F-statistic: 2.666 on 1 and 461 DF,  p-value: 0.1032
  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: Yes, after dropping the highest p value variable, the co-efficient of the other variables underwent a change.

m_full_minus_cls_credits <- lm(score ~ rank + ethnicity + gender + language + age + cls_perc_eval  + cls_students + cls_level + cls_profs  + bty_avg  + pic_outfit + pic_color, data = evals)
summary(m_full_minus_cls_credits)
## 
## Call:
## lm(formula = score ~ rank + ethnicity + gender + language + age + 
##     cls_perc_eval + cls_students + cls_level + cls_profs + bty_avg + 
##     pic_outfit + pic_color, data = evals)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.7498 -0.3200  0.1056  0.3679  0.9200 
## 
## Coefficients:
##                         Estimate Std. Error t value Pr(>|t|)    
## (Intercept)            4.3098194  0.2918733  14.766  < 2e-16 ***
## ranktenure track      -0.1957586  0.0829015  -2.361 0.018635 *  
## ranktenured           -0.1809000  0.0647027  -2.796 0.005398 ** 
## ethnicitynot minority  0.0429967  0.0778938   0.552 0.581229    
## gendermale             0.2366593  0.0524895   4.509 8.33e-06 ***
## languagenon-english   -0.2589399  0.1133484  -2.284 0.022810 *  
## age                   -0.0090463  0.0031973  -2.829 0.004873 ** 
## cls_perc_eval          0.0059006  0.0015636   3.774 0.000182 ***
## cls_students           0.0002954  0.0003829   0.771 0.440863    
## cls_levelupper        -0.0065495  0.0565243  -0.116 0.907807    
## cls_profssingle       -0.0427280  0.0525927  -0.812 0.416974    
## bty_avg                0.0315543  0.0177371   1.779 0.075917 .  
## pic_outfitnot formal  -0.1362125  0.0751223  -1.813 0.070467 .  
## pic_colorcolor        -0.2091633  0.0728769  -2.870 0.004297 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.5077 on 449 degrees of freedom
## Multiple R-squared:  0.1531, Adjusted R-squared:  0.1286 
## F-statistic: 6.243 on 13 and 449 DF,  p-value: 7.671e-11
  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: Taking out cls_profs variable, improved the R2 square the most.

m_full_minus_cls_profs <- 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_minus_cls_profs)
## 
## 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
  1. Verify that the conditions for this model are reasonable using diagnostic plots.
plot(m_full_minus_cls_profs)

hist(m_full_minus_cls_profs$residuals)

plot(evals$score ~ evals$gender)

plot(evals$score ~ evals$rank)

plot(evals$score ~ evals$ethnicity)

plot(evals$score ~ evals$language)

  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: No, I believe having each course as an observation wouldn’t have an impact since the evaluations from each course will be different.

  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: Considering the stats presented by the model, we can derive to a conclusion that characteristics of a professor associated with a higher score would include being a male, non-white, english speaking with a moderate beauty score.

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

Answer: No, it cannot be generalized since the sample was only from one university but not diverisified through different geographies and population demographics. Therefore, it’s not feasible to apply the result of such observation to entire population.