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

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, and can be found here . We have observations on 21 different variables, some categorical and some numerical. The meaning of each variable can be found here.

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.

  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?

  3. Excluding score, select two other variables and describe their relationship with each other using an appropriate visualization.

Simple linear regression

The fundamental phenomenon suggested by the study is that better looking teachers are evaluated more favourably. Let’s create a scatterplot to see if this appears to be the case, using score as the y-variable and bty_avg as the x-variable.

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?

  1. What was misleading about the initial scatterplot?

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, uncheck the Show confidence interval box in the Add regression line.

  1. Let’s see if the apparent trend in the plot is something more than natural variation. Fit a linear regression model 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?

  2. 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).

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.

Create a scatterplot using bty_f1lower as your x-variable and bty_avg as your y-variable. Also determine the correlation coefficient for these two variables.

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 by creating a descriptive statistics analysis, then selecting all the variables from bty_f1lower through bty_avg (you can shift-click the select them all). Then open the plots menu and select scatter plots. This will create a scatter plot for each pair of variables.

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.

We’re going to see a small problem if we do this directly. Try to create a Linear Regression analysis, add score as the dependent variable, and add both bty_avg and gender as covariates. We can’t add gender because JASP will only accept scale variables as covariates. (You can also tell this by looking at the bottom corner of the box to see the scale icon.) We’ll need to create a new variable to be able to include gender.

Go back to your data, and add a new variable, which we’ll call gender_male. The particular direction we do this doesn’t matter, but we’ll code males as 1 and females as 0, so we can consider the new column to be essentially a yes/no question whether the individual identifies as male. (Such variables are often referred to as “dummy” variables.) We could reverse these roles and create a gender_female and would come to the same eventual conclusions.

For the definition of the new variable, we can use ifElse(gender=male,1,0) to achieve this.

If we then return to our linear regression analysis, we can now include both bty_avg and gender_male as covariates.

  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.

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

Because of the way we encoded gender into the gender_male variable, 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 male professors? (Hint: For male professors, 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?

The decision to call the indicator variable gendermale instead of genderfemale has no deeper meaning. Often software will indicate these types of variables by coding a \(0\) to whichever category comes first alphabetically, so we are matching that convention.

  1. We’ll next create a new model with gender removed and rank added in. Note that the rank variable has three levels: teaching, tenure track, tenured. This means that we will need to add two new columns instead of just one. We can select one of the three levels to be our “baseline” and the two variables will each be comparisons with our baseline. To be consistent, let’s pick teaching to be the baseline, and create rank_tenure_track and rank_tenured to be the comparisons with teaching. Add both of these new variables as covariates along with bty_avg. What can you learn from your new model?

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, and average beauty rating. You will need to create new variables for each of these variables that are nominal so that we can create our regression.

  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.

Let’s create the model using a linear regression analysis.

  1. Check your suspicions from the previous exercise. Include the model output in your response.

  2. Interpret the coefficient associated with the ethnicity variable.

  3. Drop one variable at a time and peek at the adjusted \(R^2\). Removing which variable increases adjusted \(R^2\) the most? Drop the variable with the highest p-value and re-fit the model. Did the coefficients and significance of the other explanatory variables change with this variable removed? (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?

  4. Using this method (called backward-selection) and adjusted \(R^2\) 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.

Note: JASP can actually perform this backward-selection process for you. In the linear regression analysis, the pull-down menu for Method allows you to select “Backward”, and the Model Specification menu allows you to select the criterion for which variables are dropped. We are opting to do this by hand to better understand the process.

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

  2. 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?

  3. 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.

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

References

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