Multiple linear regression

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 a slightly 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")

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. There is no control group and the sample is only from the University of Texas at Austin so no causal relationship can be established and it cannot be generalized. I’m not sure why they only asked 6 students about the professor’s appearance? Beauty is very subjective so an average beauty score seems like it would be more representative of the population and you might find a difference between perceived beauty at different course “levels”? Perhaps a student’s academic performance impacts how beautiful they perceive the teacher to be? And I’m not sure exactly what they mean by “upper and lower level”? I guess that is referring to the academic difficulty level of the class. So a better way to word the question would be, “Is there a correlation between perceived teacher beauty and course evaluation scores at the University of Texas at Austin?”

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?

   hist(evals$score, xlab="Average Professor Evaluation Score",
   main="Histogram of Professor Evaluation Scores")

   

   summary(evals$score)

   ##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
   ##   2.300   3.800   4.300   4.175   4.600   5.000

   The distribution of score is left skewed with a median of 4.3 and mean of 4.175. I’m not surprised by this result. I would expect most teachers to get mostly positive evaluations with a much smaller proportion receiving more negative reviews. I think students in general don’t want to speak badly of their teachers even when they were not happy with a course, so this is about what I would expect to see. The minimum average score of 2.3 is only about half of the maximum possible indicating that even the worst rated teachers were not rated very badly across the board by all students.

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

   

   

   Ethnicity does not appear to have much of an influence on average beauty rating. Both boxplots have similar means and ranges. There are fewer data points overall for minority professors, but the spread and variance looks very similar.

   

   

   Age however, does appear to have an influence on average beauty rating. The boxplots and scatterplot both show a decline in mean beauty rating as age increases although the trend is not as easy to see in the scatterplot as it is in the boxplots until you add the linear regression line.

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?

4. 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), ylab="Evaluation Score", xlab="Average Beauty Rating",
    main="Evaluation Scores and Average Beauty Rating")

   

   The original scatterplot made it very difficult to see that there were many overlapping datapoints. So the plot made it appear that there were less datapoints in total and less clustered in specific areas than were actually there.

5. 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)
   plot(jitter(evals$score) ~ jitter(evals$bty_avg), ylab="Evaluation Score", xlab="Average Beauty Rating",
    main="Evaluation Scores and Average Beauty Rating")
   abline(m_bty)

   

   m_bty$coefficients

   ## (Intercept)     bty_avg 
   ##  3.88033795  0.06663704

   \[\widehat{\text{score}} = 3.88 + 0.07 \times \text{bty_avg}\]

   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

   Average beauty rating is a statistically significant predictor of evaluation score, and with a slopw of 0.067 we would expect to see an increase of 0.067 in teh professor’s evaluation score for each increase of 1 point on the beauty rating scale. Considering that the beauty rating scale and the eval score scale are only 1-10 and 1-5 respectively, that is a very large increase.

6. 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).

   Linearity and Constant Variability

   

   The conditions of linearity and constant variability seem to be met. There is no apparent pattern in the residuals and they do not fan out or change much in variability.

   Nearly Normal Residuals

   

   

   The residuals have a significant left skew that may mean a least squares regression is not 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)

## 
## 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

7. 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.

   Linearity and Constant Variability

   

   The model seems to meet the condtitions for linearity and constant variability. There is no apparent change in variability in the scatterplot of residuals or in the boxplots for gender. The residuals in the scatterplot also show no apparent pattern that would indicate a non-linear relationaship.

   Nearly Normal Residuals

   

   The residuals in our multiple regression model show even more left skew than our original linear model. Not really sure if they meet this condition or not?

   

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

   The addition of gender has increased the parameter estimate for bty_avg from a slope of 0.06664 to 0.07416 and increased the amount of variability explained by the model from about 3.3% to about 5.5%. 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)

9. 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?

   \[\widehat{score} = 3.75 + 0.07 \times bty\_avg + 0.17\]

   For two professors with the same beauty rating, the male professor tends to have a higher course evaluation score.

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.)

10. 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.

    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

    R seems to handle multiple levels by coding the first level alphabetically as zero, so it has no slope and each of the others are treated as binary variables with a value of one if positive. Each additional level after the first is given a slope (coefficient) in the model. I had to look this up in “Practical Statistics for Data Scientists” by Peter Bruce and Andrew Bruce to be confident that categorical variables with multiple levels are handle this way.

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.

11. 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 would expect that cls_perc_eval or “percent of students in class who completed evaluation.” would not have any correlation with the prefessor’s evaluation score, so the should have the highest p-value.

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)

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

    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

    Wow! Big surprise for me. It never occurred to me that there would be a correlation between cls_perc_eval and score. But looking back no it now I guess it makes sense that students who are not satisfied with a course or professor would be less likely to complete the evaluations.

13. Interpret the coefficient associated with the ethnicity variable.

    The coefficient associated with the ethnicity indicates that all other things being equal, we would expect a .12 increase in the score for professors who are not members of a minority group. This cold indicate some racial bias, however with a p-value of 0.11 we cannot be confident that this finding is not due to random sampling.

14. 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?

    m_full2 <- 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_full2)

    ## 
    ## 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

    m_full$coefficients[c(1:10, 12:15)] - m_full2$coefficients

    ##           (Intercept)      ranktenure track           ranktenured 
    ##          7.961761e-03          8.133220e-05          4.512112e-05 
    ## ethnicitynot minority            gendermale   languagenon-english 
    ##         -3.952838e-03          8.249882e-04         -1.521743e-03 
    ##                   age         cls_perc_eval          cls_students 
    ##         -8.003969e-06          3.847644e-05         -1.408227e-05 
    ##        cls_levelupper cls_creditsone credit               bty_avg 
    ##         -1.234699e-04         -4.076375e-03          1.704157e-04 
    ##  pic_outfitnot formal        pic_colorcolor 
    ##         -4.358943e-03          1.789689e-03

    Yes, the coefficients and significance of the other explanatory variables did change when cls_profs was dropped from the model. If they had not changed that would have indicated that there was no colinnearity at all between cls_profs and the other predictor variables.

15. 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.

    m_best <- lm(score ~ ethnicity + gender + language + age + cls_perc_eval 
             + cls_credits + bty_avg + pic_color, data = evals)
    summary(m_best)

    ## 
    ## Call:
    ## lm(formula = score ~ ethnicity + gender + language + age + cls_perc_eval + 
    ##     cls_credits + bty_avg + pic_color, data = evals)
    ## 
    ## Residuals:
    ##      Min       1Q   Median       3Q      Max 
    ## -1.85320 -0.32394  0.09984  0.37930  0.93610 
    ## 
    ## Coefficients:
    ##                        Estimate Std. Error t value Pr(>|t|)    
    ## (Intercept)            3.771922   0.232053  16.255  < 2e-16 ***
    ## ethnicitynot minority  0.167872   0.075275   2.230  0.02623 *  
    ## gendermale             0.207112   0.050135   4.131 4.30e-05 ***
    ## languagenon-english   -0.206178   0.103639  -1.989  0.04726 *  
    ## age                   -0.006046   0.002612  -2.315  0.02108 *  
    ## cls_perc_eval          0.004656   0.001435   3.244  0.00127 ** 
    ## cls_creditsone credit  0.505306   0.104119   4.853 1.67e-06 ***
    ## bty_avg                0.051069   0.016934   3.016  0.00271 ** 
    ## pic_colorcolor        -0.190579   0.067351  -2.830  0.00487 ** 
    ## ---
    ## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    ## 
    ## Residual standard error: 0.4992 on 454 degrees of freedom
    ## Multiple R-squared:  0.1722, Adjusted R-squared:  0.1576 
    ## F-statistic:  11.8 on 8 and 454 DF,  p-value: 2.58e-15

    \[\widehat{score} = 3.772 + 0.168 \times ethnicity + 0.207 \times gender + -0.206 \times language + -0.006 \times age \\+ 0.005 \times cls\_perc\_eval + 0.505 \times cls\_credits + 0.051 \times bty\_avg + -0.191 \times pic\_color\]

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

    Linearity and Constant Variability

    

    There does seem to be some issues with the constant variability condition for the language, number of credits, and picture color variables. The difference may be due mostly to the significant difference in the number of data points for each of the two possible levels for each of those cattegorical variables. With more data points perhaps this aparent difference would be lessened.

    For the three continuous variables, age, percent of students who completed the evaluation and average beauty rating there does not seem to be any pattern that would suggest a non-linear relationship.

    Nearly Normal Residuals

    

    

    There is a significant left skew, but there do not appear to be any outliers that would be cause for concern.

17. 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?

    Yes, it possibly violates the independence condition since you would expect similar outcomes for the same professor in different courses. Possibly a better way to analyze and model the data would be to aggregate the data for each professor first so that each case represents aggragated data for one professor across all courses they taught.

18. 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 the final model, I would expect a professor at University of Texas at Austin that is associated with a high evaluation score to be a young, attractive, non-minority (white) male, who was educated in english, and teaching a one-credit course.

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

    No, I would not be comfortable generalizing my conclusions to any school other than the University of Texas at Austin. The sample was not randomized to include professor’s from any other university. There may be confounding variables not considered that contribute to the findings that are specific to the college and culture at the University of Texas at Austin.