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.

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:

    • This is an observational study.
    • No, since the study design is not an experiment, causality cannot be inferred from the results.
    • The question can be rephrased as whether differences in beauty ratings are associated with 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?

    Answer:

    • The distribution is asymmetric, with a pronounced left skew. The mean and median are 4.2 and 4.3, respectively, while the standard deviation and IQR are 0.5 and 1.8, respectively.
    • Yes the distribution is skewed to the left. Ratings are capped at 5, and it appears that most ratings are between 3.5 and 5. On the other hand, there is a tail of ratings below 3, leading to the left skew in the distribution.
    • This is somewhat to be expected, as many people when rating subjects that are other people, tend to use numerical ratings that are above the midpoint of the numerical range, e.g., above 3 on a scale of 1 to 5. This may be out of politeness or for other reasons. So the distribution ends up with a majority of ratings above average, but with the right tail capped because of the ceiling on ratings; meanwhile, there will be a left tail of ratings that are effectively not floored, since the center of the distribuion is above the midpoint of the rating scale.

    ##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   2.300   3.800   4.300   4.175   4.600   5.000
    ## [1] 0.5438645

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

    Answer:

    • Let’s take a look at beauty vs. age. We graph below a scatter plot of the bty_avg and age variables. There appears to be a weak negative relationship; older ages are associated with lower beauty scores.
    • The negative relationship between beauty and age persists for both men and women, although again it is not a strong relationship.
    • Next we look at rank vs. age, as shown in the bar chart below. Older ages above ~45 tend to be associated with tenured and teaching positions, while younger ages below ~40 tend to be associated with tenure-track and teaching positions.

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:

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?

    Answer:

    Because of fixed increments in the score (y-axis) and bty_avg (x-axis) variables, many observations coincided with the same x and y value, so were super-imposed in the initial scatterplot. As a result, the density of data points could not be observed. The jitter adds slight noise to the plotted values, so the observation density is apparent.

  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?

    Answer:

    • See the scatterplot and linear regression line below.
    • The equation for the linear model is:
      \[ \widehat{\text{score}} = 3.880 + 0.067 \cdot \text{bty_avg} \]
    • The slope of 0.067 implies that an additional point of beauty rating is associated with an additional 0.067 point of evaluation score.
    • From the regression statistics below, the coefficient of bty_avg is statistically significant with a p-value of \(\ll 0.05\), which indicates that the average beauty score is a statistically significant predictor.
    • However, bty_avg does not appear to be a practically significant predictor, as the linear model has a low \(R^2\) of only 3.5%. In other words, the linear model explains only 3.5% of the variability in the evaluation scores.
    ## 
## 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

  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).

    Answer:

    • Linearity: From the scatterplot above and the scatterplot of residuals below, it appears that there is only a very weak linear relationship between bty_avg and score. This condition is at best weakly satisfied.

    • Nearly normal residuals: The histogram of residuals and the normal probability plot of residuals below both show that the residuals are not normally distributed. This condition is not satisfied, or is at best weakly satisfied.

    • Constant variability: From the scatterplot of residuals above, the variance appears to be approximately constant, although there may be a slight decline in variance in the higher range of bty_avg (in the right tail). This condition is probably satisfied.

    • Independent observations: This condition is likely satisfied, as the dataset doesn’t include time series data, and presumably the evaluations were completed independently by the students. Also the introduction indicates that the dataset came from a large random sample.

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.

## [1] 0.8439112

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:

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.

## 
## 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.

    Answer:

    • Residuals are nearly normal: This can be assessed through a histogram of residuals and a normal probability plot of residuals, as shown below. Both plots indicate that the residuals are not normally distributed. This condition is not satisfied, or is at best weakly satisfied.

    • Variance of residuals is nearly constant: This can be evaluated using a plot of the absolute value of the residuals against their corresponding fitted values, as shown below. It appears that this condition is weakly satisfied, as the variability of the residuals appears to decline for larger fitted values (the right side of the plot).

    • Residuals are independent: This condition is likely satisfied, as the dataset doesn’t include time series data, and presumably the evaluations were completed independently by the students. This can be assessed using a plot of the residuals vs. the order of observation. From the plot below, it is apparent that there are clusters of observations that seem to be correlated. This is likely related to the issue highlighted below (in question #17), that all class observations are included for each professor. Class evaluations from different classes for the same professor are clearly associated with each other. So this condition is mostly satisfied.
    ##  [1] "1"  "2"  "3"  "4"  "5"  "6"  "7"  "8"  "9"  "10" "11" "12" "13" "14"
## [15] "15" "16" "17" "18" "19" "20"

    • Each variable is linearly related to the outcome: This can be evaluated using plots of the residuals against each predictor variable (bty_avg and gender). From the residual plots, it appears that there is a weak linear relationship between bty_avg and score, as well as between gender and score. This condition is likely satisfied.

  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?

    Answer:

    • bty_avg is still a statistically significant predictor of score, as the estimate of the related coefficient has a p-value of \(\ll 0.05\).
    • Yes the parameter estimate for bty_avg has changed from 0.067 to 0.074.

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.

  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:

    • For males, the linear model becomes:
      \[ \begin{aligned} \widehat{\text{score}} &= 3.747 + 0.074 \cdot \text{bty_avg} + 0.172 \cdot 1 \\ &= 3.919 + 0.074 \cdot \text{bty_avg} \end{aligned} \]
    • For female and male professors having the same beauty rating, male professors tend to receive higher evaluation scores on average. Evaluation scores tend to be higher for males than for females by 0.172, on average, holding bty_avg constant.

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:

    • See the linear model below.
    • For a categorical variable with \(n > 2\) levels, R treats the first level (alphabetically) as the baseline, and then handles the remaining \(n-1\) levels by creating \(n-1\) indicator variables (one for each level). In the instance of rank, the baseline is teaching, while there are indicator variables for tenure track and tenured.
    ## 
## 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.

    Answer:

    High p-values are associated with predictor variables that are not significant. Three variables that potentially are not signficant, and therefore have high p-values, are:
    • cls_students (class size)
    • cls_profs (number of professors)
    • cls_credits (number of credits).

Let’s run the model…

## 
## 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.

    Answer:

    Reviewing the model output above, it is evident from the p-values that class size and number of professors are not significant:
    • cls_students: 0.23
    • cls_profssingle: 0.78
    • cls_creditsone credit: 1.84e-05

    However, it appears that the number of credits is significant, which seems surprising. Also, I wasn’t expecting cls_level (course level) to be insignificant, with a p-value of 0.29.

  2. Interpret the coefficient associated with the ethnicity variable.

    Answer:

    The coefficient associated with ethnicitynot minority is 0.123, which indicates that when holding all other variables constant, not minority is associated with evaluations scores that are higher by 0.123, on average, than those of 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?

    Answer:

    • We drop the cls_profs variable (p-value of 0.78) and re-fit the model (see below).
    • Yes, the coefficient estimates and p-values changed from the full model to this model, although the changes were not dramatic.
    • If the other coefficient estimates don’t change when a variable is dropped, this indicates that the dropped variable was not collinear with the other explanatory variables. In this case, the coefficient estimates and p-values changed only slightly, which indicates that the cls_profs variable is not strongly collinear with the other variables.
    ## 
## 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.

    Answer:

    • Using the highest p-value > 0.05 as the criterion for backward selection, we drop the following variables in order:
      • cls_level
      • cls_students
      • rank
      • pic_outfit
    • We arrive at the final model m_f5 below.
    • The linear model m_f5 can be expressed mathematically as:
      \[ \begin{aligned} \widehat{\text{score}} = &\beta_0 + \beta_1 \cdot \text{ethnicitynot minority} + \beta_2 \cdot \text{gendermale} \\ &+ \beta_3 \cdot \text{languagenon-english} + \beta_4 \cdot \text{age} + \beta_5 \cdot \text{cls_perc_eval} \\ &+ \beta_6 \cdot \text{cls_creditsone credit} + \beta_7 \cdot \text{bty_avg} + \beta_8 \cdot \text{pic_colorcolor} \end{aligned} \] Substituting in the coefficient estimates, this becomes: \[ \begin{aligned} \widehat{\text{score}} = &3.772 + 0.168 \cdot \text{ethnicitynot minority} + 0.207 \cdot \text{gendermale} \\ &- 0.206 \cdot \text{languagenon-english} - 0.006 \cdot \text{age} + 0.005 \cdot \text{cls_perc_eval} \\ &+ 0.505 \cdot \text{cls_creditsone credit} + 0.051 \cdot \text{bty_avg} - 0.191 \cdot \text{pic_colorcolor} \end{aligned} \]
    ## 
## Call:
## lm(formula = score ~ rank + ethnicity + gender + language + age + 
##     cls_perc_eval + cls_students + cls_credits + bty_avg + pic_outfit + 
##     pic_color, data = evals)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.7761 -0.3187  0.0875  0.3547  0.9367 
## 
## Coefficients:
##                         Estimate Std. Error t value Pr(>|t|)    
## (Intercept)            4.0856255  0.2888881  14.143  < 2e-16 ***
## ranktenure track      -0.1420696  0.0818201  -1.736 0.083184 .  
## ranktenured           -0.0895940  0.0658566  -1.360 0.174372    
## ethnicitynot minority  0.1424342  0.0759800   1.875 0.061491 .  
## gendermale             0.2037722  0.0513416   3.969 8.40e-05 ***
## languagenon-english   -0.2093185  0.1096785  -1.908 0.056966 .  
## age                   -0.0087287  0.0031224  -2.795 0.005404 ** 
## cls_perc_eval          0.0053545  0.0015306   3.498 0.000515 ***
## cls_students           0.0003573  0.0003585   0.997 0.319451    
## cls_creditsone credit  0.4733728  0.1106549   4.278 2.31e-05 ***
## bty_avg                0.0410340  0.0174449   2.352 0.019092 *  
## pic_outfitnot formal  -0.1172152  0.0716857  -1.635 0.102722    
## pic_colorcolor        -0.1973196  0.0681052  -2.897 0.003948 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4975 on 450 degrees of freedom
## Multiple R-squared:  0.185,  Adjusted R-squared:  0.1632 
## F-statistic:  8.51 on 12 and 450 DF,  p-value: 1.275e-14
    ## 
## Call:
## lm(formula = score ~ rank + ethnicity + gender + language + age + 
##     cls_perc_eval + cls_credits + bty_avg + pic_outfit + pic_color, 
##     data = evals)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.78424 -0.31397  0.09261  0.35904  0.92154 
## 
## Coefficients:
##                        Estimate Std. Error t value Pr(>|t|)    
## (Intercept)            4.152893   0.280892  14.785  < 2e-16 ***
## ranktenure track      -0.142239   0.081819  -1.738 0.082814 .  
## ranktenured           -0.083092   0.065532  -1.268 0.205469    
## ethnicitynot minority  0.143509   0.075972   1.889 0.059535 .  
## gendermale             0.208080   0.051159   4.067 5.61e-05 ***
## languagenon-english   -0.222515   0.108876  -2.044 0.041558 *  
## age                   -0.009074   0.003103  -2.924 0.003629 ** 
## cls_perc_eval          0.004841   0.001441   3.359 0.000849 ***
## cls_creditsone credit  0.472669   0.110652   4.272 2.37e-05 ***
## bty_avg                0.043578   0.017257   2.525 0.011903 *  
## pic_outfitnot formal  -0.136594   0.068998  -1.980 0.048347 *  
## pic_colorcolor        -0.189905   0.067697  -2.805 0.005246 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4975 on 451 degrees of freedom
## Multiple R-squared:  0.1832, Adjusted R-squared:  0.1632 
## F-statistic: 9.193 on 11 and 451 DF,  p-value: 6.364e-15
    ## 
## 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
    ## 
## 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

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

    Answer:

    • Residuals are nearly normal: This can be assessed through a histogram of residuals and a normal probability plot of residuals, as shown below. Both plots indicate that the residuals are not normally distributed. This condition is not satisfied, or is at best weakly satisfied.

    • Variance of residuals is nearly constant: This can be evaluated using a plot of the absolute value of the residuals against their corresponding fitted values, as shown below. It appears that this condition is only very weakly satisfied, as the variability of the residuals appears to decline for larger fitted values (the right side of the plot).

    • Residuals are independent: This condition is likely satisfied, as the dataset doesn’t include time series data, and presumably the evaluations were completed independently by the students. This can be assessed using a plot of the residuals vs. the order of observation. From the plot below, it is apparent that there are clusters of observations that seem to be correlated. This is likely related to the issue highlighted below (in question #17), that all class observations are included for each professor. Class evaluations from different classes for the same professor are clearly associated with each other. So this condition is mostly satisfied.
    ##  [1] "1"  "2"  "3"  "4"  "5"  "6"  "7"  "8"  "9"  "10" "11" "12" "13" "14"
## [15] "15" "16" "17" "18" "19" "20"

    • Each variable is linearly related to the outcome: This can be evaluated using plots of the residuals against each predictor variable (ethnicity, gender, language, age, cls_perc_eval, cls_credits, bty_avg and pic_color). From the residual plots, it appears that there are weak linear relationships between each of the independent variables and score. This condition is likely satisfied. However, note that for certain variables, the variability of the residuals is not constant across the variable range (or across groups, if the variable is categorical); for instance, see ethnicity, language, cls_credits, which each show a noticeable difference in the variability of residuals across groups.

  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?

    Answer:

    • This indicates that the observations are not independent, since multiple rows are associated with the same professor. In particular, it appears that courses from the same professor are ordered together in the dataset, in which case we should review the residuals by their observation order to see if there is any pattern. This issue is analagous to the problem of time series data.

    • In terms of data collection for the analysis, it would be better to take a random sample of courses, in which case each observation would be independent.

  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.

    Answer:

    • Based on the model m_f5, the highest evaluation scores result when each contributing term is positive, which occurs when:
      • ethnicitynot minority = 1: not minority
      • gendermale = 1: male
      • languagenon-english = 0: professor received education from English-language school
      • age is low: younger age
      • cls_perc_eval is high: higher percentage of students in class completed evaluations
      • cls_creditsone credit = 1: class is one credit
      • bty_avg is high: higher average beauty rating
      • pic_colorcolor = 0: professor’s picture is black & white
    • For instance, if a professor has the above characteristics (not minority, male, English-language school, one credit, black & white picture) and where age = 35, cls_perc_eval = 100, and bty_avg = 7, then the predicted evaluation score would be 5.3 (or max’ed out at 5, if the maximum score is 5):
      \[ \begin{aligned} \widehat{\text{score}} &= 3.772 + 0.168 \cdot 1 + 0.207 \cdot 1 \\ &- 0.206 \cdot 0 - 0.006 \cdot 35 + 0.005 \cdot 100 \\ &+ 0.505 \cdot 1 + 0.051 \cdot 7 - 0.191 \cdot 0 \\ &= 5.3 \end{aligned} \]
    • At the other end of the spectrum, if a professor has the opposite characteristics (minority, female, non-English-language school, multi credit, color picture) and where age = 60, cls_perc_eval = 50, and bty_avg = 3, then the predicted evaluation score would be 3.4: \[ \begin{aligned} \widehat{\text{score}} &= 3.772 + 0.168 \cdot 0 + 0.207 \cdot 0 \\ &\text{ } - 0.206 \cdot 1 - 0.006 \cdot 60 + 0.005 \cdot 50 \\ &\text{ } + 0.505 \cdot 0 + 0.051 \cdot 3 - 0.191 \cdot 1 \\ &= 3.4 \end{aligned} \]
    ## [1] 5.299
    ## [1] 3.418

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

    Answer:

    No, the conclusions of this analysis should not be generalized to all professors at all schools, for several reasons:

    • The linear model has a poor fit, with \(R^2 = 0.16\); in other words, the model can explain only 16% of the variability in the evaluation scores.
    • The introduction indicates that six students rated the professors’ physical appearance, and these were not necessarily the same students who completed the evaluations. Views of physical appearance tend to be subjective. As a result, this analysis is associating evaluation scores from one set of students (students who took the course) with subjective views of the professor’s physical appearance from a different (small) set of students. It would be a stronger analysis to use the same students to measure both factors, that is, to use course evaluation scores and professor beauty ratings from the same set of students who took the class.
    • The dataset is specific to professors and students at the University of Texas at Austin. There may be idiosyncratic factors relating to the evaluation form (wording of questions, the rating scale, how / when evaluations are solicited, etc.), which may influence the evaluation data; such factors would have to be normalized (e.g., by using identical evaluation forms) across schools in order to generalize the conclusions.

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.