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

Load packages

In this lab, you will explore and visualize the data using the tidyverse suite of packages. The data can be found in the companion package for OpenIntro resources, openintro.

Let’s load the packages.

library(tidyverse)
library(openintro)
library(GGally)

This is the first time we’re using the GGally package. You will be using the ggpairs function from this package later in the lab.

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.

glimpse(evals)
## Rows: 463
## Columns: 23
## $ course_id     <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1…
## $ prof_id       <int> 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5,…
## $ score         <dbl> 4.7, 4.1, 3.9, 4.8, 4.6, 4.3, 2.8, 4.1, 3.4, 4.5, 3.8, 4…
## $ rank          <fct> tenure track, tenure track, tenure track, tenure track, …
## $ ethnicity     <fct> minority, minority, minority, minority, not minority, no…
## $ gender        <fct> female, female, female, female, male, male, male, male, …
## $ language      <fct> english, english, english, english, english, english, en…
## $ age           <int> 36, 36, 36, 36, 59, 59, 59, 51, 51, 40, 40, 40, 40, 40, …
## $ cls_perc_eval <dbl> 55.81395, 68.80000, 60.80000, 62.60163, 85.00000, 87.500…
## $ cls_did_eval  <int> 24, 86, 76, 77, 17, 35, 39, 55, 111, 40, 24, 24, 17, 14,…
## $ cls_students  <int> 43, 125, 125, 123, 20, 40, 44, 55, 195, 46, 27, 25, 20, …
## $ cls_level     <fct> upper, upper, upper, upper, upper, upper, upper, upper, …
## $ cls_profs     <fct> single, single, single, single, multiple, multiple, mult…
## $ cls_credits   <fct> multi credit, multi credit, multi credit, multi credit, …
## $ bty_f1lower   <int> 5, 5, 5, 5, 4, 4, 4, 5, 5, 2, 2, 2, 2, 2, 2, 2, 2, 7, 7,…
## $ bty_f1upper   <int> 7, 7, 7, 7, 4, 4, 4, 2, 2, 5, 5, 5, 5, 5, 5, 5, 5, 9, 9,…
## $ bty_f2upper   <int> 6, 6, 6, 6, 2, 2, 2, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, 9, 9,…
## $ bty_m1lower   <int> 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 7, 7,…
## $ bty_m1upper   <int> 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 6, 6,…
## $ bty_m2upper   <int> 6, 6, 6, 6, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 6, 6,…
## $ bty_avg       <dbl> 5.000, 5.000, 5.000, 5.000, 3.000, 3.000, 3.000, 3.333, …
## $ pic_outfit    <fct> not formal, not formal, not formal, not formal, not form…
## $ pic_color     <fct> color, color, color, color, color, color, color, color, …

We have observations on 21 different variables, some categorical and some numerical. The meaning of each variable can be found by bringing up the help file:

?evals

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 strictly an observational study based upon obtained data. There is nothing experimental about the study, as there is no intervention being implemented to qualify. Beauty, of course, is in the eye of the beholder and not necessarily an objective measure that is uniformly assessed by the students. Also, one would have to be sure that there is a direct pathway, not confounded, mediated, or moderated, between the perception of beauty by the students and a noted difference in the course evaluations. You would have to control for all possible confounding, and based upon this current study, I would be concerned with that presumption. I would suggest changing it the research question to be, given the possible co-variables, and all things remaining constant, is there correlation between student perceived beauty of the professor and their course evaluation scores.

  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?
hist(evals$score)

By plotting a histogram we note that there seems to be a left-hand skewing of the distribution. Immediately, that tells us that in general the students rate the professors higher overall. I am not surprised because in general students who do poorly in a particular course would tend to rate the professor lower and, since most students tend to pass courses their professor evaluations would also tend to be higher. For this particular study, we would have to determine whether “beauty” has a particular affect contributing to the distribution. We can only speak about the general population of professors and not any particular professor.

  1. Excluding score, select two other variables and describe their relationship with each other using an appropriate visualization.
ggplot(evals, aes(age, bty_avg)) + geom_point() + geom_smooth(method = "lm")

ggplot(evals, aes(age, bty_avg, color=gender)) + geom_point() + geom_smooth(method = "lm")

ggplot(evals, aes(age, bty_avg, color=gender)) + geom_point() + geom_smooth(method = "lm") + facet_wrap(~gender)

In these visualizations, we are exploring the potential relationship between age and the average beauty score of the professor. We can see by the first visualization that there seems to be a inverse relationship with age and average beauty score. When we control for gender, we see that the the curve may be slightly steeper for males vs. females, but there are also more older males over 65 than females, which could be biasing the curve downward a bit.

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:

ggplot(data = evals, aes(x = bty_avg, y = score)) +
  geom_point()

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. Replot the scatterplot, but this time use geom_jitter as your layer. What was misleading about the initial scatterplot?
ggplot(data = evals, aes(x = bty_avg, y = score)) +
  geom_jitter()

The discreteness of the data points an indication that there may be an increasing linear trend between evaluation score and the professors perceived beauty score. By using geom_jitter, a small random variation is added to the location of each data point removing a potential optical illusion based upon the discreteness. It now looks less organized.

  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. 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

Though the result is statistically significant (p < 0.00005), the effect size is relatively small. For every 1 point increase in average beauty, the evaluation score goes up by 0.067 points. Also, this model only accounts for approximately 3% of the variance noted in the response variable (score). Not a very good predictor.

Add the line of the bet fit model to your plot using the following:

ggplot(data = evals, aes(x = bty_avg, y = score)) +
  geom_jitter() +
  geom_smooth(method = "lm")

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, use se = FALSE.

ggplot(data = evals, aes(x = bty_avg, y = score)) +
  geom_jitter() +
  geom_smooth(method = "lm", se = FALSE)

  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).
ggplot(data = m_bty, aes(x = .fitted, y = .resid)) +
  geom_point() +
  geom_hline(yintercept = 0, linetype = "dashed", color="red") +
  xlab("Fitted values") +
  ylab("Residuals")

ggplot(data = m_bty, aes(x = .resid)) +
  geom_histogram(binwidth = 0.2) +
  xlab("Residuals")

ggplot(data = m_bty, aes(sample = .resid)) +
  stat_qq()

Looking at the various plots note some potential issues, but for the most part they appear to indicate normality. It is important to note the issues. The residuals plot indicates a larger number of residuals below the base line that could indicate a heavier weighting. The highest peak of the residuals plot is not centered at 0, indicating a bias in the model. This could be due to the outliers noticed in the plot. The histogram of the residuals indicate a leftward skewing which support and coincides with the residual plot. The QQ plot shows a slight concavity downward, but it seems fairly normal.

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.

ggplot(data = evals, aes(x = bty_f1lower, y = bty_avg)) +
  geom_point()

evals %>% 
  summarise(cor(bty_avg, bty_f1lower))
## # A tibble: 1 × 1
##   `cor(bty_avg, bty_f1lower)`
##                         <dbl>
## 1                       0.844

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 (columns 13 through 19) using the following command:

evals %>%
  select(contains("bty")) %>%
  ggpairs()

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.

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
  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.
ggplot(data = m_bty_gen, aes(x = .fitted, y = .resid)) +
  geom_point() +
  geom_hline(yintercept = 0, linetype = "dashed", color="red") +
  xlab("Fitted values") +
  ylab("Residuals")

ggplot(data = m_bty_gen, aes(x = .resid)) +
  geom_histogram(binwidth = 0.15) +
  xlab("Residuals")

ggplot(data = m_bty_gen, aes(sample = .resid)) +
  stat_qq()

Like the previous single variable model, we see the curves appear to fairly normal with a left hand skew and a concave downward QQ plot. We also see a farther peak to the right off 0, indicating a bias and potential outlier impact on the model. However, it appears to be fairly normal and does not violate the normality required for OLS.

  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?

Yes, bty_avg is still a predictor of score. Yes, the parameter estimate increased in magnitude, but is still relatively small. Also, the amount of variance in the response variable due to the independent variables (R squared) improved from about 3% to 5%, but is still pretty weak.

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 male and female to being an indicator variable called gendermale that takes a value of \(0\) for female professors and a value of \(1\) for male professors. (Such variables are often referred to as “dummy” variables.)

As a result, for female professors, the parameter estimate is multiplied by zero, leaving the intercept and slope form familiar from simple regression.

\[ \begin{aligned} \widehat{score} &= \hat{\beta}_0 + \hat{\beta}_1 \times bty\_avg + \hat{\beta}_2 \times (0) \\ &= \hat{\beta}_0 + \hat{\beta}_1 \times bty\_avg\end{aligned} \]

ggplot(data = evals, aes(x = bty_avg, y = score, color = pic_color)) +
 geom_smooth(method = "lm", formula = y ~ x, se = FALSE)

  1. What is the equation of the line corresponding to those with color pictures? (Hint: For those with color pictures, the parameter estimate is multiplied by 1.) For two professors who received the same beauty rating, which color picture tends to have the higher course evaluation score?

\[ \begin{aligned} \widehat{score} &= \hat{\beta}_0 + \hat{\beta}_1 \times bty\_avg + \hat{\beta}_2 \times pic\_color \\ &= \hat{\beta}_0 + \hat{\beta}_1 \times bty\_avg + \hat{\beta}_2 \times (1)\end{aligned} \]

Equation provided above. So, two people receiving an average beauty rating of 4, the person with the black and white picture would have the higher evaluation score. The person with the color picture would have a lower evaluation score. In this case, since black&white comes alphabetically before color, that is given the value 0 and color is given the value 1. But since this pushes the curve down, we can deduce that the parameter estimate (beta2) has a negative impact on the evaluation score.

The decision to call the indicator variable gendermale instead of genderfemale 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.
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 handles the new categorical variable with three levels, similar to that of two. The model is a simple univariate equation when the variable is “teaching” and then the beta takes on different value when the variable is either “tenure track” or “tenured”.

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, 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.

Without peeking, I would think a few variables might qualify. I suspect cls_level and cls_students might be highly correlated as upper division courses tend to have fewer students than lower level courses. I also suspect the cls_profs might qualify as it does not seem rational that the sections of a course being taught by a number professors would matter, as one would think that evaluations would be independent as a student would not be sitting in two sections of the same course to make comparative evaluations.

Let’s run the model…

m_full <- lm(score ~ rank + gender + ethnicity + language + age + cls_perc_eval 
             + cls_students + cls_level + cls_profs + cls_credits + bty_avg 
             + pic_outfit + pic_color, data = evals)
summary(m_full)
## 
## Call:
## lm(formula = score ~ rank + gender + ethnicity + 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    
## gendermale             0.2109481  0.0518230   4.071 5.54e-05 ***
## ethnicitynot minority  0.1234929  0.0786273   1.571  0.11698    
## 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.

While the cls_level and cls_students are not statistically significant, they are very close to each others p_values. However, they are not the highest. It does appear the cls_profs does have the highest p_value of 0.77806.

  1. Interpret the coefficient associated with the ethnicity variable.

With everything else remaining constant, a professor who is “not minority” could expect their evaluation score to increase by a value of 0.1234929; however, that result is not statistically significant (p=0.11698). Therefore, it could indicate that the effect on the evaluation score could be “0”.

  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?
m_full_minus_1 <- lm(score ~ rank + gender + ethnicity + language + age + cls_perc_eval 
             + cls_students + cls_level + cls_credits + bty_avg 
             + pic_outfit + pic_color, data = evals)
summary(m_full_minus_1)
## 
## Call:
## lm(formula = score ~ rank + gender + ethnicity + 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    
## gendermale             0.2101231  0.0516873   4.065 5.66e-05 ***
## ethnicitynot minority  0.1274458  0.0772887   1.649 0.099856 .  
## 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

While some p_values increase or decreased, the overall remaining values did not change enough to alter their original statistical significance in the full model. However, the adjusted-R square did increase slightly.

  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.
m_full_minus_all <- lm(score ~ gender + ethnicity + age + cls_perc_eval 
            + cls_credits + bty_avg 
             + pic_color, data = evals)
summary(m_full_minus_all)
## 
## Call:
## lm(formula = score ~ gender + ethnicity + age + cls_perc_eval + 
##     cls_credits + bty_avg + pic_color, data = evals)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.85434 -0.33568  0.09247  0.38288  0.93903 
## 
## Coefficients:
##                        Estimate Std. Error t value Pr(>|t|)    
## (Intercept)            3.690771   0.229181  16.104  < 2e-16 ***
## gendermale             0.201574   0.050220   4.014 6.99e-05 ***
## ethnicitynot minority  0.216955   0.071348   3.041  0.00250 ** 
## age                   -0.006034   0.002621  -2.302  0.02176 *  
## cls_perc_eval          0.004719   0.001439   3.278  0.00113 ** 
## cls_creditsone credit  0.527806   0.103839   5.083 5.44e-07 ***
## bty_avg                0.052431   0.016975   3.089  0.00213 ** 
## pic_colorcolor        -0.170149   0.066780  -2.548  0.01116 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.5008 on 455 degrees of freedom
## Multiple R-squared:  0.1649, Adjusted R-squared:  0.1521 
## F-statistic: 12.84 on 7 and 455 DF,  p-value: 4.344e-15

Score = B0 + B1 x gender + B2 x ethnicity + B3 x age + B4 x cls_per_eval + B5 x cls_credit + B6 x bty_avg + B7 x pic_color

  1. Verify that the conditions for this model are reasonable using diagnostic plots.
ggplot(data = m_full_minus_all, aes(x = .fitted, y = .resid)) +
  geom_point() +
  geom_hline(yintercept = 0, linetype = "dashed", color="red") +
  xlab("Fitted values") +
  ylab("Residuals")

ggplot(data = m_full_minus_all, aes(x = .resid)) +
  geom_histogram(binwidth = 0.2) +
  xlab("Residuals")

ggplot(data = m_full_minus_all, aes(sample = .resid)) +
  stat_qq()

While we were able to fit the model and get a result, I am becoming more concerned about the results of the diagnostic plots. The residual plot appears to be taking on a cone shaped appearance, indicating that residuals are becoming closer to 0 as the fitted values increase in magnitude and there are many higher magnitude negative outliers appearing. The histogram of the residuals indicates a left hand skew, with a peak farther to the right away from 0. Also the QQ plot seems to be taking on a greater concavity downward than our previous model runs. We may have a less than optimal model.

  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?

Absolutely. That would mean that each row (observation) may be highly correlated with one or more other rows, and be influencing the results. This may explain the disparities in noted previously with the residuals. Logically, you would have to calculate a mean for each of the variables for a particular professor and then use that in the modelling.

  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.

Based upon the model and variable significance, positive characteristics that would increase the score would be male, not minority, greater percent of class completing evaluation, higher credit course, and higher average beauty rating. The attributes that would have the least negative impact on the evaluation score would be a younger professor who had a black and white picture. However, it should be noted that two variables, though significant, had very small impacts on the overall evaluation score in this model-age and percent of class completing the evaluation.

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

Absolutely not. Besides the fact the model is not optimal, this was a single population sampled under a single university academic structure. We do not know anything about the population of students being asked to complete the evaluations, nor do we know if the academic faculty is consistent with other universities. The best we can ascertain from this modeling and analysis is that it applies only to this specific population and is not generalizable to a larger population without additional information and data sampling.