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 an observational study because the researchers simply observe and collect data on existing conditions or behaviors. They don’t manipulate any variables. Its not possible to answer the question as currently constructed. Because of the Correlation vs Causation factor, a better question would be “Is there a correlation between instructor beauty and course evaluations?”

  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?

The distribution of score is skewed to the left which suggest that the students have a tendency to score the evals above the average. I expected the opposite since from my experience students tend to rate professors lower when providing feedback

hist(evals$score, main = "Distribution of score in evals", xlab = "Score")
abline(v = mean(evals$score), col = "red", lwd = 2)
abline(v = median(evals$score), col = "blue", lwd = 2)
legend("topleft", legend = c("Mean", "Median"), col = c("red", "blue"), lwd = 2)

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

The scatterplot shows a weak negative relationship between beauty average and age. There is a considerable amount of scatter in the data points.

# Visualize the relationship between bty_avg and age of professors and added a trend line
evals |>
  ggplot(aes(x = bty_avg, y = age)) +
  geom_point() +
  geom_smooth(method = "lm") +
  labs(title = "Relationship between Beauty Avg and Age of Professors", x = "Beauty Average", y = "Age")

Simple linear regression

The fundamental phenomenon suggested by the study is that better looking teachers are evaluated more favorably. Let’s create a scatterplot to see if this appears to be the case:

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. Re-plot 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 number of data points in the initial plot seem to be way less than actual observations. There are 463 observations in the data-frame. The jitter plot seems to be more accurate because it adds some random jitter to the data points assisting with the overlapping.

  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?

The equation for the linear model is score = β₀ + β₁ * bty_avg + ε. The average beauty score seems to be a statistically significant predictor.

# Fit a linear model
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

Add the line of the best 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).

It seems that the variance of the residuals might not be constant across the range of fitted values. The errors for the higher predicted scores have a smaller variance compared to errors for lower predicted scores

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

We can see deviations from the diagonal line in the QQ-plot. The residuals seem to curve away from the line at the tails suggesting a slightly more extreme than expected normal distribution

# Use residual plots to evaluate whether the conditions of least squares regression are resonable
qqnorm(residuals(m_bty))
qqline(residuals(m_bty))

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

{r})

  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.

There seems to be a linear relationship between the gender and score and the bty_avg. As established in the previous exercises, there is a relationship between the variables.

# Determine if there is a linear relationship between bty and score
evals |>
  ggplot(aes(x = score, y = bty_avg, color = gender)) +
  geom_point() +
  geom_smooth(method = "lm") +
  labs(title = "Relationship between Beauty Avg and Score and Gender", x = "Beauty Average", y = "Score")

# Determine if residuals are normally distributed
qqnorm(residuals(m_bty_gen))
qqline(residuals(m_bty_gen))

# Determine if the variablility of the residuals is nearly constantby comparing the residuals to the fitted values
ggplot(data = m_bty_gen, aes(x = .fitted, y = .resid)) +
  geom_point() +
  geom_hline(yintercept = 0, linetype = "dashed") +
  xlab("Fitted Values") +
  ylab("Residuals") +
  geom_jitter()

  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?

bty_avg is still a significant predictor of score. The addition of gender made the p-value smaller so it adds significance.

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 re-codes 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?

Score = C1 + C2 * bty_avg for color photos and for non-color photos Score = C1 + C2 * bty_avg - C3. This indicates that for two professors who received the same beauty rating, the picture with color tends to have a higher score.

m_bty_pic <- lm(score ~ bty_avg + pic_color, data = evals)
summary(m_bty_pic)
## 
## Call:
## lm(formula = score ~ bty_avg + pic_color, data = evals)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.8892 -0.3690  0.1293  0.4023  0.9125 
## 
## Coefficients:
##                Estimate Std. Error t value Pr(>|t|)    
## (Intercept)     4.06318    0.10908  37.249  < 2e-16 ***
## bty_avg         0.05548    0.01691   3.282  0.00111 ** 
## pic_colorcolor -0.16059    0.06892  -2.330  0.02022 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.5323 on 460 degrees of freedom
## Multiple R-squared:  0.04628,    Adjusted R-squared:  0.04213 
## F-statistic: 11.16 on 2 and 460 DF,  p-value: 1.848e-05

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.

It creates one less dummy variable than the number of categories in the original variable. In this case, rank has 3 levels, so R creates 2 dummy variables (rank_tenure track and ranktenured). Rank_teaching is not represented.

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

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.

I would expect the language of the university where the professor got their degree to have the highest p-value because it doesn’t seem like it would have any association with score.

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.

The highest p value goes to the cls_profssingle variable that was created by the model. This makes sense because the number of professors teaching sections in the course should not have any association with the score

library(broom)
tidy(m_full) %>%
  arrange(desc(p.value)) # Sort by p-value (descending)
## # A tibble: 15 × 5
##    term                   estimate std.error statistic  p.value
##    <chr>                     <dbl>     <dbl>     <dbl>    <dbl>
##  1 cls_profssingle       -0.0147    0.0520      -0.282 7.78e- 1
##  2 cls_levelupper         0.0605    0.0576       1.05  2.94e- 1
##  3 cls_students           0.000455  0.000377     1.20  2.29e- 1
##  4 ranktenured           -0.0973    0.0663      -1.47  1.43e- 1
##  5 pic_outfitnot formal  -0.113     0.0739      -1.53  1.28e- 1
##  6 ethnicitynot minority  0.123     0.0786       1.57  1.17e- 1
##  7 ranktenure track      -0.148     0.0821      -1.80  7.28e- 2
##  8 languagenon-english   -0.230     0.111       -2.06  3.97e- 2
##  9 bty_avg                0.0400    0.0175       2.29  2.27e- 2
## 10 age                   -0.00901   0.00314     -2.87  4.27e- 3
## 11 pic_colorcolor        -0.217     0.0715      -3.04  2.52e- 3
## 12 cls_perc_eval          0.00533   0.00154      3.46  5.90e- 4
## 13 gendermale             0.211     0.0518       4.07  5.54e- 5
## 14 cls_creditsone credit  0.502     0.116        4.33  1.84e- 5
## 15 (Intercept)            4.10      0.291       14.1   1.32e-37
  1. Interpret the coefficient associated with the ethnicity variable.

The coefficient associated with the ethnicity variable is .123 with a p value of 0.17. The coefficient indicates that on average, professors identified as part of the “not minority” group are predicted to have a score .123 higher than professors in the minority category, when all other variables in the model are held constant.The high p-value indicates that it is not statistically significant to claim that professors that are “not minority” have higher scores than professors who are minorities.

  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?

By removing the cls_prof variable the other coefficients and p-values for the remaining variables hardly changed. This indicates that the dropped variable likely wasn’t collinear with the other explanatory variables.

m_full2 <- 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_full2)
## 
## 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
tidy(m_full2) %>%
  arrange(desc(p.value))
## # A tibble: 14 × 5
##    term                   estimate std.error statistic  p.value
##    <chr>                     <dbl>     <dbl>     <dbl>    <dbl>
##  1 cls_levelupper         0.0606    0.0575        1.05 2.92e- 1
##  2 cls_students           0.000469  0.000374      1.25 2.10e- 1
##  3 ranktenured           -0.0974    0.0663       -1.47 1.42e- 1
##  4 pic_outfitnot formal  -0.108     0.0722       -1.50 1.34e- 1
##  5 ethnicitynot minority  0.127     0.0773        1.65 9.99e- 2
##  6 ranktenure track      -0.148     0.0820       -1.80 7.23e- 2
##  7 languagenon-english   -0.228     0.111        -2.05 4.05e- 2
##  8 bty_avg                0.0399    0.0175        2.28 2.30e- 2
##  9 age                   -0.00900   0.00313      -2.87 4.26e- 3
## 10 pic_colorcolor        -0.219     0.0711       -3.08 2.21e- 3
## 11 cls_perc_eval          0.00529   0.00153       3.45 6.07e- 4
## 12 gendermale             0.210     0.0517        4.07 5.66e- 5
## 13 cls_creditsone credit  0.506     0.115         4.40 1.33e- 5
## 14 (Intercept)            4.09      0.289        14.1  7.53e-38
  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.

\[ \begin{aligned} \widehat{score} &= \hat{\beta}_0 + \hat{\beta}_1 \times language \\ &+ \hat{\beta}_2 \times ethnicity \\ &+\hat{\beta}_3 \times age \\ &+ \hat{\beta}_4 \times pic\_color \\ &+ \hat{\beta}_5 \times bty\_avg \\ &+ \hat{\beta}_6 \times cls\_perc\_eval \\ &+ \hat{\beta}_8 \times gender \\ &+ \hat{\beta}_9 \times cls\_credits \end{aligned} \]

# Run the model selection function
m_best <- model.select(m_full, sig = 0.05, verbose = TRUE)
## -------------STEP  1 -------------
##  The drop statistics : 
## Single term deletions
## 
## Model:
## score ~ rank + gender + ethnicity + language + age + cls_perc_eval + 
##     cls_students + cls_level + cls_profs + cls_credits + bty_avg + 
##     pic_outfit + pic_color
##               Df Sum of Sq    RSS     AIC F value    Pr(>F)    
## <none>                     111.08 -630.90                      
## rank           2    0.8930 111.98 -631.19  1.8007 0.1663804    
## gender         1    4.1085 115.19 -616.09 16.5694 5.544e-05 ***
## ethnicity      1    0.6117 111.70 -630.36  2.4668 0.1169791    
## language       1    1.0557 112.14 -628.52  4.2576 0.0396509 *  
## age            1    2.0456 113.13 -624.45  8.2499 0.0042688 ** 
## cls_perc_eval  1    2.9698 114.06 -620.69 11.9769 0.0005903 ***
## cls_students   1    0.3599 111.44 -631.40  1.4513 0.2289607    
## cls_level      1    0.2740 111.36 -631.76  1.1052 0.2936925    
## cls_profs      1    0.0197 111.11 -632.82  0.0795 0.7780566    
## cls_credits    1    4.6495 115.73 -613.92 18.7510 1.839e-05 ***
## bty_avg        1    1.2967 112.38 -627.53  5.2294 0.0226744 *  
## pic_outfit     1    0.5768 111.66 -630.50  2.3262 0.1279153    
## pic_color      1    2.2893 113.37 -623.46  9.2328 0.0025162 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## --------
## Term dropped in step 1 : cls_profs 
## --------
## 
## -------------STEP  2 -------------
##  The drop statistics : 
## Single term deletions
## 
## Model:
## score ~ rank + gender + ethnicity + language + age + cls_perc_eval + 
##     cls_students + cls_level + cls_credits + bty_avg + pic_outfit + 
##     pic_color
##               Df Sum of Sq    RSS     AIC F value    Pr(>F)    
## <none>                     111.11 -632.82                      
## rank           2    0.8939 112.00 -633.11  1.8063 0.1654529    
## gender         1    4.0895 115.19 -618.08 16.5265 5.665e-05 ***
## ethnicity      1    0.6728 111.78 -632.02  2.7191 0.0998556 .  
## language       1    1.0442 112.15 -630.49  4.2199 0.0405303 *  
## age            1    2.0422 113.15 -626.39  8.2529 0.0042616 ** 
## cls_perc_eval  1    2.9502 114.06 -622.69 11.9224 0.0006072 ***
## cls_students   1    0.3893 111.49 -633.20  1.5733 0.2103843    
## cls_level      1    0.2752 111.38 -633.67  1.1121 0.2922000    
## cls_credits    1    4.7999 115.90 -615.24 19.3974 1.329e-05 ***
## bty_avg        1    1.2872 112.39 -629.49  5.2018 0.0230315 *  
## pic_outfit     1    0.5574 111.66 -632.50  2.2527 0.1340803    
## pic_color      1    2.3457 113.45 -625.15  9.4795 0.0022052 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## --------
## Term dropped in step 2 : cls_level 
## --------
## 
## -------------STEP  3 -------------
##  The drop statistics : 
## Single term deletions
## 
## Model:
## score ~ rank + gender + ethnicity + language + age + cls_perc_eval + 
##     cls_students + cls_credits + bty_avg + pic_outfit + pic_color
##               Df Sum of Sq    RSS     AIC F value    Pr(>F)    
## <none>                     111.38 -633.67                      
## rank           2    0.8140 112.19 -634.30  1.6443 0.1943109    
## gender         1    3.8989 115.28 -619.74 15.7525 8.400e-05 ***
## ethnicity      1    0.8698 112.25 -632.07  3.5142 0.0614914 .  
## language       1    0.9015 112.28 -631.94  3.6423 0.0569659 .  
## age            1    1.9342 113.31 -627.70  7.8147 0.0054040 ** 
## cls_perc_eval  1    3.0290 114.41 -623.25 12.2380 0.0005148 ***
## cls_students   1    0.2459 111.63 -634.65  0.9934 0.3194514    
## cls_credits    1    4.5296 115.91 -617.22 18.3006 2.306e-05 ***
## bty_avg        1    1.3694 112.75 -630.02  5.5329 0.0190925 *  
## pic_outfit     1    0.6618 112.04 -632.93  2.6736 0.1027219    
## pic_color      1    2.0777 113.46 -627.12  8.3942 0.0039478 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## --------
## Term dropped in step 3 : cls_students 
## --------
## 
## -------------STEP  4 -------------
##  The drop statistics : 
## Single term deletions
## 
## Model:
## score ~ rank + gender + ethnicity + language + age + cls_perc_eval + 
##     cls_credits + bty_avg + pic_outfit + pic_color
##               Df Sum of Sq    RSS     AIC F value    Pr(>F)    
## <none>                     111.63 -634.65                      
## rank           2    0.7892 112.42 -635.39  1.5943 0.2041875    
## gender         1    4.0945 115.72 -619.97 16.5430 5.613e-05 ***
## ethnicity      1    0.8832 112.51 -633.00  3.5683 0.0595353 .  
## language       1    1.0338 112.66 -632.38  4.1769 0.0415580 *  
## age            1    2.1163 113.74 -627.96  8.5504 0.0036285 ** 
## cls_perc_eval  1    2.7922 114.42 -625.21 11.2814 0.0008493 ***
## cls_credits    1    4.5163 116.14 -618.29 18.2472 2.368e-05 ***
## bty_avg        1    1.5783 113.20 -630.15  6.3770 0.0119026 *  
## pic_outfit     1    0.9700 112.60 -632.65  3.9191 0.0483466 *  
## pic_color      1    1.9477 113.57 -628.64  7.8693 0.0052456 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## --------
## Term dropped in step 4 : rank 
## --------
## 
## -------------STEP  5 -------------
##  The drop statistics : 
## Single term deletions
## 
## Model:
## score ~ gender + ethnicity + language + age + cls_perc_eval + 
##     cls_credits + bty_avg + pic_outfit + pic_color
##               Df Sum of Sq    RSS     AIC F value    Pr(>F)    
## <none>                     112.42 -635.39                      
## gender         1    4.0577 116.47 -620.97 16.3513 6.184e-05 ***
## ethnicity      1    1.1790 113.59 -632.56  4.7508 0.0297983 *  
## language       1    1.3403 113.75 -631.90  5.4010 0.0205673 *  
## age            1    1.6847 114.10 -630.50  6.7888 0.0094749 ** 
## cls_perc_eval  1    2.9147 115.33 -625.54 11.7455 0.0006655 ***
## cls_credits    1    6.1208 118.54 -612.84 24.6649 9.681e-07 ***
## bty_avg        1    1.8553 114.27 -629.81  7.4762 0.0064972 ** 
## pic_outfit     1    0.7141 113.13 -634.46  2.8775 0.0905102 .  
## pic_color      1    1.7841 114.20 -630.10  7.1895 0.0076009 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## --------
## Term dropped in step 5 : pic_outfit 
## --------
## 
## -------------STEP  6 -------------
##  The drop statistics : 
## Single term deletions
## 
## Model:
## score ~ gender + ethnicity + language + age + cls_perc_eval + 
##     cls_credits + bty_avg + pic_color
##               Df Sum of Sq    RSS     AIC F value    Pr(>F)    
## <none>                     113.13 -634.46                      
## gender         1    4.2526 117.38 -619.37 17.0660 4.299e-05 ***
## ethnicity      1    1.2393 114.37 -631.41  4.9734  0.026228 *  
## language       1    0.9862 114.11 -632.44  3.9576  0.047259 *  
## age            1    1.3350 114.46 -631.03  5.3574  0.021080 *  
## cls_perc_eval  1    2.6225 115.75 -625.85 10.5243  0.001265 ** 
## cls_credits    1    5.8690 119.00 -613.04 23.5530 1.674e-06 ***
## bty_avg        1    2.2663 115.39 -627.28  9.0950  0.002707 ** 
## pic_color      1    1.9951 115.12 -628.36  8.0068  0.004867 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
summary(m_best)
## 
## Call:
## lm(formula = score ~ gender + ethnicity + 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 ***
## gendermale             0.207112   0.050135   4.131 4.30e-05 ***
## ethnicitynot minority  0.167872   0.075275   2.230  0.02623 *  
## 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
  1. Verify that the conditions for this model are reasonable using diagnostic plots.

The standard regression assumptions include the following about residuals/errors:

  • The error has a normal distribution (normality assumption).

  • The errors have mean zero.

  • The errors have same but unknown variance (homoscedasticity assumption).

  • The error are independent of each other (independent errors assumption).

# Apply the diagnostic plots to the best model
par(mfrow = c(2, 2))
plot(m_best)

# Import olsrr package
if(!require("olsrr")) {install.packages("olsrr"); library("olsrr")}
par(mfrow = c(2,2 ))

# Residual QQ Plot - Graph for detecting violation of normality assumption.
ols_plot_resid_qq(m_best)

# Residual vs Fitted Values Plot - Graph for detecting violation of linearity assumption.
ols_plot_resid_fit(m_best)

# Residual Histogram - Graph for detecting violation of normality assumption.
ols_plot_resid_hist(m_best)

  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?

This could potentially add independence errors. Courses taught by the same professor could affect the scores of all their courses

  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.

Professor Characteristics:

  • Gender: The analysis suggests a slight positive association between being male and receiving a higher score.

  • Ethnicity: The analysis shows a slight positive association between not belonging to a minority group and receiving a higher score.

  • Age: There’s a weak negative association between age and score, suggesting a slight decrease in score with increasing age. However, the magnitude of this effect is likely small.

Course Characteristics:

  • Language: Courses taught in English are predicted to have slightly higher scores compared to those taught in non-English languages.

  • Course Evaluations: Courses with a higher percentage of students giving positive evaluations (cls_perc_eval) tend to have higher scores. This suggests that student perception of the course plays a significant role in the evaluation.

  • Course Credits: One-credit courses are predicted to have higher scores compared to courses with a different number of credits. .

Other Considerations:

  • The beauty rating (bty_avg) and picture color (pic_color) showed some association with scores as well.
  1. 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 because the data used in the model comes from a single university. This can lead to biased conclusions. Also there can be confounding variables such as class size, course difficulty, department etc.