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.

Study Type: This is an observational study, meaning the researchers just collected data without trying to change or influence anything. Here, they observed course evaluations and professors’ appearance ratings but didn’t assign or control any conditions. So, this study isn’t designed to determine cause and effect.

Original Research Question: The original question was whether “beauty leads directly to the differences in course evaluations.” In other words, it asks if physical appearance has a direct impact on how students rate their professors.

Limitations of the Study Design: Since this is an observational study, we can’t actually say if beauty causes differences in course evaluations. Observational studies let us look for patterns and associations, but they don’t prove causation. Other factors, like teaching style, personality, or course difficulty, could also impact evaluations and might even be linked to appearance ratings, which makes it hard to isolate a direct effect.

Rephrasing the Question: A better way to phrase the question would be:

“Is there an association between professors’ physical appearance and their course evaluations?” This version focuses on exploring a possible link, without suggesting that appearance directly causes a change in evaluations, which is a more realistic goal for this kind of study.

  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 right skew reflects a generally positive trend in how students rate courses, which aligns with typical patterns in student feedback.

# Load necessary libraries
library(ggplot2)

# Plotting the histogram and density overlay for the `score` variable
ggplot(evals, aes(x = score)) +
  geom_histogram(aes(y = ..density..), bins = 30, fill = "skyblue", color = "black", alpha = 0.7) +
  geom_density(color = "darkblue", size = 1) +
  labs(title = "Distribution of Course Evaluation Scores",
       x = "Score",
       y = "Density") +
  theme_minimal()

  1. Excluding score, select two other variables and describe their relationship with each other using an appropriate visualization.
# Load necessary library
library(ggplot2)

# Scatter plot for `bty_avg` vs `age`
ggplot(evals, aes(x = age, y = bty_avg)) +
  geom_point(color = "steelblue", alpha = 0.6) +
  geom_smooth(method = "lm", color = "darkred", se = FALSE) +
  labs(title = "Relationship Between Age and Average Beauty Rating",
       x = "Professor's Age",
       y = "Average Beauty Rating") +
  theme_minimal()

it would suggest that younger professors tend to receive slightly higher beauty ratings.

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()

# Scatter plot with jitter for `bty_avg` vs `age`
ggplot(evals, aes(x = age, y = bty_avg)) +
  geom_jitter(color = "steelblue", alpha = 0.6, width = 0.3, height = 0.3) +
  geom_smooth(method = "lm", color = "darkred", se = FALSE) +
  labs(title = "Relationship Between Age and Average Beauty Rating (with Jitter)",
       x = "Professor's Age",
       y = "Average Beauty Rating") +
  theme_minimal()

Using geom_jitter in the scatter plot spreads out points that might otherwise sit right on top of each other. This is helpful because, in the initial scatter plot, many points likely overlapped, especially if a lot of professors had similar ages or received similar appearance ratings. When points stack up like that, it can look like there are fewer data points than there actually are or that the data is more spread out than it is.

By adding a small random shift (jitter) to each point, we can see the true density of the data. This makes it easier to spot clusters and gives us a much clearer picture of where ratings and ages tend to concentrate, showing us more accurately how age and bty_avg relate to each other.

  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?

Insert your answer here

# Fitting a linear model to predict `score` based on `bty_avg`
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

Statistically significant? Yes, likely. Practically significant? Not really—the effect on scores is too minor to make a big difference.

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).
# Fit the linear model
m_bty <- lm(score ~ bty_avg, data = evals)

# Plot each diagnostic plot separately
par(mfrow = c(1, 1))  # Reset to one plot per screen
plot(m_bty, which = 1)  # Residuals vs Fitted

plot(m_bty, which = 2)  # Normal Q-Q

plot(m_bty, which = 3)  # Scale-Location

plot(m_bty, which = 5)  # Residuals vs Leverage

# Set smaller margins
par(mfrow = c(2, 2), mar = c(4, 4, 2, 2)) 
plot(m_bty)

Residuals vs. Fitted Values Plot Purpose: This plot shows how the errors (residuals) are spread out across the predicted scores. It helps us check if the relationship we modeled is actually linear. What to Look For: Ideally, we want the residuals to be scattered randomly around zero without any clear pattern. Interpretation: If the points are scattered randomly, it suggests that a linear model is a good fit for this data. However, if we see a distinct shape, like a curve, it may mean that a non-linear model would be a better choice. 2. Normal Q-Q Plot Purpose: This plot checks if the residuals follow a normal distribution, which is one of the assumptions of linear regression. What to Look For: If the residuals are normally distributed, the points should fall along a straight line. Interpretation: Small deviations from the line are generally okay, but larger deviations, especially at the ends (tails), may indicate that the residuals are not perfectly normal. If the residuals deviate significantly, it could affect our confidence in the model’s results. 3. Scale-Location Plot (or Spread-Location Plot) Purpose: This plot checks if the residuals have a constant spread, or “variance,” across all predicted values, which is known as homoscedasticity. What to Look For: Ideally, we want the residuals to have a random, even spread around a horizontal line. Interpretation: If the spread of residuals is fairly consistent across the plot, it suggests that our model’s predictions are equally reliable across the range of fitted values. However, if the spread is increasing or decreasing, it suggests heteroscedasticity (non-constant variance), which can affect the accuracy of our model. 4. Residuals vs. Leverage Plot Purpose: This plot identifies any data points that may have an unusually large influence on the model, known as high-leverage points. What to Look For: Watch for points in the corners of the plot that may have both high leverage and large residuals. Interpretation: If any points stand out, especially in the corners, they could be influential outliers that may have a disproportionate impact on the model’s results. It’s important to consider whether these points are affecting the overall reliability of the model. Overall Together, these plots help us check if the main assumptions of linear regression are met: a linear relationship, normally distributed residuals, consistent spread of residuals, and no overly influential outliers. If these assumptions look reasonable, we can feel more confident about using this model’s results.

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.
# Fit the model with average beauty score and gender as predictors
m_bty_gen <- lm(score ~ bty_avg + gender, data = evals)
# Plot diagnostic plots for the model with gender included
par(mfrow = c(2, 2))  # Arrange plots in a 2x2 grid
plot(m_bty_gen)

1. Residuals vs. Fitted Values Plot Purpose: Checks for linearity and whether the residuals (errors) are scattered randomly around zero. What to Look For: Ideally, residuals should be randomly scattered around zero without any clear pattern. This suggests that the relationship between score, bty_avg, and gender is adequately linear. If a pattern (e.g., a curve) appears, it could mean that the model does not fully capture the true relationship. 2. Normal Q-Q Plot Purpose: Assesses whether the residuals follow a normal distribution. What to Look For: Residuals should fall along a straight line if they’re normally distributed. Small deviations from this line are generally acceptable, but large deviations may indicate non-normality. Non-normality in residuals can affect the reliability of the model’s p-values and confidence intervals. 3. Scale-Location Plot Purpose: Checks for homoscedasticity, which means the residuals should have a constant spread across all levels of fitted values. What to Look For: The residuals should have a consistent spread around a horizontal line. If the spread of residuals systematically increases or decreases, it indicates heteroscedasticity, meaning the model’s accuracy may vary across fitted values. This could impact the model’s reliability. 4. Residuals vs. Leverage Plot Purpose: Identifies any high-leverage points that might have a large impact on the model. What to Look For: Points in the upper corners, especially those with large residuals, could be influential outliers. Such points may disproportionately affect the model’s results, so it’s essential to check if any data points are unduly influencing the model. Summary If these diagnostic plots show that the residuals are appropriately linear, normally distributed, consistently spread, and free from overly influential points, the conditions for the model are likely reasonable. This would mean we can trust the p-values and parameter estimates for bty_avg and gender.

  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?
# Summary of the model with `bty_avg` and `gender` as predictors
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

Summary bty_avg remains a significant predictor of score, even after accounting for gender, though it explains only a small portion of the overall variability. Gender also has a significant effect on score, with male professors, on average, receiving slightly higher scores. This suggests that both beauty and gender have independent effects on professor evaluations, although the overall explanatory power of these variables is modest.

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?
# Fit the model with `bty_avg` and `pic_color` as predictors
m_bty_pic <- lm(score ~ bty_avg + pic_color, data = evals)

# View the summary to see the coefficients
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
# Plot with separate regression lines for color and non-color pictures
ggplot(data = evals, aes(x = bty_avg, y = score, color = pic_color)) +
  geom_point(alpha = 0.6) +
  geom_smooth(method = "lm", formula = y ~ x, se = FALSE) +
  labs(title = "Course Evaluation Score vs. Beauty Rating by Picture Type",
       x = "Average Beauty Rating (bty_avg)",
       y = "Course Evaluation Score (score)",
       color = "Picture Type") +
  theme_minimal()

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.
# Fit the model with `bty_avg` and `rank` as predictors, removing `gender`
m_bty_rank <- lm(score ~ bty_avg + rank, data = evals)

# View the summary to examine how R handles `rank`
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

when a categorical variable has more than two levels, R automatically sets one level as the reference and creates a separate dummy variable for each of the other levels. This allows us to compare each non-reference level’s effect relative to the reference category.

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.

since outfit likely does not affect a student’s experience or learning outcome, it’s reasonable to assume it won’t be significantly associated with the professor’s score, resulting in a high p-value in the model.

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.
# Fit the full 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)

# View the model summary
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

Our initial guess about pic_outfit having the highest p-value was close but not accurate. The variable cls_profssingle actually has the highest p-value, suggesting it has the weakest association with professor scores in this model. This result aligns with the notion that certain aspects, like the number of professors, may have little impact on how students rate their professors compared to other factors.

  1. Interpret the coefficient associated with the ethnicity variable.

The coefficient for ethnicity (specifically, ethnicitynot minority) is 0.1235. This tells us that, on average, professors who are classified as “not minority” tend to receive course evaluation scores that are about 0.12 points higher than those classified as “minority,” all else being equal.

However, the p-value for this coefficient is 0.117, which is above the usual cutoff of 0.05 for statistical significance. This means that the small difference in scores we see here could simply be due to chance, rather than a meaningful effect of ethnicity on professor scores. In short, while the model shows a slight increase in scores for non-minority professors, it’s not a strong enough effect to be considered statistically significant.

  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?
# Re-fit the model without `cls_profssingle`
m_full_refined <- lm(score ~ rank + gender + ethnicity + language + age + 
                     cls_perc_eval + cls_students + cls_level + cls_credits + 
                     bty_avg + pic_outfit + pic_color, data = evals)

# View the summary to examine any changes
summary(m_full_refined)
## 
## 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

The removal of cls_profssingle did not meaningfully impact the results, suggesting it was an independent predictor with minimal collinearity with other variables. This result reinforces that cls_profssingle was not a valuable predictor of score in this context.

  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.
# Full model with all variables
m_full <- lm(score ~ rank + gender + ethnicity + language + age + cls_perc_eval +
             cls_students + cls_level + cls_credits + bty_avg + pic_outfit + pic_color, data = evals)

# Perform backward selection using step() function
m_best <- step(m_full, direction = "backward", criterion = "p-value")
## Start:  AIC=-632.82
## 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
## - cls_level      1    0.2752 111.38 -633.67
## - cls_students   1    0.3893 111.49 -633.20
## - rank           2    0.8939 112.00 -633.11
## <none>                       111.11 -632.82
## - pic_outfit     1    0.5574 111.66 -632.50
## - ethnicity      1    0.6728 111.78 -632.02
## - language       1    1.0442 112.15 -630.49
## - bty_avg        1    1.2872 112.39 -629.49
## - age            1    2.0422 113.15 -626.39
## - pic_color      1    2.3457 113.45 -625.15
## - cls_perc_eval  1    2.9502 114.06 -622.69
## - gender         1    4.0895 115.19 -618.08
## - cls_credits    1    4.7999 115.90 -615.24
## 
## Step:  AIC=-633.67
## 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
## - cls_students   1    0.2459 111.63 -634.65
## - rank           2    0.8140 112.19 -634.30
## <none>                       111.38 -633.67
## - pic_outfit     1    0.6618 112.04 -632.93
## - ethnicity      1    0.8698 112.25 -632.07
## - language       1    0.9015 112.28 -631.94
## - bty_avg        1    1.3694 112.75 -630.02
## - age            1    1.9342 113.31 -627.70
## - pic_color      1    2.0777 113.46 -627.12
## - cls_perc_eval  1    3.0290 114.41 -623.25
## - gender         1    3.8989 115.28 -619.74
## - cls_credits    1    4.5296 115.91 -617.22
## 
## Step:  AIC=-634.65
## score ~ rank + gender + ethnicity + language + age + cls_perc_eval + 
##     cls_credits + bty_avg + pic_outfit + pic_color
## 
##                 Df Sum of Sq    RSS     AIC
## - rank           2    0.7892 112.42 -635.39
## <none>                       111.63 -634.65
## - ethnicity      1    0.8832 112.51 -633.00
## - pic_outfit     1    0.9700 112.60 -632.65
## - language       1    1.0338 112.66 -632.38
## - bty_avg        1    1.5783 113.20 -630.15
## - pic_color      1    1.9477 113.57 -628.64
## - age            1    2.1163 113.74 -627.96
## - cls_perc_eval  1    2.7922 114.42 -625.21
## - gender         1    4.0945 115.72 -619.97
## - cls_credits    1    4.5163 116.14 -618.29
## 
## Step:  AIC=-635.39
## score ~ gender + ethnicity + language + age + cls_perc_eval + 
##     cls_credits + bty_avg + pic_outfit + pic_color
## 
##                 Df Sum of Sq    RSS     AIC
## <none>                       112.42 -635.39
## - pic_outfit     1    0.7141 113.13 -634.46
## - ethnicity      1    1.1790 113.59 -632.56
## - language       1    1.3403 113.75 -631.90
## - age            1    1.6847 114.10 -630.50
## - pic_color      1    1.7841 114.20 -630.10
## - bty_avg        1    1.8553 114.27 -629.81
## - cls_perc_eval  1    2.9147 115.33 -625.54
## - gender         1    4.0577 116.47 -620.97
## - cls_credits    1    6.1208 118.54 -612.84
summary(m_best)
## 
## Call:
## lm(formula = score ~ gender + ethnicity + 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 ***
## gendermale             0.202597   0.050102   4.044 6.18e-05 ***
## ethnicitynot minority  0.163818   0.075158   2.180 0.029798 *  
## 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
  1. Verify that the conditions for this model are reasonable using diagnostic plots.
# Display diagnostic plots for the final model
par(mfrow = c(2, 2))  # Arrange plots in a 2x2 grid
plot(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?

since courses are grouped by professor, we need to consider the potential correlation between courses taught by the same individual. Ignoring this structure could violate the independence assumption of linear regression, potentially leading to biased results. A mixed-effects model would likely provide a more accurate approach by properly accounting for the non-independence of observations due to the repeated measures on professors.

  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 on the final model, here’s a profile of a professor and course at the University of Texas at Austin that would likely receive a high evaluation score:

Professor Characteristics:

Higher Beauty Rating: Professors with higher beauty scores tend to receive better evaluations. Male: Male professors are associated with slightly higher scores than female professors. English as Primary Language: Professors who are native English speakers tend to have higher scores, possibly due to clearer communication. Younger Age: Younger professors are associated with slightly higher scores on average. Color Photo: Professors with color photos in the evaluation system tend to receive better ratings. Course Characteristics:

High Engagement: Courses where a large percentage of students participate in evaluations tend to have higher scores, suggesting that engaged students are more likely to rate the course positively. One-Credit Course: One-credit courses are linked with higher evaluations, which could be due to these courses being more focused or less intensive. In summary, a high evaluation score is more likely for a younger, attractive, male professor who is a native English speaker and has a color photo in the system, teaching a one-credit course with high student engagement. These factors collectively contribute to higher ratings in the model’s predictions.

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

I wouldn’t feel comfortable generalizing these conclusions to professors at other universities, and here’s why:

Specific to UT Austin: The data is only from the University of Texas at Austin, so the results reflect that university’s unique culture, student body, and expectations. Other universities might have different dynamics that influence how students rate professors.

Different University Environments: Universities vary a lot—they can be public or private, research-focused or teaching-focused, and they’re spread across different regions and cultures. These differences mean that what affects professor ratings at one university may not apply at another.

Some Factors Might Not Apply Elsewhere: Certain factors in this study, like beauty ratings or the use of color photos, may not be relevant at other universities or might not even be collected in the same way. Plus, things like gender or age might influence ratings differently depending on the cultural context.

Just Observations, Not Cause and Effect: Since this is observational data, we’re seeing associations rather than proven cause-and-effect relationships. For example, while higher beauty ratings are linked to higher scores, this doesn’t mean that appearance directly causes better ratings in every context.

In short, these findings are specific to UT Austin and might not hold true in other settings. For a broader application, we’d need to see if similar patterns appear across a range of universities and contexts.