Multiple linear regression

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Grading the professor

Many college courses conclude by giving students the opportunity to evaluate the course and the instructor anonymously. However, the use of these student evaluations as an indicator of course quality and teaching effectiveness is often criticized because these measures may reflect the influence of non-teaching related characteristics, such as the physical appearance of the instructor. The article titled, “Beauty in the classroom: instructors’ pulchritude and putative pedagogical productivity” (Hamermesh and Parker 2005) found that instructors who are viewed to be better looking receive higher instructional ratings.

In this lab we 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 we will explore the data using the dplyr package and visualize it using the ggplot2 package for data visualization. The data can be found in the companion package for this course, statsr.

Let’s load the packages.

library(statsr)
library(dplyr)
library(ggplot2)

## Warning: package 'ggplot2' was built under R version 3.3.2

library(GGally)

## Warning: package 'GGally' was built under R version 3.3.2

This is the first time we’re using the GGally package. We 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. (This is a slightly modified version of the original data set that was released as part of the replication data for Data Analysis Using Regression and Multilevel/Hierarchical Models (Gelman and Hill 2007).) The result is a data frame where each row contains a different course and columns represent variables about the courses and professors.

Let’s load the data:

data(evals)

variable	description
score	average professor evaluation score: (1) very unsatisfactory - (5) excellent.
rank	rank of professor: teaching, tenure track, tenured.
ethnicity	ethnicity of professor: not minority, minority.
gender	gender of professor: female, male.
language	language of school where professor received education: english or non-english.
age	age of professor.
cls_perc_eval	percent of students in class who completed evaluation.
cls_did_eval	number of students in class who completed evaluation.
cls_students	total number of students in class.
cls_level	class level: lower, upper.
cls_profs	number of professors teaching sections in course in sample: single, multiple.
cls_credits	number of credits of class: one credit (lab, PE, etc.), multi credit.
bty_f1lower	beauty rating of professor from lower level female: (1) lowest - (10) highest.
bty_f1upper	beauty rating of professor from upper level female: (1) lowest - (10) highest.
bty_f2upper	beauty rating of professor from second upper level female: (1) lowest - (10) highest.
bty_m1lower	beauty rating of professor from lower level male: (1) lowest - (10) highest.
bty_m1upper	beauty rating of professor from upper level male: (1) lowest - (10) highest.
bty_m2upper	beauty rating of professor from second upper level male: (1) lowest - (10) highest.
bty_avg	average beauty rating of professor.
pic_outfit	outfit of professor in picture: not formal, formal.
pic_color	color of professor’s picture: color, black & white.

1. Is this an observational study or an experiment? A
   1. Observational study
   2. Experiment
2. The original research question posed in the paper is whether beauty leads directly to the differences in course evaluations. Given the study design, should the question be rephrased? If so, how? B
   1. No, the question is worded accurately.
   2. Yes, revise wording to “Is there an association between beauty and course evaluations?”
   3. Yes, revise wording to “Does beauty score increase the professor’s course evaluations?”
   4. Yes, revise wording to “Does beauty score decrease the professor’s course evaluations?”

Exploring the data

3. Which of the following statements is false about the distribution of score? D
   1. The median of the distribution is 4.3.
   2. 25% of the students gave their professors a score of over 4.6.
   3. 11 of students gave a professor a score below 3.
   4. The left skewness of the data suggests that the students are less likely to rate the professors highly.

# EDA
summary(evals$score)

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

# Number of cases with score below 3
evals %>%
    filter(score < 3) %>%
    nrow()

## [1] 11

# Shape of distribution
hist(evals$score)

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

# Average beauty rating of professor and ethnicity
plot(evals$bty_avg ~ evals$ethnicity)

# Average beauty rating of professor and percent of students completed evaluation
plot(evals$bty_avg ~ evals$cls_perc_eval)

# Ethnicity of professor and level of class
plot(evals$ethnicity ~ evals$cls_level)

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 we draw conclusions about the trend, compare the number of observations in the data frame with the approximate number of points on the scatterplot. Is anything awry?

Exercise: Replot the scatterplot, but this time replace the geom_point() layer with a geom_jitter() layer. (Use ?geom_jitter to learn more.) What was misleading about the initial scatterplot?

# What does `geom_jitter` do?
?geom_jitter

#Replot the scatterplot
ggplot(data=evals, aes(x=bty_avg, y=score))+
    geom_jitter()

Let’s see if the apparent trend in the plot is something more than natural variation. Fit a linear model called m_bty to predict average professor score by average beauty rating and add the line to your plot using the following. If you do not remember how to do this, refer to the previous lab.

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

# The last line can be replaced with 'stat_smooth.'

The blue line is the model. The shaded gray area around the line tells us about the variability we might expect in our 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)

# The last line can be replaced with 'stat_smooth.'

Exercise: Print a summary of the linear model, write out the equation, and interpret the slope.
The equation: (\(\hat{y}\)) = 3.88 + 0.07 x bty_avg
Interpretation of the slope: The model predicts that 1 point increases/decreases in beauty rating increases/decreases 0.07 point in score on the average

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

anova(m_bty)

## Analysis of Variance Table
## 
## Response: score
##            Df  Sum Sq Mean Sq F value    Pr(>F)    
## bty_avg     1   4.786  4.7859  16.731 5.083e-05 ***
## Residuals 461 131.868  0.2860                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

4. Average beauty score is a statistically significant predictor of evaluation score. A
   1. True
   2. False
5. Use residual plots to evaluate whether the conditions of least squares regression are reasonable. Which of the following statements is an incorrect analysis of the residual plots and conditions? B
   1. Linear association: The residuals plot shows a random scatter.
   2. Constant variance of residuals: No fan shape in residuals plot.
   3. Nearly normal residuals: Residuals are right skewed, but the sample size is large, so this may not be an important violation of conditions.
   4. Independent observations: Classes sampled randomly, no order effect.

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

qqnorm(m_bty$residuals)
qqline(m_bty$residuals)

hist(m_bty$residuals)

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_jitter()

evals %>% 
  summarise(cor(bty_avg, bty_f1lower))

## # A tibble: 1 × 1
##   `cor(bty_avg, bty_f1lower)`
##                         <dbl>
## 1                   0.8439112

As expected the relationship is quite strong - after all, the average score is calculated using the individual scores. We can actually take a look at the relationships between all beauty variables (columns 13 through 19) using the following command:

ggpairs(evals, columns = 13:19)

These variables are collinear (correlated), and adding more than one of these variables to the model would not add much value to the model. In this application and with these highly-correlated predictors, it is reasonable to use the average beauty score as the single representative of these variables.

In order to see if beauty is still a significant predictor of professor score after we’ve accounted for the gender of the professor, we can add the gender term into the model.

m_bty_gen <- lm(score ~ bty_avg + gender, data = evals)
summary(m_bty_gen)

## 
## Call:
## lm(formula = score ~ bty_avg + gender, data = evals)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.8305 -0.3625  0.1055  0.4213  0.9314 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  3.74734    0.08466  44.266  < 2e-16 ***
## bty_avg      0.07416    0.01625   4.563 6.48e-06 ***
## gendermale   0.17239    0.05022   3.433 0.000652 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.5287 on 460 degrees of freedom
## Multiple R-squared:  0.05912,    Adjusted R-squared:  0.05503 
## F-statistic: 14.45 on 2 and 460 DF,  p-value: 8.177e-07

anova(m_bty_gen)

## Analysis of Variance Table
## 
## Response: score
##            Df  Sum Sq Mean Sq F value    Pr(>F)    
## bty_avg     1   4.786  4.7859  17.123 4.168e-05 ***
## gender      1   3.293  3.2934  11.783 0.0006518 ***
## Residuals 460 128.575  0.2795                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

6. P-values and parameter estimates should only be trusted if the conditions for the regression are reasonable. Using diagnostic plots, we can conclude that the conditions for this model are reasonable. B
   1. True
   2. False

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

hist(m_bty_gen$residuals)

qqnorm(m_bty_gen$residuals)
qqline(m_bty_gen$residuals)

Exercise: Print a summary of the multiple linear regression model. Is bty_avg still a significant predictor of score? Has the addition of gender to the model changed the parameter estimate for bty_avg?
The p-value for bty_avg is close to zero and is still a sginificant predictor of score
The addition of gender to the model increases the parameter estimaye for bty_avg

summary(m_bty)

## 
## Call:
## lm(formula = score ~ bty_avg, data = evals)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.9246 -0.3690  0.1420  0.3977  0.9309 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  3.88034    0.07614   50.96  < 2e-16 ***
## bty_avg      0.06664    0.01629    4.09 5.08e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.5348 on 461 degrees of freedom
## Multiple R-squared:  0.03502,    Adjusted R-squared:  0.03293 
## F-statistic: 16.73 on 1 and 461 DF,  p-value: 5.083e-05

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

Note that the estimate for gender is now called gendermale. You’ll see this name change whenever you introduce a categorical variable. The reason is that R recodes gender from having the values of female and male to being an indicator variable called gendermale that takes a value of \(0\) for females and a value of \(1\) for males. (Such variables are often referred to as “dummy” variables.)

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

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

7. For two professors (one male and one female) who received the same beauty rating, the male professor is predicted to have the higher course evaluation score than the female. B
   1. True
   2. False

The decision to call the indicator variable gendermale instead ofgenderfemale has no deeper meaning. R simply codes the category that comes first alphabetically as a \(0\). (You can change the reference level of a categorical variable, which is the level that is coded as a 0, using therelevel function. Use ?relevel to learn more.)

Exercise: 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

8. Which of the following is the correct order of the three levels of rank if we were to order them from lowest predicted course evaluation score to highest predicted course evaluation score? C
   1. Teaching, Tenure Track, Tenured
   2. Tenure track, Tenured
   3. Tenure Track, Tenured, Teaching
   4. Teaching, Tenured, Tenure Track

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.