title: “Multiple linear regression” output: statsr:::statswithr_lab

references:

• id: Hamermesh2005 title: Beauty in the Classroom - Instructors' Pulchritude and Putative Pedagogical Productivity author:
  • family: Hamermesh given: Daniel S.
  • family: Parker given: Amy volume: 24 URL: 'http://www.sciencedirect.com/science/article/pii/S0272775704001165' DOI: 10.1016/j.econedurev.2004.07.013 publisher: Economics of Education Review ISSN: 0272-7757 issue: 4 page: 369-376 type: article-journal issued: year: 2005 month: 8
• id: Gelman2007 title: Data Analysis Using Regression and Multilevel/Hierarchical Models author:
  • family: Gelman given: Andrew
  • family: Hill given: Jennifer publisher: Cambridge University Press city: type: book issued: year: 2007 edition: 1 ISBN: 9780521686891 —

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” [@Hamermesh2005] 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)
library(GGally)

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 [@Gelman2007].) 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)

1. Is this an observational study or an experiment?

   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?

   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

1. Which of the following statements is false about the distribution of score?
   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.
   “{r score-dist} # type your code for Question 3 here, and Knit

<div id="exercise">
**Exercise:** Excluding `score`, select two other variables and describe their 
relationship using an appropriate visualization (scatterplot, side-by-side boxplots, 
or mosaic plot).
</div>
```{r two-vars-eda}
# type your code for the Exercise here, and Knit

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?

# type your code for the Exercise here, and Knit

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 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)

# type your code for the Exercise here, and Knit

1. Average beauty score is a statistically significant predictor of evaluation score.

   1. True
   2. False

2. 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?

   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.

   # type your code for the Question 5 here, and Knit
   

## 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.

```{r bty-rel}
ggplot(data = evals, aes(x = bty_f1lower, y = bty_avg)) +
  geom_jitter()
evals %>% 
  summarise(cor(bty_avg, bty_f1lower))

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)

1. 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.
   1. True
   2. False
   ”{r diag-mlr-model} # type your code for the Exercise here, and Knit

<div id="exercise">
**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`?
</div>
```{r summary-mlr-model}
# type your code for the Exercise here, and Knit

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.

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

1. 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.
   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.)

# type your code for the Exercise here, and Knit

1. 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?
   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.

Prediction

Suppose we want to use the model we created earlier, m_bty_gen to predict the evaluation score for a professor, Dr. Hypo Thetical, who is a male tenure track professor with an average beauty of 3.

If we wanted to do this by hand, we would simply plug in these values into the linear model.

We can also calculate the predicted value in R.

First, we need to create a new data frame for this professor.

newprof <- data.frame(gender = "male", bty_avg = 3)

Note that I didn't need to add rank = "tenure track" to this data frame since this variable is not used in our model.

Then, I can do the prediction using the predict function:

predict(m_bty_gen, newprof)

We can also construct a prediction interval around this prediction, which will provide a measure of uncertainty around the prediction.

predict(m_bty_gen, newprof, interval = "prediction", level = 0.95)

Hence, the model predicts, with 95% confidence, that a male professor with an average beauty score of 3 is expected to have an evaluation score between 3.1 and 5.18.

The search for the best model

We will start with a full model that predicts professor score based on rank, ethnicity, gender, language of the university where they got their degree, age, proportion of students that filled out evaluations, class size, course level, number of professors, number of credits, average beauty rating, outfit, and picture color.

# type your code for the Exercise here, and Knit

m1 <- lm(score ~ ethnicity + gender + language + age + cls_perc_eval 
             + cls_students + cls_level + cls_profs + cls_credits + bty_avg, data = evals)
summary(m1)$adj.r.squared

m2 = lm(score ~ rank + gender + language + age + cls_perc_eval + 
    cls_students + cls_level + cls_profs + cls_credits + bty_avg, data = evals)
summary(m2)$adj.r.squared

# type your code for the Exercise here, and Knit