```{r global_options, include=FALSE}
knitr::opts_chunk$set(eval = TRUE, message = FALSE, warning = FALSE)
```

## 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.

```{r load-packages, message=FALSE}
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`.

```{r}
glimpse(evals)
```

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:

```{r help-evals, eval=FALSE}
?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.

**Insert your answer here**

2.  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?

**Insert your answer here**

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

**Insert your answer here**

## 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:

```{r scatter-score-bty_avg}
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?

4.  Replot the scatterplot, but this time use `geom_jitter` as your layer. What 
    was misleading about the initial scatterplot?

```{r scatter-score-bty_avg-jitter}
ggplot(data = evals, aes(x = bty_avg, y = score)) +
  geom_jitter()
```

**Insert your answer here**

5.  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**

Add the line of the bet fit model to your plot using the following:
    
```{r scatter-score-bty_avg-line-se}
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`.

```{r scatter-score-bty_avg-line}
ggplot(data = evals, aes(x = bty_avg, y = score)) +
  geom_jitter() +
  geom_smooth(method = "lm", se = FALSE)
```

6.  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).

**Insert your answer here**

## 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_point()

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. You can actually look at the relationships between all beauty variables (columns 13 through 19) using the following command:

```{r bty-rels}
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.

```{r scatter-score-bty_avg_pic-color}
m_bty_gen <- lm(score ~ bty_avg + gender, data = evals)
summary(m_bty_gen)
```

7.  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.

**Insert your answer here**

8.  Is `bty_avg` still a significant predictor of `score`? Has the addition
    of `gender` to the model changed the parameter estimate for `bty_avg`?

**Insert your answer here**

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.

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

<!-- We can plot this line and the line corresponding to those with color pictures
with the following  -->
<!-- custom function. -->

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

9.  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?

**Insert your answer here**

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 the`relevel()` function. Use `?relevel` to learn more.)

10. 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`.

**Insert your answer here**

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.

11. 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.

**Insert your answer here**

Let's run the model...

```{r m_full, tidy = FALSE}
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)
```

12. Check your suspicions from the previous exercise. Include the model output
    in your response.

**Insert your answer here**

13. Interpret the coefficient associated with the ethnicity variable.

**Insert your answer here**

14. 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?

**Insert your answer here**

15. 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.

**Insert your answer here**

16. Verify that the conditions for this model are reasonable using diagnostic 
    plots.

**Insert your answer here**

17. 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?

**Insert your answer here**

18. 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.

**Insert your answer here**

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

**Insert your answer here**


* * *
