multiple_regression.Rmd

---
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
---

<div class="instructions">
Complete all **Exercises**, and submit answers to **Questions** on the Coursera 
platform.
</div>

## 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.

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

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

<div class="question">
 Is this an observational study or an experiment?

* Observational study 
* Experiment 
</div>

<div class="question">
 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? 

* No, the question is worded accurately. 
* Yes, revise wording to "Is there an association between beauty and course evaluations?" 
* Yes, revise wording to "Does beauty score increase the professor's course evaluations?"  
* Yes, revise wording to "Does beauty score decrease the professor's course evaluations?" 
</div>

## Exploring the data

<div class="question">
 Which of the following statements is **false** about the distribution of `score`? 

* The median of the distribution is 4.3. 
* 25\% of the students gave their professors a score of over 4.6.  
* 11 of students gave a professor a score below 3.  
* The left skewness of the data suggests that the students are less likely to rate the professors highly. 
</div>
```{r score-dist}
# type your code for Question 3 here, and Knit

```

<div class="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:

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

<div class="exercise">
**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?
</div>
```{r jitter}
# 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.

```{r scatter-score-bty_avg-line}
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`.

```{r scatter-score-bty_avg-line-no-se}
ggplot(data = evals, aes(x = bty_avg, y = score)) +
  geom_jitter() +
  geom_smooth(method = "lm", se = FALSE)
```

<div class="exercise">
**Exercise:** Print a summary of the linear model, write out the equation, and 
interpret the slope. 
</div>

```{r summary-model}
# type your code for the Exercise here, and Knit

```

<div class="question">
 Average beauty score is a statistically significant predictor of evaluation score. 
 
* True 
* False 
</div>

<div class="question">
 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? 

* Linear association: The residuals plot shows a random scatter. 
* Constant variance of residuals: No fan shape in residuals plot. 
* Nearly normal residuals: Residuals are right skewed, but the sample size is large, so this may not be an important violation of conditions. 
* Independent observations: Classes sampled randomly, no order effect. 
</div>

```{r diag-score-bty_avg}
# 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:

```{r bty-rels, fig.width = 12, fig.height= 5}
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.

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

<div class="question">
 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. 

* True 
* False 
</div>
</div>
```{r diag-mlr-model}
# type your code for the Exercise here, and Knit

```

<div class="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.

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

<div class="question">
 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.

* True 
* False 
</div>

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.)

<div class="exercise">
**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.
</div>

```{r new-mlr-model}
# type your code for the Exercise here, and Knit

```

<div class="question">
 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? 

* Teaching, Tenure Track, Tenured 
* Tenure track, Tenured 
* Tenure Track, Tenured, Teaching 
* Teaching, Tenured, Tenure Track 
</div>

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.
```{r new-prof}
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:
```{r new-prof-predict}
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.
```{r new-prof-predict-interval}
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.

<div id="exercise>
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.
</div>

Let's run the model...

```{r m_full, tidy = FALSE}
m_full <- lm(score ~ rank + ethnicity + gender + language + age + cls_perc_eval 
             + cls_students + cls_level + cls_profs + cls_credits + bty_avg 
             + pic_outfit + pic_color, data = evals)
summary(m_full)
```

<div id="exercise>
**Exercise:** Check your suspicions from the previous exercise. Include the model 
output in your response.
</div>

<div class="question">
 Which of the following is the correct intrepetation of the coefficient associated with the ethnicity variable. <br> Non-minority professors are expected on average to score ... 

* 0.12 points lower than minority professors, all else held constant. 
* 0.12 points higher than minority professors, all else held constant.  
* 0.02 points lower than minority professors, all else held constant. 
* 0.02 points higher than minority professors, all else held constant. 
</div>

<div class="exercise">
**Exercise:** 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?
</div>
```{r p-val-select}
# type your code for the Exercise here, and Knit

```

Now we try a different model selection method: adjusted $R^2$. Create a new model, 
`m1`, where you remove `rank` from the list of explanatory variables. Check out the 
adjusted $R^2$ of this new model and compare it to the adjusted $R^2$ of the full model.

```{r eval=FALSE}
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
```

Then, try dropping the next variable from the full model (`ethnicity`):

```{r eval=FALSE}
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
```

<div class="exercise">
**Exercise:** Repeat this process until you have tried removing each variable 
from the full model at a time, and determine removal of which variable yields 
the highest improvement in the adjusted $R^2$.
</div>

```{r adj-rsq-select}
# type your code for the Exercise here, and Knit

```

<div class="question">
 Elimination of which variable from the full model yielded the highest adjusted R-squared? 

* `bty_avg` 
* `cls_profs`  
* `cls_students` 
* `rank` 
</div>

To complete the model selection we would continue removing variables one at a 
time until removal of another variable did not increase adjusted $R^2$.

<div class="exercise">
**Exercise:** 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?
</div>

<div class="exercise">
**Exercise:** 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.
</div>

<div class="exercise">
**Exercise:** Would you be comfortable generalizing your conclusions to apply 
to professors generally (at any university)?  Why or why not?
</div>

<div id="license">
This is a product of OpenIntro that is released under a [Creative Commons Attribution-ShareAlike 3.0 Unported](http://creativecommons.org/licenses/by-sa/3.0). 
This lab was written by Mine &Ccedil;etinkaya-Rundel and Andrew Bray.
</div>

## References