---
title: "Markdown example: multiple linear regression"
author: "Open Intro (modified by Nick Horton and Caroline Kusiak)"
date: "June 20, 2016"
output:
  pdf_document: default
  html_document:
    css: ../lab.css
    highlight: pygments
    theme: cerulean
  
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
---


```{r opts, include=FALSE}
require(mosaic)
trellis.par.set(theme=theme.mosaic()) # change default color scheme for lattice
```


## 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.



## 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.

```{r load-data, message=FALSE}
library(mosaic)
evals <- read.csv("http://www.amherst.edu/~nhorton/workshop/evals.csv",
  stringsAsFactors=FALSE)
```

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.
`age`            | age of professor.
`cls_students`   | total number of students in class.
`cls_level`      | class level: lower, upper.
`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.

## 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, can you rephrase the question?

SOLUTION: YOUR SOLUTION GOES 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?

```{r}
favstats(~ score, data=evals)
```

SOLUTION: YOUR SOLUTION GOES HERE.  Note that there are n=`r nrow(evals)` observations
in the dataset.


## 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, fig.height=2.7}
xyplot(score ~ bty_avg, data = evals)
```


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?

3.  Replot the scatterplot, but this time use the function `jitter()` on the
    $y$ or the $x$-coordinate. (Use `?jitter` to learn more.) What was
    misleading about the initial scatterplot?  (Note that this code block won't run until you change eval=FALSE to eval=TRUE.)

```{r eval=FALSE, fig.height=2.7}
xyplot(jitter(score) ~ jitter(bty_avg), data=evals)
```

4.  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 `type=c("p", "r")` argument or the `plotModel()` function. 
    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?

```{r}
m_bty <- lm(score ~ bty_avg, data = evals)
msummary(m_bty)
plotModel(mod = m_bty)
```


5.  Use residual plots to evaluate whether the conditions of least squares
    regression are reasonable. Provide plots and comments for each one.

```{r}
# mplot(m_bty, which=1)    # remove first comment character to run command
```


## Multiple linear regression

The data set contains several variables pertaining to a professor's beauty score: This is based on the average of six students' ratings. 

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)
msummary(m_bty_gen)
```

Let's also display this in a better way using the `xtable` package.

```{r,results="asis",echo=FALSE}
library(xtable)
options(xtable.comment = FALSE)
todisplay <- xtable(m_bty_gen)
print(todisplay)
```

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

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.

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

We can plot this data and the line corresponding to males with the
following code.

```{r twoLines, fig.height=2.8}
plotModel(m_bty_gen)
```

SOLUTION:

7.  What is the equation of the line corresponding to males? (*Hint:* For
    males, the parameter estimate is multiplied by 1.) For two professors
    who received the same beauty rating, which gender tends to have the
    higher course evaluation score?

SOLUTION:

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.)

8. 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`.

SOLUTION:


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, 
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.
Let's find the coefficients for the following model...

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

SOLUTION:

10. Interpret the coefficient associated with the ethnicity variable.

SOLUTION:

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

SOLUTION:

12. 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?

SOLUTION:

13. 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.

SOLUTION:

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

SOLUTION:

<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 (and updated by Caroline Kusiak and Nicholas Horton).
</div>

## References
