---
title: "SDM4 in R: Linear Regression (Chapter 7)"
author: "Nicholas Horton (nhorton@amherst.edu)"
date: "January 2, 2017"
output: 
  html_document:
    fig_height: 5
    fig_width: 7
  pdf_document:
    fig_height: 3
    fig_width: 5
  word_document:
    fig_height: 4
    fig_width: 6
---


```{r, include=FALSE}
# Don't delete this chunk if you are using the mosaic package
# This loads the mosaic and dplyr packages
require(mosaic)
```

```{r, include=FALSE}
# Some customization.  You can alter or delete as desired (if you know what you are doing).

# This changes the default colors in lattice plots.
trellis.par.set(theme=theme.mosaic())  

# knitr settings to control how R chunks work.
require(knitr)
opts_chunk$set(
  tidy=FALSE,     # display code as typed
  size="small"    # slightly smaller font for code
)
```

## Introduction and background 

This document is intended to help describe how to undertake analyses introduced 
as examples in the Fourth Edition of \emph{Stats: Data and Models} (2014) by De Veaux, Velleman, and Bock.
More information about the book can be found at http://wps.aw.com/aw_deveaux_stats_series.  This
file as well as the associated R Markdown reproducible analysis source file used to create it can be found at http://nhorton.people.amherst.edu/sdm4.

This work leverages initiatives undertaken by Project MOSAIC (http://www.mosaic-web.org), an NSF-funded effort to improve the teaching of statistics, calculus, science and computing in the undergraduate curriculum. In particular, we utilize the `mosaic` package, which was written to simplify the use of R for introductory statistics courses. A short summary of the R needed to teach introductory statistics can be found in the mosaic package vignettes (http://cran.r-project.org/web/packages/mosaic).

## Chapter 7: Linear Regression

### Section 7.1: Least squares: the line of best fit

Figure 7.2 (page 183) displays a scatterplot of the Burger King data with a superimposed regression line.

```{r message=FALSE}
library(mosaic); library(readr)
options(digits=3)
BK <- read_csv("http://nhorton.people.amherst.edu/sdm4/data/Burger_King_Items.csv")
names(BK)  
xyplot(Fat ~ Protein, ylab="Fat (gm)", xlab="Protein (gm)", type=c("p", "r"), data=BK)
```

We can calculate the residual for a particular value with 31 grams of protein.

```{r}
BKmod <- lm(Fat ~ Protein, data=BK)
BKfun <- makeFun(BKmod)
BKfun(31)    # predicted value for a item with 31 grams of protein
```

### Section 7.2 The linear model

```{r}
coef(BKmod)
BKfun(0)
BKfun(32) - BKfun(31)

### Section 7.3 Finding the least squares line

```{r}
sx <- sd(~ Protein, data=BK); sx
sy <- sd(~ Fat, data=BK); sy
r <- cor(Protein ~ Fat, data=BK); r   # same as cor(Fat ~ Protein)!
r*sy/sx
coef(BKmod)[2]
```


### Section 7.4 Regression to the mean

### Section 7.5 Examining the residuals

Figure 7.5 (page 193) displays the scatterplot of residuals as a function of the amount of protein.

The `msummary()` function generates a lot of output (much of which won't be familiar).
```{r}
xyplot(resid(BKmod) ~ Protein, type=c("p", "r"), data=BK)
msummary(BKmod)
```

The residual standard error of 10.6 grams matches the value reported on page 194.

### Section 7.6 R-squared: variation accounted for by the model

```{r}
rsquared(BKmod)
```

### Section 7.7 Regression assumptions and conditions