---
title: "SDM4 in R: Analysis of Variance (Chapter 26)"
author: "Nicholas Horton (nhorton@amherst.edu) and Sarah McDonald"
date: "June 13, 2018"
output: 
  pdf_document:
    fig_height: 2.8
    fig_width: 7
  html_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)
options(digits = 5)
```

```{r, include = FALSE}
# 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 *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).
A paper describing the mosaic approach was published in the *R Journal*: https://journal.r-project.org/archive/2017/RJ-2017-024.

## Chapter 26: Analysis of variance

### Section 26.1: Testing whether the means of several groups are zero

The graph in Figure 26.1 (page 747) can be generated using the `bwplot()` function.

```{r}
Soap <- read.csv("http://nhorton.people.amherst.edu/sdm4/data/Bacterial_Soap.csv")
gf_boxplot(Bacterial.Counts ~ Method, data = Soap)
```

The example on page 750 considers the outcomes in hand volumes for three treatments
post surgery.
```{r warning = FALSE}
Contrast <- read.csv("http://nhorton.people.amherst.edu/sdm4/data/Contrast_baths.csv")
gf_boxplot(Hand.Vol.Chg ~ Treatment, data = Contrast)
```

The summary statistics at the bottom of page 751 can be calculated using `favstats()`.

```{r}
favstats(Bacterial.Counts ~ Method, data = Soap)
```

### Section 26.2: The ANOVA table

The `aov()` function can be used to fit an analysis of variance model.

```{r}
aovmod <- aov(Bacterial.Counts ~ Method, data = Soap)
summary(aovmod)
```

This model has 3 degrees of freedom for the model (numerator) and 28 degrees of
freedom for the error (denominator).  The `xpf()` function can replicate the calculation of the exact p-value (and generate Figure 26.4, page 754).

```{r}
xpf(7.0636, df1 = 3, df2 = 28)
```


The treatment means can be generated using `model.tables()` (see page 757).

```{r}
model.tables(aovmod)
```

The residual standard deviation can be calculated (page 759).

```{r}
n <- 32
k <- 4
sp <- sqrt(sum(resid(aovmod)^2/(n-k)))
sp
sqrt(1410)
```

We can also see how the results are equivalent when fitting a regression model
with indicators.

```{r}
lmmod <- lm(Bacterial.Counts ~ Method, data = Soap)
msummary(lmmod)
```