---
title: "Analysis of Deviance"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Analysis of Deviance}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

# Classic One-Way ANOVA

As an introduction, lets start with one way ANOVA. Here three random variables following a normal distribution with a common standard deviation are created. For this test, the null hypothesis is

$$ H_{0}: \mu_0 = \mu_1 = \mu_2  $$

```{r part1}
library(LRTesteR)

set.seed(123)
x <- c(
  rnorm(n = 50, mean = 1, sd = 1),
  rnorm(n = 50, mean = 3, sd = 1),
  rnorm(n = 50, mean = 5, sd = 1)
)
fctr <- c(rep(1, 50), rep(2, 50), rep(3, 50))
fctr <- factor(fctr, levels = c("1", "2", "3"))
gaussian_mu_one_way(x = x, fctr = fctr, conf.level = 0.95)
```
# Nonparametric One-Way ANOVA

One-way analysis without assuming the data is normally distributed.

```{r part2}
empirical_mu_one_way(x = x, fctr = fctr, conf.level = 0.95)
```

# Cauchy Random Variables
Here two random variables following a Cauchy distribution with a common location and different scales are created. For this test, the null hypothesis is 

$$ H_{0}: \gamma_0 = \gamma_1  $$

```{r part3}
set.seed(1)
x <- c(rcauchy(n = 50, location = 2, scale = 1), rcauchy(n = 50, location = 2, scale = 3))
fctr <- c(rep(1, 50), rep(2, 50))
fctr <- factor(fctr, levels = c("1", "2"))
cauchy_scale_one_way(x = x, fctr = fctr, conf.level = 0.95)
```



# Poisson Random Variables
Here three poisson random variables with different lambdas are created. The null hypothesis is 

$$ H_{0}: \lambda_0 = \lambda_1 = \lambda_2  $$

```{r part4}
set.seed(1)
x <- c(rpois(n = 50, lambda = 1), rpois(n = 50, lambda = 2), rpois(n = 50, lambda = 3))
fctr <- c(rep(1, 50), rep(2, 50), rep(3, 50))
fctr <- factor(fctr, levels = c("1", "2", "3"))
poisson_lambda_one_way(x = x, fctr = fctr, conf.level = 0.95)
```

# Mathematical Details

All one way tests have a null hypothesis the groups share a common value of the parameter. The alternative is at least one group's parameter is unequal to the others.  If the test involves a nuisance parameter, it is assumed equal across groups for parametric tests. All functions apply the Bonferroni correction to the set of confidence intervals.
