---
title: "Analysis of Variance"
author: "Renae L. Shrum"
output: 
  beamer_presentation: 
    colortheme: beaver
    fonttheme: professionalfonts
    theme: Szeged
---

## Anova Terms III
$t=\text{\# treatment groups}$  
$n_i=i^{th} \text{group sample size}$  
$N=\sum n_i=\text{total observations}$  
$s_i^2=i^{th} \text{treatment group variance}$  
$\bar y_{i\cdot}=i^{th} \text{treatment group mean}$  
$\bar y_{\cdot\cdot}=\text{grand mean}$  
$df_{Tr}=$ treatment degrees of freedom  
$df_E=$ Error degrees of freedom  
$df_{Total}=$ Total degrees of freedom  
$SSTr=$ Sum of squares treatment  
$SSE=$ Sum of squares error  
$TSS=$ Total sum of squares  
$MSTr=$ Mean square treatment  
$MSE=$ Mean square error  

## Anova Formulas
$df_{Tr}=df_1=t-1$  
$df_E=df_2=N-t$  
$df_{Total}=df_1+df_2=N-1$  
$\bar y_{\cdot\cdot}=\frac{\sum y_{ij}}{N}=\frac{\sum \bar y_{i\cdot}}{t}$  
$SSTR=\sum n_i(\bar y_{i\cdot}-\bar y_{\cdot\cdot})^2$  
$=n_1(\bar y_1-\bar y_{\cdot\cdot})^2+n_2(\bar y_2-\bar y_{\cdot\cdot})^2+\cdots+n_t(\bar y_t-\bar y_{\cdot\cdot})^2$  
$SSE=\sum(y_{ij}-\bar y_{i\cdot})^2=\sum s_i^2(n_i-1)$  
$=s_1^2(n_1-1)+s_2^2(n_2-1)+\cdots+s_t^2(n_t-1)$  
$TSS=SSTR+SSE$  
$MSTr=\frac{SSTr}{df_{Tr}}=\frac{SSTR}{t-1}$  
$MSE=\frac{SSE}{df_E}=\frac{SSE}{N-t}$  
$F=\frac{MSTr}{MSE}$

## Anova table

| Source of Variation | df   | SS   | MS   | F                | pvalue            |
| :-------------------|:-----|:-----|:-----|:-----------------|:------------------|
| Treatment           | $t-1$| SSTr | MSTr |$\frac{MSTr}{MSE}$|$P(F\geq F_{calc})$|
| Error               | $N-t$| SSE  | MSE  |                  |                   |
| Total               | $N-1$| TSS  |      |                  |                   |