11.4 Chapter 11 Formulas

One-Way ANOVA

$H_{0}: \mu_{1} = \mu_{2} = \mu_{3} = \ldots = \mu_{k}$

f $H_{1}:$ At least one mean is different.

Source	$SS$ = Sum of Squares	$df$	$MS$ = Mean Square	F
Between (Factor)	$\sum n_{i} \left(\bar{x}_{i} - \bar{x}_{GM}\right)^{2}$	$k-1$	$MSB = \frac{SSB}{k-1}$	$F = \frac{MSB}{MSW}$
Within (Error)	$\sum \left(n_{i} - 1\right) s_{i}^{2}$	$N-k$	$MSW = \frac{SSW}{N-k}$
Total	SST	$N-1$

$\bar{x}_{i}$ = sample mean from the $i^{th}$ group

$n_{i}$ = sample size of the $i^{th}$ group

$k$ = number of groups

$s_{i}^{2}$ = sample variance from the $i^{th}$ group

$N = n_{1} + n_{2} + \ldots + n_{k}$

$\bar{x}_{GM} = \frac{\sum x_{i}}{N}$

Bonferroni Test

$H_{0}: \mu_{i} = \mu_{j}$

$H_{1}: \mu_{i} \neq \mu_{j}$

Bonferroni test statistic: $t = \dfrac{\bar{x}_{i} - \bar{x}_{j}}{\sqrt{ \left(MSW \left(\frac{1}{n_{i}} + \frac{1}{n_{j}}\right) \right)}}$

Multiply p-value by $m = {}_k C_{2}$ , divide area for critical value by $m = {}_k C_{2}$ .

Two-Way ANOVA

Row Effect (Factor A):	$H_{0}:$ The row variable has no effect on the average ___________________. $H_{1}:$ The row variable has an effect on the average ___________________.
Column Effect (Factor B):	$H_{0}$ : The column variable has no effect on the average ___________________. $H_{1}$ : The column variable has an effect on the average ___________________.
Interaction Effect (A×B):	$H_{0}:$ There is no interaction effect between row variable and column variable on the average ___________________. $H_{1}:$ There is an interaction effect between row variable and column variable on the average ___________________.

Source	$SS$	$df$	$MS$	F
$A$ (row factor)	$SS_{A}$	$a-1$	$MS_{A} = \frac{SS_{A}}{df_{A}}$	$F_{A} = \frac{MS_{A}}{MSE}$
$B$ (column factor)	$SS_{B}$	$b-1$	$MS_{B} = \frac{SS_{B}}{df_{B}}$	$F_{B} = \frac{MS_{B}}{MSE}$
$A \times B$ (interaction)	$SS_{A \times B}$	$(a-1)(b-1)$	$MS_{A \times B} = \frac{SS_{A \times B}}{df_{A \times B}}$	$F_{A \times B} = \frac{MS_{A \times B}}{MSE}$
Error (within)	$SSE$	$ab(n-1)$	$MSE = \frac{SSE}{df_{E}}$
Total	$SST$	$N-1$

Adapted from Mostly Harmless Statistics by Rachel Webb (Portland State University, https://mostlyharmlessstat.wixsite.com/webpage), © Rachel Webb, licensed under CC BY-SA 4.0. Changes were made. License: CC-BY-SA-4.0.