11.4: Chapter 11 Formulas
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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 | \(SST\) | \(N-1\) |