11.4: Chapter 11 Formulas

Mostly Harmless Statistics · LibreTexts

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    From LibreTexts, under CC BY-SA 4.0, as the source states it; its equations are written in TeX, which the PDF shows typeset. This page at LibreTexts.

    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 \(SST\) \(N-1\)    

    This page titled 11.4: Chapter 11 Formulas was last modified on Sun, 12 Mar 2023 17:54:58 GMT and is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Rachel Webb.