Two-way analysis of variance: Difference between revisions

Revision as of 23:36, 15 November 2021

Revision as of 15:58, 19 December 2021

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Specifically, the mean of the response variable is modeled as a [[linear combination]] of the explanatory variables:

<math>\mu_{ijkij} = \mu + \alpha_i + \beta_j + \gamma_{ijkij}</math>,

where <math>\mu</math> is the grand mean, <math>\alpha_i</math> is the additive main effect of level <math>i </math> from the first factor (''i''-th row in the contigency table), <math>\beta_j</math> is the additive main effect of level <math>j</math> from the second factor (''j''-th column in the contingency table) and <math>\gamma_{ijkij}</math> is the non-additive interaction effect of treatment <math>(i,j)</math> for samples <math>k=1,...,n_ij</math> from both factors (cell at row ''i'' and column ''j'' in the contingency table).

Another equivalent way of describing the two-way ANOVA is by mentioning that, besides the variation explained by the factors, there remains some [[statistical noise]]. This amount of unexplained variation is handled via the introduction of one random variable per data point, <math>\epsilon_{ijk}</math>, called [[Errors and residuals in statistics|error]]. These <math>n</math> random variables are seen as deviations from the means, and are assumed to be independent and normally distributed: