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12 resid2way - Residual Analysis for two-way ANOVA The...

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Residual Analysis for two-way ANOVA The twoway model with K replicates, including inter- action, is Y ijk = μ ij + ijk = μ + α i + β j + γ ij + ijk with i = 1 , . . . , I , j = 1 , . . . , J , k = 1 , . . . , K . In carrying out the F tests for interaction, and for the main effects of factors A and B, we have assumed that ijk are as sample from N (0 , σ 2 ) . Among other things, this means that: the distribution of the errors (and in particular, the variance σ 2 ) does not differ depending on the level of factor A, the level of factor B, or the mean of the response ( μ ij = μ + α i + β j + γ ij ) the errors are a sample from a normal distribution If these assumptions hold, then the p-values for the tests of interaction and main effects are valid. If the as- sumptions do not hold, then the p-values may substan- tially over- or under-estimate the evidence against the null hypotheses. Residuals are usually defined as the difference “data- prediction”. In the twoway anova model with interaction, the pre- dicted value of Y ijk is ˆ μ ij , and so the residuals are r ijk = Y ijk - ˆ μ ij = Y ijk - ¯ Y ij.
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(Another way of writing the residual for the twoway model with interaction is r ijk = Y ijk - ˆ μ - ˆ α i - ˆ β j - ˆ γ ij .) If the sample size is moderately large, the residuals should be approximately equal to the errors ijk , and so we use the residuals (which are known to us) in place of the errors
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