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I want to run a Two-Way ANOVA test on an experiment which has a two-by-three configuration of experimental conditions, and one hundred data for each experimental condition, hence, 600 total data points.

None of these data, per Shapiro-Wilk, allow us to accept the hypothesis of normality, so I want to run something like a Box-Cox or a Yeo-Johnson to achieve normality. (In fact, it will be Yeo-Johnson, because some of these data have meaningful zero values and not negative values, but I think my question generalizes to both techniques.)

Does one perform six separate Box-Cox (or Yeo-Johnson) transformations, one for the data of each experimental condition? Or does one combine the data into a single set, perform a single transform, and then split the data back into six sets?

My initial intuition was strongly the former, since six Shapiro-Wilk tests are performed, yes? But on inspecting the results of the Yeo-Johnson transformations, I saw considerable changes in the variances-- in one case an extreme change-- which did not seem to track one another, so I could no longer support the "approximately equal variances" assumption for the transformed data.

Am I missing something, or will these transforms sometimes help to satisfy one assumption by sacrificing another?

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    $\begingroup$ Some points that I think relate more to premises of the question -- 1. Monotonic transformations might move your spike at zero about, but it will stay together. 2. If you want to interpret interaction terms (as is generally the case with two way ANOVA) this will be quite tricky in the presence of a nonlinear transformation when you want to transform back. $\endgroup$
    – Glen_b
    Nov 5 '21 at 3:36
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Using statistical tests to decide on how to do the final analysis is generally a bad idea. Tests do not have power of 1.0 for detecting non-normality, and then there is the absence of evidence is not evidence of absence problem.

Transforming different subsets of Y differently is a recipe for not being able to interpret any of the results.

Transformations such as Box-Cox are not flexible enough and are highly dependent on zero being a special origin for the measurements.

Normality is assessed on model residuals, not on raw data.

If you want to use a robust and powerful model that is invariant to transformations of Y and resistent to outliers, use a semiparametric ordinal regression model, which will also handle interaction elegantly. See examples in the Nonparametrics chapter of BBR.

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