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My stats questions concerns the appropriateness of using different transformations of the same variable within different analyses (within the same larger study).

My research involves emotional expression as a dependent variable. I'm looking at 1) baseline differences between healthy controls (HCs) and individuals with a disease and 2) treatment effects within the disease group (Treatment Groups: HC-no-treatment, Disease-no-treatment, Disease-treatment 1, Disease-treatment 2). Each subject was assessed at baseline, post-treatment, and follow-up.

Research Aim 1. Baseline Differences Between Health Status Groups (Healthy Controls and Disease).
When looking at the emotion expression data, at baseline, by health status group (2: healthy controls and Disease), the raw data is not normally distributed (based on Shapiro-Wilk). Because square root transformation normalizes the distribution, I was planning on using a t-test (as opposed to Mann-Whitney) to assess baseline differences between the healthy control and disease groups.

Research Aim 2. Treatment Effects. When looking at the emotion expression data, by treatment group (HC, disease-no-treatment, disease-treatment1, disease-treatment2) at each time (baseline, post, follow-up), the raw is not normally distributed (based on Shapiro-Wilk). For these groupings, the square root transformation does NOT normalize the distribution. However, a log10 transformation does normalize the distribution.

Main Question: Can I use a square root transformation for Aim 1 and a different transformation for Aim 2 (e.g.,log10 in a mixed model ANOVA, or raw data if I end up using nonparametric stats for Aim 2)?

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    $\begingroup$ "the raw data is not normally distributed" - why would this matter? What assumes raw data is normal? $\endgroup$
    – Glen_b
    Mar 21 '13 at 23:07
  • $\begingroup$ By "raw data" I meant dependent variable, without applying any transformation. Does the t-test not assume normality of the dependent variable? $\endgroup$
    – dav
    Apr 9 '13 at 20:45
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    $\begingroup$ It assumes the distribution of the dependent variable in the populations are conditionally normal - what is it that is being checked for normality? What kind(s) of non-normality do you observe? Note that when you transform the data, you're no longer estimating differences in means. Even if the t-test were unsuitable, depending on what you want to achieve you may be better off doing something other than transformation. $\endgroup$
    – Glen_b
    Apr 9 '13 at 21:16
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You should probably do something else than a transformation. You might do transformations as you describe, it is not necessarily an error, but, depending on your audience, it might be hard to defend. Further:

  • Can you interpret difference from baseline on a square root scale? Can your interpret it? As an alternative, you could use bootstrap for tests or confidence intervals.

  • Will treatment effects be interpretable on the log scale? ... Better to use an interpretable scale, and then maybe bootstrap for tests/confidence intervals.

See also the comments under the question.

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