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I had planned to perform a 3 x 2 repeated measures ANOVA before I realized that all the variables are distributed in a bimodal, U-shaped distribution where 0 and 1 are the modes. The high occurrence of 0 and 1 are meaningful to the analysis and therefore it may be inappropriate to transform the data.

Why is the variable bounded? The outcome variable is a gaze behavior. Zero indicates the absence of a behavior, and 1 indicates that the behavior is strong.

What values are within the variable? There are 0 and 1, and lots of values in between.

Considering that 1) the data violate ANOVA assumptions of normality of residuals; 2) I prefer not to transform the data; 3) sample size is small (n = 30, within-subjects), I am now planning to proceed with a non-parametric permutation test.

Would my understanding be correct?

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  • $\begingroup$ What is the outcome variable? Why is it bounded between 0 and 1? $\endgroup$
    – dipetkov
    Commented Apr 7, 2023 at 16:26
  • $\begingroup$ The outcome variable is a gaze behavior, calculated as = (looking time to Area1) / (looking time to Area1 + looking time to Area2). Zero indicates the absence of a behavior, and 1 indicates that the behavior is strong. $\endgroup$
    – Coloane
    Commented Apr 7, 2023 at 17:11
  • $\begingroup$ Does the outcome variable take exact 0 and/or 1 values? $\endgroup$
    – dipetkov
    Commented Apr 7, 2023 at 17:17
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    $\begingroup$ Thank you for the additional information. (It's best to edit the body of your question to include these details; they are quite relevant). In that case beta regression as suggested by @rep_ho sounds like an appropriate model to consider. See beta-regression. $\endgroup$
    – dipetkov
    Commented Apr 7, 2023 at 17:24
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    $\begingroup$ PS: Note that the histogram doesn't tell us if there are exact 0s and 1s, or just values that are very close to 0 and 1. $\endgroup$
    – dipetkov
    Commented Apr 7, 2023 at 17:26

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  1. Yes, the data are not suitable for ANOVA for the reasons you've mentioned
  2. Transformation won't help, because you will still have a hard limit at some value, so your days won't be normal anyway.
  3. Monte Carlo permutation test, or some rank based nonparametric test should be fine.
  4. You might also use parametric generalized linear models with an appropriate link function, maybe a beta regression.
  5. If your data are like that because of censoring, then you might need to do something else, but i don't know much about that
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