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I have an experiment design with one "within subject" factor and one "between-subject" factor. So I have to use mixed-model ANOVA, but empirical distributions are not approximated by any good family. Thus I have to implement some nonparametric version of ANOVA. The only one is known to me is due to Brunner and Langer[1], with zero hypothesis on equivalence of marginal distrbutions. I would like to compare medians differences and report something related to them as effect size. As far as I understand, after Brunner and Langer ANOVA I should use stochastic dominance posthoc test and the effect size related to it. Thus can't say anything about medians.

Is there any nonparametric version of ANOVA with hypothesis on median?

Could I make some posthoc test on median differences after Brunner and Langer ANOVA?

[1] Brunner E, Domhof S, Langer F (2002). Nonparametric Analysis of longitudinal Data in Factorial Experiments. John Wiley & Sons, New York.

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The usual ANOVA is an OLS regression model that estimates the mean of conditional distributions. The counterpart to OLS that estimates conditional medians is quantile regression for the median, which uses absolute loss instead of square loss to estimate the parameters. If your interest is explicitly in the medians, this seems like a viable path forward.

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  • $\begingroup$ Thank you! Excuse me for not being precise enough. I can try to estimate mean also. The essential problem is that the conditional distributions doesn't fit to normal or any other known distribution. (What's worse variances of conditional distributions are essentially different) Does the quantile regression solves the problem of nonparametricity as well? $\endgroup$
    – Vash
    Jun 29 at 17:34
  • $\begingroup$ @Vash Then it sounds like you might be interested in an extension of Kruskal-Wallis called proportional odds ordinal regression. $\endgroup$
    – Dave
    Jun 29 at 17:50
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Quantile regression has been mentioned. Another viable option may be aligned ranks transformation anova (ART anova). It's a nonparametric approach. In some software implementations it can handle fixed and random effects and post-hoc tests.

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