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When conducting a t-test we can always use a Welch approximation of the df, and let go of the assumption of equal variances.

Is there something similar for (even one way) anova? (any R implementation of that?)

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The latest version of ez lets you pass a white.adjust argument to car::Anova(), which implements a correction for heteroscedasticity. See ?car::Anova() for details.

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There is a function named oneway.test() in the base stats package, which implements Welch correction for a one-way ANOVA. Its use is similar to the standard t.test() function. It is also referred to as O'Brien transformation (Biometrics 40 (1984), 1079--1087) and might be applied with two or more independent samples.

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