<p>This is a follow-up question I have after reviewing this post:&nbsp; <a href="https://stats.stackexchange.com/questions/336459/difference-in-means-statistical-test-for-non-normal-heteroscedastic-data?noredirect=1#comment635980_336459" target="_blank">Difference in means statistical test for non-normal, heteroscedastic data?</a>.</p>
<p>To be clear, I am asking from a pragmatic perspective (not to suggest that theoretical responses are not welcome).&nbsp; When normality among groups <em>is</em> present (different from the title of the question referenced above), but the group variances are substantively different, ¿what is the worst that a researcher might observe?</p>
<p>In my experience, the issue that arises the most with this scenario is &ldquo;strange&rdquo; patterns in the <em>post hoc</em> comparisons.&nbsp; (This has been observed both in my published work, but also in pedagogic settings...happy to provide details of this in the comments below.)&nbsp; What I have observed is something akin to this:&nbsp; You have three groups with $M_1 < M_2 < M_3$.&nbsp; The (omnibus) ANOVA gives $p<\alpha$, and the pairwise $t$-tests suggest $M_2$ is statistically significantly different from the other two groups...but $M_1$ and $M_3$ are not statistically significantly different.&nbsp; Part of my question is if this is what others have observed, but also, ¿what other issues have you observed with comparable scenarios?</p>
<p>A quick review of my reference texts suggest ANOVA is rather robust to mild to moderate violations of the homoscedasticity assumption, and even more so with large sample sizes.&nbsp; However, these references do not specifically state (1)&nbsp;what could go wrong or (2)&nbsp;what might happen with a large number of groups.</p>
<p>As always, thanks for any insights from personal experience that you can share.</p>