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I have data for three unequal groups: $N = 44$, $N = 354$ and $N = 347$. Is it possible to compare all three groups running a one-way ANOVA or is the first group too small?

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The unevenness of sample size of itself is not an issue when the assumptions are satisfied. The test is still completely valid.

However, it does reduce level-robustness to heteroskedasticity substantially - if the sample sizes are equal (or very close to it) then the test is robust to heteroskedasticity (at least, in that the level isn't much affected). With very different sample sizes, the Welch adjustment to degrees of freedom is a safer choice.

If you have the ability to choose the sample sizes, power is better when the sample sizes are equal or nearly equal. See gung's answer here for details on this issue.

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