ANOVA – Are Unequal Groups a Problem for One-Way ANOVA?

anovaunbalanced-classes

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?

Best Answer

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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ANOVA – Alternatives to One-Way ANOVA When Dealing With Unequal Variance

@JeremyMiles is right. First, there's a rule of thumb that the ANOVA is robust to heterogeneity of variance so long as the largest variance is not more than 4 times the smallest variance. Furthermore, the general effect of heterogeneity of variance is to make the ANOVA less efficient. That is, you would have lower power. Since you have a significant effect anyway, there is less reason to be concerned here.

Update:

I demonstrate my point about lower efficiency / power here: Efficiency of beta estimates with heteroscedasticity
I have a comprehensive overview of strategies for dealing with problematic heteroscedasticity in one-way ANOVAs here: Alternatives to one-way ANOVA for heteroscedastic data

Solved – post hoc test for 3 way anova

Easiest may be to perform regression where you can choose whichever interactions you want:

lm(y ~ temperature + quality + gender, mydata)

or:

lm(y ~ temperature*quality + gender, mydata)

or:

lm(y ~ temperature*quality*gender, mydata)

or:

lm(y ~ temperature*quality*gender - quality*gender, mydata)

Best Answer

Related Solutions

ANOVA – Alternatives to One-Way ANOVA When Dealing With Unequal Variance

Solved – post hoc test for 3 way anova

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