When Unequal Sample Sizes Are and Are NOT a Problem in ANOVA

by Karen Grace-Martin

In your statistics class, your professor made a big deal about unequal sample sizes in one-way Analysis of Variance (ANOVA) for two reasons.

1. Because she was making you calculate everything by hand.  Sums of squares require a different formula if sample sizes are unequal, but SPSS (and other statistical software) will automatically use the right formula.

2. Nice properties in ANOVA such as the Grand Mean being the intercept in an effect-coded regression model don’t hold when data are unbalanced.  Instead of the grand mean, you need to use a weighted mean.  That’s not a big deal if you’re aware of it.

The only practical issue in one-way ANOVA is that very unequal sample sizes can affect the homogeneity of variance assumption.  ANOVA is considered robust to moderate departures from this assumption, but the departure needs to stay smaller when the sample sizes are very different.  According to Keppel (1993), there isn’t a good rule of thumb for the point at which unequal sample sizes make heterogeneity of variance a problem.

Real issues with unequal sample sizes do occur in factorial ANOVA, if the sample sizes are confounded in the two (or more) factors.  For example, in a two-way ANOVA, let’s say that your two independent variables (factors) are age (young vs. old) and marital status (married vs. not).  If there are twice as many young people as old and the young group has a much larger percentage of singles than the older group, the effect of marital status cannot be distinguished from the effect of age.

Power is based on the smallest sample size, so while it doesn’t hurt power to have more observations in the larger group, it doesn’t help either.

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