How To Do Anova With Unequal Sample Size

In your statistics form, your professor made a big bargain about unequal sample sizes in i-manner Analysis of Variance (ANOVA) for two reasons.

i. Because she was making yous summate everything by hand. Sums of squares crave a different formula* if sample sizes are diff, but statistical software will automatically use the right formula. So we're not as well concerned. We're definitely using software.

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

But there are a few real issues with diff sample sizes in ANOVA. They don't invalidate an analysis, but it's important to be aware of them as you lot're interpreting your output.

Two Practical Issues for Unequal Sample Sizes in One-Way ANOVA

1. Supposition Robustness with Unequal Samples

The main practical issue in one-way ANOVA is that unequal sample sizes impact the robustness of the equal variance assumption.

ANOVA is considered robust to moderate departures from this assumption. Merely that's not true when the sample sizes are very unlike. According to Keppel (1993), there is no skillful dominion of thumb for how unequal the sample sizes need to be for heterogeneity of variance to exist a problem.

And so if you have equal variances in your groups and diff sample sizes, no trouble. If you accept unequal variances and equal sample sizes, no trouble.

The only problem is if you take unequal variances and diff sample sizes.

two. Ability with Unequal samples

The statistical power of a hypothesis test that compares groups is highest when groups have equal sample sizes.

Ability is based on the smallest sample size, so while it doesn't injure power to accept more observations in the larger grouping, it doesn't help either.

So if you have a specific number of individuals to randomly assign to groups, you lot'll have the most ability if you assign them every bit.

If your grouping is a natural one, you're not making decisions based on a total number of individuals. Information technology's very common to just happen to go a larger sample of one group compared to the others.

That doesn't bias your exam or requite you wrong results. It just means the power you have is based on the smaller sample.

So if yous take xxx individuals with Treatment A and 40 individuals with Handling B and 300 controls, that'south fine. Information technology's but that you could have stopped with 30 controls. The extra 270 didn't assist the power of this particular exam.

Yes, this all holds true for independent samples t-tests

Independent samples t-tests are essentially a simplificiation of a i-way ANOVA for only two groups. In fact, if you lot run your t-exam as an ANOVA, you'll go the aforementioned p-value. And the between-groups F statistic will be the square of the t statistic yous got in your t-test.

(Really, endeavour it…. pretty cool, huh?)

This means they piece of work the same way. Unbalanced t-tests have the same applied issues with unequal samples, but it doesn't otherwise bear on the validity or bias in the test.

Issues in Factorial ANOVA

Factorial ANOVA includes all those ANOVA models with more than than one crossed cistron. It generally involves 1 or more interaction terms.

Real issues with unequal sample sizes do occur in factorial ANOVA in one situation: when the sample sizes are confounded in the ii (or more) factors. Permit'south unpack this.

For example, in a two-way ANOVA, allow'southward say that your two contained variables (factors) are Historic period (young vs. former) and Marital Status (married vs. not).

Let's say at that place are twice every bit many immature people as quondam. So unequal sample sizes.

And say the younger group has a much larger percentage of singles than the older group. In other words, the two factors are not independent of each other. The effect of marital status cannot be distinguished from the upshot of age.

So yous may get a large mean difference between the marital statuses, only it'due south really existence driven by age.

What nigh Chi Square Tests?

(This article is near ANOVA (and t-tests), merely I've updated to include Chi-Square tests after getting a lot of questions).

There are a number of unlike chi-square tests, only the two that can seem concerning in this context are the Chi-Square Test of Independence and The Chi-Foursquare Test of Homogeneity. Both have two categorical variables. Both count the the frequencies of the combinations of these categories.

They calculate the test statistic the same mode. Without getting into the math, it'south basically a comparison of the actual frequencies of the combinations with the frequencies you'd expect under the null hypothesis.

And luckily, unequal sample sizes do not bear on the ability to summate that chi-square test statistic. Information technology's pretty rare to have equal sample sizes, in fact. The expected values take the sample sizes into business relationship. Then no problems at all here.

That said, when there is a third variable involved, y'all can have an effect with Simpson's Paradox. You may or may not accept collected that third variable, so information technology's worth thinking well-nigh whether at that place could be something else that is creating an association in a combination of two groups of that third variable that doesn't exist in each group alone.

But that's not really an upshot with unequal sample sizes. That's an issue of omitting an of import variable from an assay.

Updated Dec 18, 2020 to add more detail

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