#### Unequal sample sizes (2004-11-24)

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For some reasons, it seems to unnerve people when the sample size in the treatment and
control group are not the same. They worry about whether the tests that they would run on the
data would be valid or not. As a general rule, there is no reason that you cannot analyze
data with unequal sample sizes. There are a few procedures, such as the Tukey follow-up test
in Analysis of Variance, that require some adaptations. You also may not have as much
precision as you'd like to have. The rule of thumb is that precision is determined
predominantly by the group with the smaller sample size. Finally, some violations of
assumptions (such as unequal variances) may be more serious when the samples sizes are
unequal.

There are some situations where you might deliberately recruit unequal sample sizes. This
occurs most commonly when it costs much less to recruit patients from one group than from
another. Suppose the treatment group requires ten thousand dollars worth of therapy and the
control group only requires a hundred dollars worth of therapy. For a fixed budget, you would
be better off recruiting more control patients. With a bit of calculus you can show that the
optimal degree of unbalancedness is related to the square root of the ratio of costs.