Statistical Methods in Medical Research 4ed

Unequal-sized groups
It is usually optimal when comparing two groups to have equal numbers in each
group. But sometimes the number available in one group may be restricted, e.g.
for a rare disease in a case±control study (§19.4). In this case the power can be
increased to a limited extent by having more in the other group. If one group
contains m subjects and the other rm, then the study is approximately equivalent
to a study with n in each group where
That is,
2 1 1
ˆ ‡
n m rm :
…r ‡ 1†n
m ˆ : …4:43†
2r
This expression is derived by equating the expressions for the standard error of
the difference between two means used in a two-sample t test, and is exact for a
continuous variable and approximate for the comparison of two proportions.
Fleiss et al. (1980) give a formula for the general case of comparing two proportions
where the two samples are not of equal size, and their formula is suitable
for the inverse problem of estimating power from known sample sizes.
The total number of subjects in the study is
which is a minimum for r ˆ 1.
Other considerations
…r ‡ 1† 2
4r
4.6 Sample-size determination 145
The situations considered in this section are relatively simpleÐthose of comparing
two groups without any complicating features. The determination of sample
size is often facilitated by the use of tables and, in addition to those mentioned
earlier, the book by Lemeshow et al. (1990) contains a number of sets of tables.
Even in these simple situations it is necessary to have a reasonable idea of the
likely form of the data to be collected before sample size can be estimated. For
example, when comparing means it is necessary to have an estimate of the
standard deviation, or in comparing proportions the approximate size of one
of the proportions is required at the planning stage. Such information may be
available from earlier studies using the same variables or may be obtained from a
pilot study. In more complicated situations more information is required but
(4.41) can be used in principle, provided that it is possible to find an expression
for s, the standard deviation relevant to the comparison of interest.
2n,