Statistical Methods in Medical Research 4ed

138 Analysing means and proportions
be made? Other things being equal, the greater the sample size or the larger the
experiment, the more precise will be the estimates of the parameters and their
differences. The difficulty lies in deciding what degree of precision to aim for. An
increase in the size of a survey or of an experiment costs more money and takes
more time. Sometimes a limit is imposed by financial resources or by the time
available, and the investigator will wish to make as many observations as the
resources permit, allowing in the budget for the time and cost of the processing
and analysis of the data. In other situations there will be no obvious limit, and
the investigator will have to balance the benefits of increased precision against
the cost of increased data collection or experimentation. In some branches of
technology the whole problem can be looked at from a purely economic point
of view, but this will rarely be possible in medical research since the benefit of
experimental or survey information is so difficult to measure financially. The
determination of sample size is thus likely to be an adaptive process requiring
subjective judgement. The balancing of precision against availability of resources
may take place by trial and error, until a solution is found which satisfies both
requirements. In many situations there may be a range of acceptable solutions,
so that a degree of arbitrariness remains in the choice of sample size.
In any review of these problems at the planning stage it is likely to be important
to relate the sample size to a specified degree of precision. We shall consider first
the problem of comparing the means of two populations, m1 and m2, assuming that
they have the same known standard deviation, s, and that two equal random
samples of size n are to be taken. If the standard deviations are known to be
different, the present results may be thought of as an approximation (taking s2 to
be the mean of the two variances). If the comparison is of two proportions, p1 and
p2, s may be taken approximately to be the pooled value
f1 2 p1…1
p
‰ p1†‡p2…1 p2† Šg:
We now consider three ways in which the precision may be specified.
1 Given standard error. Suppose it is required that the standard error of the
difference between the observed means, x1 x2, is less than e; equivalently
the width of the 95% confidence intervals might be specified to be not wider
than 2e. This implies
p
s …2=n† < e
or
n > 2s 2 =e 2 : …4:36†
If the requirement is that the standard error of the mean of one sample shall
be less than e, the corresponding inequality for n is
n > s 2 =e 2 : …4:37†