Statistical Methods in Medical Research 4ed

144 Analysing means and proportions
z2b ˆ 0 540:
From Table A1 this corresponds to a power of
1 0 2946 ˆ 0 7054 or 71%:
(b) What size of mean difference between the groups could be detected with 80% power?
Substituting in (4.41) with d1 as unknown gives
50 ˆ 2
…1 96 ‡ 0 842†…0 5†
and therefore d1 ˆ 0 28. A mean difference of 0 28 l could be detected.
Note that, if the sample size has already been determined for a specified d1,
the revised value is quickly obtained by noting that the difference that can be detected is
proportional to the reciprocal of the square root of the sample size. Thus, since for
d1 ˆ 0 25 the sample size was calculated as 62 8, then for a sample size of 50 we have
d1 ˆ 0 25 …
p 62 8=50† ˆ0 28.
For the inverse problem of determining the power or the size of effect that
could be detected with a given sample size, the formulae (4.41) and (4.42) can be
used but the tables are less convenient. Since it is often required to consider what
can be achieved with a range of sample sizes, it is convenient to be able to
calculate approximate solutions more simply. This can be achieved by noting
that, in (4.41), d1 is proportional to 1= n
p and z2a ‡ z2b is proportional to n
p .
These relationships apply exactly for a continuous variable but also form a
reasonable approximation for comparing proportions.
Example 4.18
In Example 4.15 suppose only 600 patients are available so that n ˆ 300.
(a) What size of difference could be detected with 90% power?
Approximately
p
d1 ˆ 0 10 …459 4=300†
ˆ 0 124:
Using (4.42) gives dt ˆ 0 126 so the approximation is good. An increase in success rate
to about 37 6% could be detected.
(b) What would be the power to detect a difference of 0 1?
Approximately
p
1 96 ‡ z2b ˆ…1 96 ‡ 1 282† …300=459 4†
:
ˆ 2 620
Therefore z2b ˆ 0 660 and the revised power is 74.5%.
Using (4.42) gives z2b ˆ 0 626 and a revised power of 73 4%.
d1
2
,