Statistical Methods in Medical Research 4ed

The 95% confidence limits for m1 m2 are
19 0 …2 110†…10 04†
ˆ 19 0 21 2
ˆ 2 2 and 40 2:
The range of likely values for m 1
m 2 is large. If the experimenter feels dissatisfied with
this range of uncertainty, the easiest remedy is to repeat the experiment with more
observations. Higher values of n1 and n2 will tend to decrease the standard error of
x1 x2 and hence increase the precision of the comparison.
It might be tempting to apply the unpaired method to paired data. This
would be incorrect because systematic differences between pairs would not be
eliminated but would form part of the variance used in the denominator of the t
statistic. Thus, using the unpaired method for paired data would lead to a less
sensitive analysis except in cases where the pairing proved ineffective.
Unequal variances
In other situations it may be either clear that the variances differ considerably or
prudent to assume that they may do so. One possible approach, in the first case,
is to work with a transformed scale of measurement (§10 8). If the means, as well
as the variances, differ, it may be possible to find a transformed scale, such as the
logarithm of the original measurement, on which the means differ but the
variances are similar. On the other hand, if the original means are not too
different, it will usually be difficult to find a transformation that substantially
reduces the disparity between the variances.
In these situations the main defect in the methods based on the t distribution
is the use of a pooled estimate of variance. It is better to estimate the standard
error of the difference between the two means as
s
SE…x1 x2† ˆ
s 2 1
n1
‡ s2 2
n2
A significance test of the null hypothesis may be based on the statistic
d ˆ x1 x2
s ,
s 2 1
n1
4.3 Comparison of two means 109
‡ s2 2
n2
which is approximately a standardized normal deviate if n1 and n2 are reasonably
large. Similarly, approximate 100 (1 a)% confidence limits are given by
x1 x2 zaSE…x1 x2†,
where za is the appropriate standardized normal deviate corresponding to the
two-sided probability a.
: