Statistical Methods in Medical Research 4ed

2 Given difference to be significant. We might require that, if x1 x2 is greater
in absolute value than some value d1, then it shall be significant at some
specified level (say at a two-sided 2a level). Denote by z2a the standardized
normal deviate exceeded (in either direction) with probability 2a (for
2a ˆ 0 05, z2a ˆ 1 96). Then
p
d1 > z2as …2=n†
or
n > 2 z2as
d1
2
: …4:38†
3 Given power against specified difference. Criterion 2 is defined in terms of a
given observed difference, d. The true difference, d, may be either less or
greater than d, and it seems preferable to base the requirement on the value of
d. It might be possible to specify a value of d, say d1, which one did not wish
to overlook, in the sense that if d > d1 one would like to get a significant
result at, say, the two-sided 2a level. However, a significant difference cannot
be guaranteed. Sampling fluctuations may lead to a value of jdj much less
than jdj and not significantly different from zero. The probability of this is
denoted by b and referred to as the Type II error rate, that is the probability
of failing to detect a real difference (false negative). The significance level is
referred to as the Type I error rate, that is the probability of incorrectly
rejecting the null hypothesis (false positive). While the Type I error rate is
controlled at a low value by choice of significance level during analysis, the
Type II error rate can only be controlled at the design stage. One might
specify that the Type II error rate be no greater than some low value, or
equivalently that the probability of correctly detecting the difference as
significant, 1 b, should be not less than some high value. This value is
called the power of the study.
The situation is represented in Fig. 4.9. Positive values of d are significant
at the stated level if
d > z2aSE…d†, …4:39†
that is, d is to the right of the point A. For a power > 1 b the point A must
be to the left of a point cutting off a one-sided probability of b on the righthand
distribution. That is,
or
z2aSE…d† < d1 z2bSE…d†
4.6 Sample-size determination 139
d1 > …z2a ‡ z2b†SE…d†: …4:40†