124 BIOLOGICAL EFFECTS OF LOW LEVEL EXPOSURESon the numerical values of p0 and p,, even when these are equal. For
example, the probability of an H or L result is zero when the study condi
tions impose a zero disease rate for both groups. It is also zero when the
disease is universally prevalent in both groups.
Thus, in order to calculate the probability of a study outcome (a,c) land
ing in the H, N, or L zones, it is necessary to specify the rates Pj and p0 as
well as the corresponding sample sizes and n0. Since a and c are indepen
dent and binomially distributed, the probability of any particular pair (a,c)
in the ac rectangle is the product of the corresponding binomial
probabilities:
where s = n, - a and t = n0 - c. The probability that a study outcome pair
falls in the L region, say, is then the sum of all such probabilities over all
pairs (a,c) in the L region.
The probabilities of a study result falling in the L region have been
calculated for a variety of scenarios and appear in Table 7.3 (2.5% in each
tail) and Table 7.4 (5% in each tail), rounded to the nearest percent. The
probabilities along the diagonals correspond to L study results when, in
fact, the actual effect of the substance is N (so such a result is a FL). These
probabilities, all less than 2.5% by design, vary in Table 7.3A from 0%
when Pi = p0 = 0.05 to 2% when Pi = p0 = 0.50. The probabilities above
the diagonal, all much smaller still, correspond to L study results when, in
fact, the substance is harmful (FL again). The probabilities below the diago
nal correspond to L study results when, indeed, the substance is beneficial
(TL).
The percents below the diagonal are thus the sensitivities for detecting a
true beneficial effect. They increase as the difference p0 - P! increases, and
for fixed p0 - tend to increase as P! and p0 both decrease. The tables show
the probability of a true beneficial result increases with sample size. The
diagonal entries in the tables tend to approach their nominal values of 2.5
and 5% as the sample size increases, because the larger samples lessen the
effect of the discreteness of the observations. For a fixed total sample size,
maximum sensitivities for detecting true H or L effects are achieved when
the sample sizes n, and n0 are equal.
All the above analyses have been done under the assumption that the
sample sizes n, and n0 for the exposed and nonexposed groups were deter
mined in advance of the actual study. The analyses are actually applicable to
a much wider class of studies and hold when the sample sizes m, and m0 for
disease cases and noncases are determined in advance, instead of the num
bers of exposed and unexposed subjects, because the chi-square analysis is
appropriate for both cases. Instead of ac rectangles for cases there are now
ab rectangles for exposed subjects. And instead of the null hypothesis being