122 BIOLOGICAL EFFECTS OF LOW LEVEL EXPOSURESAssume the number of subjects in each group is 25: n0 = nj = 25. Setting
the expression for chi-square equal to 2.706:
determines, for any value of c, those integral values of a for which the study
result (a,c) will just be statistically significantly high or low, given the
assumption that the null hypothesis, Pj = p0, is true.
Assigning a value of zero to c in the equation yields two quadratic equa
tions in a —one with the odds ratio lower than unity, indicating some benefi
cial effect, and the other with the odds ratio exceeding unity, a harmful
effect. The first equation has only imaginary solutions, indicating that there
is no possible value for a that could significantly improve on an observed
rate of zero in the nonexposed group. The second equation has the admissi
ble solution a = 5, indicating that 5 or more diseased subjects in the
exposed group would be significantly higher than the zero number in the
comparison group.
This process is repeated with c = 1, yielding an imaginary solution again
in the first case, and the admissible solution a = 6 in the second. The
process is repeated again with c = 2, then again with c = 3, and c = 4. In
all these instances the spontaneous rate is too low, and no L result is
possible. At c = 5 though, a value of zero for a would just yield a signifi
cant low result. This process continues until finally c = 20 is reached. Here
it is found that the minimum value of a required to achieve a significant H
result is 25, the entire exposed group. When c is 21 or more there is no
admissible value for a that will provide a significant H result.
The results for all possible combinations of a and c may be viewed graphi
cally by constructing a 26-by-26 grid, with c on the horizontal axis and a on
the vertical. By the process just described, this ac square is partitioned into
three regions, H, N, and L, in Figure 7.1, according to the nature of the
result. It should be kept in mind that these regions, H, N, and L, have been
constructed under the assumption that the disease rates are the same in both
groups. This assumption establishes the distribution for chi-square. The N
region is concentrated around the diagonal of the square, where a = c. The
H region has a larger than c, and the L region has a smaller than c. By the
manner in which the significant values for a were obtained, the probability
that a lies in the H region is not more than 5% for any value of c. This
probability is not exactly 5% because of the discrete nature of the observa
tions. Indeed, in many cases the probability is considerably smaller than
5%. Thus the “target” 5% is only a nominal figure. Similar remarks apply
to the L region. In contrast, the probability of a result (a,c) falling in the N
region is at least 90% when the null hypothesis is true.
From Figure 7.1, it is not possible to obtain an L result when there are
fewer than 5 disease cases in the unexposed group. Thus, studies aimed at
finding beneficial effects with only 25 subjects per group have very poor