Normal Approximation Invalid for Large Data Set of Leukemia Deaths Among Hiroshima Atomic Bomb Survivors

A cohort of Hiroshima atomic bomb survivors was followed to determine the relationship between deaths from leukemia during 1950-1970 and estimated radiation dosage from the bombing. Subjects were stratified according to their age at the time of the bombing. Below, we tabulate a subset of the data; children in the 0-9 age group exposed to radiation doses ranging from 0 to 99 rads. Two additional dose groups, 100-199 rads and 200+ rads, are excluded for reasons documented on page 4-110 of the StatXact manual. (The full data set is on page 285 of Categorical Data Analysis, Alan Agresti, 1990, John Wiley & Sons).

In absolute terms, the leukemia death rates are rather low. Only 11 deaths were observed in a cohort of size 19461, amounting to a death rate of 0.06%. However the rates increase from 0% in the lowest dose group to 0.14% in the highest. It is therefore interesting to ask whether this increasing trend is real, or merely due to chance fluctuations in the data. Our intuition cannot help much with these extremely low death rates, and we must resort to a formal statistical test of significance. One way to determine if there is a statistically significant association between leukemia deaths and radiation exposure is to perform the Cochran-Armitage trend test (Breslow and Day, Statistical Methods in Cancer Research, 1980, page 148). The test statistic is of the form

where wj is the mid-range of the radiation dose, and Xj is the number of leukemia deaths, in the jth dose group. For these data w1 = 0 rads, w2 = 4.5 rads, w3 = 30 rads, and w4 = 75 rads. Prior to StatXact the only way to perform this trend test was to assume that T is normally distributed. Below is a plot of the exact distribution of T, computed by StatXact.

Notice that the distribution of T is not even close to normal. Its distinct values are unequally spaced, the distribution has an unusually long right tall, and it is multimodal. Not surprisingly the exact and asymptotic p-values for the Cochran-Armitage trend test differ. The results are tabulated below:

Conclusions:
1. The one-sided exact p-value is above the usual 0.05 cut-off, unlike its asymptotic approximation.
2. The extremely long and narrow right tail of the exact distribution is reflected in the fact that the one and two-sided exact p-values are very close.
3. The usual practice of doubling the one-sided p-value to obtain a two-sided p-value is not appropriate for an asymmetric distribution; in this case it yields a two-sided p-value of 0.1306 where the actual two-sided p-value is 0.0682.