Cumulative Lognormal Distributions of Dose-Response vs. Dose Distributions

A review of the author’s findings over four decades will show that the lognormal probability density function can be fit to many types of positive-variate radiation measurement and response data. The cumulative lognormal plot on probability vs. logarithmic coordinate graph paper can be shown to be useful in comparing trends in exposure distributions or responses under differing conditions or experimental parameters. For variates that can take on only positive values, such a model is more natural than the “normal”(Gaussian) model. Such modeling can also be helpful in elucidating underlying mechanisms that cause the observed data distributions. It is important, however, to differentiate between the cumulative plot of a dose distribution, in which successive percentages of data are not statistically independent, and the plots of dose-response data for which independent groups of animals or persons are irradiated or observed for selected doses or dose intervals. While independent response points can often be best fitted by appropriate regression methods, the density functions for cumulative dose or concentration distributions must be fit by particular maximum likelihood estimates from the data. Also, as indicated in the texts by D. J. Finney and by R. O. Gilbert, for example, a simple plot of such data on available probability (or probit) vs. log scale graph paper will quickly show whether an adequate representation of the data is a lognormal function. Processes that naturally generate lognormal variates are sometimes estimated by statistics that follow the lognormal straight line for a cumulative plot on a probability vs. log scale; on the other hand, sometimes the statistics of interpretation follow such a line only over a certain range. Reported examples of lognormal occupational exposure distributions include those in some facilities in which roundoff biases were removed for some years. However, for a number of exposure distributions at licensed facilities in the United States, the cumulative exposure distributions curved upward above about 1 rem, showing the pressure of the 5 rem limit in constraining the “natural” distribution of occupational exposure. The United Nations Scientific Committee (UNSCEAR) adopted this type of display in some of its reports. Kumazawa and associates (1981, 1982) fitted some of these distributions by a function named “the hybrid lognormal”, which has been used to describe exposure distributions in Canada (Sont and Ashmore 1988). Examples of the suitability of the lognormal doseresponse function for animal data on lethality and carcinogenesis have been reported earlier by the author. In 1998, the close representation of a lognormal fit to the excess absolute mortality from solid cancers was reported by the author for the Hiroshima-Nagasaki cohorts reported by UNSCEAR. The close representation of a two-stage model of carcinogenesis by families of lognormal functions has also been reported. In 1999, the author showed that the deviation (in the low range) from lognormality of plutonium in urine measured by fission track analysis can be explained as the result of convoluting observed lognormal human sample data with the randomly varying and also lognormally distributed tracks of the subtracted reagent blanks. The sum or difference of two lognormally distributed variates is not lognormal; yet, in the higher range of interpreted plutonium activity in urine samples – well above the range of variation of the blanks — the “true” lognormality of excreted plutonium can be exhibited. Thus, reasons for the departure from an actual lognormal distribution of a fundamental quantity of interest can often be explained by examining the actual measurements and calculations leading to the interpeted results. A sample of these phenomena, as observed by the author, are presented and discussed in this paper.