The distribution of cancer deaths in time. A survey test of the lognormal model

ONE of the keys to understanding of the behavior of cancer in humans is quantitation of the observations. The disease, even of a single site, is extremely variable. One needs to know how to describe this viariability by efficient statistics. Only then can one focus down on behavioral differences associated with such things as treatment differences, histological differences in tumors or the elusive host resistance. For efficient statistical description, and particularly for efficient comparisons, one needs a distribution model that in a very few parameters accurately and fully describes any set of observations. Most " error " theory for instance supposes the " normal " distribution for random deviations. There is substantial feeling that this destribution is not the basic one for many biological events (Aitchison and Brown, 1963; Gaddum, 1945). In particular, the distribution of deaths from a given disease plotted from onset are substantially skew. To describe events such as these fully, other models are needed. For cancer death times, two have been prominently advocated, the exponential (radioactive decay is a paradigm) and the lognormal. The latter says that the skewness is removed and normality achieved when death distribution is plotted not against time but against the logarithm of time. In 1948 and 1949, Boag published the major series of papers supporting the lognormal model. He had first tested five groups of patients who had been treated for particular types of cancer but had died of their disease. He showed excellent agreement between the data and the lognormal model. Later he added data on three series of untreated patients but was forced to point out that the agreement was poorer. Tivey (1954) later applied the model to many series of leukemia patients and also found that it fitted the data well (on the assumption that all patients died or would die of their cancers). Boag in his presentations advocated that the lognormal model be extended to describe death times in total patient populations by making allowances both for cures and for competing deaths from other causes. While he gave extensive details on the appropriate calculations, he presented no information as to how often his more complete treatment would fit real sets of data. Neither did Tivey's work bear on the more general situation. Hence there does not seem to be actual evidence that the lognormal model does in fact apply to typical clinical series. Neither are there indications …