Statistical analysis of aberrant crypt assays for colon cancer promotion studies

The size of aberrant crypt foci in rodent colons is a short-term assay for the promotion of colon carcinogenesis. An approximate model for the number of aberrant crypts in a focus is derived by assuming a branching process with a gamma lifetime. Maximum likelihood estimation can be implemented using symbolic manipulation packages. Although in most cases the model provides a reasonable fit for the distribution of the number of aberrant crypts per focus, there is some evidence of lack of fit. A more robust analysis can be based on summarizing the data for each animal by their mean number of aberrant crypts per focus. The branching process with gamma lifetime is used to obtain expressions for the functional relation between the variance of the number of aberrant crypts per focus and their mean. It is shown that in most cases this relation is quadratic. The quadratic relation also holds for other lifetime distributions. The parameters of the quadratic relation can be estimated by pseudolikelihood. An experiment for assessing the promotional effect of the compound HMF is used for illustration.