The Variance of the Number of Effects in an Epidemiological Cohort - The Role of Dose Uncertainty

Abstract

Two basic formulas, for the mean and variance of the number of effects in an epidemiological cohort, are de- rived. The formula for variance shows "extra-binomial variation" or "overdispersion" when there is correlated uncertainty of the probability of an effect. The formulas were validated by a numerical Monte Carlo study. The method of including "epistemic" uncertainty discussed by Hofer (E. Hofer, Health Physics, 2007) is generalized to include separately uncer- tainty from a Bayesian posterior distribution when the prior is known, and uncertainty of the prior. In this note, two basic formulas, for the mean and vari- ance of the number of effects in an epidemiological cohort, are derived. It is conventionally assumed that the number of effects has either a Poisson or binomial distribution (1). The formula for the variance given here reduces to the binomial result in the case of no correlations of the probability of an effect, but shows "extra-binomial variation" or "overdisper- sion" when there are correlations. In practice, the variance could be calculated using the method advocated by Hofer (2), where, for each individual in an epidemiological cohort, some number j = 1…M of al- ternate realizations of the dose and hence the probability of an effect, taking into account possible correlations, are gen- erated using Monte Carlo. This method is generalized here to include separately uncertainty from a Bayesian posterior distribution when the prior is known, and uncertainties caused by lack of knowledge of the prior. This is because, within a linear dose-effect response model, the average num- ber of effects is proportional to the posterior-average- collec- tive dose, and the important uncertainty is that of the poste- rior-average-collective dose caused by lack of knowledge of the prior.