The Discrete Weibull Distribution: An Alternative for Correlated Counts with Confirmation for Microbial Counts in Water

Distributions of pathogen counts in treated water over time are highly skewed, power-law-like, and discrete. Over long periods of record, a long tail is observed, which can strongly determine the long-term mean pathogen count and associated health effects. Such distributions have been modeled with the Poisson lognormal (PLN) computed (not closed-form) distribution, and a new discrete growth distribution (DGD), also computed, recently proposed and demonstrated for microbial counts in water (Risk Analysis 29, 841-856). In this article, an error in the original theoretical development of the DGD is pointed out, and the approach is shown to support the closed-form discrete Weibull (DW). Furthermore, an information-theoretic derivation of the DGD is presented, explaining the fit shown for it to the original nine empirical and three simulated (n = 1,000) long-term waterborne microbial count data sets. Both developments result from a theory of multiplicative growth of outcome size from correlated, entropy-forced cause magnitudes. The predicted DW and DGD are first borne out in simulations of continuous and discrete correlated growth processes, respectively. Then the DW and DGD are each demonstrated to fit 10 of the original 12 data sets, passing the chi-square goodness-of-fit test (alpha = 0.05, overall p = 0.1184). The PLN was not demonstrated, fitting only 4 of 12 data sets (p = 1.6 x 10-8), explained by cause magnitude correlation. Results bear out predictions of monotonically decreasing distributions, and suggest use of the DW for inhomogeneous counts correlated in time or space. A formula for computing the DW mean is presented.