The Poisson distribution is commonly assumed as the error structure for count data; however, empirical data may exhibit greater variability than expected based on a given statistical model. Greater variability could point to model misspecification, such as missing crucial information about the epidemiology of the disease or changes in population behavior. When the mechanism producing the apparent overdispersion is unknown, it is typically assumed that the variance in the data exceeds the mean (by some scaling factor). Thus, a probability distribution that allows for overdispersion (negative binomial, for example) may better represent the data. Here, we utilize simulation studies to assess how misspecifying the error structure affects parameter estimation results, specifically bias and uncertainty, as a function of the level of random noise in the data. We compare results for two parameter estimation methods: nonlinear least squares and maximum likelihood estimation with Poisson error structure. We analyze two phenomenological models the generalized growth model and generalized logistic growth model to assess how results of parameter estimation are affected by the level of overdispersion underlying in the data. We use simulation to obtain confidence intervals and mean squared error of parameter estimates. We also analyze the impact of the amount of data, or ascending phase length, on the results of the generalized growth model for increasing levels of overdispersion. The results show a clear pattern of increasing uncertainty, or confidence interval width, as the overdispersion in the data increases. While maximum likelihood estimation consistently yields narrower confidence intervals and smaller mean squared error, differences between the two methods were minimal and not practically significant. At moderate levels of overdispersion, both estimation methods yielded similar performance. Importantly, it is shown that issues of parameter uncertainty and bias in the presence of overdispersion can be mitigated with the inclusion of more data.
Keywords: epidemiological models; generalized growth model; overdispersion; parameter estimation; parameter uncertainty; phenomenological models.