Generalized linear models (GLMs) extend general linear regression models to the analysis of data with non-normally distributed error structures arising from the exponential family of probability distributions. Discrete (count) distributions within this family include the binomial, Poisson, and negative binomial distribution with known overdispersion parameter k. Generalized linear mixed models (GLMMs) are GLMs that include both fixed and random effects. Fixed effects are represented by a measured explanatory variable (covariate) and are quantified by regression coefficients. By contrast, random effects embody the unmeasured or unmeasurable characteristics of a unit of observation which induce correlation (clustering) among data collected from the same unit; e.g. numbers of mosquitoes collected from the same household. Random effects are quantified in terms of variability (variance) among data collected from distinct units of observation.
Polynomial distributed lag models (PDLMs) are suitable for analysing data where one or more explanatory variables exert a lagged effect on the collected (response) data; e.g. rainfall at some point in the past affects mosquito abundance now. Moreover, PDLMs assume that this effect is distributed over the entire lag period; e.g. rainfall over the past several weeks affects mosquito abundance now. Treating every point in the past as a separate explanatory variable with its own unique coefficient becomes infeasible for all but very short lag periods; it is impractical to estimate a large number of coefficients of often highly correlated explanatory variables (e.g. rainfall yesterday is correlated with rainfall today). This problem is avoided by PDLMs using a polynomial functional form with ample flexibility to capture the shape of the distributed effect.