Relationships between disease incidence at two levels in a spatial hierarchy

Relationships between disease incidence measured at two levels in a spatial hierarchy are derived. These relationships are based on the properties of the binomial distribution, the beta-binomial distribution, and an empirical power-law relationship that relates observed variance to theoretical binomial variance of disease incidence. Data sets for demonstrating and testing these relationships are based on observations of the incidence of grape downy mildew, citrus tristeza, and citrus scab. Disease incidence at the higher of the two scales is shown to be an asymptotic function of incidence at the lower scale, the degree of aggregation at that scale, and the size of the sampling unit. For a random pattern, the relationship between incidence measured at two spatial scales does not depend on any unknown parameters. In that case, an equation for estimating an approximate variance of disease incidence at the lower of the two scales from incidence measurements made at the higher scale is derived for use in the context of sampling. It is further shown that the effect of aggregation of incidence at the lower of the two scales is to reduce the rate of increase of disease incidence at the higher scale.