Ripley's K-function: Statistic

H_o	The distribution of disease cases is a spatial Poisson point process, where L(h) = h.
H_a	The distribution of disease cases is clustered, at some scales L(h)>h.

Test statistic

Ripley's K-function compares the pattern of the data to that produced by a homogeneous Poisson point process, where cases are considered "events." The expected number of other cases within a fixed distance (h) of one case is , where is the intensity, or mean number of cases per unit area.

K(h) can be estimated by the following formula (from Bailey and Gatrell 1995)

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Where R is the area of the region of interest, n is the total number of cases in region R, dij is the distance between the i^th and j^th cases , and I_h(dij) is the indicator function which is 1 if dij < h and 0 otherwise. Essentially, it sums the cases within distance h of each location in the dataset (each i). wij is an edge correction factor, the conditional probability that a case is observed in the region, given that it is dij from the event i.