Mantel: Statistic

H_o	The times of occurrence of the health events are distributed randomly across the case locations. This is another way of saying the time distances between pairs of cases are independent of the spatial distances between pairs of cases.
H_a	Pairs of cases near in space tend to be near in time.

Test Statistic

Mantel's test statistic, Z, is the sum, across all case pairs, of the time distance multiplied by the spatial distance. Z is also called the Mantel product.

Where

N is the number of cases
sij is distance between i and j in space, s is the average space distance, and ss is the standard deviation of sij
tij is distance between i and j in time, t is the average space distance, and st is the standard deviation of tij.

ClusterSeer uses the standardized version of the Mantel product, r. r is a measure of matrix correlation with range -1 < r < 1. It is easier to interpret than Z. Both r and Z become large when the time distances are linearly dependent on the space distances.

Significance

Although Mantel (1967) provides an approximation for the variance of Z under the null hypothesis of no association between space and time, the usual approach is to generate the distribution of Z using Monte Carlo simulations, permuting the elements of one of the distance matrices while holding the other constant. This is equivalent to repeatedly scrambling the time observations across the locations, and calculating Z each time. This is done repeatedly to generate a distribution of Z under the null hypothesis.