Knox method

The Knox method quantifies space-time interaction based on critical space and time distances.

Space and time distances between pairs of cases

• space and time distance matrices calculated from the Distance Method
• optionally, can provide a point file for a linkage map

Number of replicate runs to determine the null distribution of the test statistic

Choice of link display, near in space only, time only, or both space and time

Critical space and time distances

• in general the critical distances selected should be consistent with the disease hypothesis under investigation
• although, when knowledge of the underlying disease process is absent, mean geographic and time distances determined from the data can be used as critical distances
• systematically varying the critical distances to identify those values that maximize the test statistic can provide insight into the spatial and temporal scale of the disease process, but precludes any formal evaluation of statistical significance because of multiple tests
• do not chose critical distances outside the minimum and maximum distances observed in the data

H₀: whether cases are near in space is independent of whether they are near in time

H_a: pairs of cases near in space tend to be near in time

Test Statistic: The test statistic, X is a count of those pairs of cases that are separated by less than the critical space and time distances:

Where N is the number of cases; s_ij is the space adjacency value, 1 if the distance between cases i and j is less than the critical space distance and 0 otherwise; t_ij is the time adjacency value, 1 if the waiting time between cases i and j is less than the critical time distance and 0 otherwise.

Pairs of cases will be near to one another when interaction is present, and the test statistic will be large. The null distribution of X is constructed under an approximate randomization which permutes the row-column elements of the time adjacency matrix while holding the space adjacencies constant, a number of times specified by the user. The probability value is the proportion of the upper right hand tail of the null distribution whose X values are as large or larger than the test statistic.

A 2x2 contingency table cross-categorizing pairs as close or far in space and time

• those link types selected by the user are displayed on a map of case locations

The test statistic, Knox X, and its significance

• resolution of p-value depends on the number of runs
• p-value is a one-tailed probability corresponding to space-time interaction

A histogram of the frequency distribution of X under H₀ of no association between the space and time adjacencies

Jacquez, G.M. 1994, User manual for Stat!: Statistical software for the clustering of health events, BioMedware, Ann Arbor, MI.