A test for temporal clustering within a single time series or in several time series simultaneously. Currently the software provides two tests for temporal clustering; within a single time series and across all several time series simultaneously.

Data Requirements

Counts of cases over several time periods

this test can only be used when there are from 5 to 10 time periods
cannot use rates, counts are required, therefore the method is biased by changes in population size through time
cannot use the method when the expected number of cases in each interval is smaller than 2, the total number of cases in the time series must be greater than twice the number of time intervals

Analysis

H₀: cases occur at random over the t time periods

H_a: cases do not occur randomly through time

Test Statistic: Define t to be the number of time periods and n to be the number of cases summed over the t time periods. The test statistic, A, is the number of time periods with at least the number of cases expected in the absence of clustering,

cases:

A can be written as a normal deviate which is expected to be normally distributed with a mean of 0 and unit variance:

A will be small when cases cluster in one or a few time periods. P-values are evaluated by comparing z to the percentiles of the normal distribution.

When analyzing s time series simultaneously an overall p-value is obtained as a Chi square with one degree of freedom:

This allows testing for simultaneous clustering in several time series.

Output

For each time series:

A, and its expectation E(A) and variance Var(A)

z-score and its one-tailed p-value

Plot of A against its expectation

dashed line of the function A=E(A) describing where observations would be plotted under the null hypothesis of cases occuring at random over the t time periods

Reference

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