Larsen's test
  • temporal
  • global
  • multiple time series
    • Indications/Recommendations for use: This test measures dispersion of cases about a central time period. Rare data are required because some of the time periods must have 0 cases. Larsen's test can be used to answer two questions: `Within an area, do occupied time cells tend to occur in a sequence?' (addressed using the z-score); and `Is there an unusual pattern over time which may not necessarily be the same over the individual areas?' (addressed using the overall z-score). (More).
    Description: Larsen's test statistic K is sensitive to a unimodal clustering of occupied cells. A unimodal cluster occurs when occupied time cells tend to occur in a sequence, one right after the other, and arise when an initial case is quickly followed by the appearance of daughter cases. This may occur after an ephemeral exposure to an infectious agent or irritant, and with certain behavior-mediated phenomena such as copy-cat suicides. This module provides two tests for temporal clustering; within the individual spatial units (using a z-score as described later) and across all spatial units simultaneously (using an overall z-score).
    Test statistic: The test statistic, K, measures the tendency of time periods with at least one case to form a single cluster in time (shown at right).

    Here

    • yi is the time assigned to the ith cell in which a case occurred.
    • The index r+1 is the `central most' time cell that contained a case.
    • m is the number of time periods with at least one case.
    Null Hypothesis: Cases occur randomly through time. t is the total number of time periods.
    Alternative Hypothesis: Cases cluster about a single point in time.
    GeoMed Inputs: Counts of cases in consecutive time periods (a time series).
    GeoMed Outputs: For each time series:
    • K, and its expectation E(K) and variance Var(K)
    • z-score and its one-tailed p-value
    Group z-score and p-value.

    Plot K against its expectation E(K):

    • each time series is one point on the graph
    • dashed line of the function K=E(K) describing where observations would be plotted under the null hypothesis of a random distribution of occupied intervals across the time series
    The mean and variance of Larsen's K is used to evaluate significance.
    Example Analysis Reference: Larsen, R. J., C. L. Holmes and C. W. Heath. 1973. A statistical test for measuring unimodal clustering: a description of the test and of its application to cases of acute leukemia in metropolitan Atlanta, Georgia. Biometrics 29:301-309.
    Website maintained by Andy Long. Comments appreciated.
    longa@nku.edu