Cuzick and Edwards' test when exact locations are unknown.

Cuzick and Edwards (JR Stat Soc [B] 1990;52:73-104) have proposed a case-control test to detect spatial clustering. The test statistic is the sum, over all cases, of the number of each case's k nearest neighbors that also are cases. Their approach is attractive in that it accounts for geographic variation in population density and because it allows one to account for confounders, both known and unknown, through the judicious selection of controls. However, the test assumes case locations are known exactly, when, in practice, case locations are usually approximated by the centers of areas such as census tracts and zip code zones. In such situations, "ties" arise when cases and controls are assigned to the same area, and the loss of information precludes calculation of the test statistic. The author's approach enumerates the ways in which the ties may be resolved to obtain upper and lower bounds on the exact, unobserved, test statistic. The null hypothesis of no clustering is rejected when the upper and lower bounds are significant, and it is accepted when they are not significant. Judgment is withheld when the upper bound is significant but the lower bound is not significant. This approach allows Cuzick and Edwards' test to be used with inexact locations typical of most cluster investigations.