Properties of Tango's index for detecting clustering in time

Tango proposed an index for detecting disease clustering in time applicable to grouped data obtained from a population that remains fairly stable over the study period. In this paper, we show that Tango's index is a two-dimensional U-statistic having an asymptotic normal distribution. To apply this result in the finite sampling situation, an Edgeworth expansion is used and is shown to be at least as accurate as Tango's best result in approximating the tails of his test statistic under the null hypothesis. This is extended to show that the Edgeworth expansion can be used to approximate the power of Tango's test statistic under selected alternatives to randomness. A power study based on simulations is conducted to compare the power of Tango's index to that of three of its competitors.