A Generalization of Hewitt's Test for Seasonality

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Abstract
Hewitt's statistic for seasonality in monthly data is the maximal rank sum among all possible rank sums derived using consecutive 6-month periods. In this paper, Hewitt's test is extended to include those instances where 3, 4, or 5-month pulses or periods of raised incidence are hypothesized. Monte Carlo methods are used to derive the approximate distribution of the test statistic under the null hypothesis, when the length of the hypothesized period is k = 3, 4, or 5. A combinatorial method is used to derive exact levels for the test statistic. The test is applied to monthly data on adolescent suicide. Finally, the power of the test is compared with the X2 statistic using Monte Carlo simulation. The distribution of the test statistic was found and used to test the null hypothesis of no seasonal variation in monthly adolescent suicides, using a period of k = 3 months. The null hypothesis was rejected, indicating seasonality in the data. Monte Carlo simulations show the test statistic to be more powerful than the X2 statistic when sample sizes are small. This generalization of Hewitt's test should be most useful in those instances where the researcher wishes to carry out a quick and simple test of the null hypothesis of no seasonality against the alternative of a predetermined 3, 4, or 5-month period of raised incidence. When there is no a priori hypothesis about the appropriate length of period, the test may be used for different k values, in a more exploratory fashion.

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