On Edwards' criterion of seasonality and a non-parametric alternative.

Abstract

There are a few conditions?hay fever will serve as an example?for which it has always been possible to assert, without fear of contradiction, that their pattern of occurrence was a seasonal one. However, when it is not subjectively clear that a seasonal pattern exists, considerable difficulty can be experienced. Applied to data in the form of monthly frequencies of new cases, the common x2 test with 11 degrees of freedom will not be sensitive to seasonal fluctuations of moderate amplitude unless the sample size is very large. Moreover, when this test does detect a significant amount of variation between months it will still be left to the investigator to judge whether the variation is of a seasonal character. When a prior hypothesis is available it is, of course, easy to apply a more efficient test; for example, if it has previously been suggested that a disease tends to occur more frequently from April to September then one could test whether the proportion of cases in a fresh sample that occurs within this period exceeds 0-501 (=183/365-25). In the situation more usually encountered there is no prior hypothesis, and the same limited sample of cases must be used both to support a test of the null hypothesis and to specify a likely type of seasonality. The first convenient method of dealing with this situation was devised by Edwards (1958, 1961), who proposed restricting consideration to patterns that could be tolerably well described by a simple harmonic curve. His method of analysis yields estimates of two parameters of this curve and simultaneously provides a test criterion whose dis tribution, on the null hypothesis, approximates to that of x2 with 2 degrees of freedom. The method has recently been criticized (Wehrung and Hay, 1970) on the ground that it may be sensitive to some kinds of cyclic variation that are clearly not of simple harmonic form. So far from being a fault, this seems to us to be a virtue of the method, and one that tends to weaken the case for introducing a non-parametric test like the one proposed below. That Edwards' criterion may also be sensitive to occasional, non recurrent outbreaks is a risk acknowledged by its author, and one less likely to afflict a non-parametric test. Both types of test are liable to give a false positive result if used on data that are subject to any strong secular influence, since this will suggest the presence of a 'seasonal' peak in September-October or in March-April, depending on whether the trend is rising or falling. We were first led to reconsider the performance of Edwards' criterion after having used it, perhaps in judiciously, as a screening device to pick out, from among a large number of congenital cardiac mal formations, those particular defects in which the in fluence of seasonally varying factors could be re garded as most plausible. A conspicuous feature of this attempted screening was that a much higher proportion of apparently significant results was declared when the sample of cases was small. It was at first hoped that smallness of a diagnostic subgroup (in a total series of over 10,000 affected children) might connote aetiological homogeneity, and hence a relatively good chance of detecting genuine seasonal variation, but prudence obviously required that we should examine the adequacy of the x2 approxima tion for the smaller sample sizes. Edwards' original papers did not refer to the question of requisite sample size, and the note by Smith (1961), which accompanied the second paper, only indicated that it should be 'reasonable' and 'sufficiently large'. We therefore set up a Monte Carlo experiment using pseudo-Poisson variate values in sets of 12, and carried out a thousand runs for each of seven values of the 'true' mean monthly frequency, corresponding to sample sizes from 192 down to 24. A selection from the computer output is shown in Table I, where it will be seen that there was some excess of apparently significant results at all the sample sizes considered, but that the approxi mation was very good down to sample size 96. Even for samples as small as 48 the criterion might be considered a serviceable one, though the frequency of values beyond the upper 10% and 5% point of the x2 distribution was about twice, and that of values beyond the 1 % point about three times what it 174