Avoiding erroneously high levels of detection in combinations of semi-independent tests

Abstract

A randomization procedure is proposed which allows statistical tests to be combined into a single test to maintain specified and acceptable levels of false detection. This method was applied to the problem of detecting density dependence in 135 unpublished time-series (of ≥10 generations) from insect populations, and to simulated density-dependent and density-independent data, so that the correctness of observed levels of detection from the published data could be verified. To allow the application of the randomization procedure to Bulmer's (1975) tests and Varley and Gradwell's (1960) test, these were recast as randomization tests. The randomization procedure was tested with 39 combinations of tests for density dependence (and limitation/attraction); it generally producedcombined tests with levels of detection that were intermediate between detection levels of the constituent tests (and hence was limite by these). The specified rate of false detection (5%) was never exceeded (by more than 1%) when combined tests were applied to time-series from a random-walk model. Two different combinations of tests produced levels of detection from the published time-series which were slightly greater than their constituent tests when they were combined into single tests. These were the randomized form of Bulmer's (1975) first test with the tests of Pollard et al. (1987) and Reddingius and den Boer (1989) with the randomized form of Bulmer's second test. The combination of Bulmer's first and Pollard et al.'s test produced a greater level of detection (21.5%) than any other single test or combination of tests. These results were confirmed by the analysis of modelled density dependent data. Although the increase in power of combinations of tests over single tests is small with the data we used, the combined tests (listed above) had rates of detection that were less influenced by the form of data (of two forms of density-dependent data) than were their constituent tests. Hence, it appears that the combined tests are of greater generality than single test statistics. The method presented here for combining several statistical tests into a single randomization test is applicable in many other areas of ecology where we wish to apply several tests and take the most probable result of these; and if the tests being conducted are, or can be expressed as, randomization tests.