Tests for density dependence revisited.

We have examined a number of statistical issues associated with methods for evaluating different tests of density dependence. The lack of definitive standards and benchmarks for conducting simulation studies makes it difficult to assess the performance of various tests. The biological researcher has a bewildering choice of statistical tests for testing density dependence and the list is growing. The most recent additions have been based on computationally intensive methods such as permutation tests and boot-strapping. We believe the computational effort and time involved will preclude their widespread adoption until: (1) these methods have been fully explored under a wide range of conditions and shown to be demonstrably superior than other, simpler methods, and (2) general purpose software is made available for performing the calculations. We have advocated the use of Bulmer's (first) test as a de facto standard for comparative studies on the grounds of its simplicity, applicability, and satisfactory performance under a variety of conditions. We show that, in terms of power, Bulmer's test is robust to certain departures from normality although, as noted by other authors, it is affected by temporal trends in the data. We are not convinced that the reported differences in power between Bulmer's test and the randomisation test of Pollard et al. (1987) justifies the adoption of the latter. Nor do we believe a compelling case has been established for the parametric bootstrap likelihood ratio test of Dennis and Taper (1994). Bulmer's test is essentially a test of the serial correlation in the (log) abundance data and is affected by the presence of autocorrelated errors. In such cases the test cannot distinguish between the autoregressive effect in the errors and a true density dependent effect in the time series data. We suspect other tests may be similarly affected, although this is an area for further research. We have also noted that in the presence of autocorrelation, the type I error rates can be substantially different from the assumed level of significance, implying that in such cases the test is based on a faulty significance region. We have indicated both qualitatively and quantitatively how autoregressive error terms can affect the power of Bulmer's test, although we suggest that more work is required in this area. These apparent inadequacies of Bulmer's test should not be interpreted as a failure of the statistical procedure since the test was not intended to be used with autocorrelated error terms.