Previous studies have suggested that unbiased random walks may serve as appropriate null hypotheses for the detection of pattern in stratophenetic series. While numerous processes that influence the perceived temporal morphological evolution of a species may yield stratophenetic patterns that conform to the model of a random walk, use of the model as a null hypothesis raises several concerns. First, unbiased random walks are only a subset of a much larger set of random motions, including biased and fractional random walks. Some of these motions could also serve as appropriate null models for stratophenetic patterns. Second, due in part to the fractal nature of random walks, many types of time series begin to resemble random walks statistically as sampling resolution decreases. Therefore, indiscriminate support for unbiased random walks as null hypotheses of stratophenetic pattern leads inevitably to the commitment of Type II errors (incorrect failure to reject a null hypothesis). In this paper we simulate different hypothetical patterns of microevolution using various random walk models and apply the test of the null hypothesis. The frequency of Type II errors increases as stratigraphic completeness decreases, but at a currently unknown rate. Moreover, the test is insensitive to nongradual patterns of anagenesis. We also demonstrate that a previously published approach is closely related to a standard method of fractal time series analysis and represents a good qualitative test of evolutionary pattern. The statistical variation underlying this method, however, is currently unknown, and further work is required to make it a robust quantitative test.

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