A Critical Look at Directional Random Walk Modeling of Sparse Fossil Data

The general random walk model (GRW) of Hunt is used to infer directional evolution in mean trait values from sparse fossil data. Such evolutions are modeled as the accumulated result of small steps with mean step sizes and step variances. As shown in simulations and real data cases, the mean step sizes are often easy to estimate from data, except for cases where the mean step size is small compared to the step variance. The step variances, on the other hand, can be estimated reasonably well only when the mean trait values have small measurement errors, but even here the step variance estimation may be difficult. For fossil data with realistic measurement errors, the step variances appear to be extremely difficult to find, and they are often found to be negative. They must then be set to zero, such that GRW collapses into deterministic walk processes plus sampling errors. As a result of poor step variance estimates, the directional evolution may be both under- and overestimated as compared with generalized least squares (GLS) results, which gives the best linear unbiased estimator (BLUE) of the evolutionary slope. Here, I study this problem through simulations as well as in four real data cases. Based on weighted mean square error (WMSE) comparisons, my conclusion is that GLS in cases with large measurement errors is the best method for inference of directional evolution. When the step variances must be set to zero, GLS is simplified into weighted least squares (WLS) estimation.