Comparison of directional random walk and weighted least squares modeling of sparse fossil data

The general random walk model (GRW) of Hunt (2006) is used to infer directional evolution in mean trait values from sparse fossil data by modeling phenotypic change as the accumulated result of small steps with mean step sizes and step variances. Using simulations and real data cases, Ergon (2026) showed that the step variances can be estimated reasonably well only when the mean trait values have small measurement errors, while for fossil data with realistic measurement errors they appear to be extremely difficult to find, and they are often found to be negative. In the simulations Ergon (2026) assumed that the true phenotypic mean values were known. Here, I essentially repeat these simulations under the assumption that only mean trait values with large measurement errors are known, and based on weighted mean squared error (WMSE) comparisons the conclusion is that weighted least squares (WLS) is a better method than GRW. A second conclusion is that WLS is a better method also in the possibly rare cases with large measurement errors where the GRW parameters are estimated well. The GRW method is simply not flexible enough to handle such cases. A third conclusion is that Akaike Information Criterion (AIC) results for GRW models with large measurement errors relative to the step variance may be overly optimistic.