Maximum likelihood vs. minimum distance: searching for hills in the plain.

of deleterious spontaneous mutation from simulated tances have a random component. To deal with this commutation accumulation data, Garcı́a-Dorado and Galponent objectively, after a minimum was identified for lego (2003) found that ML estimates of the deleterious the distance, we used regression to test the significance mutation rate (U) had higher mean square errors (MSE) of the increase in distance within an interval (UMD U due to occasional ML outlier estimates. Thus, relative Uup). This regression interval should be inferred from MSE1/2 for ML U estimates was about six times larger inspection of a previous profile obtained using a difthan that for MD U estimates obtained under comparaferent seed for the generation of random variables. ble conditions, although the MSEs for the estimates of For our particular sets of parameters, it turned out that the average mutational effect were very similar. In his (UMD U 3UMD) was an appropriate interval, as it Letter to the Editor, Keightley (2004, p. 551) claims that contains a relevant proportion of the observed increase this result is due to the fact that MD nonconvergence in distance. This provides a non-ad hoc criterion to is “declared if the profile of distance . . . changes nonsigdetermine the interval for each new data set and avoids nificantly over a range of three times the parameter repeating each profile with a new random seed to test value” and that the comparison excludes “all MD replithe significance of the minimum. Replicates with noncates that fail to converge and any ML replicates for significant slopes were classified as providing no global which the ML U estimate exceeds 50” and proposes that MD estimate. Keightley notes that 15 of 62 replicates this difference in the exclusion criteria allows outliers were classified as giving no global estimate using MD, for ML estimates of U (UML) but not for MD ones (UMD). while only 6 of 60 fell into this category using ML, and As explained in our original article (Garcı́a-Dorado he suggests that outliers might have been removed from and Gallego 2003, p. 810), the Fortran MD program the MD results because of the above significance test. searches the minimum distance in a grid for (U, Pa, ) To assess Keightley’s concern, we reexamined those specified by the user (where Pa is the probability that cases that did not give a global MD estimate. Twelve the mutational effect is advantageous, and is the shape of 15 gave profiles where the distance decreased with parameter of the gamma distribution assumed for the increasing U until the profile, represented at a graphical deleterious mutational effect) to obtain a profile of the precision high enough as to reveal the distance’s randistance, minimized with respect to and Pa, against U. dom dispersion, became flat (i.e., it could not be distinFor these three parameters, we started with intervals guished from horizontal at either a statistical or a visual that were iteratively widened when required. In particulevel). On the contrary, the 6 excluded ML profiles lar, the MD estimate for U (UMD) was initially searched did not become flat, but the likelihood monotonically between U/20 and 10U, but when, occasionally, the increased with U at least up to U 50. The remainminimum distance corresponded to U values close to ing 3 of 15 MD excluded replicates had a graphical (or at) the edge of this interval, the grid was moved minimum, but it was not significant. However, their accordingly until a minimum for the distance was graphexclusion scarcely affected the results as they were not ically identified (corresponding to UMD) or the distance outliers; if we assume that these minima reflect valid estiprofile became flat. mates for U and include them in our study, the average MSE1/2 for UMD increases by only 5% and remains at 0.28 times the MSE1/2 for UML. 1Corresponding author: Departamento de Genética, Facultad de BioThus, ML estimates were searched until the likelihood logı́a, Universidad Complutense de Madrid, 28040, Madrid, Spain. E-mail: [email protected] showed a maximum or while it increased with U, up to