LIKELIHOOD TESTS OF CONTINUOUS VERSUS SPECIATIONAL CHARACTER CHANGE
One can test and potentially reject a null hypothesis of continuous change by finding the most likely trees under continuous and speciational change individually. Note that the most likely continuous tree and the most likely speciational tree usually will be different, which means that the analyses treat phylogeny as an unknown parameter rather than as a model. This also means that one need not specify particular ancestor-descendant relationships. I conducted such analyses on several clades of Paleozoic gastropods. In all cases, speciational change is so much more likely than continuous change that we can reject the null hypothesis using information theory criteria. Moreover, the most likely rates of branching change are several times greater than the most likely rates of within-lineage change. This is because gastropod trees require high rates of change even though many gastropod species have long stratigraphic ranges. Thus, the best continuous change hypotheses find the large numbers of changes separating species improbable and the lack of change within those species also improbable whereas speciational change hypotheses accord high probability to both. Stratigraphic data exacerbate this, as the high sampling intensities makes long sampling gaps over which changes might have accrued unlikely.