Since Darwin, biologists have come to recognize that the theory of descent from common ancestry is very well supported by diverse lines of evidence. However, while the qualitative evidence is overwhelming, we also need formal methods for quantifying the evidential support for common ancestry (CA) over the alternative hypothesis of separate ancestry (SA). In this paper we explore a diversity of statistical methods, using data from the primates. We focus on two alternatives to CA, species SA (the separate origin of each named species) and family SA (the separate origin of each family). We implemented statistical tests based on morphological, molecular, and biogeographic data and developed two new methods: one that tests for phylogenetic autocorrelation while correcting for variation due to confounding ecological traits and a method for examining whether fossil taxa have fewer derived differences than living taxa. We overwhelmingly rejected both species and family SA, with infinitesimal p-values. We compare these results with those from two companion papers, which also found tremendously strong support for the CA of all primates, and discuss future directions and general philosophical issues that pertain to statistical testing of historical hypotheses such as CA.