Missing data in a stochastic Dollo model for binary trait data, and its application to the dating of Proto-Indo-European

Nicholls and Gray have described a phylogenetic model for trait data. They used their model to estimate branching times on Indo-European language trees from lexical data. Alekseyenko and co-workers extended the model and gave applications in genetics. We extend the inference to handle data missing at random. When trait data are gathered, traits are thinned in a way that depends on both the trait and the missing data content. Nicholls and Gray treated missing records as absent traits. Hittite has 12% missing trait records. Its age is poorly predicted in their cross-validation. Our prediction is consistent with the historical record. Nicholls and Gray dropped seven languages with too much missing data. We fit all 24 languages in the lexical data of Ringe and co-workers. To model spatiotemporal rate heterogeneity we add a catastrophe process to the model. When a language passes through a catastrophe, many traits change at the same time. We fit the full model in a Bayesian setting, via Markov chain Monte Carlo sampling. We validate our fit by using Bayes factors to test known age constraints. We reject three of 30 historically attested constraints. Our main result is a unimodal posterior distribution for the age of Proto-Indo-European centred at 8400 years before Present with 95% highest posterior density interval equal to 7100-9800 years before Present.