  Nicholls and Gray (2008) describe a phylogenetic model for trait data. They
use their model to estimate branching times on Indo-European language trees
from lexical data. Alekseyenko et al. (2008) extended the model and give
applications in genetics. In this paper 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 missing-data content. Nicholls and Gray
(2008) treat 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 (2008) dropped
seven languages with too much missing data. We fit all twenty four languages in
the lexical data of Ringe (2002). In order to model spatial-temporal 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 MCMC.
  We validate our fit using Bayes factors to test known age constraints. We
reject three of thirty historically attested constraints. Our main result is a
unimodel posterior distribution for the age of Proto-Indo-European centered at
8400 years BP with 95% HPD equal 7100-9800 years BP.
