  The asymptotic behavior of estimates and information criteria in linear
models are studied in the context of hierarchically correlated sampling units.
The work is motivated by biological data collected on species where
autocorrelation is based on the species' genealogical tree. Hierarchical
autocorrelation is also found in many other kinds of data, such as from
microarray experiments or human languages. Similar correlation also arises in
ANOVA models with nested effects. I show that the best linear unbiased
estimators are almost surely convergent but may not be consistent for some
parameters such as the intercept and lineage effects, in the context of
Brownian motion evolution on the genealogical tree. For the purpose of model
selection I show that the usual BIC does not provide an appropriate
approximation to the posterior probability of a model. To correct for this, an
effective sample size is introduced for parameters that are inconsistently
estimated. For biological studies, this work implies that tree-aware sampling
design is desirable; adding more sampling units may not help ancestral
reconstruction and only strong lineage effects may be detected with high power.
