  We propose novel methodology for testing equality of model parameters between
two high-dimensional populations. The technique is very general and applicable
to a wide range of models. The method is based on sample splitting: the data is
split into two parts; on the first part we reduce the dimensionality of the
model to a manageable size; on the second part we perform significance testing
(p-value calculation) based on a restricted likelihood ratio statistic.
Assuming that both populations arise from the same distribution, we show that
the restricted likelihood ratio statistic is asymptotically distributed as a
weighted sum of chi-squares with weights which can be efficiently estimated
from the data. In high-dimensional problems, a single data split can result in
a "p-value lottery". To ameliorate this effect, we iterate the splitting
process and aggregate the resulting p-values. This multi-split approach
provides improved p-values. We illustrate the use of our general approach in
two-sample comparisons of high-dimensional regression models ("differential
regression") and graphical models ("differential network"). In both cases we
show results on simulated data as well as real data from recent,
high-throughput cancer studies.
