  Principal variables analysis (PVA) is a technique for selecting a subset of
variables that capture as much of the information in a dataset as possible.
Existing approaches for PVA are based on the Pearson correlation matrix, which
is not well-suited to describing the relationships between non-Gaussian
variables. We propose a generalized approach to PVA enabling the use of
different types of correlation, and we explore using Spearman, Gaussian copula,
and polychoric correlations as alternatives to Pearson correlation when
performing PVA. We compare performance in simulation studies varying the form
of the true multivariate distribution over a wide range of possibilities. Our
results show that on continuous non-Gaussian data, using generalized PVA with
Gaussian copula or Spearman correlations provides a major improvement in
performance compared to Pearson. Meanwhile, on ordinal data, generalized PVA
with polychoric correlations outperforms the rest by a wide margin. We apply
generalized PVA to a dataset of 102 clinical variables measured on individuals
with X-linked dystonia parkinsonism (XDP), a rare neurodegenerative disorder,
and we find that using different types of correlation yields substantively
different sets of principal variables.
