Statistical Modelling 10 (2010), 133–158
Latent regression analysis
Thaddeus Tarpey
Department of Mathematics and Statistics
Wright State University
Dayton, Ohio
U.S.A.
eMail: thaddeus.tarpey@wright.edu
Eva Petkova
Department of Child and Adolescent Psychiatry
New York University
New York
and
Nathan S. Kline Institute for Psychiatric Research
New York
U.S.A.
Abstract:
Finite mixture models have come to play a very prominent role in modelling
data. The finite mixture model is predicated on the assumption that distinct
latent groups exist in the population. The finite mixture model therefore is
based on a categorical latent variable that distinguishes the different
groups. Often in practice, distinct sub-populations do not actually exist.
For example, disease severity (e.g., depression) may vary continuously and
therefore, a distinction of diseased and non-diseased may not be based on
the existence of distinct sub-populations. Thus, what is needed is a
generalization of the finite mixture’s discrete latent predictor to a
continuous latent predictor. We cast the finite mixture model as a regression
model with a latent Bernoulli predictor. A latent regression model is proposed
by replacing the discrete Bernoulli predictor by a continuous latent predictor
with a beta distribution. Motivation for the latent regression model arises
from applications where distinct latent classes do not exist, but instead
individuals vary according to a continuous latent variable. The shapes of
the beta density are very flexible and can approximate the discrete Bernoulli
distribution. Examples and a simulation are provided to illustrate the latent
regression model. In particular, the latent regression model is used to model
placebo effect among drug-treated subjects in a depression study.
Keywords:
beta distribution; EM algorithm; finite and infinite mixtures;
placebo effect; quasi-Newton algorithms; skew-normal distribution
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