A Bayesian Two-Part Latent Class Model for Longitudinal Medical
Expenditure Data: Assessing the Impact of Mental Health and Substance
Brian Neelon1,∗, A. James O’Malley2, and Sharon-Lise T. Normand2,3
1Nicholas School of the Environment, Duke University, Durham, North Carolina, U.S.A.
2Department of Health Care Policy, Harvard Medical School, Boston, Massachusetts, U.S.A.
3Department of Biostatistics, Harvard School of Public Health, Boston, Massachusetts, U.S.A.
Summary: In 2001, the U.S. Office of Personnel Management required all health plans participating
in the Federal Employees Health Benefits Program to offer mental health and substance abuse
benefits on par with general medical benefits. The initial evaluation found that, on average, parity did
not result in either large spending increases or increased service use over the four-year observational
period. However, some groups of enrollees may have benefited from parity more than others. To
address this question, we propose a Bayesian two-part latent class model to characterize the effect
of parity on mental health use and expenditures. Within each class, we fit a two-part random
effects model to separately model the probability of mental health or substance abuse use and mean
spending trajectories among those having used services. The regression coefficients and random effect
covariances vary across classes, thus permitting class-varying correlation structures between the two
components of the model. Our analysis identified three classes of subjects: a group of low spenders
that tended to be male, had relatively rare use of services, and decreased their spending pattern
over time; a group of moderate spenders, primarily female, that had an increase in both use and
mean spending after the introduction of parity; and a group of high spenders that tended to have
chronic service use and constant spending patterns. By examining the joint 95% highest probability
density regions of expected changes in use and spending for each class, we confirmed that parity had
an impact only on the moderate spender class.
Biometrics xx, 0–21DOI: xxx
Key words: Bayesian analysis; Growth mixture model; Latent class model; Mental health parity;
Semi-continuous data; Two-part model.
Bayesian Two-Part Latent Class Model1
The Federal Employees Health Benefits (FEHB) Program sponsors health insurance benefits
for more than 8.5 million federal employees and retirees, plus their spouses and dependents.
Over 250 health plans currently participate in the FEHB program. At the beginning of
2001, the U.S. Office of Personnel Management implemented a parity policy that required
all health plans participating in the FEHB Program to offer mental health and substance
abuse benefits on par with general medical benefits (U.S. OPM, 2000). An early evaluation
of the policy examined changes in total mental health expenditures, including out-of-pocket
and plan spending, from 1999 to 2002, and found that, on average, parity did not result
in either the large increases in spending predicted by opponents of parity or the increased
service use anticipated by mental health advocates (Goldman et al., 2006). Because most
of the literature on the impact of parity has focused on the average effect of the policy on
costs and access to mental health and substance abuse care, little is known about its impact
on specific enrollee subpopulations—for example, the sickest patients or those carrying the
greatest financial burden of illness.
To answer this question, there are three key features of longitudinal medical expendi-
ture data that must be addressed. The data are semi-continuous, assuming non-negative
values with a spike at zero for those who use no services, followed by a continuous, right-
skewed distribution for those who have used services. Table 1 provides a description of the
total spending data for a sample of 1581 FEHB enrollees from one state, each with four
years of data, yielding a total of 6324 observations. Over 80% of enrollees had no annual
mental health expenditures, while a small fraction had large expenditures. The percentage of
spenders increased steadily over time, while median spending increased immediately following
introduction of the parity directive and then returned to baseline levels by 2002.
[Table 1 about here.]
2 Biometrics, xx 20yy
Another important feature of the data concerns repeated measurements. In the FEHB
data, each enrollee contributes an observation for each of the four study years, introducing
within-subject correlation. Moreover, in each year, there are two outcomes per enrollee:
use of mental health/substance abuse services, and if use, the level of use as measured by
expenditures. Further, it may be reasonable to assume that the probability of some use
is correlated with the expected level of spending. An appropriate statistical model should
address these multiple sources of correlation.
One modeling strategy is to apply a longitudinal two-part model (Olsen and Schafer, 2001;
Tooze, Grunwald, and Jones, 2002; Ghosh and Albert, 2009). Two-part models are mixtures
of a point mass at zero followed by a right-skewed distribution (e.g., lognormal) for the
nonzero values. The two mixture components are modeled in stages. First, the probability
of service use is modeled via mixed effects probit or logistic regression. Next, conditional on
some usage, the expected spending level is modeled through (most commonly) a lognormal
mixed effects model. The random effects for the two components are typically assumed to
be correlated; ignoring this potential correlation can yield biased inferences (Su, Tom, and
Finally, because enrollees tend to share characteristics related to spending, it is reasonable
to assume that FEHB enrollees’ trajectories fall into a small number of classes. One natural
mechanism to handle this feature is to use latent class models, in particular latent class
“heterogeneity” or “growth mixture” models (Verbeke and Lesaffre, 1996; Muth´ en and
Shedden, 1999; Muth´ en et al., 2002). Growth mixture models (GMMs) assume that subjects
first fall into one of a finite number of latent classes characterized by a class-specific mean
trajectory; then, about these class means, subjects have their own unique longitudinal
trajectories defined by a set of random effects with class-specific variance parameters. As
such, GMMs can be viewed as finite mixtures of random effects models.
Bayesian Two-Part Latent Class Model3
Growth mixtures have become increasingly popular as a way of decomposing complex
heterogeneity in longitudinal models. Lin et al. (2000) developed a GMM to estimate class-
specific PSA trajectories among men at risk for prostate cancer. Proust-Lima, Letenneur,
and Jacqmin-Gadda (2007) proposed a GMM to jointly model a set of correlated longitudinal
biomarkers and a binary event. Lin et al. (2002) and Proust-Lima et al. (2009) developed
related models to analyze longitudinal biomarkers and a time to event. Beunckens et al.
(2008) proposed a GMM for incomplete longitudinal data. In the Bayesian setting, Lenk and
DeSarbo (2000) describe computational strategies for fitting GMMs; Elliott et al. (2005)
developed a Bayesian GMM to jointly analyze daily affect and negative event occurrences
during a 35-day study period; and recently, Leiby et al. (2009) fitted a Bayesian latent class
factor-analytic model to analyze multiple outcomes from a clinical trial evaluating a new
treatment for interstitial cytitis.
We build on this previous work to develop a Bayesian two-part growth mixture model for
characterizing the effect of parity on mental health use and expenditures. The advantages
of Bayesian inference are well-known and include elicitation of prior beliefs, avoidance of
asymptotic approximations, and, as we demonstrate below, practical estimation of parameter
contrasts and multidimensional credible regions. Within each class, we fit a probit-lognormal
model with class-specific regression coefficients and random effects. An attractive feature
of the model is that it permits the random effect covariance to vary across the classes.
For example, one class might comprise enrollees with frequent high expenditures (positive
correlation between the probability of spending and the actual amount spent), whereas
another class might comprise enrollees with frequent but modest expenditure (negative
correlation between probability of spending and amount spent).
The remainder of this paper is organized as follows: Section 2 outlines the proposed
model; Section 3 describes prior elicitation, posterior computation, model comparison, and