## Mixed-Effects ZIP Models--Mental Health Service Research

Overview

The study of mental health service utilization is compromised by the lack of adequate statistical methods. With the advent of mixed- effects regression models (Laird and Ware, 1982), the complex multi-level sampling nature of these data (i.e. longitudinal and/or clustered sampling designs) can be accommodated in the statistical analysis. This represents a major advance in this field (see Gibbons et.al. 1993 for an overview in the context of mental health research). Nevertheless, now traditional mixed-effects regression models fail to accommodate the complexities of viewing service utilization as a primary outcome measure of interest. In most cases, service visits are enumerated and used as continuous and putatively normally distributed response measure in an otherwise appropriate fixed-effects or mixed-effects regression model. Of course, an often large proportion of the subjects never utilize services, whereas a few subjects are mass consumers of services. The resulting distribution is, anything but normal and data transformations are ineffective at bringing about normality for nonnegative distributions with a probability spike at zero. Alternatives include, (1) ignore the quantitative nature of the data and analyze service use as a binary outcome; (2) Create an ordinal response variable with categories of, for example, zero visits, 1 visit, 2 visits, 3 visits, 4 or more visits; (3) model the counts as a Poisson distribution in a Poisson fixed-effects or mixed-effects regression model. These options, despite their statistical sophistication, are all limited views of the reality of service utilization data. The binary approach simply discards the quantitative information that the investigator went to the trouble to collect. The ordinal approach relies on often unrealistic or arbitrary cut-points and typically assumes that the covariates have a proportional effect over the categories. The Poisson distribution often fails to adequately fit mental health service utilization data in that it underestimates the number of subjects who do not use services. A useful alternative to these heuristic approaches is to model the data as a zero-inflated Poisson distribution (Lambert, 1992). In the context of a regression model, the zero- inflated Poisson or "ZIP" model allows one to estimate one set of regression coefficients for use or non-use of services and a separate set of regression coefficients for the amount of services used, conditional on their use. The net result is an intuitively appealing model which allows mental health services researchers to simultaneously investigate the determinants of service utilization as a binary variable and the degree to which those same or different explanatory variables predict the amount of utilization for those individuals who utilize services. The primary objective of this research is to fully extend the ZIP model to the mixed-effects case, so that analysis of longitudinal and/or clustered service utilization data is possible. In addition to development of the statistical theory and estimation procedure, we propose to develop WINDOWS based freeware to be distributed from our web site and to apply the methodology in the analysis of three large mental health services research databases.

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