Disparity estimation proceeded in three steps. The first was to fit a multivariate two-part model for mental health care expenditure (one each for total, outpatient, and prescription drug) as a function of all relevant independent covariates, including appropriate interactions. We separately modeled the probability of any expenditures and the level of expenditure conditional on positive expenditures using generalized linear models (GLM) (McCullagh and Nelder 1989 ). This avoids the potential inconsistency from fitting an OLS model to logged expenditures in the second part without adequate retransformation (Manning, 1998 (link); Mullahy 1998 (link)). For the positive part of the distribution of expenditures, we modeled the expected expenditures E(y | x, y>0) directly as μ(x′β) where μ is the link between the observed raw scale of expenditure, y, and the linear predictor x′β, where x is a vector of the predictors. The GLM also allows for heteroscedastic residual variances related to the predicted mean (Buntin and Zaslavsky 2004 (link)). The conditional variance of y given that y>0 is assumed to be a power of expected expenditures, conditional on x. Thus, we can characterize the mean and variance functions as
Using diagnostics in Manning and Mullahy (2001) (link) and Buntin and Zaslavsky (2004) (link), we identified the optimal generalized linear model to have a log link, and residual variance proportional to mean squared (λ = 2). We used the modified Hosmer-Lemeshow test to assess systematic misfit overall in terms of predicted expenditures, as well as the model misfit for major covariates.
The second step was to adjust the minority distribution of need to match the White distribution while preserving the distributions of SES variables for each racial/ethnic group using the rank-and-replace method or propensity score-based method, described below. Finally, predicted expenditures were calculated using the two-part GLM and the adjusted health status distributions, and compared across racial groups.
Using diagnostics in Manning and Mullahy (2001) (link) and Buntin and Zaslavsky (2004) (link), we identified the optimal generalized linear model to have a log link, and residual variance proportional to mean squared (λ = 2). We used the modified Hosmer-Lemeshow test to assess systematic misfit overall in terms of predicted expenditures, as well as the model misfit for major covariates.
The second step was to adjust the minority distribution of need to match the White distribution while preserving the distributions of SES variables for each racial/ethnic group using the rank-and-replace method or propensity score-based method, described below. Finally, predicted expenditures were calculated using the two-part GLM and the adjusted health status distributions, and compared across racial groups.