Estimating marginal and incremental effects in the analysis of medical expenditure panel data using marginalized two-part random-effects generalized Gamma models: Evidence from China healthcare cost data

Abstract

Conditional two-part random-effects models have been proposed for the analysis of healthcare cost panel data that contain both zero costs from the non-users of healthcare facilities and positive costs from the users. These models have been extended to accommodate more flexible data structures when using the generalized Gamma distribution to model the positive healthcare expenditures. However, a major drawback with the extended model, which is inherited from the conditional models, is that it is fairly difficult to make direct marginal inference with respect to overall healthcare costs that includes both zeros and non-zeros, or even on positive healthcare costs. In this article, we first propose two types of marginalized two-part random-effects generalized Gamma models (m2RGGMs): Type I m2RGGMs for the inference on positive healthcare costs and Type II m2RGGMs for the inference on overall healthcare costs. Then, the concepts of marginal effect and incremental effect of a covariate on overall and positive healthcare costs are introduced, and estimation of these effects is carefully discussed. Especially, we derive the variance estimates of these effects by following the delta methods and Taylor series approximations for the purpose of making marginal inference. Parameter estimates of Type I and Type II m2RGGMs are obtained through maximum likelihood estimation. An empirical analysis of longitudinal healthcare costs collected in the China Health and Nutrition Survey is conducted using the proposed methodologies.

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“…These marginalized two-part random effects models will lead to treatment effects on the overall expenditure that are homogenous over the random effects. Yet, as noted by Zhang, Liu, & Hu (2018), these models are not truly marginal models, in the sense that they estimate population-average estimates, but instead estimate treatment effects that are conditional on subject-specific random effects (Diggle et al, 2002). However, this conditional property of the model applies to all (generalized) linear mixed-effects models, and it is not necessarily something negative.…”

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“…To overcome this drawback, Zhang et al. 9 developed two types of marginalized two-part random-effects generalized Gamma models (abbreviated as “generalized Gamma m2REMs”; “m2REMs” stands for “marginalized two-part random-effects models”): the Type I generalized Gamma m2REM was constructed for marginal inference on positive healthcare expenditures and the Type II generalized Gamma m2REM was for marginal inference on overall healthcare expenditures.…”

“…This article is devoted to developing the marginalized two-part random-effects models, in which the specification of generalized Gamma distribution in Part (II) is replaced by the specification of log-normal, log-skew-normal, Gamma, and inverse Gamma distribution. The motivation to further develop these models is that, because of the particular setup in the heteroscedastic variance model for one of the two shape parameters in the generalized Gamma m2REMs in Zhang et al., 9 the generalized Gamma m2REMs do not include the m2REMs with other distribution specification as their special cases anymore. The generalized Gamma distribution has been reviewed in Zhang et al.…”

“…The generalized Gamma distribution has been reviewed in Zhang et al. 9 Note that, in the generalized Gamma distribution, there are two shape parameters σ and κ that dominate the variance function of the distribution. In Zhang et al., 9 the heteroscedastic variance among positive panel healthcare expenditures is characterized by one of the shape parameters through log(σit|Yit>0,Xit)=Xit'δ whereas the other parameter κ was left constant.…”

“…9 Note that, in the generalized Gamma distribution, there are two shape parameters σ and κ that dominate the variance function of the distribution. In Zhang et al., 9 the heteroscedastic variance among positive panel healthcare expenditures is characterized by one of the shape parameters through log(σit|Yit>0,Xit)=Xit'δ whereas the other parameter κ was left constant. In such circumstances, the condition σ=κ or σ=-κ will never be satisfied in the generalized Gamma m2REMs, unless the heteroscedastic variance is omitted.…”