Smooth quantile ratio estimation with regression: estimating medical expenditures for smoking-attributable diseases

The methodological development of this paper is motivated by a common problem in econometrics where we are interested in estimating the difference in the average expenditures between two populations, say with and without a disease, as a function of the covariates. For example, let Y(1) and Y(2) be two non-negative random variables denoting the health expenditures for cases and controls. Smooth Quantile Ratio Estimation (SQUARE) is a novel approach for estimating Delta=E[Y(1)] - E[Y(2)] by smoothing across percentiles the log-transformed ratio of the two quantile functions. Dominici et al. (2005) have shown that SQUARE defines a large class of estimators of Delta, is more efficient than common parametric and nonparametric estimators of Delta, and is consistent and asymptotically normal. However, in applications it is often desirable to estimate Delta(x)=E[Y(1)|x]--E[Y(2)|x], that is, the difference in means as a function of x. In this paper we extend SQUARE to a regression model and we introduce a two-part regression SQUARE for estimating Delta(x) as a function of x. We use the first part of the model to estimate the probability of incurring any costs and the second part of the model to estimate the mean difference in health expenditures, given that a nonzero cost is observed. In the second part of the model, we apply the basic definition of SQUARE for positive costs to compare expenditures for the cases and controls having 'similar' covariate profiles. We determine strata of cases and control with 'similar' covariate profiles by the use of propensity score matching. We then apply two-part regression SQUARE to the 1987 National Medicare Expenditure Survey to estimate the difference Delta(x) between persons suffering from smoking-attributable diseases and persons without these diseases as a function of the propensity of getting the disease. Using a simulation study, we compare frequentist properties of two-part regression SQUARE with maximum likelihood estimators for the log-transformed expenditures.