Generalized modeling approaches to risk adjustment of skewed outcomes data

Abstract

There are two broad classes of models used to address the econometric problems caused by skewness in data commonly encountered in health care applications: (1) transformation to deal with skewness (e.g., ordinary least square (OLS) on ln(y)); and (2) alternative weighting approaches based on exponential conditional models (ECM) and generalized linear model (GLM) approaches. In this paper, we encompass these two classes of models using the three parameter generalized Gamma (GGM) distribution, which includes several of the standard alternatives as special cases—OLS with a normal error, OLS for the log-normal, the standard Gamma and exponential with a log link, and the Weibull. Using simulation methods, we find the tests of identifying distributions to be robust. The GGM also provides a potentially more robust alternative estimator to the standard alternatives. An example using inpatient expenditures is also analyzed.

Introduction

Many past studies of health care costs and their responses to health insurance, treatment modalities or patient characteristics indicate that estimates of mean responses may be quite sensitive to how estimators treat the skewness in the outcome (y) and other statistical problems that are common in such data. Some of the solutions that have been used in the literature rely on transformation to deal with skewness (most commonly, ordinary least square (OLS) on ln(y)), alternative weighting approaches based on exponential conditional models (ECM) and generalized linear model (GLM) approaches, the decomposition of the response into a series of estimation models that deal with specific parts of the distribution (e.g., multi-part models), or various combinations of these. The default alternative has been to ignore the data characteristics and to apply OLS without further modification.

In two recent papers, we have explored the performance of some of the alternatives found in the literature. In Manning and Mullahy (2001), we compared models for estimating the exponential conditional mean—how the log of the expected value of y varied with observed covariates x. That analysis compared OLS on log transformed dependent variables and a range of GLM alternatives with log links under a variety of data conditions that researchers often encounter in health care cost data. In Basu et al. (2004), we compared log OLS, the Gamma with a log link, and an alternative from the survival model literature, the Cox proportional hazard (PH) regression (Cox, 1972). In both papers, we proposed a set of tests that can be employed to select among the competing estimators, because we found no single estimator dominates the other alternatives or is a close second best.

Our primary interest is in the marginal effect of a covariate x₁ on E(y|x), where x₁ could be a treatment or behavioral variable of interest, in the context of modeling a response variable y as a function of a vector x = (x₀, x₁, x₂, …, x_p,)^T of covariates in a regression model for the mean function μ(x) ≡ E(y|x).¹ In this paper, we again compare exponential conditional mean models where μ(x) is assumed to follow a functional form that is the exponentiation of the linear combination of covariates x.² If E(y|x) is an exponential conditional mean, then the marginal effect, which is nonlinear in x, is:

$$m_1(x)=\frac{\partial E(y|x)}{\partial x_1}={\beta }_1{\text{e}}^{x^{\prime }\beta }$$

where $x^{\prime }\beta ={\sum }_{k=0}^p{\beta }_k{x}_k$ and x₀ may be a vector of ones. But if we log both sides, then we can summarize the marginal effect by:

$$\frac{\partial \text{ln}(E(y|x))}{\partial x_1}=\frac{\partial \text{ln}(m_1(x))}{\partial x_1}={\beta }_1$$

In what follows, we focus on this as a summary of the response of y to x.

In Manning and Mullahy (2001), we explored the performance of alternative least squares and generalized linear model estimators for the response of the expected value of y to a set of covariates x under a range of data generating processes. No single estimator was dominant or nearly dominant under all circumstances. But two patterns were clear. First, least squares could provide biased estimates of the mean response of the (untransformed) outcome variable if there was heteroscedasticity in the log scale error. Second, the GLM models would be unbiased but could be quite imprecise if the log-scale error was symmetric but heavy tailed or if the log-scale error variance is large (>1). We proposed a set of tests that would allow analysts to choose among the competing exponential conditional mean (ECM) models.³

This paper takes a different approach. It considers the estimation of a regression model using maximum likelihood for a specific distribution—the generalized Gamma (GGM) distribution. The generalized Gamma is appealing because it includes several of the standard alternatives as special cases—OLS with a normal error, OLS for the log-normal, the standard Gamma and Exponential with a log link, and the Weibull. We see two potential advantages of implementing a regression framework based on this distribution. First, it provides nested comparisons for some alternative estimators, and hence a formal alternative to the somewhat cumbersome and incomplete testing procedure in Manning and Mullahy (2001). Second, if none of the standard approaches is appropriate for the data, then the generalized Gamma regression provides an alternative estimator that will be more efficient because it better approximates the distribution than the more restrictive alternatives.

The plan for the paper is as follows. In the next section, we describe the generalized Gamma distribution in greater detail, showing the connection of the GGM regression framework to more commonly used estimators. Section 3 describes the general modeling approaches that we consider, and our simulation framework. Section 4 summarizes the results of the simulations and examines an application: (1) a study of inpatient expenditures that we have used in previous papers. The final section contains our discussion and conclusions.