Social/health maintenance organization and fee-for-service health outcomes over time - Hospital Payment: Beyond the Prospective Payment System

Health Care Financing Review, Winter, 1993 by Kenneth G. Manton, Robert Newcomer, Gene R. Lowrimore, James C. Vertrees, Charlene Harrington

To estimate parameters for external variables (for validation), or transition rates, two steps are needed. First, equations 3 and 4 are maximized for J health variables. Then, the parameters for the J variables in equations 3 and 4 are fixed to hold constant the definition of the K classes (i.e., [[lambda].sub.kjl]) and individual scores (i.e., [g.sub.ik]). Then equation 4 is maximized to produce conditional (on case mix) maximum likelihood estimation of [[lambda].sub.knl] for the N added variables. Likelihood ratio tests can be formed to determine if external variables contain significant information not represented in case-mix groups. Mortality and coverage change probabilities may be estimated by defining transition variables for each case-mix group, i.e., [Mathematical Expression Omitted] ([I.sub.1] are time intervals, and [I.sub.2] changes in status). Transition rates are estimated in a second maximum likelihood step again with the definition of case-mix groups fixed. The [Mathematical Expression Omitted] describe discrete changes (e.g., death, coverage change) over 3 years of followup. They do not describe cohort changes.

Active Life Expectancy

The [g.sub.ik[multiplied by]t] and [[lambda].sub.kjl] describe all information on health and mortality in 3 years of followup of an initially non-institutionalized population. They do not describe age-specific survival and disability changes for a cohort of such persons. This requires solving systems of difference equations for monthly intervals, to approximate life table differential (continuous time) equations. In those calculations, two additional equations are needed. The first describes health changes among survivors t to t 1:

(5) [g.sub.ik(t 1)] = {Age[multiplied by][[beta].sub.kk]}([g.sub.ik[multiplied by]) [e.sub.ik.], where {Age[multiplied by][[beta].sub.kk]} is a matrix of age-dependent transition rates between K case-mix groups. Four [[beta].sub.kk] matrices are estimated, one each for FFS and S/HMO males and females. The definition of [g.sub.ik[multiplied by]t]s in equation 2 ensures their comparability over gender, coverage, and site (Manton, Woodbury, and Tolley, 1994).

The second describes mortality as an age-dependent quadratic function of the [g.sub.ik[multiplied by]t],

(6) [Mathematical Expression Omitted]

In equation 6 all coefficients in Q are multiplied by [e.sup.[theta]t]. [theta] is the percent per year of age increase in mortality. In equation 6, a person's risk changes as [g.sub.ik[multiplied by]t] changes according to equation 5. The performance of S/HMOs and FFS in maintaining function is described by [[beta].sub.kk]; and for survival by [Q.sub.t]. [theta] is the age-related, average effect of unobserved variables for FFS and S/HMO males and females. As information in [g.sub.ik[multiplied by]t] increases, [theta] [right arrow] 0.0 (Manton et al., to be published).

Calculating cohort life tables requires using parameters in equations 5 and 6 to solve monthly difference equations. The proportion of a cohort, I, surviving to t 1, is,