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

(7) [Mathematical Expression Omitted] where Q (t) is a vector of means of [g.sub.ik[multiplied by]t] and [V.sub.t] their covariance matrix. Equations 8 and 9 show that g [sup.*](t) and ([Mathematical Expression Omitted]) are functions of mortality ([Q.sub.t]) and case mix heterogeneity Vt) or,

(8) [Mathematical Expression Omitted] and

(9) [Mathematical Expression Omitted]

Mortality depends on g(t) and Vt. Vt has deterministic (i.e., Age [multiplied by] [[beta].sub.kk]) and stochastic components. Diffusion increases, and mortality reduces, [V.sub.t]. Diffusion must reflect the 0,1 bounds on the [g.sub.ik[multiplied by]ts. We assume that [V.sub.t] has, at most, Bernoulli variance, (g [sub.k](t 1)[multiplied by](1-g (t 1)), and that correlations of [g.sub.ik[multiplied by]t]s are constant from t to t 1. The correlation matrix R is estimated from [V.sub.t] after conditioning on age. in the diagonal matrix, S, elements are square roots of the ratios of [g.sub.ik[multiplied by]t] variances to Bernoulli limits. S projects the [g.sub.ik[multiplied by]t] to a high dimensional spherical space so that, in computations, [g.sub.ik[multiplied by]t]s are not "trapped" on "faces" of B. [W.sub.t] 1 is a diagonal matrix with elements

[Mathematical Expression Omitted] The new "constrained" variance is

(10) [V.sub.t 1] = [W.sub.t 1] [SRSWT.sub.t 1], which can be used to estimate a constrained diffusion matrix,

(11) [Mathematical Expression Omitted] where [C.sub.t] = Age [multiplied by] [[beta].sub.kk] from equation 5. Changes in the means ([g.sub.ik[multiplied by]t] (t)) of [Mathematical Expression Omitted] for survi are

(12) [Mathematical Expression Omitted]

Equations 7 through 12 are used to calculate cohort life tables for K case-mix groups. Cohort life tables differ from transition variables estimated in equation 2 because the 3-year experience of initially non-institutionalized persons of different ages is used to construct disability dynamics and mortality for the life of a cohort. Thus, there are three distinct sets of calculations. One generates the [g.sub.ik[multiplied by]t], describing cases from the pooled data using equation 2. In those calculations we may estimate 3-year transition rates. Second, [g.sub.ik[multiplied by]t]s are used to generate parameters for disability dynamics (in equation 5) and mortality (in equation 6). Those parameters are used in difference equations 7 through 12 to calculate cohort life tables. The individual components of cohort dynamics can be examined by fixing selected parameters in equations 7 through 12.

In a hazard model, the risk of an event is estimated for fixed covariates (Cupples et al., 1988). The difference equations use time-varying covariates. Thus, the difference equations produce insights about cohort dynamics that cannot be made using only a hazard function. The quadratic in equation 6, one component of the cohort calculations, is a hazard function. It is estimated by maximum likelihood and, because [mu] (the mortality rate) is estimated directly, there are no problems of interpreting coefficients as in Cox or logistic functions (e.g., including quadratic terms in a Cox model makes the hazard scale dependent)--the function changes as risk factor levels change.