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

GoM was applied to pooled HSF and CAF data so that [g.sub.ik]s could be updated for health changes. The updated scores ([g.sub.ik].t) control for health variation, over individuals and time (t), in comparing S/HMO and FFS outcomes. In "pre-post" analyses, interventions are made at fixed times and do not describe systems with voluntary enrollment or disenrollment (such as S/HMOs) well, where interventions are of variable content, duration, and timing. Variables affecting choice interact with outcome--the decision to stay enrolled is made daily, and reflects the degree of satisfaction with services and outcomes.

In GoM, J multinomial variables for each of I persons ([x.sub.ij]) are each coded as [L.sub.j] binary (0, 1) variables, [y.sub.ijl]. Continuous variables are divided into [L.sub.j] intervals and then coded in binary form. The probability of [y.sub.ijl] occurring is (site and coverage indexes suppressed),

(1) [P.sub.ijl[multiplied by]t] = Prob([y.sub.ijl[multiplied by]t = 1.0) [[sigma].sub.k]([g.sub.ik

Both the [[lambda].sub.kjl] and the [g.sub.ik] are uniquely identified if J > 2K (Woodbury, Manton, and Tolley, to be published), because selecting J variables determines the space, M, of all possible responses, [y.sub.ijl]. The solution, B, is the intersection of the probability space, LB, defined by the [p.sub.ijl[multiplied by]t] estimated in equation 1, with the a priori determined M. Extreme points of B define the [[lambda].sub.kjl]. The [g.sub.ik]s are the linear functions joining [[lambda].sub.kjl]s. The [[lambda].sub.kjl] are assumed time invariant; time is represented in [g.sub.ik[multiplied by]t].

To assure [g.sub.ik[multiplied by]t]s are comparable between FFS and S/HMO and over time, the K profiles ([[lambda].sub.kjl]) are estimated from HSF and CAF data pooled over time, site, and coverage. The likelihood for the combined data is ( indicates a index for which data is combined),

(2) [Mathematical Expression Omitted] where measurement is at time t, c refers to coverage (e.g., S/HMO or FFS), and s to site (Manton et al., 1986; 1987). A person is given a CAF when a health change is detected in an annual RE-HSF, semiannual monitoring of impaired persons, or in an S/HMO clinical visit. Because scores change at variable times, we divided each record into months (i.e., t = 1, 2,..., 36). If a CAF is administered at t 1, new [g.sub.ik[multiplied by]t 1]s are calculated if health changed. Otherwise the [g.sub.ik[multiplied by]t] are assumed constant. By using monthly histories we can estimate the time spent in specific health states (i.e., having specific [g.sub.ik[multiplied by]t] values). This deals with the variable assessment times, because how long a person remains in a case-mix class is decribed by the [g.sub.ik[multiplied by]t].

The GoM likelihood in equation 2 (suppressing indexes for coverage, time, and site) is evaluated by iteratively solving two functions (Woodbury and Clive, 1974),

(3) [Mathematical Expression Omitted]

(4) [Mathematical Expression Omitted]

Normally, terms in a likelihood for individuals are collected in an independent factor and only structural parameters (i.e., those not involving i) are estimated (Cox and Hinkley, 1974). To factor individual from structural parameters, assumptions are made about the distribution of individual parameters so that the information in structural parameters is restricted to a "small" number of data moments (e.g., the (J x (J 1)]/2 unique elements in a covariance matrix for J variables in factor analysis). In equation 3, estimation of [g.sub.ik] involves [[lambda].sub.kjl]. In equation 4 estimation of [[lambda].sub.kjl] involves [g.sub.ik]. Thus, the sets of parameters are jointly estimated. This makes [[lambda].sub.kjl] estimates robust to individual variation because they are conditioned on the [g.sub.ik] distribution. Estimates of [g.sub.ik]s do not have a prespecified distribution but produce unbiased estimates of up to the Jth order moments of the [g.sub.ik] distribution (Woodbury, Manton, and Tolley, to be published). The [[lambda].sub.kjl] estimates are consistent because equation 3 implicitly constrains the moments of the [g.sub.ik] distribution across individuals.