Identifying and accommodating statistical outliers when setting prospective payment rates for inpatient rehabilitation facilities

Health Services Research, Dec, 2004 by Susan M. Paddock, Barbara O. Wynn, Grace M. Carter, Melinda Beeuwkes Buntin

BAYESIAN ANALYSIS, COMPUTATION, AND PRIOR DISTRIBUTIONS

We reanalyze the data used to derive the facility payment adjustments for the IRF PPS using the Bayesian outlier accommodation model. Not only do we estimate regression coefficients using this model, but we also estimate how non-normal the error distribution is, given the data. To do this, we estimate the degrees of freedom of the t-distribution, v, as part of the analysis.

Several introductory articles on Bayesian methods have appeared in the health literature (Spiegelhalter et al. 1999; Harrell and Shih 2001). Below we provide a basic description of the Bayesian approach but readers who want more information may wish to consult those articles. The Bayesian approach centers on Bayes's theorem: p([theta]|]y) [alpha] f(y|[theta])p([theta]). The first step of the Bayesian analysis is the same as that of the standard, non-Bayesian analysis: one specifies a likelihood function for the data, which is f(y|[theta]) in Bayes's theorem, where [theta] represents the model parameters (i.e., the regression coefficients, the variance, [[sigma].sup.2], the degrees of freedom parameter, v, and [lambda]). The Bayesian approach treats the parameters of the model as random variables and requires that prior distributions be specified for them; these prior distribution are denoted by p([theta]) in Bayes's theorem. (1) The prior distribution for a given parameter quantifies the beliefs of the analyst about that parameter prior to the analysis. (2) The Bayesian analysis proceeds by multiplying the prior distribution and the likelihood function together to obtain the posterior distribution of the model parameters, P([theta]|y). All Bayesian statistical inference is based on the posterior distributions of the model parameters.

The prior distributions chosen for this analysis are very diffuse with high variances to reflect our desire to express ignorance prior to the analysis about the possible values for the regression parameters. The prior distribution for each regression coefficient follows a normal distribution with mean 0 and a standard deviation of 100. This prior choice reflects that we have little prior knowledge about the values of the regression coefficients. Considering that the largest standard error for the regression coefficients in Table 2(a) is 0.21, this prior distribution is indeed very vague. The inverse of the variance, [[sigma].sup.-2], follows a Gamma distribution with parameters (0.01, 0.01). These prior distribution choices have historically been standard choices in Bayesian linear regression (DeGroot 1970), though both choices are sufficiently noninformative enough that the posterior estimates are insensitive to them in this analysis.

The model is fit using three specifications to assess the sensitivity of results to the choice of the prior distribution for v. The first specification places equal prior probability on values of v that support the assumption of approximately normal errors (of standard linear regression) versus smaller values of v that suggest the use of a heavier tailed t-distribution; the second prior places probability 0.01 on v being greater than or equal to 30; and the third places probability 0.2 on such an event a priori. The three prior specifications for v that were examined are: