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

(1) [gamma] = 1/v and a Beta prior is placed on [gamma] such that the probability of v being greater than 30 is 0.5;

(2) [gamma] = 1/v and a Beta prior is placed on 7 such that the probability of v being greater than 30 is 0.01 (Verdinelli and Wasserman 1991);

(3) v can take values 2, 4, 6, 8, 10, 12, 15, 20, 30, or 50 with probability 1/10 for each possible value (Spiegelhalter et al. 1995).

We found that the posterior distributions were insensitive to these choices of priors and so only present the results based on specification (1).

The Bayesian outlier accommodation model is easy to fit in a freely available software package called WinBUGS Version 1.3 (Spiegelhalter, Thomas, and Best 2000). It is downloadable from http://www.mrc-bsu.cam. ac.uk/bugs/. A tutorial article on WinBUGS software is available from Fryback, Stout, and Rosenberg (2001). WinBUGS is especially useful for Bayesian analyses for which the posterior distributions of interest cannot be analytically derived. Within WinBUGS, one specifies the likelihood function and prior distributions and then WinBUGS derives the posterior distributions of interest by using a numerical integration technique called Markov Chain Monte Carlo (MCMC) simulation. The MCMC simulation is iterative; once starting values are specified for model parameters, MCMC simulation is used to repeatedly draw random samples from the posterior distributions of interest. Posterior means and standard deviations of model parameters can be computed directly from these MCMC samples. WinBUGS has several diagnostic tests housed within it to help the user assess whether the MCMC simulation algorithm has converged. The WinBUGS code used for the analysis can be requested from the corresponding author of this paper.

RESULTS: FULLY SPECIFIED BAYESIAN OUTLIER ACCOMMODATION MODEL

The Bayesian outlier accommodation model is fit using the same dataset used to produce Tables 2(a) and 3(a), which appear in Carter et al. (2002). Table 2(b) shows the 95 percent posterior probability intervals for the regression parameters. The 95 percent posterior probability interval is the Bayesian analogue to the 95 percent confidence interval, though note that their interpretations are different: The Bayesian 95 percent posterior probability interval for the freestanding regression coefficient can be interpreted as the probability that the freestanding- coefficient is in (0.013, 0.099) is 0.95, whereas the analogous 95 percent confidence interval is interpreted as the probability that the interval, (0.012, 0.094), contains the true, fixed value of the freestanding coefficient is 0.95. The regression coefficients of Tables 2(a) and 2(b) largely agree. The posterior mean of v is 7.9, with a 95 percent posterior probability interval of (4.7, 14.1), which provides evidence that the correct error distribution is indeed fatter-tailed than a normal (recall from Figure 1 that values of v greater than 30 indicate approximate normality of the regression errors). Hence, the two columns labeled "mean" in Table 2 differ because the assumed error distributions differ, which determines how the extreme observations affect regression parameter estimates. Though the posterior mean for low income measure is lower than the estimate in Table 2(a) (0.294 vs. 0.359), this variable is significantly related to cost in both models, as the 95 percent confidence interval of Table 2(a) and the 95 percent posterior probability interval of Table 2(b) fall above 0. One disagreement between Table 2(a) and Table 2(b) is that the indirect teaching- variable is significantly related to log(cost) in Table 2(b) but is not significant in Table 2(a); the posterior mean of the indirect teaching coefficient is 0.438 in Table 2(b), with 95 percent posterior probability interval of (0.0,52, 0.820), whereas in Table 2(a) the estimate is 0.342 with 95 percent confidence interval of (-0.0(i4, 0.748). This is particularly interesting- because characteristics that were significantly related to cost after adjusting for other facility characteristics were more carefully considered for inclusion in the payment regression model; teaching is compensated under the acute care PPS; and there were issues regarding the availability of a reliable resident count measure. In particular, full-time equivalent counts for number of residents in freestanding IRFs and in rehabilitation units are not provided on the cost report. To be comparable with the number of residents reported for a rehabilitation unit, the number of residents assigned to the routine area of a freestanding hospital was estimated by multiplying its total resident count by the ratio of resident salaries apportioned to routine areas to the total resident salaries for the facility (Carter et al. 2002).