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 OUTLIER ACCOMMODATION MODEL

We explore replacing the standard linear regression model with a Bayesian outlier accommodation model for developing payment adjustments. The Bayesian outlier accommodation model downweights the influence of statistical outlier IRFs relative to standard linear regression and allows for IRFs with unusually high variance in costs to remain in the data without unduly influencing the regression coefficients. Structurally, the Bayesian outlier accommodation model is very similar to the standard linear regression model, with the key difference being that the error term for IRF i is normally distributed with variance [[sigma].sup.2] /[[lambda].sub.i] [n.sub.i]), rather than [[sigma].sup.2] /[n.sub.i], as in standard weighted linear regression. The [[lambda].sub.i] term for IRF i indicates the degree to which IRF i is unduly influential in the standard linear regression analysis and the extent to which its contribution to the estimation of the regression parameters should be downweighted relative to the standard normal linear regression. In the standard linear regression, 2, is equal to 1 for all IRFs. If IRF i is unduly influential, then [[lambda].sub.i] is less than 1 in the Bayesian outlier accommodation model, thereby effectively reducing the IRF weight from [n.sub.i] to [[lambda].sub.i][n.sub.i] The [[lambda].sub.i] terms are unknown and, along with the regression coefficients, must be estimated from the data. In contrast to standard robust regression such as that offered in Stata's rreg, the outlier accommodation is explicitly included in the model formulation by entering these [[lambda].sub.i] terms into the model, as opposed to using user-specified tuning constants. Since each [[lambda].sub.i] is always greater than 0, no observation is ever completely deleted from the analysis.

The specification of the Bayesian outlier accommodation model is completed by specifying a distribution for the [[lambda].sub.i] The [[lambda].sub.i] terms are modeled as following a Gamma distribution with parameters (v/2, v/2) with mean 1. The complete model specification implies that log(cost) follows a t-distribution with v degrees of freedom (Verdinelli and Wasserman 1991; Gelman et al. 1996). The t-distribution is a symmetric distribution, just like the normal distribution, but it has heavier tails and allows for the existence of more unusual, higher variance data points than does the normal distribution. The degrees of freedom parameter, v, indicates how heavy tailed the distribution is relative to the normal; Figures 1(b)-(f) show the normal quantile plot for errors generated from t-distributions with 50, 30, 20, 10, and 5 degrees of freedom, denoted by "t(50)" through "t(5)" on Figure 1. The larger the degrees of freedom, the closer the t-distribution is to the normal distribution. In practice, t-distributions with degrees of freedom of 30 or greater are considered to be approximately normal. The t-distribution is a good choice for these data; Figure 1 (a) shows the regression diagnostic plot for the fully specified standard linear regression of Table 2(a), and the figure indicates that not only are there IRFs that appear to be statistical outliers, but the assumption of normally distributed errors is questionable, as Figure 1(a) more closely resembles Figures 1(e)-(f)than Figures 1(b)-(c).