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

APPROPRIATENESS OF THE NORMAL ERRORS ASSUMPTION

While the goals of the payment regression are more policy-oriented than statistical, the fully specified regression should be evaluated from a statistical viewpoint. The variables that are significant predictors of payment and thus potential payment variables are identified with the fully specified regression model.

A correct interpretation of the fully specified regression relies on basic linear regression assumptions to hold, namely that the error terms are normally distributed. Figure l(a) is a standard regression diagnostic plot that indicates whether the errors are normally distributed. If the errors are normally distributed, then the observations should fall along the solid diagonal line in that graph. However, Figure 1 (a) indicates that the normal errors assumption is questionable, given the number of observations that fall below the line on the left-hand side and fall above it on the right-hand side. This pattern indicates that an error distribution with heavier tails might be more appropriate given the data (Weisberg 1985).

When faced with the presence of statistical outlier IRFs in the data, a few options are typically employed in practice:

1. Fit the model using standard linear regression, with all observations. Potential problems with this approach are that statistical outliers could unduly influence regression coefficients and thus payment adjustments. The statistical outlier IRFs could be unusual enough that they should be excluded from the payment regression for policy reasons; for example, if the statistical outlier IRFs were not subject to the same constraints or rules as the other IRFs. For instance, IRFs that were in their phase-in periods under the former payment system during 1998 and 1999 and were thus establishing their costs could possibly be different from IRFs already subject to the payment system.

2. Delete statistical outlier IRFs from the data and then derive the facility payment adjustments. This would be reasonable if it were clear that the statistical outlier IREs do not represent the universe of facilities on which standard IRE PPS payments should be calculated, perhaps because of the phase-in period or because of reporting errors. However, deleting certain IRFs from the calculation of the facility payment adjustment without a good reason could be criticized as subjective. If there is not a clear reason for deleting them from the data, then deleting them risks losing potentially valuable information about the relationship between cost and IRF characteristics. Deleting IRFs from the analysis relies strongly on the normal errors assumption of linear regression; it may be the case that this assumption is unwarranted and an alternative model would be more appropriate.

3. Fit a statistical model to the data that is sufficiently robust to statistical outliers such that coefficient estimates are relatively insensitive to their presence in the data. Health policy practitioners find linear regression to be valuable because of its overall interpretability and ease of use (Sheingold 1990), so ideally an alternative to standard linear regression should be readily interpretable as well. Robust regression provides such an alternative. Robust regression procedures are available in some commonly used software packages, such as Stata's "rreg" procedure (Stata Corporation 2003; Hamilton 1998). The rreg procedure first deletes gross outliers from the data and then computes observation-specific weights for the contribution of each observation to the final regression; these weights take on values between 0 and 1, such that a case with weight 1 will be fully counted while a case with a weight of 0 will be deleted from the analysis. The user specifies tuning constants prior to the analysis that determine the degree to which each observation will affect the regression coefficients in the final model. Because rreg does not allow for the facility weights to be properly included in the model to reflect the per-discharge payment policy of the IRF PPS--that is, the [n.sub.i], from Equation (1)--we do not further consider it as an option for developing facility payment adjustments. We fit a Bayesian outlier accommodation model to the data, therefore, which is robust to the presence of outliers while allowing for the facility weights to be included in the analysis.