An integer programming model to limit hospital selection in studies with repeated sampling

Health Services Research, June, 1995 by Michael Shwartz, Ronald K. Klimberg, Melinda Karp, Lisa I. Iezzoni, Arlene S. Ash, Janelle Heineke, Susan M.C. Payne, Joseph D. Restuccia

Given this hospital classification, a reasonable alternative sampling strategy would be to randomly sample areas within each area category (i.e., high, medium, and low) for each condition, and then review records at the major hospitals associated with those areas, perhaps in proportion to the percentage of discharges accounted for. However, based on a simulation study described further on, we determined that sampling separately for each condition would result in more hospitals than logistically or budgetarily feasible. Therefore, we decided to concentrate the record reviews by selecting hospitals that could be used in as many of the condition-specific studies as possible. Before describing modifications to the model formulation presented earlier and results from the application, we describe the power analysis performed for this type of problem.

Power Analysis

For each of the seven selected conditions, the principal study question is the following: Does the rate of inappropriateness differ for small areas with high, medium, and low rates of hospitalization? For this type of analysis, hospital discharges are viewed as nested within small areas, and areas are treated as a random effect. The effect of hospital within area is considered negligible, since the referral patterns of local physicians are hypothesized as the principal determinants of admissions.

The parameters on which power depends are the following: [Mathematical Expression Omitted] = the variance of the average inappropriateness rates across the area categories high, medium, and low; [Mathematical Expression Omitted] = the variance of the inappropriateness rates among areas within each of the area categories; and [Mathematical Expression Omitted] = the variance of inappropriateness among discharges in the same area. Let p be the number of distinct area categories examined (which initially is three - high, medium, and low), q be the number of areas sampled per area category, and n be the number of discharges sampled from each area. The test of the hypothesis of no difference in inappropriateness among the area categories references an F-statistic with (p - 1) and p(q - 1) degrees of freedom (Winer 1971). When [Mathematical Expression Omitted] is not zero (that is, when inappropriateness does differ by area category), the computed F-statistic has noncentrality parameter [Phi], where [Mathematical Expression Omitted]. Table C.14 in Winer (1971) indicates power for different values of [Phi] as a function of the degrees of freedom of the F-statistic.

Power calculations were performed under the following assumptions: [[Sigma].sub.e] is approximately .3 (the standard deviation for a 0/1 variable, inappropriateness, with a rate of 10 percent) and [[Sigma].sub.b], is on the order of .03 (which suggests that most areas within the same category will be within .06 of the category mean). Based on power analyses of a variety of alternative sampling plans, we initially decided to select six areas per area category (q = 6) and 30 cases per area (n = 30), for a total of 540 cases per condition. This gave power of .82 to detect variability across area categories that is 80 percent or more greater than variability of areas within the same category (i.e., [[Sigma].sub.a]/[[Sigma].sub.b] = 1.8). This translates into a difference in average inappropriateness rates in which the low and high categories differ from the middle category by about .07 (compared to within-level differences among areas of .06).