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

Health services research studies frequently involve the review of medical records at a number of hospitals. With increased emphasis on outcomes research, such studies are likely to become more common. Unless performed by organizations like peer review organizations (PROs) as part of legally mandated review activities, record reviews require hospital approval, which can be difficult and time-consuming to obtain. Further, if reviews are done at the hospital, the expense of sending reviewers to a large number of geographically dispersed hospitals can be high, and the logistics of coordinating activities with many different medical record departments daunting. For these reasons, it is frequently necessary to concentrate medical record reviews among fewer hospitals than would be the case in the absence of such constraints. We have developed an integer programming model, a modification of the traditional "location" problem model (Revelle 1987), that can be used to identify a minimum set of hospitals at which to do reviews, subject to sampling design constraints derived from an analysis of statistical power.

The model is most useful in situations with the following characteristics:

1. Hospitals, or some unit with which hospitals are associated (e.g., small geographic areas), are classified into one of several categories (e.g., high, medium, and low). These categories are not necessarily exhaustive: some hospitals may not be classified into any of the categories. Further, the categories are not exclusive: a hospital may be classified into more than one category, for instance, into a high category because it treats many patients from small area A, which has a high hospital admission rate, and into a medium category because it also treats many of the patients from small geographic area B, which has a medium rate of admissions.

2. The classification is repeated several times. For example, several different diagnostic conditions may be studied and the classification done separately for each condition. The same hospital might be in a high category for condition 1, and a low category for condition 2. This situation is illustrated in Study I further on. Additionally, criteria other than condition (e.g., risk pool; see Study II) might be the basis for the repeated classifications.

3. Medical records are sampled for each condition, and analyses are performed separately for each condition.

Under these circumstances, the number of hospitals from which medical records are reviewed can be reduced by selecting hospitals that can be used in several of the different analyses. In what follows, we present the integer programming model and then briefly describe two studies in which the model has been of value. In discussing the first study, we describe power calculations for a random-effects nested design, a study design for which the model is particularly useful.

It should be noted that a key assumption underlying the value of the model is that the hypotheses being examined assume that the dominant influence on the behavior under study is not the hospital per se. In contrast, in Study I, the key factor is admitting practices of local area physicians. In Study II, we hypothesize that units within the hospital have relatively independent complication rates (e.g., high complication rates in endoscopy tell us little about the likelihood of high complication rates in invasive cardiology). As discussed in more detail in the conclusion, if the factor under study is the hospital, there really are not multiple repeated studies, as indicated in point 2 above, and an alternative sampling plan is required.

OVERVIEW OF THE HOSPITAL SELECTION MODEL

The objective of the model is to minimize the number of hospitals selected for medical record review subject to constraints imposed by consideration of statistical power, plus other constraints specific to a particular study. Here we present a general version of the model motivated by Study I, described in the next section.

Assume that small geographic areas are classified as high, medium, or low in terms of their hospital admission rates for a condition. The classification process is repeated for each of several conditions that will be studied. For each condition and each small area, there is a set of hospitals associated with the area - those most frequently used by people with that condition from that area.

Let i index hospitals, j index small geographic areas and k index conditions. For condition k; [H.sub.k] is the set of hospitals associated with areas with high hospitalization rates, [M.sub.k] is the set of hospitals associated with areas with medium hospitalization rates, and [L.sub.k] is the set of hospitals associated with areas with low hospitalization rates. [N.sub.H], [N.sub.M], and [N.sub.L] are the number of areas to be selected from the high, medium, and low categories, quantities determined as a result of an analysis of power. The Ns could differ by condition (in which case they would also be indexed by k), although we have not introduced this complexity in our applications. The unknowns are as follows: