The extremal quotient in small-area variation analysis

This article reviews the current small-area variation analysis (SAVA) approach to population-based rates of surgery, and describes a new method for ascertaining variance based on the beta-binomial probability distribution of small-area rates. The critical review of the current SAVA approach focuses (1) on how incidence rates are calculated, and (2) on how the significance of the observed magnitude between the largest and smallest rates (i.e., the external quotient) is ascertained. While reducing the problems of calculating rates by considering only certain operative procedures, the new method addresses the current inadequacies of ascertaining significant differences among small areas. Not only does it correctly assess likelihood of an extermal quotient, it also can determine the particular area's rate, producing an unlikely extermal quotient. The method evaluates the probability that the observed magnitude of the extremal quotient is due solely to chance and study design effects, and tables of these probabilities are available for the method's application. A mathematical model, based on a combination of the binomial and beta distributions, uses (1) the sample size, (2) the average of the areas' rates, (3) the variance among the rates, and (4) a specific quotient level to determine the probability of observing the quotient by chance. After computerizing this calculation, probability tables for reasonable values of these four parameters are generated. In addition to looking at just one quotient for each sample, the probability tables facilitate the easy examination of intermediate quotients when the extremal quotient is unlikely due to chance. By alternatively ignoring the highest and lowest rates, two new quotients can be produced and tested. Given that one of these two quotients is likely due to chance, the excluded rate (i.e., producing the unlikely extremal quotient) can be classified as an outliner, and the associated small area should be the focus of more detailed investigation. The probability tables reveal that the external quotient is not the appropriate statistic to be applied in studies where many small areas are to be included. The probability of seeing even a "large" extremal quotient simply by chance rapidly approaches one as the sample size increases. However, an extremal quotient modeled from a beta-binomial distribution can be useful for studies with small sample sizes (e.g., six counties). The use of this beta-binomial model for small-area rates provides a new method of designing and evaluating small-area studies where costs or domain limit the number of areas under consideration.(ABSTRACT TRUNCATED AT 400 WORDS)