On the Use of Repeated Measurements in Regression Analysis with Dichotomous Responses

Summary Although sslany snedical follow-up studies provide for repeated measurements of patient characteristics at regular intervals usual methods of analyzing these data typically make use of only one measurement of each characteristic in any single model. For example regression models for predicting death cr other morbid events often use either the first or the snost recent measurement of each characteristic. This results from the requirement in standard regression models that the nutnber af independent variables must be constant over the set of individuals and period of observation. In this paper however a method is proposedfor including these accumulating measurements in the regression analysis as they become available. The method is used to relate the occurrence of Cardiovascular Disease ( C VD) to the levels of repeated zneasurementd of serum cholesterol (SC) for participants in the Framingham Heart Study (Kannel 1976). HierarcSlical models are described in which an analysis using only the most recent SC can be tested against an analysis using only the first measmrement or using all available measurements. For the Framingham data nodels available in the more general setting give significantly betterfit than the usual models. Further in contrast with usual practice the first (baseline) measures7lent of SC is found to be more predictive than the most recent measurement. Although the regression nlethodolog.y described is directly applicable to other problems with regularly repeated measuremeslts and dichotomous response the set of hierarchical models considered will be specific to the problem under study. Many long-term studies of chronic disease provide for regular examination of patients to obtain measurements of characteristics thought to be related to the disease under investigation. But standard methods of analyzing follow-up data require that the number of measurements used in a regression analysis be constant over patients and over time. The widely used methods of Walker and Duncan (1967) and Cox (1972) and the recent work of Prentice and Gloecker (1978) and Crowley and Hu (1977) illustrate this requirement. Thus, the analysis uses only part of the accumulating data to derive regression relationships. This paper presents a method which allows the use of all available measurements in a regression analysis. The dependent variable is occurrence or non-occurrence of the event under study during each interval of follow-up. We also assume that the probability of occurrence is a multiple logistic function of the patient's measurements, but the method could be used with any regression function, including that of Prentice and Gloecker.