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Paired data occur in many experimental situations. When one views the subjects as a random sample from some large population, it may seem reasonable to model the data according to the typical one-way random effects analysis of variance (ANOVA). It is then usually of interest to estimate variance components and intraclass correlation. These estimators can be biased if key assumptions are violated, leading to erroneous interpretations and conclusions. We focus upon assumptions about the equality or inequality of means and/or variances of the two measures on each subject. In the framework of the one-way random effects ANOVA model, and three generalizations of it, we document estimators obtained as solutions to the likelihood equations. We consider the potentially serious effects of mistaken assumptions. Our findings suggest that the most general model considered is most desirable if consistent and efficient estimation of the between-subject variance component and intraclass correlation is the main goal. We also briefly connect our exposition to the study of reliability or agreement.

Statistical analysis of difference between groups in change for some variable, adjusting for initial value, is complicated by the presence of intra-individual variation in that variable. We estimate here the asymptotic bias that results from calculating the adjusted between-group difference by ordinary least squares (OLS) from observed data. We also present explicit formulae that use the OLS estimates, the difference between treatment groups in mean initial values, and a measure of the intra-individual variation to compute a corrected estimator and its variance. Alternatively, we can use OLS on transformed data to obtain unbiased estimates, in which we replace initial observed values by conditional Stein estimates of true values. We illustrate the results with data from an observational study and a clinical trial.

Many studies over the last 20 years have used logistic regression to model the relationship between the risk of developing coronary heart disease (CHD) and the levels of risk factors such as high blood pressure, high serum cholesterol, and cigarette smoking. Subsequently, several investigators have proposed the use of some of the published estimated logistic risk functions to predict risk in new populations. Because of great variation in definition of event, duration of follow-up, population characteristics, definition of risk variables, and selection of other variables in the logistic functions, direct use of such established functions would generally not have validity for the prediction of absolute risk levels. A review of fifteen of these studies indicates on the one hand generally similar results in direction and order of magnitude of effects of the major risk factors, confirming the importance of these risk factors of CHD. On the other hand the reviews indicate sufficient variation to suggest that extrapolation to new populations even to predict relative risk is not justified.