Estimation of a Simple Regression Coefficient in Samples Arising from a Sub-Sampling Procedure

Situations occasionally arise in epidemiological studies wherein a sub-sample of the entire study group is selected in an other than simple random manner and one or more specified measurements are made only on members of the sub-sample. Such selections can make it inappropriate to apply usual analysis procedures. The present paper considers two particular problems: a) Estimation of a simple regression coefficient between two variables, measured in the sub-sample only, when selection is on the basis of a correlated third variable measured on the entire group. b) Estimation of a simple regression coefficient between two variables, measured in the sub-samiple, when the dependent variable is measured on the whole group and is the basis for sub-samnpling. The usual regression coefficient estimate is shown to be biased, asymptotically unbiased estimates are developed along with their asymptotic variances and small sample behavior is investigated via some Monte Carlo studies.