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The Assessment of a Treatment Effect When a Covariate is Observed on a Subset of the Units. I. Simple Estimators ANCOVA

Rubin, Donald B.; Weisberg, Sanford.
Publication Year:
Report Number:
ETS Research Bulletin
Document Type:
Page Count:
Subject/Key Words:
Office of Education, Analysis of Covariance (ANCOVA), Mathematical Formulas, Statistical Analysis


Consider a completely randomized experiment with two groups, say treatment and control, and a covariate X that has been observed on a subset of the units. The objective is to estimate tau, the treatment effect on the variable Y which is recorded for all units. Although the average Y difference in the two groups, Y(bar, subD), is unbiased for tau, it is common to try to use X to increase the precision of the estimate of tav by forming a covariance adjustment; especially if the correlation between X and Y, rho, is large. If X is not fully observed it is not clear whether to and/or how to form the covariance adjusted estimate. In addition to Y(bar, sub D), we consider two covariance adjusted estimates of tau, (hat, sub 1s) which ignores the units without X, and tau (sub X (bar)) which fills in group means for the missing values. We compare the three estimators' variances and find the conditions under which each estimator is best. (16pp.)

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