| The effects of covariate adjustment upon the precision with which an exposure effect can be estimated are examined for the classic linear regression model and several dichotomous response variable models, specifically the logistic regression model and models based on relative risk, excess risk, and relative hazard. For the classic linear regression model, covariate adjustment can result in either increased or decreased precision, depending on the relative strengths of the covariate-response variable and the covariate-exposure variable associations. Specifically, a strong covariate-response variable association favors increased precision, while a strong covariate-exposure variable association favors decreased precision. It has been conventional wisdom to assume that this precision behavior applies to regression models in general. While, as is demonstrated, regression models based on relative and excess risk do behave in this way, the logistic regression model does not. Rather, for logistic regression, covariate adjustment always results in a loss of precision, as both a strong covariate-response variable association and a strong covariate-exposure variable association favor decreased precision. Results obtained in a special case of a relative hazard regression model suggest that these models behave in a manner similar to logistic regression.;In many situations, results pertaining to the efficiency with which a hypothesis of no exposure/treatment effect may be tested also apply to the precision with which the exposure/treatment effect can be estimated. However, in a randomized study where logistic regression models are used, covariate adjustment results in increased efficiency for tests of the hypothesis of no treatment effect, despite the associated loss of precision with which treatment effect can be estimated. In this regard, the logistic model behaves in a manner similar to classic linear regression. Other situations where testing results do not apply to precision, involving the association between two dichotomous random variables, are briefly examined. In particular, some results regarding optimal sampling schemes for testing the independence of two variables do not apply with respect to the estimation of the odds ratio. |
