Weighted Mean Impact Analysis

Abstract

Linear regression is a popular tool that is often applied to biometric and epidemiological data. It relies on strong model assumptions that are rarely satisfied. To overcome this difficulty, Brannath and Scharpenberg (2014) proposed a new population based interpretation of linear regression coefficients. The idea is to quantify how much the unconditional mean of the dependent variable Y can be changed by changing the distribution of the independent variable X. The maximum change is called "mean impact". They show that linear regression can be used to obtain a conservative estimator of the mean impact and other population association measures. This provides a clear interpretation of the linear regression coefficients also under miss-specifications of the mean structure. A disadvantage of the new association measure is its dependence on the distribution of the independent variables in the specific study population. Hence, it may be difficult to compare the results between different studies with differing covariate distributions. To overcome this difficulty we develop a method to transfer the "mean impact" from one study population to a reference population by reweighting the observations. Accordingly, we call the resulting estimator the "weighted mean impact". The weights are obtained by a simple transformation of the expectation of the covariates multiplied with the target variable. They are defined as the density of the covariables in the true population divided by the distribution of the covariables in a pseudopopulation. For the new developed weighted mean impact we show desirable asymptotic properties like consistency and asymptotic normality. Although the weights are unknown in practical applications we first consider the case of known weights to improve the understanding of the reweighting mechanisms. Subsequently, the approach is generalized to the case of unknown weights which need to be estimated. One application for the reweighting mechanisms is to solve the problem of confounding. In the context of the mean impact confounding arises if the covariates are dependent. To avoid confounding we transform the mean impact under dependent covariates into a mean impact under independent covariates by using the weighting factor. For this example the weights are the ratio of the marginal density of one of the covariates and the conditional density. For this reason Robins et al. (2000) proposed theseweights in the context ofmarginal structural models. For the weighted mean impact with unknown weights we show asymptotic properties, develop bootstrap confidence intervals and demonstrate the utility of the new method by examples and results from a simulation study.