Data integration: exploiting ratios of parameter estimates from a reduced external model

We consider the situation of estimating the parameters in a generalized linear prediction model, from an internal dataset, where the outcome variable Y is binary and there are two sets of covariates, X and Z. We have information from an external study that provides parameter estimates for a generalized linear model of Y on X. We propose a method that makes limited assumptions about the similarity of the distributions in the two study populations. The method involves orthogonalizing the Z variables and then borrowing information about the ratio of the coefficients from the external model. The method is justified based on a new result relating the parameters in a generalized linear model to the parameters in a generalized linear model with omitted covariates. The method is applicable if the regression coefficients in the Y given X model are similar in the two populations, up to an unknown scalar constant. This type of transportability between populations is something that can be checked from the available data. The asymptotic variance of the proposed method is derived. The method is evaluated in a simulation study and shown to gain efficiency compared to simple analysis of the internal dataset, and is robust compared to an alternative method of incorporating external information.