Augmenting Data With Published Results in Bayesian Linear Regression

Abstract

In most research, linear regression analyses are performed without taking into account published results (i.e., reported summary statistics) of similar previous studies. Although the prior density in Bayesian linear regression could accommodate such prior knowledge, formal models for doing so are absent from the literature. The goal of this article is therefore to develop a Bayesian model in which a linear regression analysis on current data is augmented with the reported regression coefficients (and standard errors) of previous studies. Two versions of this model are presented. The first version incorporates previous studies through the prior density and is applicable when the current and all previous studies are exchangeable. The second version models all studies in a hierarchical structure and is applicable when studies are not exchangeable. Both versions of the model are assessed using simulation studies. Performance for each in estimating the regression coefficients is consistently superior to using current data alone and is close to that of an equivalent model that uses the data from previous studies rather than reported regression coefficients. Overall the results show that augmenting data with results from previous studies is viable and yields significant improvements in the parameter estimation.

Notes

¹In this article it is assumed that only one full data set, the “current study,” is available. However, and in line with the previous paragraph on using historic data, this can straightforwardly be extended to include multiple full data sets as needed.

²Because S is also estimated from the data in the original study, it is subject to uncertainty as well. This uncertainty, however, is never reported and therefore cannot be incorporated in the present model. As S itself is not of interest, this has little impact on the results.

³Both the model in this section as well as the one in the Hierarchical Replication Model section of this article were implemented using R. However, they can also be implemented in WinBUGS, which provides a user-friendly interface for specifying the model and takes care of the sampling process without further effort from the user.

⁴During testing of the model, interactions between parameters were considered as well, but these revealed nothing of note and therefore are not discussed in the Results section.