A Two Step Model for Linear Prediction with Group Effect

In this article, we consider prediction of a univariate response from background data. The data may have a near-collinear structure and additionally group effects are assumed to exist. A two step estimation procedure is proposed. The first step is to summarize the information in the predictors via a bilinear model. The bilinear model has a Krylov structured within individual design matrix, which is the link to classical partial least squares (PLS) analysis and a between individual design matrix which handles group effects. The second step is the prediction step where a conditional expectation approach is used. The two step approach gives new insight in PLS. Explicit maximum likelihood estimator of the dispersion matrix and mean for the predictors are derived under the assumption that the covariance between the response and explanatory variable is known. It is shown that the mean square error of the two step approach is always smaller than PLS.