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 method is proposed. The first step summarizes 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 method gives new insight into PLS. Explicit maximum likelihood estimators of the dispersion matrix and mean for the predictors are derived under the assumption that the covariance between the response and explanatory variables is known. It is shown that for within-sample prediction the mean squared error of the two-step method is always smaller than PLS