Two key challenges in modern statistical applications are the large amount of information recorded per individual, and that such data are often collected in batches. These batch effects (BE) can be complex, causing distortions in mean and variance. We propose a novel sparse latent factor regression model to integrate such heterogeneous data. The model provides a tool for data exploration via dimensionality reduction and sparse low-rank covariance estimation while correcting for a range of BE. We study the use of several sparse priors (local and non-local) to learn the dimension of the latent factors. We provide a flexible methodology for sparse factor regression which is not limited to data with BE. Our model is fitted via an EM algorithm with closed-form updates, contributing a novel scalable algorithm for non-local priors. We present several bioinformatics applications. Our results show an increase in the accuracy of the dimensionality reduction, with non-local priors substantially improving the reconstruction of factor cardinality in scenarios with and without BE. The results of our analyses illustrate how failing to properly account for BE can result in unreliable inference.