Studies using assays to quantify the expression of thousands of genes on tens to thousands of cell samples have been carried out for over 20 years. Such assays are based on microarrays, DNA sequencing or other molecular technologies. All such studies involve unwanted variation, often called batch effects, associated with the cell samples and the assay process. Removing this unwanted variation is essential before the measurements can be used to address the questions that motivated the studies. Combining the results of replicate assays with measurements on negative control genes to estimate the unwanted variation and remove it has proved to be effective at this task. The main goal of this paper is to present asymptotic theory that explains this effectiveness. The approach can be widened by using pseudo-replicate sets of pseudo-samples, for use with studies having no replicate assays. Theory covering this case is also presented. The established theory is supported by results of empirical investigations, including simulation studies and a real-data example.