Most probability-based methods used to link records from two distinct data setscorresponding to the same target population do not lead to perfect linkage, i.e. there arelinkage errors in the merged data. Further, the linkage is often incomplete, in the sensethat many records in the two data sets remain unmatched at the completion of the linkageprocess. This paper introduces methods that correct for the biases due to linkage errors andincomplete linkage when carrying out regression analysis using linked data. In particular, itfocuses on the case where one of the linked data sets is a sample from the target populationand the other is a register, i.e. it covers the entire target population.