Multivariate random length data occur when we observe multiple measurements of a quantitative variable and the variable number of these measurements is also an observed outcome for each experimental unit. For example, for a patient with coronary artery disease, we may observe a number of lesions in that patient's coronary arteries, along with percentage of blockage of each lesion. Barnhart and Sampson first proposed the multiple population model to analyse multivariate random length data without covariates. This paper extends their approach to deal with multiple covariates. We propose a new multiple population regression model with covariates, and discuss the estimation issues. We analyse data from the TYPE II coronary intervention study to illustrate the methodology.