Age-period-cohort (APC) models are used to analyse data from disease registers given by age and time. When data are stratified by one further variable, for example geographical location, multivariate APC (MAPC) models can be applied to identify and estimate heterogeneous time trends across the different strata. In such models, outcomes share a set of parameters, typically the age effects, while the remaining parameters may differ across strata. In this article, we propose a conditional approach for inference to directly model relative time trends. We show that in certain situations the conditional approach can handle unmeasured confounding so that relative risks might be estimated with higher precision. Furthermore, we propose an extension for data with more stratification levels. Maximum likelihood estimation is performed using software for multinomial logistic regression. The usage of smoothing splines is suggested to stabilise estimates of relative time trends, if necessary. We apply the methodology to chronic obstructive pulmonary disease mortality data in England & Wales, stratified by three different areas and gender.