In a context of component-based multivariate model we propose to model the residual dependence of the responses. Each response of a response matrix is assumed to depend, through a Generalized Linear Model, on a set of explanatory variables, as well as on a set of additional covariates. Explanatory variables are partitioned into conceptually homogeneous variable groups, viewed as explanatory themes. Variables in themes are supposed many and redundant. Thus, generalized linear regression demands dimension reduction and regularization with respect to each theme. By contrast, additional covariates contain few variables, selected so as not to be too redundant, thus demanding no regularization. Regularization is performed searching each theme for an appropriate number of orthogonal components that both contribute to predict the responses and capture relevant structural information in themes. A small set of latent factors completes the model so as to model the covariance matrix of the linear predictors of the responses conditional on the components. To estimate the multiple-theme model, we present an algorithm combining thematic component-based model estimation and factor model estimation. This methodology is tested on simulated data and then applied to an agricultural ecology dataset.