In this paper, we propose a framework for automatic classification of patients from multimodal genetic and brain imaging data by optimally combining them. Additive models with unadapted penalties (such as the classical group lasso penalty or $L_1$-multiple kernel learning) treat all modalities in the same manner and can result in undesirable elimination of specific modalities when their contributions are unbalanced. To overcome this limitation, we introduce a multilevel model that combines imaging and genetics and that considers joint effects between these two modalities for diagnosis prediction. Furthermore, we propose a framework allowing to combine several penalties taking into account the structure of the different types of data, such as a group lasso penalty over the genetic modality and a $L_2$-penalty on imaging modalities. Finally , we propose a fast optimization algorithm, based on a proximal gradient method. The model has been evaluated on genetic (single nucleotide polymorphisms-SNP) and imaging (anatomical MRI measures) data from the ADNI database, and compared to additive models. It exhibits good performances in AD diagnosis; and at the same time, reveals relationships between genes, brain regions and the disease status.
