  Nowadays, clinical research routinely uses omics data, such as gene
expression, for predicting clinical outcomes or selecting markers.
Additionally, so-called co-data are often available, providing complementary
information on the covariates, like p-values from previously published studies
or groups of genes corresponding to pathways. Elastic net penalisation is
widely used for prediction and covariate selection. Group-adaptive elastic net
penalisation learns from co-data to improve the prediction and covariate
selection, by penalising important groups of covariates less than other groups.
Existing methods are, however, computationally expensive. Here we present a
fast method for marginal likelihood estimation of group-adaptive elastic net
penalties for generalised linear models. We first derive a low-dimensional
representation of the Taylor approximation of the marginal likelihood and its
first derivative for group-adaptive ridge penalties, to efficiently estimate
these penalties. Then we show by using asymptotic normality of the linear
predictors that the marginal likelihood for elastic net models may be
approximated well by the marginal likelihood for ridge models. The ridge group
penalties are then transformed to elastic net group penalties by using the
variance function. The method allows for overlapping groups and unpenalised
variables. We demonstrate the method in a model-based simulation study and an
application to cancer genomics. The method substantially decreases computation
time and outperforms or matches other methods by learning from co-data.
