Fast Marginal Likelihood Estimation of Penalties for Group-Adaptive Elastic Net.

Elastic net penalization is widely used in high-dimensional prediction and variable selection settings. Auxiliary information on the variables, for example, groups of variables, is often available. Group-adaptive elastic net penalization exploits this information to potentially improve performance by estimating group penalties, thereby penalizing important groups of variables less than other groups. Estimating these group penalties is, however, hard due to the high dimension of the data. Existing methods are computationally expensive or not generic in the type of response. Here we present a fast method for estimation of group-adaptive elastic net penalties for generalized linear models. We first derive a low-dimensional representation of the Taylor approximation of the marginal likelihood for group-adaptive ridge penalties, to efficiently estimate these penalties. Then we show by using asymptotic normality of the linear predictors that this marginal likelihood approximates that of elastic net models. The ridge group penalties are then transformed to elastic net group penalties by matching the ridge prior variance to the elastic net prior variance as function of the group penalties. The method allows for overlapping groups and unpenalized variables, and is easily extended to other penalties. For a model-based simulation study and two cancer genomics applications we demonstrate a substantially decreased computation time and improved or matching performance compared to other methods. Supplementary materials for this article are available online.

ecpc: an R-package for generic co-data models for high-dimensional prediction.

BACKGROUND: High-dimensional prediction considers data with more variables than samples. Generic research goals are to find the best predictor or to select variables. Results may be improved by exploiting prior information in the form of co-data, providing complementary data not on the samples, but on the variables. We consider adaptive ridge penalised generalised linear and Cox models, in which the variable-specific ridge penalties are adapted to the co-data to give a priori more weight to more important variables. The R-package ecpc originally accommodated various and possibly multiple co-data sources, including categorical co-data, i.e. groups of variables, and continuous co-data. Continuous co-data, however, were handled by adaptive discretisation, potentially inefficiently modelling and losing information. As continuous co-data such as external p values or correlations often arise in practice, more generic co-data models are needed. RESULTS: Here, we present an extension to the method and software for generic co-data models, particularly for continuous co-data. At the basis lies a classical linear regression model, regressing prior variance weights on the co-data. Co-data variables are then estimated with empirical Bayes moment estimation. After placing the estimation procedure in the classical regression framework, extension to generalised additive and shape constrained co-data models is straightforward. Besides, we show how ridge penalties may be transformed to elastic net penalties. In simulation studies we first compare various co-data models for continuous co-data from the extension to the original method. Secondly, we compare variable selection performance to other variable selection methods. The extension is faster than the original method and shows improved prediction and variable selection performance for non-linear co-data relations. Moreover, we demonstrate use of the package in several genomics examples throughout the paper. CONCLUSIONS: The R-package ecpc accommodates linear, generalised additive and shape constrained additive co-data models for the purpose of improved high-dimensional prediction and variable selection. The extended version of the package as presented here (version number 3.1.1 and higher) is available on ( https://cran.r-project.org/web/packages/ecpc/ ).

Flexible co-data learning for high-dimensional prediction.

Clinical research often focuses on complex traits in which many variables play a role in mechanisms driving, or curing, diseases. Clinical prediction is hard when data is high-dimensional, but additional information, like domain knowledge and previously published studies, may be helpful to improve predictions. Such complementary data, or co-data, provide information on the covariates, such as genomic location or P-values from external studies. We use multiple and various co-data to define possibly overlapping or hierarchically structured groups of covariates. These are then used to estimate adaptive multi-group ridge penalties for generalized linear and Cox models. Available group adaptive methods primarily target for settings with few groups, and therefore likely overfit for non-informative, correlated or many groups, and do not account for known structure on group level. To handle these issues, our method combines empirical Bayes estimation of the hyperparameters with an extra level of flexible shrinkage. This renders a uniquely flexible framework as any type of shrinkage can be used on the group level. We describe various types of co-data and propose suitable forms of hypershrinkage. The method is very versatile, as it allows for integration and weighting of multiple co-data sets, inclusion of unpenalized covariates and posterior variable selection. For three cancer genomics applications we demonstrate improvements compared to other models in terms of performance, variable selection stability and validation.