The Brier score is commonly used for evaluating probability predictions. In survival analysis, with right-censored observations of the event times, this score can be weighted by the inverse probability of censoring (IPCW) to retain its original interpretation. It is common practice to estimate the censoring distribution with the Kaplan-Meier estimator, even though it assumes that the censoring distribution is independent of the covariates. This paper investigates problems that may arise for the IPCW weighting scheme when the covariates used in the prediction model contain information about the censoring times. In particular, this may occur for administratively censored data if the distribution of the covariates varies with calendar time. For administratively censored data, we propose an alternative version of the Brier score. This administrative Brier score does not require estimation of the censoring distribution and is valid also when the censoring times can be predicted from the covariates.

BACKGROUND: Identifying gene interactions is a topic of great importance in genomics, and approaches based on network models provide a powerful tool for studying these. Assuming a Gaussian graphical model, a gene association network may be estimated from multiomic data based on the non-zero entries of the inverse covariance matrix. Inferring such biological networks is challenging because of the high dimensionality of the problem, making traditional estimators unsuitable. The graphical lasso is constructed for the estimation of sparse inverse covariance matrices in such situations, using [Formula: see text]-penalization on the matrix entries. The weighted graphical lasso is an extension in which prior biological information from other sources is integrated into the model. There are however issues with this approach, as it naïvely forces the prior information into the network estimation, even if it is misleading or does not agree with the data at hand. Further, if an associated network based on other data is used as the prior, the method often fails to utilize the information effectively. RESULTS: We propose a novel graphical lasso approach, the tailored graphical lasso, that aims to handle prior information of unknown accuracy more effectively. We provide an R package implementing the method, tailoredGlasso. Applying the method to both simulated and real multiomic data sets, we find that it outperforms the unweighted and weighted graphical lasso in terms of all performance measures we consider. In fact, the graphical lasso and weighted graphical lasso can be considered special cases of the tailored graphical lasso, and a parameter determined by the data measures the usefulness of the prior information. We also find that among a larger set of methods, the tailored graphical is the most suitable for network inference from high-dimensional data with prior information of unknown accuracy. With our method, mRNA data are demonstrated to provide highly useful prior information for protein-protein interaction networks. CONCLUSIONS: The method we introduce utilizes useful prior information more effectively without involving any risk of loss of accuracy should the prior information be misleading.

New methods for time-to-event prediction are proposed by extending the Cox proportional hazards model with neural networks. Building on methodology from nested case-control studies, we propose a loss function that scales well to large data sets and enables fitting of both proportional and non-proportional extensions of the Cox model. Through simulation studies, the proposed loss function is verified to be a good approximation for the Cox partial log-likelihood. The proposed methodology is compared to existing methodologies on real-world data sets and is found to be highly competitive, typically yielding the best performance in terms of Brier score and binomial log-likelihood. A python package for the proposed methods is available at https://github.com/havakv/pycox.

BACKGROUND: Survival prediction from high-dimensional genomic data is an active field in today's medical research. Most of the proposed prediction methods make use of genomic data alone without considering established clinical covariates that often are available and known to have predictive value. Recent studies suggest that combining clinical and genomic information may improve predictions, but there is a lack of systematic studies on the topic. Also, for the widely used Cox regression model, it is not obvious how to handle such combined models. RESULTS: We propose a way to combine classical clinical covariates with genomic data in a clinico-genomic prediction model based on the Cox regression model. The prediction model is obtained by a simultaneous use of both types of covariates, but applying dimension reduction only to the high-dimensional genomic variables. We describe how this can be done for seven well-known prediction methods: variable selection, unsupervised and supervised principal components regression and partial least squares regression, ridge regression, and the lasso. We further perform a systematic comparison of the performance of prediction models using clinical covariates only, genomic data only, or a combination of the two. The comparison is done using three survival data sets containing both clinical information and microarray gene expression data. Matlab code for the clinico-genomic prediction methods is available at http://www.med.uio.no/imb/stat/bmms/software/clinico-genomic/. CONCLUSIONS: Based on our three data sets, the comparison shows that established clinical covariates will often lead to better predictions than what can be obtained from genomic data alone. In the cases where the genomic models are better than the clinical, ridge regression is used for dimension reduction. We also find that the clinico-genomic models tend to outperform the models based on only genomic data. Further, clinico-genomic models and the use of ridge regression gives for all three data sets better predictions than models based on the clinical covariates alone.

MOTIVATION: Survival prediction from gene expression data and other high-dimensional genomic data has been subject to much research during the last years. These kinds of data are associated with the methodological problem of having many more gene expression values than individuals. In addition, the responses are censored survival times. Most of the proposed methods handle this by using Cox's proportional hazards model and obtain parameter estimates by some dimension reduction or parameter shrinkage estimation technique. Using three well-known microarray gene expression data sets, we compare the prediction performance of seven such methods: univariate selection, forward stepwise selection, principal components regression (PCR), supervised principal components regression, partial least squares regression (PLS), ridge regression and the lasso. RESULTS: Statistical learning from subsets should be repeated several times in order to get a fair comparison between methods. Methods using coefficient shrinkage or linear combinations of the gene expression values have much better performance than the simple variable selection methods. For our data sets, ridge regression has the overall best performance. AVAILABILITY: Matlab and R code for the prediction methods are available at http://www.med.uio.no/imb/stat/bmms/software/microsurv/.