In many problems involving generalized linear models, the covariates are subject to measurement error. When the number of covariates p exceeds the sample size n, regularized methods like the lasso or Dantzig selector are required. Several recent papers have studied methods which correct for measurement error in the lasso or Dantzig selector for linear models in the p > n setting. We study a correction for generalized linear models based on Rosenbaum and Tsybakov's matrix uncertainty selector. By not requiring an estimate of the measurement error covariance matrix, this generalized matrix uncertainty selector has a great practical advantage in problems involving high-dimensional data. We further derive an alternative method based on the lasso, and develop efficient algorithms for both methods. In our simulation studies of logistic and Poisson regression with measurement error, the proposed methods outperform the standard lasso and Dantzig selector with respect to covariate selection, by reducing the number of false positives considerably. We also consider classification of patients on the basis of gene expression data with noisy measurements. * oystein.sorensen@medisin.uio.no † arnoldo.frigessi@medisin.uio.no ‡ magne.thoresen@medisin.uio.no

Statistical prediction methods typically require some form of fine-tuning of tuning parameter(s), with K-fold cross-validation as the canonical procedure. For ridge regression, there exist numerous procedures, but common for all, including cross-validation, is that one single parameter is chosen for all future predictions. We propose instead to calculate a unique tuning parameter for each individual for which we wish to predict an outcome. This generates an individualized prediction by focusing on the vector of covariates of a specific individual. The focused ridge-fridge-procedure is introduced with a 2-part contribution: First we define an oracle tuning parameter minimizing the mean squared prediction error of a specific covariate vector, and then we propose to estimate this tuning parameter by using plug-in estimates of the regression coefficients and error variance parameter. The procedure is extended to logistic ridge regression by using parametric bootstrap. For high-dimensional data, we propose to use ridge regression with cross-validation as the plug-in estimate, and simulations show that fridge gives smaller average prediction error than ridge with cross-validation for both simulated and real data. We illustrate the new concept for both linear and logistic regression models in 2 applications of personalized medicine: predicting individual risk and treatment response based on gene expression data. The method is implemented in the R package fridge.

When measuring a range of genomic, epigenomic, and transcriptomic variables for the same tissue sample, an integrative approach to analysis can strengthen inference and lead to new insights. This is also the case when clustering patient samples, and several integrative cluster procedures have been proposed. Common for these methodologies is the restriction to a joint cluster structure, equal in all data layers. We instead present a clustering extension of the Joint and Individual Variance Explained algorithm (JIVE), Joint and Individual Clustering (JIC), enabling the construction of both joint and data type-specific clusters simultaneously. The procedure builds on the connection between k-means clustering and principal component analysis, and hence, the number of clusters can be determined by the number of relevant principal components. The proposed procedure is compared with iCluster, a method restricted to only joint clusters, and simulations show that JIC is clearly advantageous when both individual and joint clusters are present. The procedure is illustrated using gene expression and miRNA levels measured in breast cancer tissue from The Cancer Genome Atlas. The analysis suggests a division into three joint clusters common for both data types and two expression-specific clusters.

Regional flood frequency analysis is commonly applied in situations where there exists insufficient data at a location for a reliable estimation of flood quantiles. We develop a Bayesian hierarchical modeling framework for a regional analysis of data from 203 large catchments in Norway with the generalized extreme value distribution as the underlying model. Generalized linear models on the parameters of the generalized extreme value distribution are able to incorporate location‐specific geographic and meteorological information and thereby accommodate these effects on the flood quantiles. A Bayesian model averaging component additionally assesses model uncertainty in the effect of the proposed covariates. The resulting regional model is seen to give substantially better predictive performance than the regional model currently used in Norway.

We investigate the effect of measurement error on principal component analysis in the high‐dimensional setting. The effects of random, additive errors are characterized by the expectation and variance of the changes in the eigenvalues and eigenvectors. The results show that the impact of uncorrelated measurement error on the principal component scores is mainly in terms of increased variability and not bias. In practice, the error‐induced increase in variability is small compared with the original variability for the components corresponding to the largest eigenvalues. This suggests that the impact will be negligible when these component scores are used in classification and regression or for visualizing data. However, the measurement error will contribute to a large variability in component loadings, relative to the loading values, such that interpretation based on the loadings can be difficult. The results are illustrated by simulating additive Gaussian measurement error in microarray expression data from cancer tumours and control tissues.

Principal component analysis is a popular dimension reduction technique often used to visualize high-dimensional data structures. In genomics, this can involve millions of variables, but only tens to hundreds of observations. Theoretically, such extreme high dimensionality will cause biased or inconsistent eigenvector estimates, but in practice, the principal component scores are used for visualization with great success. In this paper, we explore when and why the classical principal component scores can be used to visualize structures in high-dimensional data, even when there are few observations compared with the number of variables. Our argument is twofold: First, we argue that eigenvectors related to pervasive signals will have eigenvalues scaling linearly with the number of variables. Second, we prove that for linearly increasing eigenvalues, the sample component scores will be scaled and rotated versions of the population scores, asymptotically. Thus, the visual information of the sample scores will be unchanged, even though the sample eigenvectors are biased. In the case of pervasive signals, the principal component scores can be used to visualize the population structures, even in extreme high-dimensional situations.

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