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Generalized linear mixed models (GLMMs) are widely used in research for their ability to model correlated outcomes with non-Gaussian conditional distributions. The proper selection of fixed and random effects is a critical part of the modeling process since model misspecification may lead to significant bias. However, the joint selection of fixed and random effects has historically been limited to lower-dimensional GLMMs, largely due to the use of criterion-based model selection strategies. Here we present the R package glmmPen, one of the first to select fixed and random effects in higher dimension using a penalized GLMM modeling framework. Model parameters are estimated using a Monte Carlo Expectation Conditional Minimization (MCECM) algorithm, which leverages Stan and RcppArmadillo for increased computational efficiency. Our package supports the Binomial, Gaussian, and Poisson families and multiple penalty functions. In this manuscript we discuss the modeling procedure, estimation scheme, and software implementation through application to a pancreatic cancer subtyping study. Simulation results show our method has good performance in selecting both the fixed and random effects in high dimensional GLMMs.

Modern biomedical datasets are increasingly high dimensional and exhibit complex correlation structures. Generalized Linear Mixed Models (GLMMs) have long been employed to account for such dependencies. However, proper specification of the fixed and random effects in GLMMs is increasingly difficult in high dimensions, and computational complexity grows with increasing dimension of the random effects. We present a novel reformulation of the GLMM using a factor model decomposition of the random effects, enabling scalable computation of GLMMs in high dimensions by reducing the latent space from a large number of random effects to a smaller set of latent factors. We also extend our prior work to estimate model parameters using a modified Monte Carlo Expectation Conditional Minimization algorithm, allowing us to perform variable selection on both the fixed and random effects simultaneously. We show through simulation that through this factor model decomposition, our method can fit high dimensional penalized GLMMs faster than comparable methods and more easily scale to larger dimensions not previously seen in existing approaches.

Multi-group data are commonly seen in practice. Such data structure consists of data from multiple groups and can be challenging to analyze due to data heterogeneity. We propose a novel Joint and Individual Component Regression (JICO) model to analyze multi-group data. In particular, our proposed model decomposes the response into shared and group-specific components, which are driven by low-rank approximations of joint and individual structures from the predictors respectively. The joint structure has the same regression coefficients across multiple groups, whereas individual structures have group-specific regression coefficients. Moreover, the choice of global and individual ranks allows our model to cover global and group-specific models as special cases. For model estimation, we formulate this framework under the representation of latent components and propose an iterative algorithm to solve for the joint and individual scores under the new representation. To construct the latent scores, we utilize the Continuum Regression (CR), which provides a unified framework that covers the Ordinary Least Squares (OLS), the Partial Least Squares (PLS), and the Principal Component Regression (PCR) as its special cases. We show that JICO attains a good balance between global and group-specific models and remains flexible due to the usage of CR. Finally, we conduct simulation studies and analysis of an Alzheimer's disease dataset to further demonstrate the effectiveness of JICO. R implementation of JICO is available online at https://github.com/peiyaow/JICO.

Multimodal data, where different types of data are collected from the same subjects, are fast emerging in a large variety of scientific applications. Factor analysis is commonly used in integrative analysis of multimodal data, and is particularly useful to overcome the curse of high dimensionality and high correlations. However, there is little work on statistical inference for factor analysis based supervised modeling of multimodal data. In this article, we consider an integrative linear regression model that is built upon the latent factors extracted from multimodal data. We address three important questions: how to infer the significance of one data modality given the other modalities in the model; how to infer the significance of a combination of variables from one modality or across different modalities; and how to quantify the contribution, measured by the goodness-of-fit, of one data modality given the others. When answering each question, we explicitly characterize both the benefit and the extra cost of factor analysis. Those questions, to our knowledge, have not yet been addressed despite wide use of factor analysis in integrative multimodal analysis, and our proposal bridges an important gap. We study the empirical performance of our methods through simulations, and further illustrate with a multimodal neuroimaging analysis.

In the genomic era, the identification of gene signatures associated with disease is of significant interest. Such signatures are often used to predict clinical outcomes in new patients and aid clinical decision-making. However, recent studies have shown that gene signatures are often not replicable. This occurrence has practical implications regarding the generalizability and clinical applicability of such signatures. To improve replicability, we introduce a novel approach to select gene signatures from multiple datasets whose effects are consistently non-zero and account for between-study heterogeneity. We build our model upon some rank-based quantities, facilitating integration over different genomic datasets. A high dimensional penalized Generalized Linear Mixed Model (pGLMM) is used to select gene signatures and address data heterogeneity. We compare our method to some commonly used strategies that select gene signatures ignoring between-study heterogeneity. We provide asymptotic results justifying the performance of our method and demonstrate its advantage in the presence of heterogeneity through thorough simulation studies. Lastly, we motivate our method through a case study subtyping pancreatic cancer patients from four gene expression studies.

Factor modeling is an essential tool for exploring intrinsic dependence structures among high-dimensional random variables. Much progress has been made for estimating the covariance matrix from a high-dimensional factor model. However, the blessing of dimensionality has not yet been fully embraced in the literature: much of the available data is often ignored in constructing covariance matrix estimates. If our goal is to accurately estimate a covariance matrix of a set of targeted variables, shall we employ additional data, which are beyond the variables of interest, in the estimation? In this paper, we provide sufficient conditions for an affirmative answer, and further quantify its gain in terms of Fisher information and convergence rate. In fact, even an oracle-like result (as if all the factors were known) can be achieved when a sufficiently large number of variables is used. The idea of utilizing data as much as possible brings computational challenges. A divide-and-conquer algorithm is thus proposed to alleviate the computational burden, and also shown not to sacrifice any statistical accuracy in comparison with a pooled analysis. Simulation studies further confirm our advocacy for the use of full data, and demonstrate the effectiveness of the above algorithm. Our proposal is applied to a microarray data example that shows empirical benefits of using more data.

Data subject to heavy-tailed errors are commonly encountered in various scientific fields, especially in the modern era with explosion of massive data. To address this problem, procedures based on quantile regression and Least Absolute Deviation (LAD) regression have been devel- oped in recent years. These methods essentially estimate the conditional median (or quantile) function. They can be very different from the conditional mean functions when distributions are asymmetric and heteroscedastic. How can we efficiently estimate the mean regression functions in ultra-high dimensional setting with existence of only the second moment? To solve this problem, we propose a penalized Huber loss with diverging parameter to reduce biases created by the traditional Huber loss. Such a penalized robust approximate quadratic (RA-quadratic) loss will be called RA-Lasso. In the ultra-high dimensional setting, where the dimensionality can grow exponentially with the sample size, our results reveal that the RA-lasso estimator produces a consistent estimator at the same rate as the optimal rate under the light-tail situation. We further study the computational convergence of RA-Lasso and show that the composite gradient descent algorithm indeed produces a solution that admits the same optimal rate after sufficient iterations. As a byproduct, we also establish the concentration inequality for estimat- ing population mean when there exists only the second moment. We compare RA-Lasso with other regularized robust estimators based on quantile regression and LAD regression. Extensive simulation studies demonstrate the satisfactory finite-sample performance of RA-Lasso.