Showing papers in "Biometrics in 2017"

Outcome-adaptive lasso: Variable selection for causal inference.

Abstract: Methodological advancements, including propensity score methods, have resulted in improved unbiased estimation of treatment effects from observational data. Traditionally, a "throw in the kitchen sink" approach has been used to select covariates for inclusion into the propensity score, but recent work shows including unnecessary covariates can impact both the bias and statistical efficiency of propensity score estimators. In particular, the inclusion of covariates that impact exposure but not the outcome, can inflate standard errors without improving bias, while the inclusion of covariates associated with the outcome but unrelated to exposure can improve precision. We propose the outcome-adaptive lasso for selecting appropriate covariates for inclusion in propensity score models to account for confounding bias and maintaining statistical efficiency. This proposed approach can perform variable selection in the presence of a large number of spurious covariates, that is, covariates unrelated to outcome or exposure. We present theoretical and simulation results indicating that the outcome-adaptive lasso selects the propensity score model that includes all true confounders and predictors of outcome, while excluding other covariates. We illustrate covariate selection using the outcome-adaptive lasso, including comparison to alternative approaches, using simulated data and in a survey of patients using opioid therapy to manage chronic pain.

A penalized framework for distributed lag non-linear models.

Abstract: Distributed lag non-linear models (DLNMs) are a modelling tool for describing potentially non-linear and delayed dependencies. Here, we illustrate an extension of the DLNM framework through the use of penalized splines within generalized additive models (GAM). This extension offers built-in model selection procedures and the possibility of accommodating assumptions on the shape of the lag structure through specific penalties. In addition, this framework includes, as special cases, simpler models previously proposed for linear relationships (DLMs). Alternative versions of penalized DLNMs are compared with each other and with the standard unpenalized version in a simulation study. Results show that this penalized extension to the DLNM class provides greater flexibility and improved inferential properties. The framework exploits recent theoretical developments of GAMs and is implemented using efficient routines within freely available software. Real-data applications are illustrated through two reproducible examples in time series and survival analysis.

Statistical significance for hierarchical clustering

Abstract: Cluster analysis has proved to be an invaluable tool for the exploratory and unsupervised analysis of high-dimensional datasets. Among methods for clustering, hierarchical approaches have enjoyed substantial popularity in genomics and other fields for their ability to simultaneously uncover multiple layers of clustering structure. A critical and challenging question in cluster analysis is whether the identified clusters represent important underlying structure or are artifacts of natural sampling variation. Few approaches have been proposed for addressing this problem in the context of hierarchical clustering, for which the problem is further complicated by the natural tree structure of the partition, and the multiplicity of tests required to parse the layers of nested clusters. In this article, we propose a Monte Carlo based approach for testing statistical significance in hierarchical clustering which addresses these issues. The approach is implemented as a sequential testing procedure guaranteeing control of the family-wise error rate. Theoretical justification is provided for our approach, and its power to detect true clustering structure is illustrated through several simulation studies and applications to two cancer gene expression datasets.

A general statistical framework for subgroup identification and comparative treatment scoring.

Abstract: Many statistical methods have recently been developed for identifying subgroups of patients who may benefit from different available treatments. Compared with the traditional outcome-modeling approaches, these methods focus on modeling interactions between the treatments and covariates while by-pass or minimize modeling the main effects of covariates because the subgroup identification only depends on the sign of the interaction. However, these methods are scattered and often narrow in scope. In this article, we propose a general framework, by weighting and A-learning, for subgroup identification in both randomized clinical trials and observational studies. Our framework involves minimum modeling for the relationship between the outcome and covariates pertinent to the subgroup identification. Under the proposed framework, we may also estimate the magnitude of the interaction, which leads to the construction of scoring system measuring the individualized treatment effect. The proposed methods are quite flexible and include many recently proposed estimators as special cases. As a result, some estimators originally proposed for randomized clinical trials can be extended to observational studies, and procedures based on the weighting method can be converted to an A-learning method and vice versa. Our approaches also allow straightforward incorporation of regularization methods for high-dimensional data, as well as possible efficiency augmentation and generalization to multiple treatments. We examine the empirical performance of several procedures belonging to the proposed framework through extensive numerical studies.

Alternative measures of between-study heterogeneity in meta-analysis: Reducing the impact of outlying studies.

Abstract: Meta-analysis has become a widely used tool to combine results from independent studies The collected studies are homogeneous if they share a common underlying true effect size; otherwise, they are heterogeneous A fixed-effect model is customarily used when the studies are deemed homogeneous, while a random-effects model is used for heterogeneous studies Assessing heterogeneity in meta-analysis is critical for model selection and decision making Ideally, if heterogeneity is present, it should permeate the entire collection of studies, instead of being limited to a small number of outlying studies Outliers can have great impact on conventional measures of heterogeneity and the conclusions of a meta-analysis However, no widely accepted guidelines exist for handling outliers This article proposes several new heterogeneity measures In the presence of outliers, the proposed measures are less affected than the conventional ones The performance of the proposed and conventional heterogeneity measures are compared theoretically, by studying their asymptotic properties, and empirically, using simulations and case studies

Simulation-based hypothesis testing of high dimensional means under covariance heterogeneity.

Abstract: In this article, we study the problem of testing the mean vectors of high dimensional data in both one-sample and two-sample cases. The proposed testing procedures employ maximum-type statistics and the parametric bootstrap techniques to compute the critical values. Different from the existing tests that heavily rely on the structural conditions on the unknown covariance matrices, the proposed tests allow general covariance structures of the data and therefore enjoy wide scope of applicability in practice. To enhance powers of the tests against sparse alternatives, we further propose two-step procedures with a preliminary feature screening step. Theoretical properties of the proposed tests are investigated. Through extensive numerical experiments on synthetic data sets and an human acute lymphoblastic leukemia gene expression data set, we illustrate the performance of the new tests and how they may provide assistance on detecting disease-associated gene-sets. The proposed methods have been implemented in an R-package HDtest and are available on CRAN.

A generalized Fellner-Schall method for smoothing parameter optimization with application to Tweedie location, scale and shape models

Abstract: We consider the optimization of smoothing parameters and variance components in models with a regular log likelihood subject to quadratic penalization of the model coefficients, via a generalization of the method of Fellner (1986) and Schall (1991). In particular: (i) we generalize the original method to the case of penalties that are linear in several smoothing parameters, thereby covering the important cases of tensor product and adaptive smoothers; (ii) we show why the method's steps increase the restricted marginal likelihood of the model, that it tends to converge faster than the EM algorithm, or obvious accelerations of this, and investigate its relation to Newton optimization; (iii) we generalize the method to any Fisher regular likelihood. The method represents a considerable simplification over existing methods of estimating smoothing parameters in the context of regular likelihoods, without sacrificing generality: for example, it is only necessary to compute with the same first and second derivatives of the log-likelihood required for coefficient estimation, and not with the third or fourth order derivatives required by alternative approaches. Examples are provided which would have been impossible or impractical with pre-existing Fellner-Schall methods, along with an example of a Tweedie location, scale and shape model which would be a challenge for alternative methods, and a sparse additive modeling example where the method facilitates computational efficiency gains of several orders of magnitude. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

Comparing large covariance matrices under weak conditions on the dependence structure and its application to gene clustering

Abstract: Comparing large covariance matrices has important applications in modern genomics, where scientists are often interested in understanding whether relationships (e.g., dependencies or co-regulations) among a large number of genes vary between different biological states. We propose a computationally fast procedure for testing the equality of two large covariance matrices when the dimensions of the covariance matrices are much larger than the sample sizes. A distinguishing feature of the new procedure is that it imposes no structural assumptions on the unknown covariance matrices. Hence, the test is robust with respect to various complex dependence structures that frequently arise in genomics. We prove that the proposed procedure is asymptotically valid under weak moment conditions. As an interesting application, we derive a new gene clustering algorithm which shares the same nice property of avoiding restrictive structural assumptions for high-dimensional genomics data. Using an asthma gene expression dataset, we illustrate how the new test helps compare the covariance matrices of the genes across different gene sets/pathways between the disease group and the control group, and how the gene clustering algorithm provides new insights on the way gene clustering patterns differ between the two groups. The proposed methods have been implemented in an R-package HDtest and are available on CRAN.

A space-time skew-t model for threshold exceedances.

Abstract: To assess the compliance of air quality regulations, the Environmental Protection Agency (EPA) must know if a site exceeds a pre-specified level. In the case of ozone, the level for compliance is fixed at 75 parts per billion, which is high, but not extreme at all locations. We present a new space-time model for threshold exceedances based on the skew-t process. Our method incorporates a random partition to permit long-distance asymptotic independence while allowing for sites that are near one another to be asymptotically dependent, and we incorporate thresholding to allow the tails of the data to speak for themselves. We also introduce a transformed AR(1) time-series to allow for temporal dependence. Finally, our model allows for high-dimensional Bayesian inference that is comparable in computation time to traditional geostatistical methods for large data sets. We apply our method to an ozone analysis for July 2005, and find that our model improves over both Gaussian and max-stable methods in terms of predicting exceedances of a high level.

Greedy outcome weighted tree learning of optimal personalized treatment rules

Abstract: Summary We propose a subgroup identification approach for inferring optimal and interpretable personalized treatment rules with high-dimensional covariates. Our approach is based on a two-step greedy tree algorithm to pursue signals in a high-dimensional space. In the first step, we transform the treatment selection problem into a weighted classification problem that can utilize tree-based methods. In the second step, we adopt a newly proposed tree-based method, known as reinforcement learning trees, to detect features involved in the optimal treatment rules and to construct binary splitting rules. The method is further extended to right censored survival data by using the accelerated failure time model and introducing double weighting to the classification trees. The performance of the proposed method is demonstrated via simulation studies, as well as analyses of the Cancer Cell Line Encyclopedia (CCLE) data and the Tamoxifen breast cancer data.

A framework for Bayesian nonparametric inference for causal effects of mediation

Abstract: We propose a Bayesian non-parametric (BNP) framework for estimating causal effects of mediation, the natural direct, and indirect, effects. The strategy is to do this in two parts. Part 1 is a flexible model (using BNP) for the observed data distribution. Part 2 is a set of uncheckable assumptions with sensitivity parameters that in conjunction with Part 1 allows identification and estimation of the causal parameters and allows for uncertainty about these assumptions via priors on the sensitivity parameters. For Part 1, we specify a Dirichlet process mixture of multivariate normals as a prior on the joint distribution of the outcome, mediator, and covariates. This approach allows us to obtain a (simple) closed form of each marginal distribution. For Part 2, we consider two sets of assumptions: (a) the standard sequential ignorability (Imai et al., 2010) and (b) weakened set of the conditional independence type assumptions introduced in Daniels et al. (2012) and propose sensitivity analyses for both. We use this approach to assess mediation in a physical activity promotion trial.

A Dirichlet-tree multinomial regression model for associating dietary nutrients with gut microorganisms.

Abstract: Summary Understanding the factors that alter the composition of the human microbiota may help personalized healthcare strategies and therapeutic drug targets. In many sequencing studies, microbial communities are characterized by a list of taxa, their counts, and their evolutionary relationships represented by a phylogenetic tree. In this article, we consider an extension of the Dirichlet multinomial distribution, called the Dirichlet-tree multinomial distribution, for multivariate, over-dispersed, and tree-structured count data. To address the relationships between these counts and a set of covariates, we propose the Dirichlet-tree multinomial regression model for which we develop a penalized likelihood method for estimating parameters and selecting covariates. For efficient optimization, we adopt the accelerated proximal gradient approach. Simulation studies are presented to demonstrate the good performance of the proposed procedure. An analysis of a data set relating dietary nutrients with bacterial counts is used to show that the incorporation of the tree structure into the model helps increase the prediction power.

Model averaged double robust estimation.

Abstract: Researchers estimating causal effects are increasingly challenged with decisions on how to best control for a potentially high-dimensional set of confounders Typically, a single propensity score model is chosen and used to adjust for confounding, while the uncertainty surrounding which covariates to include into the propensity score model is often ignored, and failure to include even one important confounder will results in bias We propose a practical and generalizable approach that overcomes the limitations described above through the use of model averaging We develop and evaluate this approach in the context of double robust estimation More specifically, we introduce the model averaged double robust (MA-DR) estimators, which account for model uncertainty in both the propensity score and outcome model through the use of model averaging The MA-DR estimators are defined as weighted averages of double robust estimators, where each double robust estimator corresponds to a specific choice of the outcome model and the propensity score model The MA-DR estimators extend the desirable double robustness property by achieving consistency under the much weaker assumption that either the true propensity score model or the true outcome model be within a specified, possibly large, class of models Using simulation studies, we also assessed small sample properties, and found that MA-DR estimators can reduce mean squared error substantially, particularly when the set of potential confounders is large relative to the sample size We apply the methodology to estimate the average causal effect of temozolomide plus radiotherapy versus radiotherapy alone on one-year survival in a cohort of 1887 Medicare enrollees who were diagnosed with glioblastoma between June 2005 and December 2009

Hypothesis testing of matrix graph model with application to brain connectivity analysis

Abstract: Brain connectivity analysis is now at the foreground of neuroscience research. A connectivity network is characterized by a graph, where nodes represent neural elements such as neurons and brain regions, and links represent statistical dependence that is often encoded in terms of partial correlation. Such a graph is inferred from the matrix-valued neuroimaging data such as electroencephalography and functional magnetic resonance imaging. There have been a good number of successful proposals for sparse precision matrix estimation under normal or matrix normal distribution; however, this family of solutions does not offer a direct statistical significance quantification for the estimated links. In this article, we adopt a matrix normal distribution framework and formulate the brain connectivity analysis as a precision matrix hypothesis testing problem. Based on the separable spatial-temporal dependence structure, we develop oracle and data-driven procedures to test both the global hypothesis that all spatial locations are conditionally independent, and simultaneous tests for identifying conditional dependent spatial locations with false discovery rate control. Our theoretical results show that the data-driven procedures perform asymptotically as well as the oracle procedures and enjoy certain optimality properties. The empirical finite-sample performance of the proposed tests is studied via intensive simulations, and the new tests are applied on a real electroencephalography data analysis.

On longitudinal prediction with time‐to‐event outcome: Comparison of modeling options

Abstract: Long-term follow-up is common in many medical investigations where the interest lies in predicting patients' risks for a future adverse outcome using repeatedly measured predictors over time. A key quantity is the likelihood of developing an adverse outcome among individuals who survived up to time s given their covariate information up to time s. Simple, yet reliable, methodology for updating the predicted risk of disease progression using longitudinal markers remains elusive. Two main approaches have been considered in the literature. One approach, based on joint modeling (JM) of failure time and longitudinal covariate process (Tsiatis and Davidian, 2004), derives such longitudinal predictive probability from the joint probability of a longitudinal marker and an event at a given time. A second approach, the partly conditional (PC) modeling (Zheng and Heagerty, 2005), directly models the predictive probability conditional on survival up to a landmark time and information accrued by that time. In this article, we propose new PC models for longitudinal prediction that are more flexible than joint modeling and improve the prediction accuracy over existing PC models. We provide procedures for making inference regarding future risk for an individual with longitudinal measures up to a given time. In addition, we conduct simulations to evaluate both JM and PC approaches in order to provide practical guidance on modeling choices. We use standard measures of predictive accuracy adapted to our setting to explore the predictiveness of the two approaches. We illustrate the performance of the two approaches on a dataset from the End Stage Renal Disease Study (ESRDS).

A generalized Levene's scale test for variance heterogeneity in the presence of sample correlation and group uncertainty

Abstract: We generalize Levene's test for variance (scale) heterogeneity between k groups for more complex data, when there are sample correlation and group membership uncertainty. Following a two-stage regression framework, we show that least absolute deviation regression must be used in the stage 1 analysis to ensure a correct asymptotic χk-12/(k-1) distribution of the generalized scale (gS) test statistic. We then show that the proposed gS test is independent of the generalized location test, under the joint null hypothesis of no mean and no variance heterogeneity. Consequently, we generalize the recently proposed joint location-scale (gJLS) test, valuable in settings where there is an interaction effect but one interacting variable is not available. We evaluate the proposed method via an extensive simulation study and two genetic association application studies.

Instrumental variables estimation of exposure effects on a time‐to‐event endpoint using structural cumulative survival models

Abstract: The use of instrumental variables for estimating the effect of an exposure on an outcome is popular in econometrics, and increasingly so in epidemiology. This increasing popularity may be attributed to the natural occurrence of instrumental variables in observational studies that incorporate elements of randomization, either by design or by nature (e.g., random inheritance of genes). Instrumental variables estimation of exposure effects is well established for continuous outcomes and to some extent for binary outcomes. It is, however, largely lacking for time-to-event outcomes because of complications due to censoring and survivorship bias. In this article, we make a novel proposal under a class of structural cumulative survival models which parameterize time-varying effects of a point exposure directly on the scale of the survival function; these models are essentially equivalent with a semi-parametric variant of the instrumental variables additive hazards model. We propose a class of recursive instrumental variable estimators for these exposure effects, and derive their large sample properties along with inferential tools. We examine the performance of the proposed method in simulation studies and illustrate it in a Mendelian randomization study to evaluate the effect of diabetes on mortality using data from the Health and Retirement Study. We further use the proposed method to investigate potential benefit from breast cancer screening on subsequent breast cancer mortality based on the HIP-study.

Bayesian response-adaptive designs for basket trials

Abstract: We develop a general class of response-adaptive Bayesian designs using hierarchical models, and provide open source software to implement them. Our work is motivated by recent master protocols in oncology, where several treatments are investigated simultaneously in one or multiple disease types, and treatment efficacy is expected to vary across biomarker-defined subpopulations. Adaptive trials such as I-SPY-2 (Barker et al., 2009) and BATTLE (Zhou et al., 2008) are special cases within our framework. We discuss the application of our adaptive scheme to two distinct research goals. The first is to identify a biomarker subpopulation for which a therapy shows evidence of treatment efficacy, and to exclude other subpopulations for which such evidence does not exist. This leads to a subpopulation-finding design. The second is to identify, within biomarker-defined subpopulations, a set of cancer types for which an experimental therapy is superior to the standard-of-care. This goal leads to a subpopulation-stratified design. Using simulations constructed to faithfully represent ongoing cancer sequencing projects, we quantify the potential gains of our proposed designs relative to conventional non-adaptive designs.

Incorporating covariates into integrated factor analysis of multi-view data.

Abstract: In modern biomedical research, it is ubiquitous to have multiple data sets measured on the same set of samples from different views (i.e., multi-view data). For example, in genetic studies, multiple genomic data sets at different molecular levels or from different cell types are measured for a common set of individuals to investigate genetic regulation. Integration and reduction of multi-view data have the potential to leverage information in different data sets, and to reduce the magnitude and complexity of data for further statistical analysis and interpretation. In this article, we develop a novel statistical model, called supervised integrated factor analysis (SIFA), for integrative dimension reduction of multi-view data while incorporating auxiliary covariates. The model decomposes data into joint and individual factors, capturing the joint variation across multiple data sets and the individual variation specific to each set, respectively. Moreover, both joint and individual factors are partially informed by auxiliary covariates via nonparametric models. We devise a computationally efficient Expectation-Maximization (EM) algorithm to fit the model under some identifiability conditions. We apply the method to the Genotype-Tissue Expression (GTEx) data, and provide new insights into the variation decomposition of gene expression in multiple tissues. Extensive simulation studies and an additional application to a pediatric growth study demonstrate the advantage of the proposed method over competing methods.

Ordinal probability effect measures for group comparisons in multinomial cumulative link models.

Abstract: Summary We consider simple ordinal model-based probability effect measures for comparing distributions of two groups, adjusted for explanatory variables. An “ordinal superiority” measure summarizes the probability that an observation from one distribution falls above an independent observation from the other distribution, adjusted for explanatory variables in a model. The measure applies directly to normal linear models and to a normal latent variable model for ordinal response variables. It equals Φ(β/2) for the corresponding ordinal model that applies a probit link function to cumulative multinomial probabilities, for standard normal cdf Φ and effect β that is the coefficient of the group indicator variable. For the more general latent variable model for ordinal responses that corresponds to a linear model with other possible error distributions and corresponding link functions for cumulative multinomial probabilities, the ordinal superiority measure equals exp(β)/[1+exp(β)] with the log–log link and equals approximately exp(β/2)/[1+exp(β/2)] with the logit link, where β is the group effect. Another ordinal superiority measure generalizes the difference of proportions from binary to ordinal responses. We also present related measures directly for ordinal models for the observed response that need not assume corresponding latent response models. We present confidence intervals for the measures and illustrate with an example.

Design and monitoring of multi-arm multi-stage clinical trials.

Abstract: Two-arm group sequential designs have been widely used for over 40 years, especially for studies with mortality endpoints. The natural generalization of such designs to trials with multiple treatment arms and a common control (MAMS designs) has, however, been implemented rarely. While the statistical methodology for this extension is clear, the main limitation has been an efficient way to perform the computations. Past efforts were hampered by algorithms that were computationally explosive. With the increasing interest in adaptive designs, platform designs, and other innovative designs that involve multiple comparisons over multiple stages, the importance of MAMS designs is growing rapidly. This article provides break-through algorithms that can compute MAMS boundaries rapidly thereby making such designs practical. For designs with efficacy-only boundaries the computational effort increases linearly with number of arms and number of stages. For designs with both efficacy and futility boundaries the computational effort doubles with successive increases in number of stages.

A fast small-sample kernel independence test for microbiome community-level association analysis.

Abstract: Summary To fully understand the role of microbiome in human health and diseases, researchers are increasingly interested in assessing the relationship between microbiome composition and host genomic data. The dimensionality of the data as well as complex relationships between microbiota and host genomics pose considerable challenges for analysis. In this article, we apply a kernel RV coefficient (KRV) test to evaluate the overall association between host gene expression and microbiome composition. The KRV statistic can capture nonlinear correlations and complex relationships among the individual data types and between gene expression and microbiome composition through measuring general dependency. Testing proceeds via a similar route as existing tests of the generalized RV coefficients and allows for rapid p-value calculation. Strategies to allow adjustment for confounding effects, which is crucial for avoiding misleading results, and to alleviate the problem of selecting the most favorable kernel are considered. Simulation studies show that KRV is useful in testing statistical independence with finite samples given the kernels are appropriately chosen, and can powerfully identify existing associations between microbiome composition and host genomic data while protecting type I error. We apply the KRV to a microbiome study examining the relationship between host transcriptome and microbiome composition within the context of inflammatory bowel disease and are able to derive new biological insights and provide formal inference on prior qualitative observations.

Adaptive contrast weighted learning for multi-stage multi-treatment decision-making.

Abstract: Dynamic treatment regimes (DTRs) are sequential decision rules that focus simultaneously on treatment individualization and adaptation over time. To directly identify the optimal DTR in a multi-stage multi-treatment setting, we propose a dynamic statistical learning method, adaptive contrast weighted learning. We develop semiparametric regression-based contrasts with the adaptation of treatment effect ordering for each patient at each stage, and the adaptive contrasts simplify the problem of optimization with multiple treatment comparisons to a weighted classification problem that can be solved by existing machine learning techniques. The algorithm is implemented recursively using backward induction. By combining doubly robust semiparametric regression estimators with machine learning algorithms, the proposed method is robust and efficient for the identification of the optimal DTR, as shown in the simulation studies. We illustrate our method using observational data on esophageal cancer.

A general instrumental variable framework for regression analysis with outcome missing not at random

Abstract: The instrumental variable (IV) design is a well-known approach for unbiased evaluation of causal effects in the presence of unobserved confounding. In this article, we study the IV approach to account for selection bias in regression analysis with outcome missing not at random. In such a setting, a valid IV is a variable which (i) predicts the nonresponse process, and (ii) is independent of the outcome in the underlying population. We show that under the additional assumption (iii) that the IV is independent of the magnitude of selection bias due to nonresponse, the population regression in view is nonparametrically identified. For point estimation under (i)-(iii), we propose a simple complete-case analysis which modifies the regression of primary interest by carefully incorporating the IV to account for selection bias. The approach is developed for the identity, log and logit link functions. For inferences about the marginal mean of a binary outcome assuming (i) and (ii) only, we describe novel and approximately sharp bounds which unlike Robins-Manski bounds, are smooth in model parameters, therefore allowing for a straightforward approach to account for uncertainty due to sampling variability. These bounds provide a more honest account of uncertainty and allows one to assess the extent to which a violation of the key identifying condition (iii) might affect inferences. For illustration, the methods are used to account for selection bias induced by HIV testing nonparticipation in the evaluation of HIV prevalence in the Zambian Demographic and Health Surveys.

A multi-dimensional functional principal components analysis of EEG data.

Abstract: The electroencephalography (EEG) data created in event-related potential (ERP) experiments have a complex high-dimensional structure. Each stimulus presentation, or trial, generates an ERP waveform which is an instance of functional data. The experiments are made up of sequences of multiple trials, resulting in longitudinal functional data and moreover, responses are recorded at multiple electrodes on the scalp, adding an electrode dimension. Traditional EEG analyses involve multiple simplifications of this structure to increase the signal-to-noise ratio, effectively collapsing the functional and longitudinal components by identifying key features of the ERPs and averaging them across trials. Motivated by an implicit learning paradigm used in autism research in which the functional, longitudinal, and electrode components all have critical interpretations, we propose a multidimensional functional principal components analysis (MD-FPCA) technique which does not collapse any of the dimensions of the ERP data. The proposed decomposition is based on separation of the total variation into subject and subunit level variation which are further decomposed in a two-stage functional principal components analysis. The proposed methodology is shown to be useful for modeling longitudinal trends in the ERP functions, leading to novel insights into the learning patterns of children with Autism Spectrum Disorder (ASD) and their typically developing peers as well as comparisons between the two groups. Finite sample properties of MD-FPCA are further studied via extensive simulations.

Clustering multivariate functional data with phase variation

Abstract: When functional data come as multiple curves per subject, characterizing the source of variations is not a trivial problem. The complexity of the problem goes deeper when there is phase variation in addition to amplitude variation. We consider clustering problem with multivariate functional data that have phase variations among the functional variables. We propose a conditional subject-specific warping framework in order to extract relevant features for clustering. Using multivariate growth curves of various parts of the body as a motivating example, we demonstrate the effectiveness of the proposed approach. The found clusters have individuals who show different relative growth patterns among different parts of the body.

Diagnosing misspecification of the random-effects distribution in mixed models.

Abstract: It is traditionally assumed that the random effects in mixed models follow a multivariate normal distribution, making likelihood-based inferences more feasible theoretically and computationally. However, this assumption does not necessarily hold in practice which may lead to biased and unreliable results. We introduce a novel diagnostic test based on the so-called gradient function proposed by Verbeke and Molenberghs (2013) to assess the random-effects distribution. We establish asymptotic properties of our test and show that, under a correctly specified model, the proposed test statistic converges to a weighted sum of independent chi-squared random variables each with one degree of freedom. The weights, which are eigenvalues of a square matrix, can be easily calculated. We also develop a parametric bootstrap algorithm for small samples. Our strategy can be used to check the adequacy of any distribution for random effects in a wide class of mixed models, including linear mixed models, generalized linear mixed models, and non-linear mixed models, with univariate as well as multivariate random effects. Both asymptotic and bootstrap proposals are evaluated via simulations and a real data analysis of a randomized multicenter study on toenail dermatophyte onychomycosis.

Pearson's chi-square test and rank correlation inferences for clustered data

Abstract: Pearson's chi-square test has been widely used in testing for association between two categorical responses. Spearman rank correlation and Kendall's tau are often used for measuring and testing association between two continuous or ordered categorical responses. However, the established statistical properties of these tests are only valid when each pair of responses are independent, where each sampling unit has only one pair of responses. When each sampling unit consists of a cluster of paired responses, the assumption of independent pairs is violated. In this article, we apply the within-cluster resampling technique to U-statistics to form new tests and rank-based correlation estimators for possibly tied clustered data. We develop large sample properties of the new proposed tests and estimators and evaluate their performance by simulations. The proposed methods are applied to a data set collected from a PET/CT imaging study for illustration.