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Fatal car crashes are the leading cause of death among teenagers in the USA. The Graduated Driver Licensing (GDL) programme is one effective policy for reducing the number of teen fatal car crashes. Our study focuses on the number of fatal car crashes in Michigan during 1990–2004 excluding 1997, when the GDL started. We use Poisson regression with spatially dependent random effects to model the county level teen car crash counts. We develop a measurement error model to account for the fact that the total teenage population in the county level is used as a proxy for the teenage driver population. To the best of our knowledge, there is no existing literature that considers adjustment for measurement error in an offset variable. Furthermore, limited work has addressed the measurement errors in the context of spatial data. In our modelling, a Berkson measurement error model with spatial random effects is applied to adjust for the error-prone offset variable in a Bayesian paradigm. The Bayesian Markov chain Monte Carlo (MCMC) sampling is implemented in rstan. To assess the consequence of adjusting for measurement error, we compared two models with and without adjustment for measurement error. We found the effect of a time indicator becomes less significant with the measurement-error adjustment. It leads to our conclusion that the reduced number of teen drivers can help explain, to some extent, the effectiveness of GDL.

Dietary data collected from 24-hour dietary recalls are observed with significant measurement errors. In the nonparametric curve estimation literature, much of the effort has been devoted to designing methods that are consistent under contamination by noise, and which have been traditionally applied for analyzing those data. However, some foods such as alcohol or fruits are consumed only episodically, and may not be consumed during the day when the 24-hour recall is administered. These so-called excess zeros make existing nonparametric estimators break down, and new techniques need to be developed for such data. We develop two new consistent semiparametric estimators of the distribution of such episodically consumed food data, making parametric assumptions only on some less important parts of the model. We establish its theoretical properties and illustrate the good performance of our fully data-driven method in simulated and real data. Supplementary materials for this article are available online.

Canonical correlation analysis investigates linear relationships between two sets of variables, but often works poorly on modern data sets due to high-dimensionality and mixed data types such as continuous, binary and zero-inflated. To overcome these challenges, we propose a semiparametric approach for sparse canonical correlation analysis based on Gaussian copula. Our main contribution is a truncated latent Gaussian copula model for data with excess zeros, which allows us to derive a rank-based estimator of the latent correlation matrix for mixed variable types without the estimation of marginal transformation functions. The resulting canonical correlation analysis method works well in high-dimensional settings as demonstrated via numerical studies, as well as in application to the analysis of association between gene expression and micro RNA data of breast cancer patients., Accepted to Biometrika. Main text: 19 pages and 3 figures. Supplementary material: 28 pages and 9 figures

In nutrition, epidemiology, and other public health fields, composite scores are a common tool used to assess a health behaviour. These composite scores compare an individual's health behaviour to an idealized standard and provide a number, often between 0 and 100, to indicate their compliance to a health behaviour. Crucially, this measure of health behaviour is applied across populations (gender, smoking status, etc.) and health outcomes (colon cancer, breast cancer, etc.) to create a single interpretable score. One such composite score is the 2005 Healthy Eating Index that breaks diet into 12 components and evaluates nutritional intake by adherence to these components. We provide a general method that can be used to reassess the importance of these 12 components using flexible non-linear models, across populations and diseases, based on an asymptotic least squares approximation. We establish oracle properties of this variable selection technique in our context, which is different from the usual one population and one disease context. Although our methods are motivated by the Healthy Eating Index, they are broad enough to be applied to any composite score and a broad range of non-linear models.

Case-controls studies are popular epidemiological designs for detecting gene–environment interactions in the etiology of complex diseases, where the genetic susceptibility and environmental exposures may often be reasonably assumed independent in the source population. Various papers have presented analytical methods exploiting gene–environment independence to achieve better efficiency, all of which require either a rare disease assumption or a distributional assumption on the genetic variables. We relax both assumptions. We construct a semiparametric estimator in case-control studies exploiting gene–environment independence, while the distributions of genetic susceptibility and environmental exposures are both unspecified and the disease rate is assumed unknown and is not required to be close to zero. The resulting estimator is semiparametric efficient and its superiority over prospective logistic regression, the usual analysis in case-control studies, is demonstrated in various numerical illustrations.

Model averaging generally provides better predictions than model selection, but the existing model averaging methods cannot lead to parsimonious models. Parsimony is an especially important property when the number of parameters is large. To achieve a parsimonious model averaging coefficient estimator, we suggest a novel criterion for choosing weights. Asymptotic properties are derived in two practical scenarios: (i) one or more correct models exist in the candidate model set and (ii) all candidate models are misspecified. Under the former scenario, it is proved that our method can put the weight one to the smallest correct model and the resulting model averaging estimators of coefficients have many zeros and thus lead to a parsimonious model. The asymptotic distribution of the estimators is also provided. Under the latter scenario, prediction is mainly focused on and we prove that the proposed procedure is asymptotically optimal in the sense that its squared prediction loss and risk are asymptotically identical to those of the best—but infeasible—model averaging estimator. Numerical analysis shows the promise of the proposed procedure over existing model averaging and selection methods.

Summary Quantile regression is a semiparametric method for modeling associations between variables. It is most helpful when the covariates have complex relationships with the location, scale, and shape of the outcome distribution. Despite the method’s robustness to distributional assumptions and outliers in the outcome, regression quantiles may be biased in the presence of measurement error in the covariates. The impact of function-valued covariates contaminated with heteroscedastic error has not yet been examined previously; although, studies have investigated the case of scalar-valued covariates. We present a two-stage strategy to consistently fit linear quantile regression models with a function-valued covariate that may be measured with error. In the first stage, an instrumental variable is used to estimate the covariance matrix associated with the measurement error. In the second stage, simulation extrapolation (SIMEX) is used to correct for measurement error in the function-valued covariate. Point-wise standard errors are estimated by means of nonparametric bootstrap. We present simulation studies to assess the robustness of the measurement error corrected for functional quantile regression. Our methods are applied to National Health and Examination Survey data to assess the relationship between physical activity and body mass index among adults in the United States.

Summary We consider the problem where the data consist of a survival time and a binary outcome measurement for each individual, as well as corresponding predictors. The goal is to select the common set of predictors which affect both the responses, and not just one of them. In addition, we develop a survival prediction model based on data integration. The paper is motivated by the Cancer Genomic Atlas databank, which is currently the largest genomics and transcriptomics database. The data contain cancer survival information along with cancer stages for each patient. Furthermore, it contains reverse phase protein array measurements for each individual, which are the predictors associated with these responses. The biological motivation is to identify the major actionable proteins associated with both survival outcomes and cancer stages. We develop a Bayesian hierarchical model to model jointly the survival time and the classification of the cancer stages. Moreover, to deal with the high dimensionality of the reverse phase protein array measurements, we use a shrinkage prior to identify significant proteins. Simulations and Cancer Genomic Atlas data analysis show that the joint integrated modelling approach improves survival prediction.

Summary We develop model averaging estimation in the linear regression model where some covariates are subject to measurement error. The absence of the true covariates in this framework makes the calculation of the standard residual-based loss function impossible. We take advantage of the explicit form of the parameter estimators and construct a weight choice criterion. It is asymptotically equivalent to the unknown model average estimator minimizing the loss function. When the true model is not included in the set of candidate models, the method achieves optimality in terms of minimizing the relative loss, whereas, when the true model is included, the method estimates the model parameter with root n rate. Simulation results in comparison with existing Bayesian information criterion and Akaike information criterion model selection and model averaging methods strongly favour our model averaging method. The method is applied to a study on health.

Summary Huntington disease is an autosomal dominant, neurodegenerative disease without clearly identified biomarkers for when motor-onset occurs. Current standards to determine motor-onset rely on a clinician’s subjective judgment that a patient’s extrapyramidal signs are unequivocally associated with Huntington disease. This subjectivity can lead to error which could be overcome using an objective, data-driven metric that determines motor-onset. Recent studies of motor-sign decline—the longitudinal degeneration of motor-ability in patients—have revealed that motor-onset is closely related to an inflection point in its longitudinal trajectory. We propose a nonlinear location-shift marker model that captures this motor-sign decline and assesses how its inflection point is linked to other markers of Huntington disease progression. We propose two estimating procedures to estimate this model and its inflection point: one is a parametric method using nonlinear mixed effects model and the other one is a multi-stage nonparametric approach, which we developed. In an empirical study, the parametric approach was sensitive to correct specification of the mean structure of the longitudinal data. In contrast, our multi-stage nonparametric procedure consistently produced unbiased estimates regardless of the true mean structure. Applying our multi-stage nonparametric estimator to Neurobiological Predictors of Huntington Disease, a large observational study of Huntington disease, leads to earlier prediction of motor-onset compared to the clinician’s subjective judgment.

Many methods have recently been proposed for efficient analysis of case-control studies of gene-environment interactions using a retrospective likelihood framework that exploits the natural assumption of gene-environment independence in the underlying population. However, for polygenic modelling of gene-environment interactions, which is a topic of increasing scientific interest, applications of retrospective methods have been limited due to a requirement in the literature for parametric modelling of the distribution of the genetic factors. We propose a general, computationally simple, semiparametric method for analysis of case-control studies that allows exploitation of the assumption of gene-environment independence without any further parametric modelling assumptions about the marginal distributions of any of the two sets of factors. The method relies on the key observation that an underlying efficient profile likelihood depends on the distribution of genetic factors only through certain expectation terms that can be evaluated empirically. We develop asymptotic inferential theory for the estimator and evaluate its numerical performance via simulation studies. An application of the method is presented.

© 2017 American Statistical Association. We consider a problem motivated by issues in nutritional epidemiology, across diseases and populations. In this area, it is becoming increasingly common for diseases to be modeled by a single diet score, such as the Healthy Eating Index, the Mediterranean Diet Score, etc. For each disease and for each population, a partially linear single-index model is fit. The partially linear aspect of the problem is allowed to differ in each population and disease. However, and crucially, the single-index itself, having to do with the diet score, is common to all diseases and populations, and the nonparametrically estimated functions of the single-index are the same up to a scale parameter. Using B-splines with an increasing number of knots, we develop a method to solve the problem, and display its asymptotic theory. An application to the NIH-AARP Study of Diet and Health is described, where we show the advantages of using multiple diseases and populations simultaneously rather than one at a time in understanding the effect of increased Milk consumption. Simulations illustrate the properties of the methods. Supplementary materials for this article are available online.

© 2017 American Statistical Association. Prediction precision is arguably the most relevant criterion of a model in practice and is often a sought after property. A common difficulty with covariates measured with errors is the impossibility of performing prediction evaluation on the data even if a model is completely given without any unknown parameters. We bypass this inherent difficulty by using special properties on moment relations in linear regression models with measurement errors. The end product is a model selection procedure that achieves the same optimality properties that are achieved in classical linear regression models without covariate measurement error. Asymptotically, the procedure selects the model with the minimum prediction error in general, and selects the smallest correct model if the regression relation is indeed linear. Our model selection procedure is useful in prediction when future covariates without measurement error become available, for example, due to improved technology or better management and design of data collection procedures. Supplementary materials for this article are available online.

We consider a problem of data integration. Consider determining which genes affect a disease. The genes, which we call predictor objects, can be measured in different experiments on the same individual. We address the question of finding which genes are predictors of disease by any of the experiments. Our formulation is more general. In a given data set, there are a fixed number of responses for each individual, which may include a mix of discrete, binary and continuous variables. There is also a class of predictor objects, which may differ within a subject depending on how the predictor object is measured, i.e., depend on the experiment. The goal is to select which predictor objects affect any of the responses, where the number of such informative predictor objects or features tends to infinity as sample size increases. There are marginal likelihoods for each way the predictor object is measured, i.e., for each experiment. We specify a pseudolikelihood combining the marginal likelihoods, and propose a pseudolikelihood information criterion. Under regularity conditions, we establish selection consistency for the pseudolikelihood information criterion with unbounded true model size, which includes a Bayesian information criterion with appropriate penalty term as a special case. Simulations indicate that data integration improves upon, sometimes dramatically, using only one of the data sources.

Frequentist standard errors are a measure of uncertainty of an estimator, and the basis for statistical inferences. Frequestist standard errors can also be derived for Bayes estimators. However, except in special cases, the computation of the standard error of Bayesian estimators requires bootstrapping, which in combination with Markov chain Monte Carlo (MCMC) can be highly time consuming. We discuss an alternative approach for computing frequentist standard errors of Bayesian estimators, including importance sampling. Through several numerical examples we show that our approach can be much more computationally efficient than the standard bootstrap.

© Institute of Mathematical Statistics, 2019. A fundamental assumption used in causal inference with observational data is that treatment assignment is ignorable given measured confounding variables. This assumption of no missing confounders is plausible if a large number of baseline covariates are included in the analysis, as we often have no prior knowledge of which variables can be important confounders. Thus, estimation of treatment effects with a large number of covariates has received considerable attention in recent years. Most existing methods require specifying certain parametric models involving the outcome, treatment and confounding variables, and employ a variable selection procedure to identify confounders. However, selection of a proper set of confounders depends on correct specification of the working models. The bias due to model misspecification and incorrect selection of confounding variables can yield misleading results. We propose a robust and efficient approach for inference about the average treatment effect via a flexible modeling strategy incorporating penalized variable selection. Specifically, we consider an estimator constructed based on an efficient influence function that involves a propensity score and an outcome regression. We then propose a new sparse sufficient dimension reduction method to estimate these two functions without making restrictive parametric modeling assumptions. The proposed estimator of the average treatment effect is asymptotically normal and semiparametrically efficient without the need for variable selection consistency. The proposed methods are illustrated via simulation studies and a biomedical application.

© 2018, International Biometric Society. We propose a copula-based approach for analyzing functional data with correlated multiple functional outcomes exhibiting heterogeneous shape characteristics. To accommodate the possibly large number of parameters due to having several functional outcomes, parameter estimation is performed in two steps: first, the parameters for the marginal distributions are estimated using the skew t family, and then the dependence structure both within and across outcomes is estimated using a Gaussian copula. We develop an estimation algorithm for the dependence parameters based on the Karhunen–Loève expansion and an EM algorithm that significantly reduces the dimension of the problem and is computationally efficient. We also demonstrate prediction of an unknown outcome when the other outcomes are known. We apply our methodology to diffusion tensor imaging data for multiple sclerosis (MS) patients with three outcomes and identify differences in both the marginal distributions and the dependence structure between the MS and control groups. Our proposed methodology is quite general and can be applied to other functional data with multiple outcomes in biology and other fields. Supplementary materials accompanying this paper appear online.

© Institute of Statistical Science. All rights reserved. This paper is dedicated to the memory of Peter G. Hall. It concerns a deceptively simple question: if one observes variables corrupted with measurement error of possibly very complex form, can one recreate asymptotically the clusters that would have been found had there been no measurement error? We show that the answer is yes, and that the solution is surprisingly simple and general. The method itself is to simulate, by computer, realizations with the same distribution as that of the true variables, and then to apply clustering to these realizations. Technically, we show that if one uses K-means clustering or any other risk minimizing clustering, and a multivariate deconvolution device with certain smoothness and convergence properties, then, in the limit, the cluster means based on our method converge to the same cluster means as if there were no measurement error. Along with the method and its technical justification, we analyze two important nutrition data sets, finding patterns that make sense nutritionally.

© 2020, © 2020 American Statistical Association. Estimating the marginal and joint densities of the long-term average intakes of different dietary components is an important problem in nutritional epidemiology. Since these variables cannot be directly measured, data are usually collected in the form of 24-hr recalls of the intakes, which show marked patterns of conditional heteroscedasticity. Significantly compounding the challenges, the recalls for episodically consumed dietary components also include exact zeros. The problem of estimating the density of the latent long-time intakes from their observed measurement error contaminated proxies is then a problem of deconvolution of densities with zero-inflated data. We propose a Bayesian semiparametric solution to the problem, building on a novel hierarchical latent variable framework that translates the problem to one involving continuous surrogates only. Crucial to accommodating important aspects of the problem, we then design a copula based approach to model the involved joint distributions, adopting different modeling strategies for the marginals of the different dietary components. We design efficient Markov chain Monte Carlo algorithms for posterior inference and illustrate the efficacy of the proposed method through simulation experiments. Applied to our motivating nutritional epidemiology problems, compared to other approaches, our method provides more realistic estimates of the consumption patterns of episodically consumed dietary components. Supplementary materials for this article, including a standardized description of the materials available for reproducing the work, are available as an online supplement.

We assessed the ability of several penalized regression methods for linear and logistic models to identify outcome-associated predictors and the impact of predictor selection on parameter inference for practical sample sizes. We studied effect estimates obtained directly from penalized methods (Algorithm 1), or by refitting selected predictors with standard regression (Algorithm 2). For linear models, penalized linear regression, elastic net, smoothly clipped absolute deviation (SCAD), least angle regression and LASSO had a low false negative (FN) predictor selection rates but false positive (FP) rates above 20 % for all sample and effect sizes. Partial least squares regression had few FPs but many FNs. Only relaxo had low FP and FN rates. For logistic models, LASSO and penalized logistic regression had many FPs and few FNs for all sample and effect sizes. SCAD and adaptive logistic regression had low or moderate FP rates but many FNs. 95 % confidence interval coverage of predictors with null effects was approximately 100 % for Algorithm 1 for all methods, and 95 % for Algorithm 2 for large sample and effect sizes. Coverage was low only for penalized partial least squares (linear regression). For outcome-associated predictors, coverage was close to 95 % for Algorithm 2 for large sample and effect sizes for all methods except penalized partial least squares and penalized logistic regression. Coverage was sub-nominal for Algorithm 1. In conclusion, many methods performed comparably, and while Algorithm 2 is preferred to Algorithm 1 for estimation, it yields valid inference only for large effect and sample sizes. Electronic supplementary material The online version of this article (doi:10.1007/s00180-016-0690-2) contains supplementary material, which is available to authorized users.

An important use of measurement error models is to correct regression models for bias due to covariate measurement error. Most measurement error models assume that the observed error-prone covariate (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$W$\end{document}) is a linear function of the unobserved true covariate (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$X$\end{document}) plus other covariates (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$Z$\end{document}) in the regression model. In this paper, we consider models for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$W$\end{document} that include interactions between \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$X$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$Z$\end{document}. We derive the conditional distribution of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$X$\end{document} given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$W$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$Z$\end{document} and use it to extend the method of regression calibration to this class of measurement error models. We apply the model to dietary data and test whether self-reported dietary intake includes an interaction between true intake and body mass index. We also perform simulations to compare the model to simpler approximate calibration models.

In this paper, the influence of measurement errors in exposure doses in a regression model with binary response is studied. Recently, it has been recognized that uncertainty in exposure dose is characterized by errors of two types: classical additive errors and Berkson multiplicative errors. The combination of classical additive and Berkson multiplicative errors has not been considered in the literature previously. In a simulation study based on data from radio-epidemiological research of thyroid cancer in Ukraine caused by the Chornobyl accident, it is shown that ignoring measurement errors in doses leads to overestimation of background prevalence and underestimation of excess relative risk. In the work, several methods to reduce these biases are proposed. They are new regression calibration, an additive version of efficient SIMEX, and novel corrected score methods.

Motivated by physical activity data obtained from the BodyMedia FIT device (www.bodymedia.com), we take a functional data approach for longitudinal studies with continuous proportional outcomes. The functional structure depends on three factors. In our three-factor model, the regression structures are specified as curves measured at various factor-points with random effects that have a correlation structure. The random curve for the continuous factor is summarized using a few important principal components. The difficulties in handling the continuous proportion variables are solved by using a quasilikelihood type approximation. We develop an efficient algorithm to fit the model, which involves the selection of the number of principal components. The method is evaluated empirically by a simulation study. This approach is applied to the BodyMedia data with 935 males and 84 consecutive days of observation, for a total of 78, 540 observations. We show that sleep efficiency increases with increasing physical activity, while its variance decreases at the same time.

Summary Analysing secondary outcomes is a common practice for case–control studies. Traditional secondary analysis employs either completely parametric models or conditional mean regression models to link the secondary outcome to covariates. In many situations, quantile regression models complement mean-based analyses and provide alternative new insights on the associations of interest. For example, biomedical outcomes are often highly asymmetric, and median regression is more useful in describing the ‘central’ behaviour than mean regressions. There are also cases where the research interest is to study the high or low quantiles of a population, as they are more likely to be at risk. We approach the secondary quantile regression problem from a semiparametric perspective, allowing the covariate distribution to be completely unspecified. We derive a class of consistent semiparametric estimators and identify the efficient member. The asymptotic properties of the resulting estimators are established. Simulation results and a real data analysis are provided to demonstrate the superior performance of our approach with a comparison with the only existing approach so far in the literature.

Motivated by analyses of DNA methylation data, we propose a semiparametric mixture model, namely the generalized exponential tilt mixture model, to account for heterogeneity between differentially methylated and non-differentially methylated subjects in the cancer group, and capture the differences in higher order moments (e.g. mean and variance) between subjects in cancer and normal groups. A pairwise pseudolikelihood is constructed to eliminate the unknown nuisance function. To circumvent boundary and non-identifiability problems as in parametric mixture models, we modify the pseudolikelihood by adding a penalty function. In addition, the test with simple asymptotic distribution has computational advantages compared with permutation-based test for high-dimensional genetic or epigenetic data. We propose a pseudolikelihood based expectation-maximization test, and show the proposed test follows a simple chi-squared limiting distribution. Simulation studies show that the proposed test controls Type I errors well and has better power compared to several current tests. In particular, the proposed test outperforms the commonly used tests under all simulation settings considered, especially when there are variance differences between two groups. The proposed test is applied to a real data set to identify differentially methylated sites between ovarian cancer subjects and normal subjects.

© 2018, © 2018 American Statistical Association. We develop a Bayes factor-based testing procedure for comparing two population means in high-dimensional settings. In ‘large-p-small-n” settings, Bayes factors based on proper priors require eliciting a large and complex p × p covariance matrix, whereas Bayes factors based on Jeffrey’s prior suffer the same impediment as the classical Hotelling T2 test statistic as they involve inversion of ill-formed sample covariance matrices. To circumvent this limitation, we propose that the Bayes factor be based on lower dimensional random projections of the high-dimensional data vectors. We choose the prior under the alternative to maximize the power of the test for a fixed threshold level, yielding a restricted most powerful Bayesian test (RMPBT). The final test statistic is based on the ensemble of Bayes factors corresponding to multiple replications of randomly projected data. We show that the test is unbiased and, under mild conditions, is also locally consistent. We demonstrate the efficacy of the approach through simulated and real data examples. Supplementary materials for this article are available online.

© 2018, Institute of Mathematical Statistics. All rights reserved. Studying the relationship between covariates based on retrospective data is the main purpose of secondary analysis, an area of increasing interest. We examine the secondary analysis problem when multiple covariates are available, while only a regression mean model is specified. Despite the completely parametric modeling of the regression mean function, the case-control nature of the data requires special treatment and semiparametric efficient estimation generates various nonparametric estimation problems with multivariate covariates. We devise a dimension reduction approach that fits with the specified primary and secondary models in the original problem setting, and use reweighting to adjust for the case-control nature of the data, even when the disease rate in the source population is unknown. The resulting estimator is both locally efficient and robust against the misspecification of the regression error distribution, which can be heteroscedastic as well as non-Gaussian. We demonstrate the advantage of our method over several existing methods, both analytically and numerically.

© 2018, Institute of Mathematical Statistics. All rights reserved. Epidemiologists often categorize a continuous risk predictor, even when the true risk model is not a categorical one. Nonetheless, such categorization is thought to be more robust and interpretable, and thus their goal is to fit the categorical model and interpret the categorical parameters. We address the question: with measurement error and categorization, how can we do what epidemiologists want, namely to estimate the parameters of the categorical model that would have been estimated if the true predictor was observed? We develop a general methodology for such an analysis, and illustrate it in linear and logistic regression. Simulation studies are presented and the methodology is applied to a nutrition data set. Discussion of alternative approaches is also included.

Exposure measurement error can result in a biased estimate of the association between an exposure and outcome. When the exposure–outcome relationship is linear on the appropriate scale (e.g. linear, logistic) and the measurement error is classical, that is the result of random noise, the result is attenuation of the effect. When the relationship is non-linear, measurement error distorts the true shape of the association. Regression calibration is a commonly used method for correcting for measurement error, in which each individual’s unknown true exposure in the outcome regression model is replaced by its expectation conditional on the error-prone measure and any fully measured covariates. Regression calibration is simple to execute when the exposure is untransformed in the linear predictor of the outcome regression model, but less straightforward when non-linear transformations of the exposure are used. We describe a method for applying regression calibration in models in which a non-linear association is modelled by transforming the exposure using a fractional polynomial model. It is shown that taking a Bayesian estimation approach is advantageous. By use of Markov chain Monte Carlo algorithms, one can sample from the distribution of the true exposure for each individual. Transformations of the sampled values can then be performed directly and used to find the expectation of the transformed exposure required for regression calibration. A simulation study shows that the proposed approach performs well. We apply the method to investigate the relationship between usual alcohol intake and subsequent all-cause mortality using an error model that adjusts for the episodic nature of alcohol consumption.

© 2018 American Statistical Association. We consider the problem of multivariate density deconvolution when interest lies in estimating the distribution of a vector valued random variable X but precise measurements on X are not available, observations being contaminated by measurement errors U. The existing sparse literature on the problem assumes the density of the measurement errors to be completely known. We propose robust Bayesian semiparametric multivariate deconvolution approaches when the measurement error density of U is not known but replicated proxies are available for at least some individuals. Additionally, we allow the variability of U to depend on the associated unobserved values of X through unknown relationships, which also automatically includes the case of multivariate multiplicative measurement errors. Basic properties of finite mixture models, multivariate normal kernels, and exchangeable priors are exploited in novel ways to meet modeling and computational challenges. Theoretical results showing the flexibility of the proposed methods in capturing a wide variety of data-generating processes are provided. We illustrate the efficiency of the proposed methods in recovering the density of X through simulation experiments. The methodology is applied to estimate the joint consumption pattern of different dietary components from contaminated 24 h recalls. Supplementary materials for this article are available online.

© 2015, Springer Science+Business Media New York. In truncated polynomial spline or B-spline models where the covariates are measured with error, a fully Bayesian approach to model fitting requires the covariates and model parameters to be sampled at every Markov chain Monte Carlo iteration. Sampling the unobserved covariates poses a major computational problem and usually Gibbs sampling is not possible. This forces the practitioner to use a Metropolis–Hastings step which might suffer from unacceptable performance due to poor mixing and might require careful tuning. In this article we show for the cases of truncated polynomial spline or B-spline models of degree equal to one, the complete conditional distribution of the covariates measured with error is available explicitly as a mixture of double-truncated normals, thereby enabling a Gibbs sampling scheme. We demonstrate via a simulation study that our technique performs favorably in terms of computational efficiency and statistical performance. Our results indicate up to 62 and 54 % increase in mean integrated squared error efficiency when compared to existing alternatives while using truncated polynomial splines and B-splines respectively. Furthermore, there is evidence that the gain in efficiency increases with the measurement error variance, indicating the proposed method is a particularly valuable tool for challenging applications that present high measurement error. We conclude with a demonstration on a nutritional epidemiology data set from the NIH-AARP study and by pointing out some possible extensions of the current work.

© 2015, © American Statistical Association, Institute of Mathematical Statistics, and Interface Foundation of North America. Recent years have seen active developments of various penalized regression methods, such as LASSO and elastic net, to analyze high-dimensional data. In these approaches, the direction and length of the regression coefficients are determined simultaneously. Due to the introduction of penalties, the length of the estimates can be far from being optimal for accurate predictions. We introduce a new framework, regression by projection, and its sparse version to analyze high-dimensional data. The unique nature of this framework is that the directions of the regression coefficients are inferred first, and the lengths and the tuning parameters are determined by a cross-validation procedure to achieve the largest prediction accuracy. We provide a theoretical result for simultaneous model selection consistency and parameter estimation consistency of our method in high dimension. This new framework is then generalized such that it can be applied to principal components analysis, partial least squares, and canonical correlation analysis. We also adapt this framework for discriminant analysis. Compared with the existing methods, where there is relatively little control of the dependency among the sparse components, our method can control the relationships among the components. We present efficient algorithms and related theory for solving the sparse regression by projection problem. Based on extensive simulations and real data analysis, we demonstrate that our method achieves good predictive performance and variable selection in the regression setting, and the ability to control relationships between the sparse components leads to more accurate classification. In supplementary materials available online, the details of the algorithms and theoretical proofs, and R codes for all simulation studies are provided.

Summary We study the regression relationship between covariates in case–control data: an area known as the secondary analysis of case–control studies. The context is such that only the form of the regression mean is specified, so that we allow an arbitrary regression error distribution, which can depend on the covariates and thus can be heteroscedastic. Under mild regularity conditions we establish the theoretical identifiability of such models. Previous work in this context has either specified a fully parametric distribution for the regression errors, specified a homoscedastic distribution for the regression errors, has specified the rate of disease in the population (we refer to this as the true population) or has made a rare disease approximation. We construct a class of semiparametric estimation procedures that rely on none of these. The estimators differ from the usual semiparametric estimators in that they draw conclusions about the true population, while technically operating in a hypothetical superpopulation. We also construct estimators with a unique feature, in that they are robust against the misspecification of the regression error distribution in terms of variance structure, whereas all other non-parametric effects are estimated despite the biased samples. We establish the asymptotic properties of the estimators and illustrate their finite sample performance through simulation studies, as well as through an empirical example on the relationship between red meat consumption and hetero-cyclic amines. Our analysis verified the positive relationship between red meat consumption and two forms of hetro-cyclic amines, indicating that increased red meat consumption leads to increased levels of MeIQx and PhIP, both being risk factors for colorectal cancer. Computer software as well as data to illustrate the methodology are available from http://www.stat.tamu.edu/~carroll/matlab__programs/software.php .

© 2017, Institute of Mathematical Statistics. All rights reserved. While popular, single index models and additive models have potential limitations, a fact that leads us to propose SiAM, a novel hybrid combination of these two models. We first address model identifiability under general assumptions. The result is of independent interest. We then develop an estimation procedure by using splines to approximate unknown functions and establish the asymptotic properties of the resulting estimators. Furthermore, we suggest a two-step procedure for establishing confidence bands for the nonparametric additive functions. This procedure enables us to make global inferences. Numerical experiments indicate that SiAM works well with finite sample sizes, and are especially robust to model structures. That is, when the model reduces to either single-index or additive scenario, the estimation and inference results are comparable to those based on the true model, while when the model is misspecified, the superiority of our method can be very great.

© 2017, Institute of Mathematical Statistics. All rights reserved. We introduce a general single index semiparametric measurement error model for the case that the main covariate of interest is measured with error and modeled parametrically, and where there are many other variables also important to the modeling. We propose a semiparametric bias-correction approach to estimate the effect of the covariate of interest. The resultant estimators are shown to be root-n consistent, asymptotically normal and locally efficient. Comprehensive simulations and an analysis of an empirical data set are performed to demonstrate the finite sample performance and the bias reduction of the locally efficient estimators.

© 2014, © 2014 American Statistical Association, Institute of Mathematical Statistics, and Interface Foundation of North America. We consider the problem of estimating the density of a random variable when precise measurements on the variable are not available, but replicated proxies contaminated with measurement error are available for sufficiently many subjects. Under the assumption of additive measurement errors this reduces to a problem of deconvolution of densities. Deconvolution methods often make restrictive and unrealistic assumptions about the density of interest and the distribution of measurement errors, for example, normality and homoscedasticity and thus independence from the variable of interest. This article relaxes these assumptions and introduces novel Bayesian semiparametric methodology based on Dirichlet process mixture models for robust deconvolution of densities in the presence of conditionally heteroscedastic measurement errors. In particular, the models can adapt to asymmetry, heavy tails, and multimodality. In simulation experiments, we show that our methods vastly outperform a recent Bayesian approach based on estimating the densities via mixtures of splines. We apply our methods to data from nutritional epidemiology. Even in the special case when the measurement errors are homoscedastic, our methodology is novel and dominates other methods that have been proposed previously. Additional simulation results, instructions on getting access to the dataset and R programs implementing our methods are included as part of online supplementary materials.

We provide evidence that a root-mean-square test of goodness-of-fit can be significantly more powerful than state-of-the-art exact tests in detecting deviations from Hardy-Weinberg equilibrium. Unlike Pearson's chi-square test, the log--likelihood-ratio test, and Fisher's exact test, which are sensitive to relative discrepancies between genotypic frequencies, the root-mean-square test is sensitive to absolute discrepancies. This can increase statistical power, as we demonstrate using benchmark datasets and through asymptotic analysis. With the aid of computers, exact P-values for the root-mean-square statistic can be calculated eeffortlessly, and can be easily implemented using the author's freely available code., 29 pages, 6 figures

We describe a new approach to analyze chirp syllables of free-tailed bats from two regions of Texas in which they are predominant: Austin and College Station. Our goal is to characterize any systematic regional differences in the mating chirps and assess whether individual bats have signature chirps. The data are analyzed by modeling spectrograms of the chirps as responses in a Bayesian functional mixed model. Given the variable chirp lengths, we compute the spectrograms on a relative time scale interpretable as the relative chirp position, using a variable window overlap based on chirp length. We use 2D wavelet transforms to capture correlation within the spectrogram in our modeling and obtain adaptive regularization of the estimates and inference for the regions-specific spectrograms. Our model includes random effect spectrograms at the bat level to account for correlation among chirps from the same bat, and to assess relative variability in chirp spectrograms within and between bats. The modeling of spectrograms using functional mixed models is a general approach for the analysis of replicated nonstationary time series, such as our acoustical signals, to relate aspects of the signals to various predictors, while accounting for between-signal structure. This can be done on raw spectrograms when all signals are of the same length, and can be done using spectrograms defined on a relative time scale for signals of variable length in settings where the idea of defining correspondence across signals based on relative position is sensible.

The Lasso shrinkage procedure achieved its popularity, in part, by its tendency to shrink estimated coefficients to zero, and its ability to serve as a variable selection procedure. Using data-adaptive weights, the adaptive Lasso modified the original procedure to increase the penalty terms for those variables estimated to be less important by ordinary least squares. Although this modified procedure attained the oracle properties, the resulting models tend to include a large number of "false positives" in practice. Here, we adapt the concept of local false discovery rates (lFDRs) so that it applies to the sequence, λn, of smoothing parameters for the adaptive Lasso.We define the lFDR for a given λn to be the probability that the variable added to the model by decreasing λn to λn - δ is not associated with the outcome, where δ is a small value. We derive the relationship between the lFDR and λn, show lFDR =1 for traditional smoothing parameters, and show how to select λn so as to achieve a desired lFDR.We compare the smoothing parameters chosen to achieve a specified lFDR and those chosen to achieve the oracle properties, as well as their resulting estimates for model coefficients, with both simulation and an example from a genetic study of prostate specific antigen. © The Author 2013. Published by Oxford University Press. All rights reserved.

Sedentary behavior has already been associated with mortality, cardiovascular disease, and cancer. Questionnaires are an affordable tool for measuring sedentary behavior in large epidemiological studies. Here, we introduce and evaluate two statistical methods for quantifying measurement error in questionnaires. Accurate estimates are needed for assessing questionnaire quality. The two methods would be applied to validation studies that measure a sedentary behavior by both questionnaire and accelerometer on multiple days. The first method fits a reduced model by assuming the accelerometer is without error, while the second method fits a more complete model that allows both measures to have error. Because accelerometers tend to be highly accurate, we show that ignoring the accelerometer's measurement error, can result in more accurate estimates of measurement error in some scenarios. In this manuscript, we derive asymptotic approximations for the Mean-Squared Error of the estimated parameters from both methods, evaluate their dependence on study design and behavior characteristics, and offer an R package so investigators can make an informed choice between the two methods. We demonstrate the difference between the two methods in a recent validation study comparing Previous Day Recalls (PDR) to an accelerometer-based ActivPal.

We study additive function-on-function regression where the mean response at a particular time point depends on the time point itself as well as the entire covariate trajectory. We develop a computationally efficient estimation methodology based on a novel combination of spline bases with an eigenbasis to represent the trivariate kernel function. We discuss prediction of a new response trajectory, propose an inference procedure that accounts for total variability in the predicted response curves, and construct pointwise prediction intervals. The estimation/inferential procedure accommodates realistic scenarios such as correlated error structure as well as sparse and/or irregular designs. We investigate our methodology in finite sample size through simulations and two real data applications., Comment: 26 pages, 4 figures

In recent years, genome-wide association studies (GWAS) and gene-expression profiling have generated a large number of valuable datasets for assessing how genetic variations are related to disease outcomes. With such datasets, it is often of interest to assess the overall effect of a set of genetic markers, assembled based on biological knowledge. Genetic marker-set analyses have been advocated as more reliable and powerful approaches compared with the traditional marginal approaches (Curtis and others, 2005. Pathways to the analysis of microarray data. TRENDS in Biotechnology 23, 429-435; Efroni and others, 2007. Identification of key processes underlying cancer phenotypes using biologic pathway analysis. PLoS One 2, 425). Procedures for testing the overall effect of a marker-set have been actively studied in recent years. For example, score tests derived under an Empirical Bayes (EB) framework (Liu and others, 2007. Semiparametric regression of multidimensional genetic pathway data: least-squares kernel machines and linear mixed models. Biometrics 63, 1079-1088; Liu and others, 2008. Estimation and testing for the effect of a genetic pathway on a disease outcome using logistic kernel machine regression via logistic mixed models. BMC bioinformatics 9, 292-2; Wu and others, 2010. Powerful SNP-set analysis for case-control genome-wide association studies. American Journal of Human Genetics 86, 929) have been proposed as powerful alternatives to the standard Rao score test (Rao, 1948. Large sample tests of statistical hypotheses concerning several parameters with applications to problems of estimation. Mathematical Proceedings of the Cambridge Philosophical Society, 44, 50-57). The advantages of these EB-based tests are most apparent when the markers are correlated, due to the reduction in the degrees of freedom. In this paper, we propose an adaptive score test which up-or down-weights the contributions from each member of the marker-set based on the Z-scores of their effects. Such an adaptive procedure gains power over the existing procedures when the signal is sparse and the correlation among the markers is weak. By combining evidence from both the EB-based score test and the adaptive test, we further construct an omnibus test that attains good power in most settings. The null distributions of the proposed test statistics can be approximated well either via simple perturbation procedures or via distributional approximations. Through extensive simulation studies, we demonstrate that the proposed procedures perform well in finite samples. We apply the tests to a breast cancer genetic study to assess the overall effect of the FGFR2 gene on breast cancer risk. © 2012 The Author.

Covariate measurement error and missing responses are typical features in longitudinal data analysis. There has been extensive research on either covariate measurement error or missing responses, but relatively little work has been done to address both simultaneously. In this paper, we propose a simple method for the marginal analysis of longitudinal data with time-varying covariates, some of which are measured with error, while the response is subject to missingness. Our method has a number of appealing properties: assumptions on the model are minimal, with none needed about the distribution of the mismeasured covariate; implementation is straightforward and its applicability is broad. We provide both theoretical justification and numerical results.

We propose a multiple imputation estimator for parameter estimation in a quantile regression model when some covariates are missing at random. The estimation procedure fully utilizes the entire dataset to achieve increased efficiency, and the resulting coefficient estimators are root-n consistent and asymptotically normal. To protect against possible model misspecification, we further propose a shrinkage estimator, which automatically adjusts for possible bias. The finite sample performance of our estimator is investigated in a simulation study. Finally, we apply our methodology to part of the Eating at American's Table Study data, investigating the association between two measures of dietary intake. 2012 Biometrika Trust2012 © 2012 Biometrika Trust.

We consider the problem of testing for a constant nonparametric effect in a general semiparametric regression model when there is a potential for interaction between the parametrically and nonparametrically modeled variables. The work was originally motivated by a unique testing problem in genetic epidemiology (Chatterjee et al., 2006) that involved a typical generalized linear model but with an additional term reminiscent of the Tukey 1-degree-of-freedom formulation, and their interest was in testing for main effects of the genetic variables, while gaining statistical power by allowing for a possible interaction between genes and the environment. Later work (Maity et al., 2009) involved the possibility of modeling the environmental variable nonparametrically, but they focused on whether there was a parametric main effect for the genetic variables. In this paper, we consider the complementary problem, where the interest is in testing for the main effect of the nonparametrically modeled environmental variable. We derive a generalized likelihood ratio test for this hypothesis, show how to implement it, and provide evidence that our method can improve statistical power when compared to standard partially linear models with main effects only. We use the method for the primary purpose of analyzing data from a case-control study of colorectal adenoma. © 2010 Elsevier B.V.

There has been great public health interest in estimating usual, i.e., long-term average, intake of episodically consumed dietary components that are not consumed daily by everyone, e.g., fish, red meat and whole grains. Short-term measurements of episodically consumed dietary components have zero-inflated skewed distributions. So-called two-part models have been developed for such data in order to correct for measurement error due to within-person variation and to estimate the distribution of usual intake of the dietary component in the univariate case. However, there is arguably much greater public health interest in the usual intake of an episodically consumed dietary component adjusted for energy (caloric) intake, e.g., ounces of whole grains per 1000 kilo-calories, which reflects usual dietary composition and adjusts for different total amounts of caloric intake. Because of this public health interest, it is important to have models to fit such data, and it is important that the model-fitting methods can be applied to all episodically consumed dietary components. We have recently developed a nonlinear mixed effects model (Kipnis, et al., 2010), and have fit it by maximum likelihood using nonlinear mixed effects programs and methodology (the SAS NLMIXED procedure). Maximum likelihood fitting of such a nonlinear mixed model is generally slow because of 3-dimensional adaptive Gaussian quadrature, and there are times when the programs either fail to converge or converge to models with a singular covariance matrix. For these reasons, we develop a Monte-Carlo (MCMC) computation of fitting this model, which allows for both frequentist and Bayesian inference. There are technical challenges to developing this solution because one of the covariance matrices in the model is patterned. Our main application is to the National Institutes of Health (NIH)-AARP Diet and Health Study, where we illustrate our methods for modeling the energy-adjusted usual intake of fish and whole grains. We demonstrate numerically that our methods lead to increased speed of computation, converge to reasonable solutions, and have the flexibility to be used in either a frequentist or a Bayesian manner. © 2011 Berkeley Electronic Press. All rights reserved.

Summary We consider functional measurement error models, i.e. models where covariates are measured with error and yet no distributional assumptions are made about the mismeasured variable. We propose and study a score-type local test and an orthogonal series-based, omnibus goodness-of-fit test in this context, where no likelihood function is available or calculated—i.e. all the tests are proposed in the semiparametric model framework. We demonstrate that our tests have optimality properties and computational advantages that are similar to those of the classical score tests in the parametric model framework. The test procedures are applicable to several semiparametric extensions of measurement error models, including when the measurement error distribution is estimated non-parametrically as well as for generalized partially linear models. The performance of the local score-type and omnibus goodness-of-fit tests is demonstrated through simulation studies and analysis of a nutrition data set.

The authors consider the analysis of hierarchical longitudinal functional data based upon a functional principal components approach. In contrast to standard frequentist approaches to selecting the number of principal components, the authors do model averaging using a Bayesian formulation. A relatively straightforward reversible jump Markov Chain Monte Carlo formulation has poor mixing properties and in simulated data often becomes trapped at the wrong number of principal components. In order to overcome this, the authors show how to apply Stochastic Approximation Monte Carlo (SAMC) to this problem, a method that has the potential to explore the entire space and does not become trapped in local extrema. The combination of reversible jump methods and SAMC in hierarchical longitudinal functional data is simplified by a polar coordinate representation of the principal components. The approach is easy to implement and does well in simulated data in determining the distribution of the number of principal components, and in terms of its frequentist estimation properties. Empirical applications are also presented.

In simulation studies for discriminant analysis, misclassification errors are often computed using the Monte Carlo method, by testing a classifier on large samples generated from known populations. Although large samples are expected to behave closely to the underlying distributions, they may not do so in a small interval or region, and thus may lead to unexpected results. We demonstrate with an example that the LDA misclassification error computed via the Monte Carlo method may often be smaller than the Bayes error. We give a rigorous explanation and recommend a method to properly compute misclassification errors.

Local polynomial estimators are popular techniques for nonparametric regression estimation and have received great attention in the literature. Their simplest version, the local constant estimator, can be easily extended to the errors-in-variables context by exploiting its similarity with the deconvolution kernel density estimator. The generalization of the higher order versions of the estimator, however, is not straightforward and has remained an open problem for the last 15 years. We propose an innovative local polynomial estimator of any order in the errors-in-variables context, derive its design-adaptive asymptotic properties and study its finite sample performance on simulated examples. We provide not only a solution to a long-standing open problem, but also provide methodological contributions to error-in-variable regression, including local polynomial estimation of derivative functions.