A latent factor linear mixed model for high‐dimensional longitudinal data analysis

High‐dimensional longitudinal data involving latent variables such as depression and anxiety that cannot be quantified directly are often encountered in biomedical and social sciences. Multiple responses are used to characterize these latent quantities, and repeated measures are collected to capture their trends over time. Furthermore, substantive research questions may concern issues such as interrelated trends among latent variables that can only be addressed by modeling them jointly. Although statistical analysis of univariate longitudinal data has been well developed, methods for modeling multivariate high‐dimensional longitudinal data are still under development. In this paper, we propose a latent factor linear mixed model (LFLMM) for analyzing this type of data. This model is a combination of the factor analysis and multivariate linear mixed models. Under this modeling framework, we reduced the high‐dimensional responses to low‐dimensional latent factors by the factor analysis model, and then we used the multivariate linear mixed model to study the longitudinal trends of these latent factors. We developed an expectation–maximization algorithm to estimate the model. We used simulation studies to investigate the computational properties of the expectation–maximization algorithm and compare the LFLMM model with other approaches for high‐dimensional longitudinal data analysis. We used a real data example to illustrate the practical usefulness of the model. Copyright © 2013 John Wiley & Sons, Ltd.     

Combining stratified and unstratified log‐rank tests in paired survival data

The log‐rank test is the most widely used nonparametric method for testing treatment differences in survival between two treatment groups due to its efficiency under the proportional hazards model. Most previous work on the log‐rank test has assumed that the samples from the two treatment groups are independent. This assumption is not always true. In multi‐center clinical trials, survival times of patients in the same medical center may be correlated due to factors specific to each center. For such data, we can construct both stratified and unstratified log‐rank tests. These two tests turn out to have very different powers for correlated samples. An appropriate linear combination of these two tests may give a more powerful test than either of the individual test. Under a bivariate frailty model, we obtain closed‐form asymptotic local alternative distributions and the correlation coefficient between these two tests. Based on these results we construct an optimal linear combination of the two test statistics to maximize the local power. Simulation studies with Hougaard's model confirm our construction. We also study the robustness of the combined test by simulations. Copyright © 2010 John Wiley & Sons, Ltd.     

Multiple imputation in the presence of non‐normal data

Multiple imputation (MI) is becoming increasingly popular for handling missing data. Standard approaches for MI assume normality for continuous variables (conditionally on the other variables in the imputation model). However, it is unclear how to impute non‐normally distributed continuous variables. Using simulation and a case study, we compared various transformations applied prior to imputation, including a novel non‐parametric transformation, to imputation on the raw scale and using predictive mean matching (PMM) when imputing non‐normal data. We generated data from a range of non‐normal distributions, and set 50% to missing completely at random or missing at random. We then imputed missing values on the raw scale, following a zero‐skewness log, Box–Cox or non‐parametric transformation and using PMM with both type 1 and 2 matching. We compared inferences regarding the marginal mean of the incomplete variable and the association with a fully observed outcome. We also compared results from these approaches in the analysis of depression and anxiety symptoms in parents of very preterm compared with term‐born infants. The results provide novel empirical evidence that the decision regarding how to impute a non‐normal variable should be based on the nature of the relationship between the variables of interest. If the relationship is linear in the untransformed scale, transformation can introduce bias irrespective of the transformation used. However, if the relationship is non‐linear, it may be important to transform the variable to accurately capture this relationship. A useful alternative is to impute the variable using PMM with type 1 matching. Copyright © 2016 John Wiley & Sons, Ltd.     

A three‐level mixed‐effects location scale model with an application to ecological momentary assessment data

In studies using ecological momentary assessment (EMA), or other intensive longitudinal data collection methods, interest frequently centers on changes in the variances, both within‐subjects and between‐subjects. For this, Hedeker et al. (Biometrics 2008; 64: 627–634) developed an extended two‐level mixed‐effects model that treats observations as being nested within subjects and allows covariates to influence both the within‐subjects and between‐subjects variance, beyond their influence on means. However, in EMA studies, subjects often provide many responses within and across days. To account for the possible systematic day‐to‐day variation, we developed a more flexible three‐level mixed‐effects location scale model that treats observations within days within subjects, and allows covariates to influence the variance at the subject, day, and observation level (over and above their usual effects on means) using a log‐linear representation throughout. We provide details of a maximum likelihood solution and demonstrate how SAS PROC NLMIXED can be used to achieve maximum likelihood estimates in an alternative parameterization of our proposed three‐level model. The accuracy of this approach using NLMIXED was verified by a series of simulation studies. Data from an adolescent mood study using EMA were analyzed to demonstrate this approach. The analyses clearly show the benefit of the proposed three‐level model over the existing two‐level approach. The proposed model has useful applications in many studies with three‐level structures where interest centers on the joint modeling of the mean and variance structure. Copyright © 2012 John Wiley & Sons, Ltd.     

A Bayesian generalized age–period–cohort power model for cancer projections

Age–period–cohort (APC) models are the state of art in cancer projections, assessing past and recent trends and extrapolating mortality or incidence data into the future. Nordpred is a well‐established software, assuming a Poisson distribution for the counts and a log‐link or power‐link function with fixed power; however, its predictive performance is poor for sparse data. Bayesian models with log‐link function have been applied, but they can lead to extreme estimates. In this paper, we address criticisms of the aforementioned models by providing Bayesian formulations based on a power‐link and develop a generalized APC power‐link model, which assumes a random rather than fixed power parameter. In addition, a power model with a fixed power parameter of five was formulated in the Bayesian framework. The predictive performance of the new models was evaluated on Swiss lung cancer mortality data using model‐based estimates of observed periods. Results indicated that the generalized APC power‐link model provides best estimates for male and female lung cancer mortality. The gender‐specific models were further applied to project lung cancer mortality in Switzerland during the periods 2009–2013 and 2014–2018. Copyright © 2014 John Wiley & Sons, Ltd.     

Analysis of multi‐arm tumor growth trials in xenograft animals using phase change adaptive piecewise quadratic models

Xenograft trials allow tumor growth in human cell lines to be monitored over time in a mouse model. We consider the problem of inferring the effect of treatment combinations on tumor growth. A piecewise quadratic model with flexible phase change locations is proposed to model the effect of change in therapy over time. Each piece represents a growth phase, with phase changes in response to change in treatment. Piecewise slopes represent phase‐specific (log) linear growth rates and curvature parameters represent departure from linear growth. Trial data are analyzed in two stages: (i) subject‐specific curve fitting (ii) analysis of slope and curvature estimates across subjects. A least‐squares approach with penalty for phase change point location is proposed for curve fitting. In simulation studies, the method is shown to give consistent estimates of slope and curvature parameters under independent and AR (1) measurement error. The piecewise quadratic model is shown to give excellent fit (median R²=0.98) to growth data from a six armed xenograft trial on a lung carcinoma cell line. Copyright © 2010 John Wiley & Sons, Ltd.     

Latent time‐varying factors in longitudinal analysis: a linear mixed hidden Markov model for heart rates

Longitudinal data are often segmented by unobserved time‐varying factors, which introduce latent heterogeneity at the observation level, in addition to heterogeneity across subjects. We account for this latent structure by a linear mixed hidden Markov model. It integrates subject‐specific random effects and Markovian sequences of time‐varying effects in the linear predictor. We propose an expectation?‐maximization algorithm for maximum likelihood estimation, based on data augmentation. It reduces to the iterative maximization of the expected value of a complete likelihood function, derived from an augmented dataset with case weights, alternated with weights updating. In a case study of the Survey on Stress Aging and Health in Russia, the model is exploited to estimate the influence of the observed covariates under unobserved time‐varying factors, which affect the cardiovascular activity of each subject during the observation period. Copyright © 2014 John Wiley & Sons, Ltd.     

A generalized partially linear mean‐covariance regression model for longitudinal proportional data,with applications to the analysis of quality of life data from cancer clinical trials

Motivated by the analysis of quality of life data from a clinical trial on early breast cancer, we propose in this paper a generalized partially linear mean‐covariance regression model for longitudinal proportional data, which are bounded in a closed interval. Cholesky decomposition of the covariance matrix for within‐subject responses and generalized estimation equations are used to estimate unknown parameters and the nonlinear function in the model. Simulation studies are performed to evaluate the performance of the proposed estimation procedures. Our new model is also applied to analyze the data from the cancer clinical trial that motivated this research. In comparison with available models in the literature, the proposed model does not require specific parametric assumptions on the density function of the longitudinal responses and the probability function of the boundary values and can capture dynamic changes of time or other interested variables on both mean and covariance of the correlated proportional responses. Copyright © 2017 John Wiley & Sons, Ltd.     

How to control for unmeasured confounding in an observational time‐to‐event study with exposure incidence information: the treatment choice Cox model

In an observational study of the effect of a treatment on a time‐to‐event outcome, a major problem is accounting for confounding because of unknown or unmeasured factors. We propose including covariates in a Cox model that can partially account for an unknown time‐independent frailty that is related to starting or stopping treatment as well as the outcome of interest. These covariates capture the times at which treatment is started or stopped and so are called treatment choice (TC) covariates. Three such models are developed: first, an interval TC model that assumes a very general form for the respective hazard functions of starting treatment, stopping treatment, and the outcome of interest and second, a parametric TC model that assumes that the log hazard functions for starting treatment, stopping treatment, and the outcome event include frailty as an additive term. Finally, a hybrid TC model that combines attributes from the parametric and interval TC models. As compared with an ordinary Cox model, the TC models are shown to substantially reduce the bias of the estimated hazard ratio for treatment when data are simulated from a realistic Cox model with residual confounding due to the unobserved frailty. The simulations also indicate that the bias decreases or levels off as the sample size increases. A TC model is illustrated by analyzing the Women's Health Initiative Observational Study of hormone replacement for post‐menopausal women. Published 2017. This article has been contributed to by US Government employees and their work is in the public domain in the USA.     

Incorporating baseline measurements into the analysis of crossover trials with time‐to‐event endpoints

Two‐period two‐treatment (2×2) crossover designs are commonly used in clinical trials. For continuous endpoints, it has been shown that baseline (pretreatment) measurements collected before the start of each treatment period can be useful in improving the power of the analysis. Methods to achieve a corresponding gain for censored time‐to‐event endpoints have not been adequately studied. We propose a method in which censored values are treated as missing data and multiply imputed using prespecified parametric event time models. The event times in each imputed data set are then log‐transformed and analyzed using a linear model suitable for a 2×2 crossover design with continuous endpoints, with the difference in period‐specific baselines included as a covariate. Results obtained from the imputed data sets are synthesized for point and confidence interval estimation of the treatment ratio of geometric mean event times using model averaging in conjunction with Rubin's combination rule. We use simulations to illustrate the favorable operating characteristics of our method relative to two other methods for crossover trials with censored time‐to‐event data, ie, a hierarchical rank test that ignores the baselines and a stratified Cox model that uses each study subject as a stratum and includes period‐specific baselines as a covariate. Application to a real data example is provided.     

Comparison of hypertabastic survival model with other unimodal hazard rate functions using a goodness‐of‐fit test

We studied the problem of testing a hypothesized distribution in survival regression models when the data is right censored and survival times are influenced by covariates. A modified chi‐squared type test, known as Nikulin‐Rao‐Robson statistic, is applied for the comparison of accelerated failure time models. This statistic is used to test the goodness‐of‐fit for hypertabastic survival model and four other unimodal hazard rate functions. The results of simulation study showed that the hypertabastic distribution can be used as an alternative to log‐logistic and log‐normal distribution. In statistical modeling, because of its flexible shape of hazard functions, this distribution can also be used as a competitor of Birnbaum‐Saunders and inverse Gaussian distributions. The results for the real data application are shown. Copyright © 2017 John Wiley & Sons, Ltd.     

Estimation for zero‐inflated over‐dispersed count data model with missing response

In this paper, we develop estimation procedure for the parameters of a zero‐inflated over‐dispersed/under‐dispersed count model in the presence of missing responses. In particular, we deal with a zero‐inflated extended negative binomial model in the presence of missing responses. A weighted expectation maximization algorithm is used for the maximum likelihood estimation of the parameters involved. Some simulations are conducted to study the properties of the estimators. Robustness of the procedure is shown when count data follow other over‐dispersed models, such as the log‐normal mixture of the Poisson distribution or even from a zero‐inflated Poisson model. An illustrative example and a discussion leading to some conclusions are given. Copyright © 2016 John Wiley & Sons, Ltd.     

Joint Modeling of Genetic Association and Population Stratification Using Latent Class Models

We show how latent class log‐linear models can be used to test for an association between a candidate gene and a disease phenotype in a stratified population when the stratification is unobserved. The stratification may arise because of several ethnic groups or immigration and may lead to spurious associations between several loci and the disease. The information about the stratification is drawn from additional markers that are chosen to be independent of the disease and unlinked to the candidate gene and to each other within each population stratum. We use the EM algorithm to simultaneously estimate all the model parameters, including proportions of individuals in the latent population strata. The latent class model is used to test the phenotype association of single nucleotide polymorphism markers in four candidate regions in population‐based case‐control data selected from simulated Genetic Analysis Workshop (GAW) 12 population isolate 30. The analysis clearly demonstrates how the number of false positive associations can be reduced when the model accounts for population stratification. © 2001 Wiley‐Liss, Inc.     

Meta‐analysis of studies with bivariate binary outcomes: a marginal beta‐binomial model approach

When conducting a meta‐analysis of studies with bivariate binary outcomes, challenges arise when the within‐study correlation and between‐study heterogeneity should be taken into account. In this paper, we propose a marginal beta‐binomial model for the meta‐analysis of studies with binary outcomes. This model is based on the composite likelihood approach and has several attractive features compared with the existing models such as bivariate generalized linear mixed model (Chu and Cole, 2006) and Sarmanov beta‐binomial model (Chen et al., 2012). The advantages of the proposed marginal model include modeling the probabilities in the original scale, not requiring any transformation of probabilities or any link function, having closed‐form expression of likelihood function, and no constraints on the correlation parameter. More importantly, because the marginal beta‐binomial model is only based on the marginal distributions, it does not suffer from potential misspecification of the joint distribution of bivariate study‐specific probabilities. Such misspecification is difficult to detect and can lead to biased inference using currents methods. We compare the performance of the marginal beta‐binomial model with the bivariate generalized linear mixed model and the Sarmanov beta‐binomial model by simulation studies. Interestingly, the results show that the marginal beta‐binomial model performs better than the Sarmanov beta‐binomial model, whether or not the true model is Sarmanov beta‐binomial, and the marginal beta‐binomial model is more robust than the bivariate generalized linear mixed model under model misspecifications. Two meta‐analyses of diagnostic accuracy studies and a meta‐analysis of case–control studies are conducted for illustration. Copyright © 2015 John Wiley & Sons, Ltd.     

Capture-recapture using multiple data sources: estimating the prevalence of diabetes

Objective: To examine the potential for using multiple list sources and capture‐recapture methods for estimating the prevalence of diagnosed diabetes. Method: A model‐averaging procedure using an adjusted Akaike's Information Criterion (QAICc) was used to combine capture‐recapture estimates from log‐linear models obtained from simultaneously analysing four sources of data. The method was illustrated using four separate lists of patients with diabetes, resident in Otago, New Zealand. Results: Eighteen candidate models with a QAICc weight of more than 0.01 were obtained. A total of 5,716 individuals were enrolled on one or more of the four lists, of whom 379 (6.6%) appeared on all four lists and 1,670 (29.2%) appeared on one list only. The model‐averaged estimate of the total number of people with diagnosed diabetes was 6,721 (95% CI: 6,097, 7,346). The age‐standardised prevalence was 3.70% (95% CI: 3.36–4.04%) for the total population and 4.45% (95% CI: 4.03–4.86) for adults aged 15+ years. Conclusions: Estimated diabetes prevalence was consistent with national survey results. Capture‐recapture methods, combined with model averaging, are a cheap, efficient tool to estimate the prevalence of diagnosed diabetes. Implications: This method provides a relatively easy way to estimate the prevalence of diagnosed diabetes using routinely collected diabetes information, thus providing the opportunity to monitor the diabetes epidemic and inform planning decisions and resource allocation.     

Logistic‐AFT location‐scale mixture regression models with nonsusceptibility for left‐truncated and general interval‐censored data

In conventional survival analysis there is an underlying assumption that all study subjects are susceptible to the event. In general, this assumption does not adequately hold when investigating the time to an event other than death. Owing to genetic and/or environmental etiology, study subjects may not be susceptible to the disease. Analyzing nonsusceptibility has become an important topic in biomedical, epidemiological, and sociological research, with recent statistical studies proposing several mixture models for right‐censored data in regression analysis. In longitudinal studies, we often encounter left, interval, and right‐censored data because of incomplete observations of the time endpoint, as well as possibly left‐truncated data arising from the dissimilar entry ages of recruited healthy subjects. To analyze these kinds of incomplete data while accounting for nonsusceptibility and possible crossing hazards in the framework of mixture regression models, we utilize a logistic regression model to specify the probability of susceptibility, and a generalized gamma distribution, or a log‐logistic distribution, in the accelerated failure time location‐scale regression model to formulate the time to the event. Relative times of the conditional event time distribution for susceptible subjects are extended in the accelerated failure time location‐scale submodel. We also construct graphical goodness‐of‐fit procedures on the basis of the Turnbull–Frydman estimator and newly proposed residuals. Simulation studies were conducted to demonstrate the validity of the proposed estimation procedure. The mixture regression models are illustrated with alcohol abuse data from the Taiwan Aboriginal Study Project and hypertriglyceridemia data from the Cardiovascular Disease Risk Factor Two‐township Study in Taiwan. Copyright © 2013 John Wiley & Sons, Ltd.     

Objective Bayesian model selection for Cox regression

There is now a large literature on objective Bayesian model selection in the linear model based on the g‐prior. The methodology has been recently extended to generalized linear models using test‐based Bayes factors. In this paper, we show that test‐based Bayes factors can also be applied to the Cox proportional hazards model. If the goal is to select a single model, then both the maximum a posteriori and the median probability model can be calculated. For clinical prediction of survival, we shrink the model‐specific log hazard ratio estimates with subsequent calculation of the Breslow estimate of the cumulative baseline hazard function. A Bayesian model average can also be employed. We illustrate the proposed methodology with the analysis of survival data on primary biliary cirrhosis patients and the development of a clinical prediction model for future cardiovascular events based on data from the Second Manifestations of ARTerial disease (SMART) cohort study. Cross‐validation is applied to compare the predictive performance with alternative model selection approaches based on Harrell's c‐Index, the calibration slope and the integrated Brier score. Finally, a novel application of Bayesian variable selection to optimal conditional prediction via landmarking is described. Copyright © 2016 John Wiley & Sons, Ltd.     

A model‐informed rank test for right‐censored data with intermediate states

The generalized Wilcoxon and log‐rank tests are commonly used for testing differences between two survival distributions. We modify the Wilcoxon test to account for auxiliary information on intermediate disease states that subjects may pass through before failure. For a disease with multiple states where patients are monitored periodically but exact transition times are unknown (e.g. staging in cancer), we first fit a multi‐state Markov model to the full data set; when censoring precludes the comparison of survival times between two subjects, we use the model to estimate the probability that one subject will have survived longer than the other given their censoring times and last observed status, and use these probabilities to compute an expected rank for each subject. These expected ranks form the basis of our test statistic. Simulations demonstrate that the proposed test can improve power over the log‐rank and generalized Wilcoxon tests in some settings while maintaining the nominal type 1 error rate. The method is illustrated on an amyotrophic lateral sclerosis data set. Copyright © 2015 John Wiley & Sons, Ltd.     

Analysis of crossover designs with nonignorable dropout

This article addresses the analysis of crossover designs with nonignorable dropout. We study nonreplicated crossover designs and replicated designs separately. With a primary objective of comparing the treatment mean effects, we jointly model the longitudinal measures and discrete time to dropout. We propose shared‐parameter models and mixed‐effects selection models. We adapt a linear‐mixed effects model as the conditional model for the longitudinal outcomes. We invoke a discrete‐time hazards model with a complementary log‐log link function for the conditional distribution of time to dropout. We apply maximum likelihood for parameter estimation. We perform simulation studies to investigate the robustness of our proposed approaches under various missing data mechanisms. We then apply the approaches to two examples with a continuous outcome and one example with a binary outcome using existing software. We also implement the controlled multiple imputation methods as a sensitivity analysis of the missing data assumption.