Multiple Longitudinal Outcomes Research Articles

Studying the association Between longitudinal non-dense breast tissue and breast cancer risk: a joint modelling approach.

Conflicting results have appeared in the literature on whether the amount of non-dense, adipose tissue in the breast is a risk factor or a protective factor for breast cancer, and biological hypotheses supporting both have been proposed. We suggest here that limitations in study design and statistical methodology could potentially explain the inconsistent results. Specifically, we exploit recent advances in methodology and software developed for the joint analysis of multiple longitudinal outcomes and time-to-event data to jointly analyze dense and non-dense tissue trajectories and the risk of breast cancer in a large, Swedish, screening cohort. We also perform extensive sensitivity analyses by mimicking analyses/designs of previously published studies, e.g. ignoring available longitudinal data. Overall, we did not find strong evidence supporting an association between non-dense tissue and the risk of incident breast cancer. We hypothesize that (1) previous studies have not been able to isolate the effect of non-dense tissue from dense tissue or adipose tissue elsewhere in the body, that (2) estimates of the effect of non-dense tissue on risk are strongly sensitive to modeling assumptions, or that (3) the effect size of non-dense tissue on breast cancer risk is likely to be small/not clinically relevant.

Optimizing dynamic predictions from joint models using super learning.

Joint models for longitudinal and time-to-event data are often employed to calculate dynamic individualized predictions used in numerous applications of precision medicine. Two components of joint models that influence the accuracy of these predictions are the shape of the longitudinal trajectories and the functional form linking the longitudinal outcome history to the hazard of the event. Finding a single well-specified model that produces accurate predictions for all subjects and follow-up times can be challenging, especially when considering multiple longitudinal outcomes. In this work, we use the concept of super learning and avoid selecting a single model. In particular, we specify a weighted combination of the dynamic predictions calculated from a library of joint models with different specifications. The weights are selected to optimize a predictive accuracy metric using V-fold cross-validation. We use as predictive accuracy measures the expected quadratic prediction error and the expected predictive cross-entropy. In a simulation study, we found that the super learning approach produces results very similar to the Oracle model, which was the model with the best performance in the test datasets. All proposed methodology is implemented in the freely available R package JMbayes2.

Semiparametric multivariate joint model for skewed-longitudinal and survival data: A Bayesian approach.

Joint models and statistical inference for longitudinal and survival data have been an active area of statistical research and have mostly coupled a longitudinal biomarker-based mixed-effects model with normal distribution and an event time-based survival model. In practice, however, the following issues may standout: (i) Normality of model error in longitudinal models is a routine assumption, but it may be unrealistically violating data features of subject variations. (ii) Data collected are often featured by the mixed types of multiple longitudinal outcomes which are significantly correlated, ignoring their correlation may lead to biased estimation. Additionally, a parametric model specification may be inflexible to capture the complicated patterns of longitudinal data. (iii) Missing observations in the longitudinal data are often encountered; the missing measures are likely to be informative (nonignorable) and ignoring this phenomenon may result in inaccurate inference. Multilevel item response theory (MLIRT) models have been increasingly used to analyze the multiple longitudinal data of mixed types (ie, continuous and categorical) in clinical studies. In this article, we develop an MLIRT-based semiparametric joint model with skew-t distribution that consists of an extended MLIRT model for the mixed types of multiple longitudinal data and a Cox proportional hazards model, linked through random-effects. A Bayesian approach is employed for joint modeling. Simulation studies are conducted to assess performance of the proposed models and method. A real example from primary biliary cirrhosis clinical study is analyzed to estimate parameters in the joint model and also evaluate sensitivity of parameter estimates for various plausible nonignorable missing data mechanisms.

BAYESIAN INFERENCE AND DYNAMIC PREDICTION FOR MULTIVARIATE LONGITUDINAL AND SURVIVAL DATA.

Alzheimer's disease (AD) is a complex neurological disorder impairing multiple domains such as cognition and daily functions. To better understand the disease and its progression, many AD research studies collect multiple longitudinal outcomes that are strongly predictive of the onset of AD dementia. We propose a joint model based on a multivariate functional mixed model framework (referred to as MFMM-JM) that simultaneously models the multiple longitudinal outcomes and the time to dementia onset. We develop six functional forms to fully investigate the complex association between longitudinal outcomes and dementia onset. Moreover, we use the Bayesian methods for statistical inference and develop a dynamic prediction framework that provides accurate personalized predictions of disease progressions based on new subject-specific data. We apply the proposed MFMM-JM to two large ongoing AD studies: the Alzheimer's Disease Neuroimaging Initiative (ADNI) and National Alzheimer's Coordinating Center (NACC), and identify the functional forms with the best predictive performance. our method is also validated by extensive simulation studies with five settings.

Bayesian joint modelling of multiple longitudinal outcomes and dependent competing risks using bivariate Marshall–Olkin weibull distribution

Joint modelling of multiple longitudinal variables and time to an event under two dependent competing risks using bivariate Marshall–Olkin Weibull distribution is investigated. This joint modelling, enables researchers to consider the possible simultaneous time of failure under both risks. Considering dependent competing risks which is a more realistic assumption is another advantage. The shared parameter approach is applied, which links the multivariate mixed-effects model, considered as the sub-model for multiple longitudinal variables on the bivariate Marshall–Olkin Weibull distribution as the survival sub-model under two dependent competing risks. Bayesian inference is an asset to overcome computational challenges. A simulation study and a real application are presented to show the performances of the presented joint model.

A two-level copula joint model for joint analysis oflongitudinal and competing risks data.

In this article, we propose a two-level copula joint model to analyze clinical data with multiple disparate continuous longitudinal outcomes and multiple event-times in the presence of competing risks. At the first level, we use a copula to model the dependence between competing latent event-times, in the process constructing the submodel for the observed event-time, and employ the Gaussian copula to construct the submodel for the longitudinal outcomes that accounts for their conditional dependence; these submodels are glued together at the second level via the Gaussian copula to construct a joint model that incorporates conditional dependence between the observed event-time and the longitudinal outcomes. To have the flexibility to accommodate skewed data and examine possibly different covariate effects on quantiles of a non-Gaussian outcome, we propose linear quantile mixed models for the continuous longitudinal data. We adopt a Bayesian framework for model estimation and inference via Markov Chain Monte Carlo sampling. We examine the performance of the copula joint model through a simulation study and show that our proposed method outperforms the conventional approach assuming conditional independence with smaller biases and better coverage probabilities of the Bayesian credible intervals. Finally, we carry out an analysis of clinical data on renal transplantation for illustration.

Assessing mediational processes using piecewise linear growth curve models with individual measurement occasions.

Longitudinal processes often unfold concurrently where the growth patterns of two or more longitudinal outcomes are associated. Additionally, if the study under investigation is long, the growth curves may exhibit nonconstant change with respect to time. Multiple existing studies have developed multivariate growth models with nonlinear functional forms to explore joint development where two longitudinal records are correlated over time. However, the relationship between multiple longitudinal outcomes may also be unidirectional. Accordingly, it is of interest to estimate regression coefficients of such unidirectional paths. One statistical tool for such analyses is longitudinal mediation models. In this study, we develop two models to evaluate mediational processes where the linear-linear piecewise functional form is utilized to capture the change patterns. We define the mediational process as either the baseline covariate or the change in covariate influencing the change in the mediator, which, in turn, affects the change in the outcome. We present the proposed models through simulation studies and real-world data analyses. Our simulation studies demonstrate that the proposed mediational models can provide unbiased and accurate point estimates with target coverage probabilities with a 95% confidence interval. The empirical analyses demonstrate that the proposed models can estimate covariates' direct and indirect effects on the change in the outcome. We also provide the corresponding code for the proposed models.

Bayesian Joint Modeling of Multivariate Longitudinal and Survival Data With an Application to Diabetes Study.

Joint models of longitudinal and time-to-event data have received a lot of attention in epidemiological and clinical research under a linear mixed-effects model with the normal assumption for a single longitudinal outcome and Cox proportional hazards model. However, those model-based analyses may not provide robust inference when longitudinal measurements exhibit skewness and/or heavy tails. In addition, the data collected are often featured by multivariate longitudinal outcomes which are significantly correlated, and ignoring their correlation may lead to biased estimation. Under the umbrella of Bayesian inference, this article introduces multivariate joint (MVJ) models with a skewed distribution for multiple longitudinal exposures in an attempt to cope with correlated multiple longitudinal outcomes, adjust departures from normality, and tailor linkage in specifying a time-to-event process. We develop a Bayesian joint modeling approach to MVJ models that couples a multivariate linear mixed-effects (MLME) model with the skew-normal (SN) distribution and a Cox proportional hazards model. Our proposed models and method are evaluated by simulation studies and are applied to a real example from a diabetes study.

Deep learning for the dynamic prediction of multivariate longitudinal and survival data.

The joint model for longitudinal and survival data improves time-to-event predictions by including longitudinal outcome variables in addition to baseline covariates. However, in practice, joint models may be limited by parametric assumptions in both the longitudinal and survival submodels. In addition, computational difficulties arise when considering multiple longitudinal outcomes due to the large number of random effects to be integrated out in the full likelihood. In this article, we discuss several recent machine learning methods for incorporating multivariate longitudinal data for time-to-event prediction. The presented methods use functional data analysis or convolutional neural networks to model the longitudinal data, both of which scale well to multiple longitudinal outcomes. In addition, we propose a novel architecture based on the transformer neural network, named TransformerJM, which jointly models longitudinal and time-to-event data. The prognostic abilities of each model are assessed and compared through both simulation and real data analysis on Alzheimer's disease datasets. Specifically, the models were evaluated based on their ability to dynamically update predictions as new longitudinal data becomes available. We showed that TransformerJM improves upon the predictive performance of existing methods across different scenarios.

Bayesian inference and dynamic prediction of multivariate joint model with functional data: An application to Alzheimer's disease.

Alzheimer's disease (AD) is a severe neurodegenerative disorder impairing multiple domains, for example, cognition and behavior. Assessing the risk of AD progression and initiating timely interventions at early stages are critical to improve the quality of life for AD patients. Due to the heterogeneous nature and complex mechanisms of AD, one single longitudinal outcome is insufficient to assess AD severity and disease progression. Therefore, AD studies collect multiple longitudinal outcomes, including cognitive and behavioral measurements, as well as structural brain images such as magnetic resonance imaging (MRI). How to utilize the multivariate longitudinal outcomes and MRI data to make efficient statistical inference and prediction is an open question. In this article, we propose a multivariate joint model with functional data (MJM-FD) framework that relates multiple correlated longitudinal outcomes to a survival outcome, and use the scalar-on-function regression method to include voxel-based whole-brain MRI data as functional predictors in both longitudinal and survival models. We adopt a Bayesian paradigm to make statistical inference and develop a dynamic prediction framework to predict an individual's future longitudinal outcomes and risk of a survival event. We validate the MJM-FD framework through extensive simulation studies and apply it to the motivating Alzheimer's Disease Neuroimaging Initiative (ADNI) study.

Joint modeling of multivariate nonparametric longitudinal data and survival data: A local smoothing approach.

In many clinical studies, evaluating the association between longitudinal and survival outcomes is of primary concern. For analyzing data from such studies, joint modeling of longitudinal and survival data becomes an appealing approach. In some applications, there are multiple longitudinal outcomes whose longitudinal pattern is difficult to describe by a parametric form. For such applications, existing research on joint modeling is limited. In this article, we develop a novel joint modeling method to fill the gap. In the new method, a local polynomial mixed-effects model is used for describing the nonparametric longitudinal pattern of the multiple longitudinal outcomes. Two model estimation procedures, that is, the local EM algorithm and the local penalized quasi-likelihood estimation, are explored. Practical guidelines for choosing tuning parameters and for variable selection are provided. The new method is justified by some theoretical arguments and numerical studies.

Graphical models for mean and covariance of multivariate longitudinal data.

Joint mean-covariance modeling of multivariate longitudinal data helps to understand the relative changes among multiple longitudinally measured and correlated outcomes. A key challenge in the analysis of multivariate longitudinal data is the complex covariance structure. This is due to the contemporaneous and cross-temporal associations between multiple longitudinal outcomes. Graphical and data-driven tools that can aid in visualizing the dependence patterns among multiple longitudinal outcomes are not readily available. In this work, we show the role of graphical techniques: profile plots, and multivariate regressograms, in developing mean and covariance models for multivariate longitudinal data. We introduce an R package MLGM (Multivariate Longitudinal Graphical Models) to facilitate visualization and modeling mean and covariance patterns. Through two real studies, microarray data from the T-cell activation study and Mayo Clinic's primary biliary cirrhosis of the liver study, we show the key features of MLGM. We evaluate the finite sample performance of the proposed mean-covariance estimation approach through simulations.

Fast approximate inference for multivariate longitudinal data.

Collecting information on multiple longitudinal outcomes is increasingly common in many clinical settings. In many cases, it is desirable to model these outcomes jointly. However, in large data sets, with many outcomes, computational burden often prevents the simultaneous modeling of multiple outcomes within a single model. We develop a mean field variational Bayes algorithm, to jointly model multiple Gaussian, Poisson, or binary longitudinal markers within a multivariate generalized linear mixed model. Through simulation studies and clinical applications (in the fields of sight threatening diabetic retinopathy and primary biliary cirrhosis), we demonstrate substantial computational savings of our approximate approach when compared to a standard Markov Chain Monte Carlo, while maintaining good levels of accuracy of model parameters.

Joint model for survival and multivariate sparse functional data with application to a study of Alzheimer's Disease.

Studies of Alzheimer's disease (AD) often collect multiple longitudinal clinical outcomes, which are correlated and predictive of AD progression. It is of great scientific interest to investigate the association between the outcomes and time to AD onset. We model the multiple longitudinal outcomes as multivariate sparse functional data and propose a functional joint model linking multivariate functional data to event time data. In particular, we propose a multivariate functional mixed model to identify the shared progression pattern and outcome-specific progression patterns of the outcomes, which enables more interpretable modeling of associations between outcomes and AD onset. The proposed method is applied to the Alzheimer's Disease Neuroimaging Initiative study (ADNI) and the functional joint model sheds new light on inference of five longitudinal outcomes and their associations with AD onset. Simulation studies also confirm the validity of the proposed model. Data used in preparation of this article were obtained from the ADNIdatabase.

Merlin—A unified modeling framework for data analysis and methods development in Stata

The challenges in statistics and data science are rapidly growing because access to a multitude of data types continues to increase, as well as the sheer quantity of data. Analysts are now presented with multivariate data, sometimes measured repeatedly, and often requiring the ability to model nonlinear relationships and hierarchical structures. In this article, I present the merlin command, which attempts to provide an extremely general framework for data analysis. From simple settings such as fitting a linear regression model or a Weibull survival model to more complex settings such as fitting a three-level logistic mixed-effects model or a multivariate joint model of multiple longitudinal outcomes (of different types) and a recurrent event and survival with nonlinear effects, merlin can fit them all. I will take a single dataset and attempt to show you the full range of capabilities of merlin and discuss some future directions for the implementation in Stata.

Pairwise estimation of multivariate longitudinal outcomes in a Bayesian setting with extensions to the joint model

Multiple longitudinal outcomes are theoretically easily modelled via extension of the generalized linear mixed effects model. However, due to computational limitations in high dimensions, in practice these models are applied only in situations with relatively few outcomes. We adapt the solution proposed by Fieuws and Verbeke (2006) to the Bayesian setting: fitting all pairwise bivariate models instead of a single multivariate model, and combining the Markov Chain Monte Carlo (MCMC) realizations obtained for each pairwise bivariate model for the relevant parameters. We explore importance sampling as a method to more closely approximate the correct multivariate posterior distribution. Simulation studies show satisfactory results in terms of bias, RMSE and coverage of the 95% credible intervals for multiple longitudinal outcomes, even in scenarios with more limited information and non-continuous outcomes, although the use of importance sampling is not successful. We further examine the incorporation of a time-to-event outcome, proposing the use of Bayesian pairwise estimation of a multivariate GLMM in an adaptation of the corrected two-stage estimation procedure for the joint model for multiple longitudinal outcomes and a time-to-event outcome ( Mauff et al., 2020 , Statistics and Computing). The method does not work as well in the case of the corrected two-stage joint model; however, the results are promising and should be explored further.

Functional survival forests for multivariate longitudinal outcomes: Dynamic prediction of Alzheimer's disease progression.

The random survival forest (RSF) is a non-parametric alternative to the Cox proportional hazards model in modeling time-to-event data. In this article, we developed a modeling framework to incorporate multivariate longitudinal data in the model building process to enhance the predictive performance of RSF. To extract the essential features of the multivariate longitudinal outcomes, two methods were adopted and compared: multivariate functional principal component analysis and multivariate fast covariance estimation for sparse functional data. These resulting features, which capture the trajectories of the multiple longitudinal outcomes, are then included as time-independent predictors in the subsequent RSF model. This non-parametric modeling framework, denoted as functional survival forests, is better at capturing the various trends in both the longitudinal outcomes and the survival model which may be difficult to model using only parametric approaches. These advantages are demonstrated through simulations and applications to the Alzheimer's Disease Neuroimaging Initiative.

Systematic review of statistical methods for safety data in malaria chemoprevention in pregnancy trials

BackgroundDrug safety assessments in clinical trials present unique analytical challenges. Some of these include adjusting for individual follow-up time, repeated measurements of multiple outcomes and missing data among others. Furthermore, pre-specifying appropriate analysis becomes difficult as some safety endpoints are unexpected. Although existing guidelines such as CONSORT encourage thorough reporting of adverse events (AEs) in clinical trials, they provide limited details for safety data analysis. The limited guidelines may influence suboptimal analysis by failing to account for some analysis challenges above. A typical example where such challenges exist are trials of anti-malarial drugs for malaria prevention during pregnancy. Lack of proper standardized evaluation of the safety of antimalarial drugs has limited the ability to draw conclusions about safety. Therefore, a systematic review was conducted to establish the current practice in statistical analysis for preventive antimalarial drug safety in pregnancy.MethodsThe search included five databases (PubMed, Embase, Scopus, Malaria in Pregnancy Library and Cochrane Central Register of Controlled Trials) to identify original English articles reporting Phase III randomized controlled trials (RCTs) on anti-malarial drugs for malaria prevention in pregnancy published from January 2010 to July 2019.ResultsEighteen trials were included in this review that collected multiple longitudinal safety outcomes including AEs. Statistical analysis and reporting of the safety outcomes in all the trials used descriptive statistics; proportions/counts (n = 18, 100%) and mean/median (n = 2, 11.1%). Results presentation included tabular (n = 16, 88.9%) and text description (n = 2, 11.1%). Univariate inferential methods were reported in most trials (n = 16, 88.9%); including Chi square/Fisher’s exact test (n = 12, 66.7%), t test (n = 2, 11.1%) and Mann–Whitney/Wilcoxon test (n = 1, 5.6%). Multivariable methods, including Poisson and negative binomial were reported in few trials (n = 3, 16.7%). Assessment of a potential link between missing efficacy data and safety outcomes was not reported in any of the trials that reported efficacy missing data (n = 7, 38.9%).ConclusionThe review demonstrated that statistical analysis of safety data in anti-malarial drugs for malarial chemoprevention in pregnancy RCTs is inadequate. The analyses insufficiently account for multiple safety outcomes potential dependence, follow-up time and informative missing data which can compromise anti-malarial drug safety evidence development, based on the available data.

Joint models with multiple longitudinal outcomes and a time-to-event outcome: a corrected two-stage approach

Joint models for longitudinal and survival data have gained a lot of attention in recent years, with the development of myriad extensions to the basic model, including those which allow for multivariate longitudinal data, competing risks and recurrent events. Several software packages are now also available for their implementation. Although mathematically straightforward, the inclusion of multiple longitudinal outcomes in the joint model remains computationally difficult due to the large number of random effects required, which hampers the practical application of this extension. We present a novel approach that enables the fitting of such models with more realistic computational times. The idea behind the approach is to split the estimation of the joint model in two steps: estimating a multivariate mixed model for the longitudinal outcomes and then using the output from this model to fit the survival submodel. So-called two-stage approaches have previously been proposed and shown to be biased. Our approach differs from the standard version, in that we additionally propose the application of a correction factor, adjusting the estimates obtained such that they more closely resemble those we would expect to find with the multivariate joint model. This correction is based on importance sampling ideas. Simulation studies show that this corrected two-stage approach works satisfactorily, eliminating the bias while maintaining substantial improvement in computational time, even in more difficult settings.

Fast Covariance Estimation for Multivariate Sparse Functional Data.

Covariance estimation is essential yet underdeveloped for analysing multivariate functional data. We propose a fast covariance estimation method for multivariate sparse functional data using bivariate penalized splines. The tensor‐product B‐spline formulation of the proposed method enables a simple spectral decomposition of the associated covariance operator and explicit expressions of the resulting eigenfunctions as linear combinations of B‐spline bases, thereby dramatically facilitating subsequent principal component analysis. We derive a fast algorithm for selecting the smoothing parameters in covariance smoothing using leave‐one‐subject‐out cross‐validation. The method is evaluated with extensive numerical studies and applied to an Alzheimer's disease study with multiple longitudinal outcomes.