A Bayesian Shrinkage Model for Incomplete Longitudinal Binary Data with Application to the Breast Cancer Prevention Trial

Applications of Bayesian shrinkage prior models in clinical research with categorical responses

BACKGROUND

Prediction and classification algorithms are commonly used in clinical research for identifying patients susceptible to clinical conditions such as diabetes, colon cancer, and Alzheimer's disease. Developing accurate prediction and classification methods benefits personalized medicine. Building an excellent predictive model involves selecting the features that are most significantly associated with the outcome. These features can include several biological and demographic characteristics, such as genomic biomarkers and health history. Such variable selection becomes challenging when the number of potential predictors is large. Bayesian shrinkage models have emerged as popular and flexible methods of variable selection in regression settings. This work discusses variable selection with three shrinkage priors and illustrates its application to clinical data such as Pima Indians Diabetes, Colon cancer, ADNI, and OASIS Alzheimer's real-world data.

METHODS

A unified Bayesian hierarchical framework that implements and compares shrinkage priors in binary and multinomial logistic regression models is presented. The key feature is the representation of the likelihood by a Polya-Gamma data augmentation, which admits a natural integration with a family of shrinkage priors, specifically focusing on Horseshoe, Dirichlet Laplace, and Double Pareto priors. Extensive simulation studies are conducted to assess the performances under different data dimensions and parameter settings. Measures of accuracy, AUC, brier score, L1 error, cross-entropy, and ROC surface plots are used as evaluation criteria comparing the priors with frequentist methods as Lasso, Elastic-Net, and Ridge regression.

RESULTS

All three priors can be used for robust prediction on significant metrics, irrespective of their categorical response model choices. Simulation studies could achieve the mean prediction accuracy of 91.6% (95% CI: 88.5, 94.7) and 76.5% (95% CI: 69.3, 83.8) for logistic regression and multinomial logistic models, respectively. The model can identify significant variables for disease risk prediction and is computationally efficient.

CONCLUSIONS

The models are robust enough to conduct both variable selection and prediction because of their high shrinkage properties and applicability to a broad range of classification problems.

Bayesian sensitivity analyses for longitudinal data with dropouts that are potentially missing not at random: A high dimensional pattern-mixture model

Randomized clinical trials with outcome measured longitudinally are frequently analyzed using either random effect models or generalized estimating equations. Both approaches assume that the dropout mechanism is missing at random (MAR) or missing completely at random (MCAR). We propose a Bayesian pattern-mixture model to incorporate missingness mechanisms that might be missing not at random (MNAR), where the distribution of the outcome measure at the follow-up time t k , conditional on the prior history, differs across the patterns of missing data. We then perform sensitivity analysis on estimates of the parameters of interest. The sensitivity parameters relate the distribution of the outcome of interest between subjects from a missing-data pattern at time t k with that of the observed subjects at time t k . The large number of the sensitivity parameters is reduced by treating them as random with a prior distribution having some pre-specified mean and variance, which are varied to explore the sensitivity of inferences. The missing at random (MAR) mechanism is a special case of the proposed model, allowing a sensitivity analysis of deviations from MAR. The proposed approach is applied to data from the Trial of Preventing Hypertension.

A Semiparametric Bayesian Approach to Dropout in Longitudinal Studies with Auxiliary Covariates

We develop a semiparametric Bayesian approach to missing outcome data in longitudinal studies in the presence of auxiliary covariates. We consider a joint model for the full data response, missingness and auxiliary covariates. We include auxiliary covariates to "move" the missingness "closer" to missing at random (MAR). In particular, we specify a semiparametric Bayesian model for the observed data via Gaussian process priors and Bayesian additive regression trees. These model specifications allow us to capture non-linear and non-additive effects, in contrast to existing parametric methods. We then separately specify the conditional distribution of the missing data response given the observed data response, missingness and auxiliary covariates (i.e. the extrapolation distribution) using identifying restrictions. We introduce meaningful sensitivity parameters that allow for a simple sensitivity analysis. Informative priors on those sensitivity parameters can be elicited from subject-matter experts. We use Monte Carlo integration to compute the full data estimands. Performance of our approach is assessed using simulated datasets. Our methodology is motivated by, and applied to, data from a clinical trial on treatments for schizophrenia.

A Bayesian Vector Autoregressive Model with Nonignorable Missingness in Dependent Variables and Covariates: Development, Evaluation, and Application to Family Processes

Intensive longitudinal designs involving repeated assessments of constructs often face the problems of nonignorable attrition and selected omission of responses on particular occasions. However, time series models, such as vector autoregressive (VAR) models, are often fit to these data without consideration of nonignorable missingness. We introduce a Bayesian model that simultaneously represents the over-time dependencies in multivariate, multiple-subject time series data via a VAR model, and possible ignorable and nonignorable missingness in the data. We provide software code for implementing this model with application to an empirical data set. Moreover, simulation results comparing the joint approach with two-step multiple imputation procedures are included to shed light on the relative strengths and weaknesses of these approaches in practical data analytic scenarios.

Bayesian Approaches for Missing Not at Random Outcome Data: The Role of Identifying Restrictions

Missing data is almost always present in real datasets, and introduces several statistical issues. One fundamental issue is that, in the absence of strong uncheckable assumptions, effects of interest are typically not nonparametrically identified. In this article, we review the generic approach of the use of identifying restrictions from a likelihood-based perspective, and provide points of contact for several recently proposed methods. An emphasis of this review is on restrictions for nonmonotone missingness, a subject that has been treated sparingly in the literature. We also present a general, fully-Bayesian, approach which is widely applicable and capable of handling a variety of identifying restrictions in a uniform manner.

Bayesian methods for nonignorable dropout in joint models in smoking cessation studies

Inference on data with missingness can be challenging, particularly if the knowledge that a measurement was unobserved provides information about its distribution. Our work is motivated by the Commit to Quit II study, a smoking cessation trial that measured smoking status and weight change as weekly outcomes. It is expected that dropout in this study was informative and that patients with missed measurements are more likely to be smoking, even after conditioning on their observed smoking and weight history. We jointly model the categorical smoking status and continuous weight change outcomes by assuming normal latent variables for cessation and by extending the usual pattern mixture model to the bivariate case. The model includes a novel approach to sharing information across patterns through a Bayesian shrinkage framework to improve estimation stability for sparsely observed patterns. To accommodate the presumed informativeness of the missing data in a parsimonious manner, we model the unidentified components of the model under a non-future dependence assumption and specify departures from missing at random through sensitivity parameters, whose distributions are elicited from a subject-matter expert.

A Flexible Bayesian Approach to Monotone Missing Data in Longitudinal Studies with Nonignorable Missingness with Application to an Acute Schizophrenia Clinical Trial

We develop a Bayesian nonparametric model for a longitudinal response in the presence of nonignorable missing data. Our general approach is to first specify a working model that flexibly models the missingness and full outcome processes jointly. We specify a Dirichlet process mixture of missing at random (MAR) models as a prior on the joint distribution of the working model. This aspect of the model governs the fit of the observed data by modeling the observed data distribution as the marginalization over the missing data in the working model. We then separately specify the conditional distribution of the missing data given the observed data and dropout. This approach allows us to identify the distribution of the missing data using identifying restrictions as a starting point. We propose a framework for introducing sensitivity parameters, allowing us to vary the untestable assumptions about the missing data mechanism smoothly. Informative priors on the space of missing data assumptions can be specified to combine inferences under many different assumptions into a final inference and accurately characterize uncertainty. These methods are motivated by, and applied to, data from a clinical trial assessing the efficacy of a new treatment for acute Schizophrenia.

Fully Bayesian inference under ignorable missingness in the presence of auxiliary covariates

In order to make a missing at random (MAR) or ignorability assumption realistic, auxiliary covariates are often required. However, the auxiliary covariates are not desired in the model for inference. Typical multiple imputation approaches do not assume that the imputation model marginalizes to the inference model. This has been termed "uncongenial" [Meng (1994, Statistical Science 9, 538-558)]. In order to make the two models congenial (or compatible), we would rather not assume a parametric model for the marginal distribution of the auxiliary covariates, but we typically do not have enough data to estimate the joint distribution well non-parametrically. In addition, when the imputation model uses a non-linear link function (e.g., the logistic link for a binary response), the marginalization over the auxiliary covariates to derive the inference model typically results in a difficult to interpret form for the effect of covariates. In this article, we propose a fully Bayesian approach to ensure that the models are compatible for incomplete longitudinal data by embedding an interpretable inference model within an imputation model and that also addresses the two complications described above. We evaluate the approach via simulations and implement it on a recent clinical trial.

Sensitivity analysis for nonignorable missingness and outcome misclassification from proxy reports

Researchers often recruit proxy respondents, such as relatives or caregivers, for epidemiologic studies of older adults when study participants are unable to provide self-reports (eg, because of illness or cognitive impairment). In most studies involving proxy-reported outcomes, proxies are recruited only to report on behalf of participants who have missing self-reported outcomes; thus, either a proxy report or participant self-report, but not both, is available for each participant. When outcomes are binary and investigators conceptualize participant self-reports as gold standard measures, substituting proxy reports in place of missing participant self-reports in statistical analysis can introduce misclassification error and lead to biased parameter estimates. However, excluding observations from participants with missing self-reported outcomes may also lead to bias. We propose a pattern-mixture model that uses error-prone proxy reports to reduce selection bias from missing outcomes, and we describe a sensitivity analysis to address bias from differential outcome misclassification. We perform model estimation with high-dimensional (eg, continuous) covariates using propensity-score stratification and multiple imputation. We apply the methods to the Second Cohort of the Baltimore Hip Studies, a study of elderly hip fracture patients, to assess the relation between type of surgical treatment and perceived physical recovery. Simulation studies show that the proposed methods perform well. We provide SAS programs in the eAppendix (http://links.lww.com/EDE/A646) to enhance the methods' accessibility.

A note on MAR, identifying restrictions, model comparison, and sensitivity analysis in pattern mixture models with and without covariates for incomplete data

Pattern mixture modeling is a popular approach for handling incomplete longitudinal data. Such models are not identifiable by construction. Identifying restrictions is one approach to mixture model identification (Little, 1995, Journal of the American Statistical Association 90, 1112-1121; Little and Wang, 1996, Biometrics 52, 98-111; Thijs et al., 2002, Biostatistics 3, 245-265; Kenward, Molenberghs, and Thijs, 2003, Biometrika 90, 53-71; Daniels and Hogan, 2008, in Missing Data in Longitudinal Studies: Strategies for Bayesian Modeling and Sensitivity Analysis) and is a natural starting point for missing not at random sensitivity analysis (Thijs et al., 2002, Biostatistics 3, 245-265; Daniels and Hogan, 2008, in Missing Data in Longitudinal Studies: Strategies for Bayesian Modeling and Sensitivity Analysis). However, when the pattern specific models are multivariate normal, identifying restrictions corresponding to missing at random (MAR) may not exist. Furthermore, identification strategies can be problematic in models with covariates (e.g., baseline covariates with time-invariant coefficients). In this article, we explore conditions necessary for identifying restrictions that result in MAR to exist under a multivariate normality assumption and strategies for identifying sensitivity parameters for sensitivity analysis or for a fully Bayesian analysis with informative priors. In addition, we propose alternative modeling and sensitivity analysis strategies under a less restrictive assumption for the distribution of the observed response data. We adopt the deviance information criterion for model comparison and perform a simulation study to evaluate the performances of the different modeling approaches. We also apply the methods to a longitudinal clinical trial. Problems caused by baseline covariates with time-invariant coefficients are investigated and an alternative identifying restriction based on residuals is proposed as a solution.