Data augmentation priors for Bayesian and semi-Bayes analyses of conditional-logistic and proportional-hazards regression

An intuitive framework for Bayesian posterior simulation methods

Purpose

Bayesian inference has become popular. It offers several pragmatic approaches to account for uncertainty in inference decision-making. Various estimation methods have been introduced to implement Bayesian methods. Although these algorithms are powerful, they are not always easy to grasp for non-statisticians. This paper aims to provide an intuitive framework of four essential Bayesian computational methods for epidemiologists and other health researchers. We do not cover an extensive mathematical discussion of these approaches, but instead offer a non-quantitative description of these algorithms and provide some illuminating examples.

Materials and methods

Bayesian computational methods, namely importance sampling, rejection sampling, Markov chain Monte Carlo, and data augmentation are presented.

Results and conclusions

The substantial amount of research published on Bayesian inference has highlighted its popularity among researchers, while the basic concepts are not always straightforward for interested learners. We show that alternative approaches such as a weighted prior approach, which are intuitively appealing and easy-to-understand, work well in the case of low-dimensional problems and appropriate prior information. Otherwise, MCMC is a trouble-free tool in those cases.

Bias in odds ratios from logistic regression methods with sparse data sets

BACKGROUND

Logistic regression models are widely used to evaluate the association between a binary outcome and a set of covariates. However, when there are few study participants at the outcome and covariate levels, the models lead to bias of the odds ratio (OR) estimated using the maximum likelihood (ML) method. This bias is known as sparse data bias, and the estimated OR can yield impossibly large values because of data sparsity. However, this bias has been ignored in most epidemiological studies.

METHODS

We review several methods for reducing sparse data bias in logistic regression. The primary aim is to evaluate the Bayesian methods in comparison with the classical methods, such as the ML, Firth's, and exact methods using a simulation study. We also apply these methods to a real data set.

RESULTS

Our simulation results indicate that the bias of the OR from the ML, Firth's, and exact methods is considerable. Furthermore, the Bayesian methods with hyper-g prior modeling of the prior covariance matrix for regression coefficients reduced the bias under the null hypothesis, whereas the Bayesian methods with log F-type priors reduced the bias under the alternative hypothesis.

CONCLUSION

The Bayesian methods using log F-type priors and hyper-g prior are superior to the ML, Firth's, and exact methods when fitting logistic models to sparse data sets. The choice of a preferable method depends on the null and alternative hypothesis. Sensitivity analysis is important to understand the robustness of the results in sparse data analysis.

Bayesian methods for clinicians

Background: The Bayesian methods have received more attention in medical research. It is considered as a natural paradigm for dealing with applied problems in the sciences and also an alternative to the traditional frequentist approach. However, its concept is somewhat difficult to grasp by nonexperts. This study aimed to explain the foundational ideas of the Bayesian methods through an intuitive example in medical science and to illustrate some simple examples of Bayesian data analysis and the interpretation of results delivered by Bayesian analyses. In this study, data sparsity, as a problem which could be solved by this approach, was presented through an applied example. Moreover, a common sense description of Bayesian inference was offered and some illuminating examples were provided for medical investigators and nonexperts. Methods: Data augmentation prior, MCMC, and Bayes factor were introduced. Data from the Khuzestan study, a 2-phase cohort study, were applied for illustration. Also, the effect of vitamin D intervention on pregnancy outcomes was studied. Results: Unbiased estimate was obtained by the introduced methods. Conclusion: Bayesian and data augmentation as the advanced methods provide sufficient results and deal with most data problems such as sparsity.

Catalytic prior distributions with application to generalized linear models

We propose a strategy for building prior distributions that stabilize the estimation of complex “working models” when sample sizes are too small for standard statistical analysis. The stabilization is achieved by supplementing the observed data with a small amount of synthetic data generated from the predictive distribution of a simpler model. This class of prior distributions is easy to use and allows direct statistical interpretation.

A catalytic prior distribution is designed to stabilize a high-dimensional “working model” by shrinking it toward a “simplified model.” The shrinkage is achieved by supplementing the observed data with a small amount of “synthetic data” generated from a predictive distribution under the simpler model. We apply this framework to generalized linear models, where we propose various strategies for the specification of a tuning parameter governing the degree of shrinkage and study resultant theoretical properties. In simulations, the resulting posterior estimation using such a catalytic prior outperforms maximum likelihood estimation from the working model and is generally comparable with or superior to existing competitive methods in terms of frequentist prediction accuracy of point estimation and coverage accuracy of interval estimation. The catalytic priors have simple interpretations and are easy to formulate.

A Bayesian Analysis of Prenatal Maternal Factors Predicting Nonadherence to Infant HIV Medication in South Africa

While efforts to prevent mother-to-child transmission of HIV been successful in some districts in South Africa, rates remain unacceptably high in others. This study utilized Bayesian logistic regression to examine maternal-level predictors of adherence to infant nevirapine prophylaxis, including intimate partner violence, maternal adherence, HIV serostatus disclosure reaction, recency of HIV diagnosis, and depression. Women (N = 303) were assessed during pregnancy and 6 weeks postpartum. Maternal adherence to antiretroviral therapy during pregnancy predicted an 80% reduction in the odds of infant nonadherence [OR 0.20, 95% posterior credible interval (.11, .38)], and maternal prenatal depression predicted an increase [OR 1.04, 95% PCI (1.01, 1.08)]. Results suggest that in rural South Africa, failure to provide medication to infants may arise from shared risk factors with maternal nonadherence. Intervening to increase maternal adherence and reduce depression may improve adherence to infant prophylaxis and ultimately reduce vertical transmission rates.

Risk factors for positioning-related somatosensory evoked potential changes in 3946 spinal surgeries

The goal of this study was to evaluate the risk factors associated with positioning-related SSEP changes (PRSC). The study investigated the association between 18 plausible risk factors and the occurrence of intraoperative PRSC. Risk factors investigated included demographic variables, comorbidities, and procedure related variables. All patients were treated by the University of Pittsburgh Medical Center from 2010 to 2012. We used univariate and multivariate statistical methods. 69 out of the 3946 (1.75%) spinal surgeries resulted in PRSC changes. The risk of PRSC was increased for women (p < 0.001), patients older than 65 years of age (p = 0.01), higher BMI (p < 0.001) patients, smokers (p < 0.001), and patients with hypertension (p < 0.001). No associations were found between PRSC and age greater than 80 years, diabetes mellitus, cardiovascular disease, and peripheral vascular disease. Three surgical situations were associated with PRSC including abnormal baselines (p < 0.001), patients in the "superman" position (p < 0.001), and patients in surgical procedures that extended over 200 min (p = 0.03). Patients with higher BMIs and who are undergoing spinal surgery longer than 200 min, with abnormal baselines, must be positioned with meticulous attention. Gender, hypertension, and smoking were also found to be risk factors from their odds ratios.

Machine Learning for Fetal Growth Prediction

Birthweight is often used as a proxy for fetal weight. Problems with this practice have recently been brought to light. We explore whether data available at birth can be used to predict estimated fetal weight using linear and quantile regression, random forests, Bayesian additive regression trees, and generalized boosted models. We train and validate each approach using 18,517 pregnancies (31,948 ultrasound visits) from the Magee-Womens Obstetric Maternal and Infant data and 240 pregnancies in a separate dataset of high-risk pregnancies. We also quantify the relation between smoking and small-for-gestational-age birth, defined as a birthweight in the lower 10th percentile of a population birthweight standard and estimated and predicted fetal weight standard. Using mean squared error and median absolute deviation criteria, quantile regression performed best among the regression-based approaches, but generalized boosted models performed best overall. Using the birthweight standard, smoking during pregnancy increased the risk of small-for-gestational-age 3.84-fold (95% CI: 2.70, 5.47). This ratio dropped to 1.65 (95% CI: 1.50, 1.81) when using the correct fetal weight standard, which was no different from the machine learning-based predicted standards, but higher than the regression-based predicted standards. Machine learning algorithms show promise in recovering missing fetal weight information. See video abstract at, http://links.lww.com/EDE/B314.

Hierarchical Semi-Bayes Methods for Misclassification in Perinatal Epidemiology

BACKGROUND

Validation data are used to estimate the extent of misclassification in epidemiologic studies. In the Penn MOMS cohort, prepregnancy body mass index is subject to misclassification, and validation data are available to estimate the extent of misclassification. We use these data to estimate the association between maternal prepregnancy body mass index and early preterm (<32 weeks) birth using a semi-Bayes hierarchical model, allowing for more flexible adjustment for misclassification.

METHODS

We propose a two-stage model that first fits a Bayesian hierarchical model for the bias parameters in the validation study. This model shrinks bias parameters in different groups toward one another in an effort to gain precision and improve mean squared error. In the second stage, we draw random samples from the posterior distribution of the bias parameters to implement a probabilistic bias analysis adjusting for exposure misclassification in a frequentist outcome model.

RESULTS

Bias parameters from the hierarchical model were often more substantively reasonable and often had smaller variance. Adjusting results for misclassification generally attenuated the strength of the unadjusted associations. After adjusting for misclassification, underweight mothers were not at increased risk of early preterm birth relative to normal weight mothers. Severely obese mothers had an increased risk of early preterm birth relative to normal weight mothers.

CONCLUSIONS

The two-stage semi-Bayesian hierarchical model borrowed strength between group-specific bias parameters to adjust for exposure misclassification. Model results support evidence of an increased risk of early preterm birth among severely obese mothers, relative to normal weight mothers.

Estimating multiple time-fixed treatment effects using a semi-Bayes semiparametric marginal structural Cox proportional hazards regression model

Marginal structural models for time-fixed treatments fit using inverse-probability weighted estimating equations are increasingly popular. Nonetheless, the resulting effect estimates are subject to finite-sample bias when data are sparse, as is typical for large-sample procedures. Here we propose a semi-Bayes estimation approach which penalizes or shrinks the estimated model parameters to improve finite-sample performance. This approach uses simple symmetric data-augmentation priors. Limited simulation experiments indicate that the proposed approach reduces finite-sample bias and improves confidence-interval coverage when the true values lie within the central "hill" of the prior distribution. We illustrate the approach with data from a nonexperimental study of HIV treatments.

Case-control study of cumulative cigarette tar exposure and lung and upper aerodigestive tract cancers

The development of comprehensive measures for tobacco exposure is crucial to specify effects on disease and inform public health policy. In this population-based case-control study, we evaluated the associations between cumulative lifetime cigarette tar exposure and cancers of the lung and upper aerodigestive tract (UADT). The study included 611 incident cases of lung cancer; 601 cases of UADT cancers (oropharyngeal, laryngeal and esophageal cancers); and 1,040 cancer-free controls. We estimated lifetime exposure to cigarette tar based on tar concentrations abstracted from government cigarette records and self-reported smoking histories derived from a standardized questionnaire. We analyzed the associations for cumulative tar exposure with lung and UADT cancer, overall and according to histological subtype. Cumulative tar exposure was highly correlated with pack-years among ever smoking controls (Pearson coefficient = 0.90). The adjusted odds ratio (95% confidence limits) for the estimated effect of about 1 kg increase in tar exposure (approximately the interquartile range in all controls) was 1.61 (1.50, 1.73) for lung cancer and 1.21 (1.13, 1.29) for UADT cancers. In general, tar exposure was more highly associated with small, squamous and large cell lung cancer than adenocarcinoma. With additional adjustment for pack-years, positive associations between tar and lung cancer were evident, particularly for small cell and large cell subtypes. Therefore, incorporating the composition of tobacco carcinogens in lifetime smoking exposure may improve lung cancer risk estimation. This study does not support the claim of a null or inverse association between "low exposure" to tobacco smoke and risk of these cancer types.

MicroRNA-related polymorphisms and non-Hodgkin lymphoma susceptibility in the Multicenter AIDS Cohort Study

BACKGROUND

MicroRNAs, small non-coding RNAs involved in gene regulation, are implicated in lymphomagenesis. We evaluated whether genetic variations in microRNA coding regions, binding sites, or biogenesis genes (collectively referred to as miRNA-SNPs) were associated with risk of AIDS-associated non-Hodgkin lymphoma (AIDS-NHL), and serum levels of four lymphoma-related microRNAs.

METHODS

Twenty-five miRNA-SNPs were genotyped in 180 AIDS-NHL cases and 529 HIV-infected matched controls from the Multicenter AIDS Cohort Study (MACS), and real-time polymerase chain reaction was used to quantify serum microRNA levels. Adjusted odds ratios (ORs) estimated using conditional logistic regression evaluated associations between miRNA-SNPs and AIDS-NHL risk. A semi-Bayes shrinkage approach was employed to reduce likelihood of false-positive associations. Adjusted mean ratios (MR) calculated using linear regression assessed associations between miRNA-SNPs and serum microRNA levels.

RESULTS

DDX20 rs197412, a non-synonymous miRNA biogenesis gene SNP, was associated with AIDS-NHL risk (OR=1.34 per minor allele; 95% CI: 1.02-1.75), and higher miRNA-222 serum levels nearing statistical significance (MR=1.21 per minor allele; 95% CI: 0.98-1.49). MiRNA-196a2 rs11614913 was associated with decreased central nervous system (CNS) AIDS-NHL (CT vs. CC OR=0.52; 95% CI: 0.27-0.99). The minor allele of HIF1A rs2057482, which creates a miRNA-196a2 binding site, was associated with systemic AIDS-NHL risk (OR=1.73 per minor allele; 95% CI: 1.12-2.67), and decreased CNS AIDS-NHL risk (OR=0.49 per minor allele; 95% CI: 0.25-0.94).

CONCLUSIONS

This study suggests that a few miRNA-SNPs are associated with AIDS-NHL risk and may modulate miRNA expression. These results support a role for miRNA in AIDS-NHL and may highlight pathways to be targeted for risk stratification or therapeutics.

Markov chain Monte Carlo: an introduction for epidemiologists

Markov Chain Monte Carlo (MCMC) methods are increasingly popular among epidemiologists. The reason for this may in part be that MCMC offers an appealing approach to handling some difficult types of analyses. Additionally, MCMC methods are those most commonly used for Bayesian analysis. However, epidemiologists are still largely unfamiliar with MCMC. They may lack familiarity either with he implementation of MCMC or with interpretation of the resultant output. As with tutorials outlining the calculus behind maximum likelihood in previous decades, a simple description of the machinery of MCMC is needed. We provide an introduction to conducting analyses with MCMC, and show that, given the same data and under certain model specifications, the results of an MCMC simulation match those of methods based on standard maximum-likelihood estimation (MLE). In addition, we highlight examples of instances in which MCMC approaches to data analysis provide a clear advantage over MLE. We hope that this brief tutorial will encourage epidemiologists to consider MCMC approaches as part of their analytic tool-kit.

Advances in meta-analysis: examples from internal medicine to neurology

OBJECTIVE

We review the state of the art in meta-analysis and data pooling following the evolution of the statistical models employed.

METHODS

Starting from a classic definition of meta-analysis of published data, a set of apparent antinomies which characterized the development of the meta-analytic tools are reconciled in dichotomies where the second term represents a possible generalization of the first one. Particular attention is given to the generalized linear mixed models as an overall framework for meta-analysis. Bayesian meta-analysis is discussed as a further possibility of generalization for sensitivity analysis and the use of priors as a data augmentation approach.

RESULTS

We provide relevant examples to underline how the need for adequate methods to solve practical issues in specific areas of research have guided the development of advanced methods in meta-analysis.

CONCLUSIONS

We show how all the advances in meta-analysis naturally merge into the unified framework of generalized linear mixed models and reconcile apparently conflicting approaches. All these complex models can be easily implemented with the standard commercial software available.

Tobacco smoking, NBS1 polymorphisms, and survival in lung and upper aerodigestive tract cancers with semi-Bayes adjustment for hazard ratio variation

PURPOSE

Although single nucleotide polymorphisms (SNPs) of NBS1 have been associated with susceptibility to lung and upper aerodigestive tract (UADT) cancers, their relations to cancer survival and measures of effect are largely unknown.

METHODS

Using follow-up data from 611 lung cancer cases and 601 UADT cancer cases from a population-based case-control study in Los Angeles, we prospectively evaluated associations of tobacco smoking and 5 NBS1 SNPs with all-cause mortality. Mortality data were obtained from the Social Security Death Index. We used Cox regression to estimate adjusted hazard ratios (HR) for main effects and ratios of hazard ratios (RHR) derived from product terms to assess hazard ratio variations by each SNP. Bayesian methods were used to account for multiple comparisons.

RESULTS

We observed 406 (66 %) deaths in lung cancer cases and 247 (41 %) deaths in UADT cancer cases with median survival of 1.43 and 1.72 years, respectively. Ever tobacco smoking was positively associated with mortality for both cancers. We observed an upward dose-response association between smoking pack-years and mortality in UADT squamous cell carcinoma. The adjusted HR relating smoking to mortality in non-small cell lung cancer (NSCLC) was greater for cases with the GG genotype of NBS1 rs1061302 than for cases with AA/AG genotypes (semi-Bayes adjusted RHR = 1.97; 95 % limits = 1.14, 3.41).

CONCLUSIONS

A history of tobacco smoking at cancer diagnosis was associated with mortality among patients with lung cancer or UADT squamous cell carcinoma. The HR relating smoking to mortality appeared to vary with the NBS1 rs1061302 genotype among NSCLC cases.

Sensitivity analyses for sparse-data problems-using weakly informative bayesian priors

Sparse-data problems are common, and approaches are needed to evaluate the sensitivity of parameter estimates based on sparse data. We propose a Bayesian approach that uses weakly informative priors to quantify sensitivity of parameters to sparse data. The weakly informative prior is based on accumulated evidence regarding the expected magnitude of relationships using relative measures of disease association. We illustrate the use of weakly informative priors with an example of the association of lifetime alcohol consumption and head and neck cancer. When data are sparse and the observed information is weak, a weakly informative prior will shrink parameter estimates toward the prior mean. Additionally, the example shows that when data are not sparse and the observed information is not weak, a weakly informative prior is not influential. Advancements in implementation of Markov Chain Monte Carlo simulation make this sensitivity analysis easily accessible to the practicing epidemiologist.

Spatially dependent polya tree modeling for survival data

With the proliferation of spatially oriented time-to-event data, spatial modeling in the survival context has received increased recent attention. A traditional way to capture a spatial pattern is to introduce frailty terms in the linear predictor of a semiparametric model, such as proportional hazards or accelerated failure time. We propose a new methodology to capture the spatial pattern by assuming a prior based on a mixture of spatially dependent Polya trees for the baseline survival in the proportional hazards model. Thanks to modern Markov chain Monte Carlo (MCMC) methods, this approach remains computationally feasible in a fully hierarchical Bayesian framework. We compare the spatially dependent mixture of Polya trees (MPT) approach to the traditional spatial frailty approach, and illustrate the usefulness of this method with an analysis of Iowan breast cancer survival data from the Surveillance, Epidemiology, and End Results (SEER) program of the National Cancer Institute. Our method provides better goodness of fit over the traditional alternatives as measured by log pseudo marginal likelihood (LPML), the deviance information criterion (DIC), and full sample score (FSS) statistics.

Combining previously published studies with current data in Bayesian logistic regression model: an example for identifying susceptibility genes related to lung cancer in humans

A general analysis method is proposed that utilizes meta-analysis to incorporate similar studies in addition to our current investigation in order to obtain informative prior effect parameters in a logistic regression model. It is common in epidemiological studies that data from similar previous studies are available. The case of gene susceptibility association with increased lung cancer frequency was used to demonstrate this methodology. Results of Markov chain Monte Carlo (MCMC) iterations provided a more precise estimation of the regression coefficient in a logistic model with informative prior distribution compared to the noninformative prior distribution model. In situations where similar historical data are available, it is proposed to include as much relevant information as previously published results in the analysis of current data.

Prior data for non-normal priors

Data augmentation priors facilitate contextual evaluation of prior distributions and the generation of Bayesian outputs from frequentist software. Previous papers have presented approximate Bayesian methods using 2x2 tables of 'prior data' to represent lognormal relative-risk priors in stratified and regression analyses. The present paper describes extensions that use the tables to represent generalized-F prior distributions for relative risks, which subsume lognormal priors as a limiting case. The method provides a means to increase tail-weight or skew the prior distribution for the log relative risk away from normality, while retaining the simple 2x2 table form of the prior data. When prior normality is preferred, it also provides a more accurate lognormal relative-risk prior in for the 2x2 table format. For more compact representation in regression analyses, the prior data can be compressed into a single data record. The method is illustrated with historical data from a study of electronic foetal monitoring and neonatal death.

Bayesian perspectives for epidemiological research. II. Regression analysis

This article describes extensions of the basic Bayesian methods using data priors to regression modelling, including hierarchical (multilevel) models. These methods provide an alternative to the parsimony-oriented approach of frequentist regression analysis. In particular, they replace arbitrary variable-selection criteria by prior distributions, and by doing so facilitate realistic use of imprecise but important prior information. They also allow Bayesian analyses to be conducted using standard regression packages; one need only be able to add variables and records to the data set. The methods thus facilitate the use of Bayesian solutions to problems of sparse data, multiple comparisons, subgroup analyses and study bias. Because these solutions have a frequentist interpretation as "shrinkage" (penalized) estimators, the methods can also be viewed as a means of implementing shrinkage approaches to multiparameter problems.

Bayesian statistics in medicine: a 25 year review

This review examines the state of Bayesian thinking as Statistics in Medicine was launched in 1982, reflecting particularly on its applicability and uses in medical research. It then looks at each subsequent five-year epoch, with a focus on papers appearing in Statistics in Medicine, putting these in the context of major developments in Bayesian thinking and computation with reference to important books, landmark meetings and seminal papers. It charts the growth of Bayesian statistics as it is applied to medicine and makes predictions for the future. From sparse beginnings, where Bayesian statistics was barely mentioned, Bayesian statistics has now permeated all the major areas of medical statistics, including clinical trials, epidemiology, meta-analyses and evidence synthesis, spatial modelling, longitudinal modelling, survival modelling, molecular genetics and decision-making in respect of new technologies.

Bayesian perspectives for epidemiological research: I. Foundations and basic methods

One misconception (of many) about Bayesian analyses is that prior distributions introduce assumptions that are more questionable than assumptions made by frequentist methods; yet the assumptions in priors can be more reasonable than the assumptions implicit in standard frequentist models. Another misconception is that Bayesian methods are computationally difficult and require special software. But perfectly adequate Bayesian analyses can be carried out with common software for frequentist analysis. Under a wide range of priors, the accuracy of these approximations is just as good as the frequentist accuracy of the software--and more than adequate for the inaccurate observational studies found in health and social sciences. An easy way to do Bayesian analyses is via inverse-variance (information) weighted averaging of the prior with the frequentist estimate. A more general method expresses the prior distributions in the form of prior data or 'data equivalents', which are then entered in the analysis as a new data stratum. That form reveals the strength of the prior judgements being introduced and may lead to tempering of those judgements. It is argued that a criterion for scientific acceptability of a prior distribution is that it be expressible as prior data, so that the strength of prior assumptions can be gauged by how much data they represent.

Generalized conjugate priors for Bayesian analysis of risk and survival regressions

Conjugate priors for Bayesian analyses of relative risks can be quite restrictive, because their shape depends on their location. By introducing a separate location parameter, however, these priors generalize to allow modeling of a broad range of prior opinions, while still preserving the computational simplicity of conjugate analyses. The present article illustrates the resulting generalized conjugate analyses using examples from case-control studies of the association of residential wire codes and magnetic fields with childhood leukemia.

Putting background information about relative risks into conjugate prior distributions

In Bayesian and empirical Bayes analyses of epidemiologic data, the most easily implemented prior specifications use a multivariate normal distribution for the log relative risks or a conjugate distribution for the discrete response vector. This article describes problems in translating background information about relative risks into conjugate priors and a solution. Traditionally, conjugate priors have been specified through flattening constants, an approach that leads to conflicts with the true prior covariance structure for the log relative risks. One can, however, derive a conjugate prior consistent with that structure by using a data-augmentation approximation to the true log relative-risk prior, although a rescaling step is needed to ensure the accuracy of the approximation. These points are illustrated with a logistic regression analysis of neonatal-death risk.