Although lung transplantation is an accepted therapy for end-stage disease, recipient outcomes continue to be hindered by early primary graft dysfunction (PGD) as well as late rejection and bronchiolitis obliterans syndrome (BOS). We have previously shown that the pro-inflammatory cytokine response following transplantation correlates with the severity of PGD. We hypothesized that lung-transplant recipients with an increased inflammatory response immediately following surgery would also have a greater incidence of unfavorable long-term outcomes including rejection, BOS and ultimately death.
A retrospective study of lung-transplant recipients (n = 19) for whom serial blood sampling of cytokines was performed for 24 h following transplantation between March 2002 and June 2003 at a single institution. Long-term follow-up was examined for rejection, BOS and survival.
Thirteen single and six bilateral lung recipients were examined. Eleven (58%) developed BOS and eight (42%) did not. Subgroup analysis revealed an association between elevated IL-6 concentrations 4 h after reperfusion of the allograft and development of BOS (P = 0.068). The correlation between IL-6 and survival time was found to be significant (corr = −0.46, P = 0.047), indicating that higher IL-6 response had shorter survival following transplantation.
An elevation in interleukin (IL)-6 concentration immediately following lung transplantation is associated with a trend towards development of bronchiolitis obliterans, rejection and significantly decreased survival time. Further studies are warranted to confirm the correlation between the immediate inflammatory response, PGD and BOS. Identification of patients at risk for BOS based on the cytokine response after surgery may allow for early intervention.
Transplantation; Lung transplantation; Lung other; Inflammation
Rural counties in the U.S. have higher rates of obesity, sedentary lifestyle, and associated chronic diseases than non-rural areas, yet the management of obesity in rural communities has received little attention from researchers.
To compare 2 extended-care programs for weight management with an education control group.
Design, Setting, and Participants
234 obese women from rural communities who completed an initial 6-month weight-loss program were randomized to extended-care, delivered via telephone counseling or face-to-face sessions, or to an education control group. Cooperative Extension Service offices in six medically underserved rural counties served as venues for the trial. The study was conducted from June 2003 to May 2007.
The extended-care programs entailed problem-solving counseling delivered in 26 biweekly sessions. Control group participants received 26 biweekly newsletters containing weight-control advice.
Main Outcome Measure
Change in weight from randomization.
Mean weight at study entry was 96.4 kg. Mean weight loss during the initial 6-month intervention was 10.0 kg. One year after randomization, participants in the telephone and face-to-face conditions regained less weight (means ± SE = 1.3 ± 0.7 and 1.2 ± 0.6 kg, respectively) than those in the education control group (3.7 ± 0.6 kg; Ps = 0.02 and 0.03). The beneficial effects of extended-care counseling were mediated by greater adherence to behavioral weight-management strategies, and cost analyses indicated that telephone counseling was less expensive than face-to-face intervention.
Extended care delivered either by telephone or face-to-face sessions improved the one-year maintenance of lost weight compared to education alone. Telephone counseling constitutes an effective and cost-efficient option for long-term weight management. Delivering lifestyle interventions via the existing infrastructure of the Cooperative Extension Service represents a viable means of research translation into rural communities with limited access to preventive health services.
ClinicalTrials.gov number, NCT00201006.
A major challenge following successful weight loss is continuing the behaviors required for long-term weight maintenance. This challenge may be exacerbated in rural areas with limited local support resources.
This study describes and compares program costs and cost-effectiveness for 12-month extended care lifestyle maintenance programs following an initial 6-month weight loss program.
A 1-year prospective controlled randomized clinical trial.
The study included 215 female participants age 50 or older from rural areas who completed an initial 6-month lifestyle program for weight loss. The study was conducted from June 1, 2003, to May 31, 2007.
The intervention was delivered through local Cooperative Extension Service offices in rural Florida. Participants were randomly-assigned to a 12-month extended care program using either individual telephone counseling (n=67), group face-to-face counseling (n=74), or a mail/control group (n=74).
Main Outcome Measures
Program delivery costs, weight loss, and self-reported health status were directly assessed through questionnaires and program activity logs. Costs were estimated across a range of enrollment sizes to allow inferences beyond the study sample.
Statistical Analyses Performed
Non-parametric and parametric tests of differences across groups for program outcomes were combined with direct program cost estimates and expected value calculations to determine which scales of operation favored alternative formats for lifestyle maintenance.
Median weight regain during the intervention year was 1.7 kg for participants in the face-to-face format, 2.1 kg for the telephone format, and 3.1 kg for the mail/control format. For a typical group size of 13 participants, the face-to-face format had higher fixed costs, which translated into higher overall program costs ($420 per participant) when compared to individual telephone counseling ($268 per participant) and control ($226 per participant) programs. While the net weight lost after the 12-month maintenance program was higher for the face-to-face and telephone programs compared to the control group, the average cost per expected kilogram of weight lost was higher for the face-to-face program ($47/kg) compared to the other two programs (approximately $33/kg for telephone and control).
Both the scale of operations and local demand for programs are important considerations in selecting a delivery format for lifestyle maintenance. In this study, the telephone format had a lower cost, but similar outcomes compared to the face-to-face format.
Obesity; cost-effectiveness; randomized trial; rural health
A key issue in the treatment of obesity in older adults is whether the health benefits of weight loss outweigh the potential risks with respect to musculoskeletal injury.
To compare change in weight, improvements in metabolic risk factors, and reported musculoskeletal adverse events in middle-aged (50–59 years) and older (65–74 years), obese women.
Materials and methods
Participants completed an initial 6-month lifestyle intervention for weight loss, comprised of weekly group sessions, followed by 12 months of extended care with biweekly contacts. Weight and fasting blood samples were assessed at baseline, month 6, and month 18; data regarding adverse events were collected throughout the duration of the study.
Both middle-aged (n = 162) and older (n = 56) women achieved significant weight reductions from baseline to month 6 (10.1 ± 0.68 kg and 9.3 ± 0.76 kg, respectively) and maintained a large proportion of their losses at month 18 (7.6 ± 0.87 kg and 7.6 ± 1.3 kg, respectively); there were no significant differences between the two groups with respect to weight change. Older women further experienced significant reductions in systolic blood pressure, HbA1c, and C-reactive protein from baseline to month 6 and maintained these improvements at month 18. Despite potential safety concerns, we found that older women were no more likely to experience musculoskeletal adverse events during the intervention as compared with their middle-aged counterparts.
These results suggest that older, obese women can experience significant health benefits from lifestyle treatment for obesity, including weight loss and improvements in disease risk factors. Further investigation of the impact of weight loss on additional health-related parameters and risks (eg, body composition, muscular strength, physical functioning, and injuries) in older adults is needed.
lifestyle intervention; adverse events; metabolic risk factors
Bed alarm systems intended to prevent hospital falls have not been formally evaluated.
To investigate whether an intervention aimed at increasing bed alarm use decreases hospital falls and related events.
Pair-matched, cluster randomized trial over 18 months. Nursing units were allocated by computer-generated randomization on the basis of baseline fall rates. Patients and outcome assessors were blinded to unit assignment; outcome assessors may have become unblinded. (ClinicalTrials.gov registration number: NCT00183053)
16 nursing units in an urban community hospital.
27 672 inpatients in general medical, surgical, and specialty units.
Education, training, and technical support to promote use of a standard bed alarm system (intervention units); bed alarms available but not formally promoted or supported (control units).
Pre–post difference in change in falls per 1000 patient-days (primary end point); number of patients who fell, fall-related injuries, and number of patients restrained (secondary end points).
Prevalence of alarm use was 64.41 days per 1000 patient-days on intervention units and 1.79 days per 1000 patient-days on control units (P = 0.004). There was no difference in change in fall rates per 1000 patient-days (risk ratio, 1.09 [95% CI, 0.85 to 1.53]; difference, 0.41 [CI, −1.05 to 2.47], which corresponds to a greater difference in falls in control vs. intervention units) or in the number of patients who fell, injurious fall rates, or the number of patients physically restrained on intervention units compared with control units.
The study was conducted at a single site and was slightly underpowered compared with the initial design.
An intervention designed to increase bed alarm use in an urban hospital increased alarm use but had no statistically or clinically significant effect on fall-related events or physical restraint use.
Primary Funding Source
National Institute on Aging.
Pattern mixture modeling is a popular approach for handling incomplete longitudinal data. Such models are not identifiable by construction. Identifying restrictions are one approach to mixture model identification (Little, 1995; Little and Wang, 1996; Thijs et al., 2002; Kenward et al., 2003; Daniels and Hogan, 2008) and are a natural starting point for missing not at random sensitivity analysis (Thijs et al., 2002; Daniels and Hogan, 2008). However, when the pattern specific models are multivariate normal, identifying restrictions corresponding to missing at random may not exist. Furthermore, identification strategies can be problematic in models with covariates (e.g. baseline covariates with time-invariant coefficients). In this paper, we explore conditions necessary for identifying restrictions that result in missing at random (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.
Missing at random; Non-future dependence; Deviance information criterion
Acute lung injury (ALI) is a condition characterized by acute onset of severe hypoxemia and bilateral pulmonary infiltrates. ALI patients typically require mechanical ventilation in an intensive care unit. Low tidal volume ventilation (LTVV), a time-varying dynamic treatment regime, has been recommended as an effective ventilation strategy. This recommendation was based on the results of the ARMA study, a randomized clinical trial designed to compare low vs. high tidal volume strategies (The Acute Respiratory Distress Syndrome Network, 2000) . After publication of the trial, some critics focused on the high non-adherence rates in the LTVV arm suggesting that non-adherence occurred because treating physicians felt that deviating from the prescribed regime would improve patient outcomes. In this paper, we seek to address this controversy by estimating the survival distribution in the counterfactual setting where all patients assigned to LTVV followed the regime. Inference is based on a fully Bayesian implementation of Robins’ (1986) G-computation formula. In addition to re-analyzing data from the ARMA trial, we also apply our methodology to data from a subsequent trial (ALVEOLI), which implemented the LTVV regime in both of its study arms and also suffered from non-adherence.
Bayesian inference; Causal inference; Dynamic treatment regime; G-computation formula
Missing phenotype data can be a major hurdle to mapping quantitative trait loci (QTL). Though in many cases experiments may be designed to minimize the occurrence of missing data, it is often unavoidable in practice; thus, statistical methods to account for missing data are needed. In this paper we describe an approach for conjoining multiple imputation and QTL mapping. Methods are applied to map genes associated with increased breathing effort in mice after lung inflammation due to allergen challenge in developing lines of the Collaborative Cross, a new mouse genetics resource. Missing data poses a particular challenge in this study because the desired phenotype summary to be mapped is a function of incompletely observed dose-response curves. Comparison of the multiple imputation approach to two naive approaches for handling missing data suggest that these simpler methods may yield poor results: ignoring missing data through a complete case analysis may lead to incorrect conclusions, while using a last observation carried forward procedure, which does not account for uncertainty in the imputed values, may lead to anti-conservative inference. The proposed approach is widely applicable to other studies with missing phenotype data.
multiple imputation; missing data; quantitative trait loci
The primary purpose of the present set of studies was to provide a direct comparison of the effects of the angiotensin-converting enzyme inhibitor enalapril and the angiotensin receptor blocker losartan on body composition, physical performance, and muscle quality when administered late in life to aged rats. Overall, enalapril treatment consistently attenuated age-related increases in adiposity relative to both placebo and losartan. The maximal effect was achieved after 3 months of treatment (between 24 and 27 months of age), at a dose of 40 mg/kg and was observed in the absence of any changes in physical activity, body temperature, or food intake. In addition, the reduction in fat mass was not due to changes in pathology given that enalapril attenuated age-related increases in tumor development relative to placebo- and losartan-treated animals. Both enalapril and losartan attenuated age-related decreases in grip strength, suggesting that changes in body composition appear dissociated from improvements in physical function and may reflect a differential impact of enalapril and losartan on muscle quality. To link changes in adiposity to improvements in skeletal muscle quality, we performed gene array analyses to generate hypotheses regarding cell signaling pathways altered with enalapril treatment. Based on these results, our primary follow-up pathway was mitochondria-mediated apoptosis of myocytes. Relative to losartan- and placebo-treated rats, only enalapril decreased DNA fragmentation and caspase-dependent apoptotic signaling. These data suggest that attenuation of the severity of skeletal muscle apoptosis promoted by enalapril may represent a distinct mechanism through which this compound improves muscle strength/quality.
Age-related adiposity; Body composition; Sarcopenia; Renin–angiotensin system; Physical function; Muscle quality
Obese older adults are particularly susceptible to sarcopenia and have a higher prevalence of disability than their peers of normal weight. Interventions to improve body composition in late life are crucial to maintaining independence. The main mechanisms underlying sarcopenia have not been determined conclusively, but chronic inflammation, apoptosis, and impaired mitochondrial function are believed to play important roles. It has yet to be determined whether impaired cellular quality control mechanisms contribute to this process. The objective of this study was to assess the effects of a 6-month weight loss program combined with moderate-intensity exercise on the cellular quality control mechanisms autophagy and ubiquitin-proteasome, as well as on inflammation, apoptosis, and mitochondrial function, in the skeletal muscle of older obese women. The intervention resulted in significant weight loss (8.0 ± 3.9 % vs. 0.4 ± 3.1% of baseline weight, p = 0.002) and improvements in walking speed (reduced time to walk 400 meters, − 20.4 ± 16% vs. − 2.5 ± 12%, p = 0.03). In the intervention group, we observed a three-fold increase in messenger RNA (mRNA) levels of the autophagy regulators LC3B, Atg7, and lysosome-associated membrane protein-2 (LAMP-2) compared to controls. Changes in mRNA levels of FoxO3A and its targets MuRF1, MAFBx, and BNIP3 were on average seven-fold higher in the intervention group compared to controls, but these differences were not statistically significant. Tumor necrosis factor-α (TNF-α) mRNA levels were elevated after the intervention, but we did not detect significant changes in the downstream apoptosis markers caspase 8 and 3. Mitochondrial biogenesis markers (PGC1α and TFAm) were increased by the intervention, but this was not accompanied by significant changes in mitochondrial complex content and activity. In conclusion, although exploratory in nature, this study is among the first to report the stimulation of cellular quality control mechanisms elicited by a weight loss and exercise program in older obese women.
In longitudinal clinical trials, when outcome variables at later time points are only defined for patients who survive to those times, the evaluation of the causal effect of treatment is complicated. In this paper, we describe an approach that can be used to obtain the causal effect of three treatment arms with ordinal outcomes in the presence of death using a principal stratification approach. We introduce a set of flexible assumptions to identify the causal effect and implement a sensitivity analysis for non-identifiable assumptions which we parameterize parsimoniously. Methods are illustrated on quality of life data from a recent colorectal cancer clinical trial.
Principal stratification; QOL; Ordinal data; Sensitivity analysis
Obesity and a sedentary lifestyle are associated with physical impairments and biologic changes in older adults. Weight loss combined with exercise may reduce inflammation and improve physical functioning in overweight, sedentary, older adults. This study tested whether a weight loss program combined with moderate exercise could improve physical function in obese, older adult women.
Participants (N = 34) were generally healthy, obese, older adult women (age range 55–79 years) with mild to moderate physical impairments (ie, functional limitations). Participants were randomly assigned to one of two groups for 24 weeks: (i) weight loss plus exercise (WL+E; n = 17; mean age = 63.7 years [4.5]) or (ii) educational control (n = 17; mean age = 63.7 [6.7]). In the WL+E group, participants attended a group-based weight management session plus three supervised exercise sessions within their community each week. During exercise sessions, participants engaged in brisk walking and lower-body resistance training of moderate intensity. Participants in the educational control group attended monthly health education lectures on topics relevant to older adults. Outcomes were: (i) body weight, (ii) walking speed (assessed by 400-meter walk test), (iii) the Short Physical Performance Battery (SPPB), and (iv) knee extension isokinetic strength.
Participants randomized to the WL+E group lost significantly more weight than participants in the educational control group (5.95 [0.992] vs 0.23 [0.99] kg; P < 0.01). Additionally, the walking speed of participants in the WL+E group significantly increased compared with that of the control group (reduction in time on the 400-meter walk test = 44 seconds; P < 0.05). Scores on the SPPB improved in both the intervention and educational control groups from pre- to post-test (P < 0.05), with significant differences between groups (P = 0.02). Knee extension strength was maintained in both groups.
Our findings suggest that a lifestyle-based weight loss program consisting of moderate caloric restriction plus moderate exercise can produce significant weight loss and improve physical function while maintaining muscle strength in obese, older adult women with mild to moderate physical impairments.
obesity; weight loss; physical function; oxidative stress; inflammation; walking speed
In seasonal influenza epidemics, pathogens such as respiratory syncytial virus (RSV) often co-circulate with influenza and cause influenza-like illness (ILI) in human hosts. However, it is often impractical to test for each potential pathogen or to collect specimens for each observed ILI episode, making inference about influenza transmission difficult. In the setting of infectious diseases, missing outcomes impose a particular challenge because of the dependence among individuals. We propose a Bayesian competing-risk model for multiple co-circulating pathogens for inference on transmissibility and intervention efficacies under the assumption that missingness in the biological confirmation of the pathogen is ignorable. Simulation studies indicate a reasonable performance of the proposed model even if the number of potential pathogens is misspecified. They also show that a moderate amount of missing laboratory test results has only a small impact on inference about key parameters in the setting of close contact groups. Using the proposed model, we found that a non-pharmaceutical intervention is marginally protective against transmission of influenza A in a study conducted in elementary schools.
Missing data; MCMC; Infectious disease; Competing risks; Intervention efficacy
We model sparse functional data from multiple subjects with a mixed-effects regression spline. In this model, the expected values for any subject (conditioned on the random effects) can be written as the sum of a population curve and a subject-specific deviate from this population curve. The population curve and the subject-specific deviates are both modeled as free-knot b-splines with k and k′ knots located at tk and tk′, respectively. To identify the number and location of the “free” knots, we sample from the posterior p (k, tk, k′, tk′|y) using reversible jump MCMC methods. Sampling from this posterior distribution is complicated, however, by the flexibility we allow for the model’s covariance structure. No restrictions (other than positive definiteness) are placed on the covariance parameters ψ and σ2 and, as a result, no analytical form for the likelihood p (y|k, tk, k′, tk′) exists. In this paper, we consider two approximations to p(y|k, tk, k′, tk′) and then sample from the corresponding approximations to p(k, tk, k′, tk′|y). We also sample from p(k, tk, k′, tk′, ψ, σ2|y) which has a likelihood that is available in closed form. While sampling from this larger posterior is less efficient, the resulting marginal distribution of knots is exact and allows us to evaluate the accuracy of each approximation. We then consider a real data set and explore the difference between p(k, tk, k′, tk′, ψ, σ2|y) and the more accurate approximation to p(k, tk, k′, tk′|y).
B-splines; Laplace approximation; Reversible jump MCMC; Unit-information prior
Controversy exists as to whether lifelong 40% calorie restriction (CR) enhances, has no effect on, or disrupts cognitive function during aging. Here, we report the effects of CR versus ad-lib feeding on cognitive function in male Brown Norway × Fisher344 rats across a range of ages (8–38 months), using two tasks that are differentially sensitive to age-related cognitive decline: object recognition and Morris water maze (MWM). All ages performed equally in object recognition, whereas, as a group, CR rats were impaired. In contrast, there was an age-related impairment in the MWM that was attenuated by CR as measured by time in proximity with and latency to reach the platform. Distance to the platform, a more sensitive measure, was not affected by CR. Finally, CR resulted in an overall increase in physical activity, one of several behavioral confounders to consider in the interpretation of cognitive outcomes in both tasks.
Morris water maze; Object recognition; Animal models of aging; Calorie restriction
Joint models for the association of a longitudinal binary and a longitudinal continuous process are proposed for situations in which their association is of direct interest. The models are parameterized such that the dependence between the two processes is characterized by unconstrained regression coefficients. Bayesian variable selection techniques are used to parsimoniously model these coefficients. A Markov chain Monte Carlo (MCMC) sampling algorithm is developed for sampling from the posterior distribution, using data augmentation steps to handle missing data. Several technical issues are addressed to implement the MCMC algorithm efficiently. The models are motivated by, and are used for, the analysis of a smoking cessation clinical trial in which an important question of interest was the effect of the (exercise) treatment on the relationship between smoking cessation and weight gain.
Calibrated posterior predictive p-value; Data augmentation; Dependence; Joint models; Markov chain Monte Carlo; Parameter expansion; Stochastic search variable selection
Random effects are often used in generalized linear models to explain the serial dependence for longitudinal categorical data. Marginalized random effects models (MREMs) for the analysis of longitudinal binary data have been proposed to permit likelihood-based estimation of marginal regression parameters. In this paper, we introduce an extension of the MREM to accommodate longitudinal ordinal data. Maximum marginal likelihood estimation is implemented utilizing quasi-Newton algorithms with Monte Carlo integration of the random effects. Our approach is applied to analyze the quality of life data from a recent colorectal cancer clinical trial. Dropout occurs at a high rate and is often due to tumor progression or death. To deal with progression/death, we use a mixture model for the joint distribution of longitudinal measures and progression/death times and principal stratification to draw causal inferences about survivors.
marginalized likelihood-based models; ordinal data models; dropout
In this article we consider the problem of fitting pattern mixture models to longitudinal data when there are many unique dropout times. We propose a marginally specified latent class pattern mixture model. The marginal mean is assumed to follow a generalized linear model, whereas the mean conditional on the latent class and random effects is specified separately. Because the dimension of the parameter vector of interest (the marginal regression coefficients) does not depend on the assumed number of latent classes, we propose to treat the number of latent classes as a random variable. We specify a prior distribution for the number of classes, and calculate (approximate) posterior model probabilities. In order to avoid the complications with implementing a fully Bayesian model, we propose a simple approximation to these posterior probabilities. The ideas are illustrated using data from a longitudinal study of depression in HIV-infected women.
Bayesian model averaging; Incomplete data; Latent variable; Marginal model; Random effects
Monitoring health care quality involves combining continuous and discrete outcomes measured on subjects across health care units over time. This article describes a Bayesian approach to jointly modeling multilevel multidimensional continuous and discrete outcomes with serial dependence. The overall goal is to characterize trajectories of traits of each unit. Underlying normal regression models for each outcome are used and dependence among different outcomes is induced through latent variables. Serial dependence is accommodated through modeling the pairwise correlations of the latent variables. Methods are illustrated to assess trends in quality of health care units using continuous and discrete outcomes from a sample of adult veterans discharged from 1 of 22 Veterans Integrated Service Networks with a psychiatric diagnosis between 1993 and 1998.
Bayesian hierarchical model; Correlation matrix; Informative priors; Latent variable; Mental health
Using validation sets for outcomes can greatly improve the estimation of vaccine efficacy (VE) in the field (Halloran and Longini, 2001; Halloran and others, 2003). Most statistical methods for using validation sets rely on the assumption that outcomes on those with no cultures are missing at random (MAR). However, often the validation sets will not be chosen at random. For example, confirmational cultures are often done on people with influenza-like illness as part of routine influenza surveillance. VE estimates based on such non-MAR validation sets could be biased. Here we propose frequentist and Bayesian approaches for estimating VE in the presence of validation bias. Our work builds on the ideas of Rotnitzky and others (1998, 2001), Scharfstein and others (1999, 2003), and Robins and others (2000). Our methods require expert opinion about the nature of the validation selection bias. In a re-analysis of an influenza vaccine study, we found, using the beliefs of a flu expert, that within any plausible range of selection bias the VE estimate based on the validation sets is much higher than the point estimate using just the non-specific case definition. Our approach is generally applicable to studies with missing binary outcomes with categorical covariates.
Bayesian; Expert opinion; Identifiability; Influenza; Missing data; Selection model; Vaccine efficacy
Generalized linear models with serial dependence are often used for short longitudinal series. Heagerty (2002, Biometrics 58, 342–351) has proposed marginalized transition models for the analysis of longitudinal binary data. In this article, we extend this work to accommodate longitudinal ordinal data. Fisher-scoring algorithms are developed for estimation. Methods are illustrated on quality-of-life data from a recent colorectal cancer clinical trial.
Fisher scoring; Generalized linear models; QOL
In randomized studies with missing outcomes, non-identifiable assumptions are required to hold for valid data analysis. As a result, statisticians have been advocating the use of sensitivity analysis to evaluate the effect of varying asssumptions on study conclusions. While this approach may be useful in assessing the sensitivity of treatment comparisons to missing data assumptions, it may be dissatisfying to some researchers/decision makers because a single summary is not provided. In this paper, we present a fully Bayesian methodology that allows the investigator to draw a ‘single’ conclusion by formally incorporating prior beliefs about non-identifiable, yet interpretable, selection bias parameters. Our Bayesian model provides robustness to prior specification of the distributional form of the continuous outcomes.
Dirichlet process prior; Identifiability; MCHC; Non-parametric Bayes; Selection model; Sensitivity analysis
Estimation of covariance matrices in small samples has been studied by many authors. Standard estimators, like the unstructured maximum likelihood estimator (ML) or restricted maximum likelihood (REML) estimator, can be very unstable with the smallest estimated eigenvalues being too small and the largest too big. A standard approach to more stably estimating the matrix in small samples is to compute the ML or REML estimator under some simple structure that involves estimation of fewer parameters, such as compound symmetry or independence. However, these estimators will not be consistent unless the hypothesized structure is correct. If interest focuses on estimation of regression coefficients with correlated (or longitudinal) data, a sandwich estimator of the covariance matrix may be used to provide standard errors for the estimated coefficients that are robust in the sense that they remain consistent under misspecifics tion of the covariance structure. With large matrices, however, the inefficiency of the sandwich estimator becomes worrisome. We consider here two general shrinkage approaches to estimating the covariance matrix and regression coefficients. The first involves shrinking the eigenvalues of the unstructured ML or REML estimator. The second involves shrinking an unstructured estimator toward a structured estimator. For both cases, the data determine the amount of shrinkage. These estimators are consistent and give consistent and asymptotically efficient estimates for regression coefficients. Simulations show the improved operating characteristics of the shrinkage estimators of the covariance matrix and the regression coefficients in finite samples. The final estimator chosen includes a combination of both shrinkage approaches, i.e., shrinking the eigenvalues and then shrinking toward structure. We illustrate our approach on a sleep EEG study that requires estimation of a 24 × 24 covariance matrix and for which inferences on mean parameters critically depend on the covariance estimator chosen. We recommend making inference using a particular shrinkage estimator that provides a reasonable compromise between structured and unstructured estimators.
Empirical Bayes; General linear model; Givens angles; Hierarchical prior; Longitudinal data