Bayesian Dynamic Borrowing of Historical Information with Applications to the Analysis of Large-Scale Assessments

The purpose of this paper is to demonstrate and evaluate the use of Bayesian dynamic borrowing (Viele et al, in Pharm Stat 13:41-54, 2014) as a means of systematically utilizing historical information with specific applications to large-scale educational assessments. Dynamic borrowing via Bayesian hierarchical models is a special case of a general framework of historical borrowing where the degree of borrowing depends on the heterogeneity among historical data and current data. A joint prior distribution over the historical and current data sets is specified with the degree of heterogeneity across the data sets controlled by the variance of the joint distribution. We apply Bayesian dynamic borrowing to both single-level and multilevel models and compare this approach to other historical borrowing methods such as complete pooling, Bayesian synthesis, and power priors. Two case studies using data from the Program for International Student Assessment reveal the utility of Bayesian dynamic borrowing in terms of predictive accuracy. This is followed by two simulation studies that reveal the utility of Bayesian dynamic borrowing over simple pooling and power priors in cases where the historical data is heterogeneous compared to the current data based on bias, mean squared error, and predictive accuracy. In cases of homogeneous historical data, Bayesian dynamic borrowing performs similarly to data pooling, Bayesian synthesis, and power priors. In contrast, for heterogeneous historical data, Bayesian dynamic borrowing performed at least as well, if not better, than other methods of borrowing with respect to mean squared error, percent bias, and leave-one-out cross-validation.

Towards a Stochastic Model to Simulate Grapevine Architecture: A Case Study on Digitized Riesling Vines Considering Effects of Elevated CO2

Modeling plant growth, in particular with functional-structural plant models, can provide tools to study impacts of changing environments in silico. Simulation studies can be used as pilot studies for reducing the on-field experimental effort when predictive capabilities are given. Robust model calibration leads to less fragile predictions, while introducing uncertainties in predictions allows accounting for natural variability, resulting in stochastic plant growth models. In this study, stochastic model components that can be implemented into the functional-structural plant model Virtual Riesling are developed relying on Bayesian model calibration with the goal to enhance the model towards a fully stochastic model. In this first step, model development targeting phenology, in particular budburst variability, phytomer development rate and internode growth are presented in detail. Multi-objective optimization is applied to estimate a single set of cardinal temperatures, which is used in phenology and growth modeling based on a development days approach. Measurements from two seasons of grapevines grown in a vineyard with free-air carbon dioxide enrichment (FACE) are used; thus, model building and selection are coupled with an investigation as to whether including effects of elevated CO2 conditions to be expected in 2050 would improve the models. The results show how natural variability complicates the detection of possible treatment effects, but demonstrate that Bayesian calibration in combination with mixed models can realistically recover natural shoot growth variability in predictions. We expect these and further stochastic model extensions to result in more realistic virtual plant simulations to study effects, which are used to conduct in silico studies of canopy microclimate and its effects on grape health and quality.

Parkinson's Disease Cognitive Phenotypes Show Unique Clock Drawing Features when Measured with Digital Technology

BACKGROUND

A companion paper (Crowley et al., 2020) reports on the neuroimaging and neuropsychological profiles of statistically determined idiopathic non-dementia Parkinson's disease (PD).

OBJECTIVE

The current investigation sought to further examine subtle behavioral clock drawing differences within the same PD cohort by comparing 1) PD to non-PD peers on digitally acquired clock drawing latency and graphomotor metrics, and 2) PD memory, executive, and cognitively well phenotypes on the same variables.

METHODS

230 matched participants (115 PD, 115 non-PD) completed neuropsychological tests and dCDT. Statistically-derived PD cognitive phenotypes characterized PD participants as PD low executive (PDExe; n = 25), PD low memory (PDMem; n = 34), PD cognitively well (PDWell; n = 56). Using a Bayesian framework and based on apriori hypotheses, we compared groups on: total completion time (TCT), pre-first hand latency (PFHL), post-clock face latency (PCFL), total clock face area (TCFA), and total number of pen strokes.

RESULTS

Fewer strokes and slower performance to command were associated with higher odds of PD diagnosis, while a larger clock face in the copy condition was associated with lower odds of PD diagnosis. Within PD cognitive phenotypes, slower performance (TCT, PCFL) and smaller clock face to command were associated with higher odds of being PDExe than PDWell, whereas larger clock faces associated with higher odds of being PDMem than PDWell. Longer disease duration, more pen strokes (command) and smaller clocks (command) associated with higher odds of being PDExe than PDWell.

CONCLUSION

Digitally-acquired clock drawing profiles differ between PD and non-PD peers, and distinguish PD cognitive phenotypes.

Probabilistic cause-of-disease assignment using case-control diagnostic tests: A latent variable regression approach

Optimal prevention and treatment strategies for a disease of multiple causes, such as pneumonia, must be informed by the population distribution of causes among cases, or cause-specific case fractions (CSCFs). CSCFs may further depend on additional explanatory variables. Existing methodological literature in disease etiology research does not fully address the regression problem, particularly under a case-control design. Based on multivariate binary non-gold-standard diagnostic data and additional covariate information, this article proposes a novel and unified regression modeling framework for estimating covariate-dependent CSCF functions in case-control disease etiology studies. The model leverages critical control data for valid probabilistic cause assignment for cases. We derive an efficient Markov chain Monte Carlo algorithm for flexible posterior inference. We illustrate the inference of CSCF functions using extensive simulations and show that the proposed model produces less biased estimates and more valid inference of the overall CSCFs than analyses that omit covariates. A regression analysis of pediatric pneumonia data reveals the dependence of CSCFs upon season, age, human immunodeficiency virus status and disease severity. The article concludes with a brief discussion on model extensions that may further enhance the utility of the regression model in broader disease etiology research.

Of power and despair in cetacean conservation: estimation and detection of trend in abundance with noisy and short time-series

Many conservation instruments rely on detecting and estimating a population decline in a target species to take action. Trend estimation is difficult because of small sample size and relatively large uncertainty in abundance/density estimates of many wild populations of animals. Focusing on cetaceans, we performed a prospective analysis to estimate power, type-I, sign (type-S) and magnitude (type-M) error rates of detecting a decline in short time-series of abundance estimates with different signal-to-noise ratio. We contrasted results from both unregularized (classical) and regularized approaches. The latter allows to incorporate prior information when estimating a trend. Power to detect a statistically significant estimates was in general lower than 80%, except for large declines. The unregularized approach (status quo) had inflated type-I error rates and gave biased (either over- or under-) estimates of a trend. The regularized approach with a weakly-informative prior offered the best trade-off in terms of bias, statistical power, type-I, type-S and type-M error rates and confidence interval coverage. To facilitate timely conservation decisions, we recommend to use the regularized approach with a weakly-informative prior in the detection and estimation of trend with short and noisy time-series of abundance estimates.

Uncertainty in the design stage of two-stage Bayesian propensity score analysis

The two-stage process of propensity score analysis (PSA) includes a design stage where propensity scores (PSs) are estimated and implemented to approximate a randomized experiment and an analysis stage where treatment effects are estimated conditional on the design. This article considers how uncertainty associated with the design stage impacts estimation of causal effects in the analysis stage. Such design uncertainty can derive from the fact that the PS itself is an estimated quantity, but also from other features of the design stage tied to choice of PS implementation. This article offers a procedure for obtaining the posterior distribution of causal effects after marginalizing over a distribution of design-stage outputs, lending a degree of formality to Bayesian methods for PSA that have gained attention in recent literature. Formulation of a probability distribution for the design-stage output depends on how the PS is implemented in the design stage, and propagation of uncertainty into causal estimates depends on how the treatment effect is estimated in the analysis stage. We explore these differences within a sample of commonly used PS implementations (quantile stratification, nearest-neighbor matching, caliper matching, inverse probability of treatment weighting, and doubly robust estimation) and investigate in a simulation study the impact of statistician choice in PS model and implementation on the degree of between- and within-design variability in the estimated treatment effect. The methods are then deployed in an investigation of the association between levels of fine particulate air pollution and elevated exposure to emissions from coal-fired power plants.

An Approach to Addressing Multiple Imputation Model Uncertainty Using Bayesian Model Averaging

This paper considers the problem of imputation model uncertainty in the context of missing data problems. We argue that so-called "Bayesianly proper" approaches to multiple imputation, although correctly accounting for uncertainty in imputation model parameters, ignore the uncertainty in the imputation model itself. We address imputation model uncertainty by implementing Bayesian model averaging as part of the imputation process. Bayesian model averaging accounts for both model and parameter uncertainty, and thus we argue is fully Bayesianly proper. We apply Bayesian model averaging to multiple imputation under the fully conditional specification approach. An extensive simulation study is conducted comparing our Bayesian model averaging approach against normal theory-based Bayesian imputation not accounting for model uncertainty. Across almost all conditions of the simulation study, the results reveal the extent of model uncertainty in multiple imputation and a consistent advantage to our Bayesian model averaging approach over normal-theory multiple imputation under missing-at-random and missing-completely-at random in terms of Kullback-Liebler divergence and mean squared prediction error. A small case study is also presented. Directions for future research are discussed.

Predictive probability methods for interim monitoring in clinical trials with longitudinal outcomes

In clinical research and development, interim monitoring is critical for better decision-making and minimizing the risk of exposing patients to possible ineffective therapies. For interim futility or efficacy monitoring, predictive probability methods are widely adopted in practice. Those methods have been well studied for univariate variables. However, for longitudinal studies, predictive probability methods using univariate information from only completers may not be most efficient, and data from on-going subjects can be utilized to improve efficiency. On the other hand, leveraging information from on-going subjects could allow an interim analysis to be potentially conducted once a sufficient number of subjects reach an earlier time point. For longitudinal outcomes, we derive closed-form formulas for predictive probabilities, including Bayesian predictive probability, predictive power, and conditional power and also give closed-form solutions for predictive probability of success in a future trial and the predictive probability of success of the best dose. When predictive probabilities are used for interim monitoring, we study their distributions and discuss their analytical cutoff values or stopping boundaries that have desired operating characteristics. We show that predictive probabilities utilizing all longitudinal information are more efficient for interim monitoring than that using information from completers only. To illustrate their practical application for longitudinal data, we analyze 2 real data examples from clinical trials.

Bayesian correction for covariate measurement error: A frequentist evaluation and comparison with regression calibration

Bayesian approaches for handling covariate measurement error are well established and yet arguably are still relatively little used by researchers. For some this is likely due to unfamiliarity or disagreement with the Bayesian inferential paradigm. For others a contributory factor is the inability of standard statistical packages to perform such Bayesian analyses. In this paper, we first give an overview of the Bayesian approach to handling covariate measurement error, and contrast it with regression calibration, arguably the most commonly adopted approach. We then argue why the Bayesian approach has a number of statistical advantages compared to regression calibration and demonstrate that implementing the Bayesian approach is usually quite feasible for the analyst. Next, we describe the closely related maximum likelihood and multiple imputation approaches and explain why we believe the Bayesian approach to generally be preferable. We then empirically compare the frequentist properties of regression calibration and the Bayesian approach through simulation studies. The flexibility of the Bayesian approach to handle both measurement error and missing data is then illustrated through an analysis of data from the Third National Health and Nutrition Examination Survey.

Multiple Imputation in Two-Stage Cluster Samples Using The Weighted Finite Population Bayesian Bootstrap

Multistage sampling is often employed in survey samples for cost and convenience. However, accounting for clustering features when generating datasets for multiple imputation is a nontrivial task, particularly when, as is often the case, cluster sampling is accompanied by unequal probabilities of selection, necessitating case weights. Thus, multiple imputation often ignores complex sample designs and assumes simple random sampling when generating imputations, even though failing to account for complex sample design features is known to yield biased estimates and confidence intervals that have incorrect nominal coverage. In this article, we extend a recently developed, weighted, finite-population Bayesian bootstrap procedure to generate synthetic populations conditional on complex sample design data that can be treated as simple random samples at the imputation stage, obviating the need to directly model design features for imputation. We develop two forms of this method: one where the probabilities of selection are known at the first and second stages of the design, and the other, more common in public use files, where only the final weight based on the product of the two probabilities is known. We show that this method has advantages in terms of bias, mean square error, and coverage properties over methods where sample designs are ignored, with little loss in efficiency, even when compared with correct fully parametric models. An application is made using the National Automotive Sampling System Crashworthiness Data System, a multistage, unequal probability sample of U.S. passenger vehicle crashes, which suffers from a substantial amount of missing data in "Delta-V," a key crash severity measure.

Weight Smoothing for Generalized Linear Models Using a Laplace Prior

When analyzing data sampled with unequal inclusion probabilities, correlations between the probability of selection and the sampled data can induce bias if the inclusion probabilities are ignored in the analysis. Weights equal to the inverse of the probability of inclusion are commonly used to correct possible bias. When weights are uncorrelated with the descriptive or model estimators of interest, highly disproportional sample designs resulting in large weights can introduce unnecessary variability, leading to an overall larger mean square error compared to unweighted methods. We describe an approach we term 'weight smoothing' that models the interactions between the weights and the estimators as random effects, reducing the root mean square error (RMSE) by shrinking interactions toward zero when such shrinkage is allowed by the data. This article adapts a flexible Laplace prior distribution for the hierarchical Bayesian model to gain a more robust bias-variance tradeoff than previous approaches using normal priors. Simulation and application suggest that under a linear model setting, weight-smoothing models with Laplace priors yield robust results when weighting is necessary, and provide considerable reduction in RMSE otherwise. In logistic regression models, estimates using weight-smoothing models with Laplace priors are robust, but with less gain in efficiency than in linear regression settings.

Idle thoughts of a 'well-calibrated' Bayesian in clinical drug development

The use of Bayesian approaches in the regulated world of pharmaceutical drug development has not been without its difficulties or its critics. The recent Food and Drug Administration regulatory guidance on the use of Bayesian approaches in device submissions has mandated an investigation into the operating characteristics of Bayesian approaches and has suggested how to make adjustments in order that the proposed approaches are in a sense calibrated. In this paper, I present examples of frequentist calibration of Bayesian procedures and argue that we need not necessarily aim for perfect calibration but should be allowed to use procedures, which are well-calibrated, a position supported by the guidance.

A Novel Statistical Model to Estimate Host Genetic Effects Affecting Disease Transmission

There is increasing recognition that genetic diversity can affect the spread of diseases, potentially affecting plant and livestock disease control as well as the emergence of human disease outbreaks. Nevertheless, even though computational tools can guide the control of infectious diseases, few epidemiological models can simultaneously accommodate the inherent individual heterogeneity in multiple infectious disease traits influencing disease transmission, such as the frequently modeled propensity to become infected and infectivity, which describes the host ability to transmit the infection to susceptible individuals. Furthermore, current quantitative genetic models fail to fully capture the heritable variation in host infectivity, mainly because they cannot accommodate the nonlinear infection dynamics underlying epidemiological data. We present in this article a novel statistical model and an inference method to estimate genetic parameters associated with both host susceptibility and infectivity. Our methodology combines quantitative genetic models of social interactions with stochastic processes to model the random, nonlinear, and dynamic nature of infections and uses adaptive Bayesian computational techniques to estimate the model parameters. Results using simulated epidemic data show that our model can accurately estimate heritabilities and genetic risks not only of susceptibility but also of infectivity, therefore exploring a trait whose heritable variation is currently ignored in disease genetics and can greatly influence the spread of infectious diseases. Our proposed methodology offers potential impacts in areas such as livestock disease control through selective breeding and also in predicting and controlling the emergence of disease outbreaks in human populations.

A two-step semiparametric method to accommodate sampling weights in multiple imputation

Multiple imputation (MI) is a well-established method to handle item-nonresponse in sample surveys. Survey data obtained from complex sampling designs often involve features that include unequal probability of selection. MI requires imputation to be congenial, that is, for the imputations to come from a Bayesian predictive distribution and for the observed and complete data estimator to equal the posterior mean given the observed or complete data, and similarly for the observed and complete variance estimator to equal the posterior variance given the observed or complete data; more colloquially, the analyst and imputer make similar modeling assumptions. Yet multiply imputed data sets from complex sample designs with unequal sampling weights are typically imputed under simple random sampling assumptions and then analyzed using methods that account for the sampling weights. This is a setting in which the analyst assumes more than the imputer, which can led to biased estimates and anti-conservative inference. Less commonly used alternatives such as including case weights as predictors in the imputation model typically require interaction terms for more complex estimators such as regression coefficients, and can be vulnerable to model misspecification and difficult to implement. We develop a simple two-step MI framework that accounts for sampling weights using a weighted finite population Bayesian bootstrap method to validly impute the whole population (including item nonresponse) from the observed data. In the second step, having generated posterior predictive distributions of the entire population, we use standard IID imputation to handle the item nonresponse. Simulation results show that the proposed method has good frequentist properties and is robust to model misspecification compared to alternative approaches. We apply the proposed method to accommodate missing data in the Behavioral Risk Factor Surveillance System when estimating means and parameters of regression models.

Statistical ecology comes of age

The desire to predict the consequences of global environmental change has been the driver towards more realistic models embracing the variability and uncertainties inherent in ecology. Statistical ecology has gelled over the past decade as a discipline that moves away from describing patterns towards modelling the ecological processes that generate these patterns. Following the fourth International Statistical Ecology Conference (1–4 July 2014) in Montpellier, France, we analyse current trends in statistical ecology. Important advances in the analysis of individual movement, and in the modelling of population dynamics and species distributions, are made possible by the increasing use of hierarchical and hidden process models. Exciting research perspectives include the development of methods to interpret citizen science data and of efficient, flexible computational algorithms for model fitting. Statistical ecology has come of age: it now provides a general and mathematically rigorous framework linking ecological theory and empirical data.

Bayesian Causal Mediation Analysis for Group Randomized Designs with Homogeneous and Heterogeneous Effects: Simulation and Case Study

A fully Bayesian approach to causal mediation analysis for group-randomized designs is presented. A unique contribution of this article is the combination of Bayesian inferential methods with G-computation to address the problem of heterogeneous treatment or mediator effects. A detailed simulation study shows that this approach has excellent frequentist properties, particularly in the case of small sample sizes with accurate informative priors. The simulation study also demonstrates that the proposed approach can take into account heterogeneous treatment or mediator effects without bias. A case study using data from a school-based randomized intervention designed to increase parent social capital leading to improved behavioral and academic outcomes in children is offered to illustrate the Bayesian approach to causal mediation in group-randomized designs.

Bayesian Model Averaging for Propensity Score Analysis

This article considers Bayesian model averaging as a means of addressing uncertainty in the selection of variables in the propensity score equation. We investigate an approximate Bayesian model averaging approach based on the model-averaged propensity score estimates produced by the R package BMA but that ignores uncertainty in the propensity score. We also provide a fully Bayesian model averaging approach via Markov chain Monte Carlo sampling (MCMC) to account for uncertainty in both parameters and models. A detailed study of our approach examines the differences in the causal estimate when incorporating noninformative versus informative priors in the model averaging stage. We examine these approaches under common methods of propensity score implementation. In addition, we evaluate the impact of changing the size of Occam's window used to narrow down the range of possible models. We also assess the predictive performance of both Bayesian model averaging propensity score approaches and compare it with the case without Bayesian model averaging. Overall, results show that both Bayesian model averaging propensity score approaches recover the treatment effect estimates well and generally provide larger uncertainty estimates, as expected. Both Bayesian model averaging approaches offer slightly better prediction of the propensity score compared with the Bayesian approach with a single propensity score equation. Covariate balance checks for the case study show that both Bayesian model averaging approaches offer good balance. The fully Bayesian model averaging approach also provides posterior probability intervals of the balance indices.

Phase II randomized trial comparing sequential first-line everolimus and second-line sunitinib versus first-line sunitinib and second-line everolimus in patients with metastatic renal cell carcinoma

PURPOSE

A multicenter, randomized phase II trial, RECORD-3, was conducted to compare first-line everolimus followed by sunitinib at progression with the standard sequence of first-line sunitinib followed by everolimus in patients with metastatic renal cell carcinoma.

PATIENTS AND METHODS

RECORD-3 used a crossover treatment design. The primary objective was to assess progression-free survival (PFS) noninferiority of first-line everolimus compared with first-line sunitinib. Secondary end points included combined PFS for each sequence, overall survival (OS), and safety.

RESULTS

Of 471 enrolled patients, 238 were randomly assigned to first-line everolimus followed by sunitinib, and 233 were randomly assigned to first-line sunitinib followed by everolimus. The primary end point was not met; the median PFS was 7.9 months for first-line everolimus and 10.7 months for first-line sunitinib (hazard ratio [HR], 1.4; 95% CI, 1.2 to 1.8). Among patients who discontinued first-line, 108 (45%) crossed over from everolimus to second-line sunitinib, and 99 (43%) crossed over from sunitinib to second-line everolimus. The median combined PFS was 21.1 months for sequential everolimus then sunitinib and was 25.8 months for sequential sunitinib then everolimus (HR, 1.3; 95% CI, 0.9 to 1.7). The median OS was 22.4 months for sequential everolimus and then sunitinib and 32.0 months for sequential sunitinib and then everolimus (HR, 1.2; 95% CI, 0.9 to 1.6). Common treatment-emergent adverse events during first-line everolimus or sunitinib were stomatitis (53% and 57%, respectively), fatigue (45% and 51%, respectively), and diarrhea (38% and 57%, respectively).

CONCLUSION

Everolimus did not demonstrate noninferiority compared with sunitinib as a first-line therapy. The trial results support the standard treatment paradigm of first-line sunitinib followed by everolimus at progression.

Accommodating missingness when assessing surrogacy via principal stratification

BACKGROUND

When an outcome of interest in a clinical trial is late-occurring or difficult to obtain, surrogate markers can extract information about the effect of the treatment on the outcome of interest. Understanding associations between the causal effect (CE) of treatment on the outcome and the causal effect of treatment on the surrogate is critical to understanding the value of a surrogate from a clinical perspective.

PURPOSE

Traditional regression approaches to determine the proportion of the treatment effect explained by surrogate markers suffer from several shortcomings: they can be unstable and can lie outside the 0-1 range. Furthermore, they do not account for the fact that surrogate measures are obtained post randomization, and thus, the surrogate-outcome relationship may be subject to unmeasured confounding.

METHODS

to avoid these problems are of key importance. Methods Frangakis and Rubin suggested assessing the CE within prerandomization 'principal strata' defined by the counterfactual joint distribution of the surrogate marker under the different treatment arms, with the proportion of the overall outcome CE attributable to subjects for whom the treatment affects the proposed surrogate as the key measure of interest. Li et al. developed this 'principal surrogacy' approach for dichotomous markers and outcomes, utilizing Bayesian methods that accommodated nonidentifiability in the model parameters. Because the surrogate marker is typically observed early, outcome data are often missing. Here, we extend Li et al. to accommodate missing data in the observable final outcome under ignorable and nonignorable settings. We also allow for the possibility that missingness has a counterfactual component, a feature that previous literature has not addressed.

RESULTS

We apply the proposed methods to a trial of glaucoma control comparing surgery versus medication, where intraocular pressure (IOP) control at 12 months is a surrogate for IOP control at 96 months. We also conduct a series of simulations to consider the impacts of nonignorability, as well as sensitivity to priors and the ability of the decision information criterion (DIC) to choose the correct model when parameters are not fully identified.

LIMITATIONS

Because model parameters cannot be fully identified from data, informative priors can introduce nontrivial bias in moderate sample size settings, while more noninformative priors can yield wide credible intervals.

CONCLUSIONS

Assessing the linkage between CEs of treatment on a surrogate marker and CEs of a treatment on an outcome is important to understanding the value of a marker. These CEs are not fully identifiable; hence, we explore the sensitivity and identifiability aspects of these models and show that relatively weak assumptions can still yield meaningful results.

Model feedback in Bayesian propensity score estimation

Methods based on the propensity score comprise one set of valuable tools for comparative effectiveness research and for estimating causal effects more generally. These methods typically consist of two distinct stages: (1) a propensity score stage where a model is fit to predict the propensity to receive treatment (the propensity score), and (2) an outcome stage where responses are compared in treated and untreated units having similar values of the estimated propensity score. Traditional techniques conduct estimation in these two stages separately; estimates from the first stage are treated as fixed and known for use in the second stage. Bayesian methods have natural appeal in these settings because separate likelihoods for the two stages can be combined into a single joint likelihood, with estimation of the two stages carried out simultaneously. One key feature of joint estimation in this context is "feedback" between the outcome stage and the propensity score stage, meaning that quantities in a model for the outcome contribute information to posterior distributions of quantities in the model for the propensity score. We provide a rigorous assessment of Bayesian propensity score estimation to show that model feedback can produce poor estimates of causal effects absent strategies that augment propensity score adjustment with adjustment for individual covariates. We illustrate this phenomenon with a simulation study and with a comparative effectiveness investigation of carotid artery stenting versus carotid endarterectomy among 123,286 Medicare beneficiaries hospitlized for stroke in 2006 and 2007.