Control for Confounding in the Presence of Measurment Error in Hierarchical Models


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Using this approach, we established a safety threshold for the dabigatran case study that was within range of the findings from the benchmark approach 1. Comparing the safety threshold with post-approval studies of dabigatran versus warfarin indicated that no safety statement can be made.

Bayesian evidence synthesis for exploring generalizability of treatment effects: a case study of combining randomized and non-randomized results in diabetes. In this paper, we present a unified modeling framework to combine aggregated data from randomized controlled trials RCTs with individual participant data IPD from observational studies. Rather than simply pooling the available evidence into an overall treatment effect, adjusted for potential confounding, the intention of this work is to explore treatment effects in specific patient populations reflected by the IPD.

We present a new Bayesian hierarchical meta-regression model, which combines submodels, representing different types of data into a coherent analysis. Predictors of baseline risk are estimated from the individual data. Simultaneously, a bivariate random effects distribution of baseline risk and treatment effects is estimated from the combined individual and aggregate data. Therefore, given a subgroup of interest, the estimated treatment effect can be calculated through its correlation with baseline risk.

We highlight different types of model parameters: those that are the focus of inference e. The impact of risk factors on the amount of time taken to reach an endpoint is a common parameter of interest. Hazard ratios are often estimated using a discrete-time approximation, which works well when the by-interval event rate is low. However, if the intervals are made more frequent than the observation times, missing values will arise.

We investigated common analytical approaches, including available-case AC analysis, last observation carried forward LOCF , and multiple imputation MI , in a setting where time-dependent covariates also act as mediators. We generated complete data to obtain monthly information for all individuals, and from the complete data, we selected "observed" data by assuming that follow-up visits occurred every 6 months.

Meta-Regression

MI proved superior to LOCF and AC analyses when only data on confounding variables were missing; AC analysis also performed well when data for additional variables were missing completely at random. We applied the 3 approaches to data from the Canadian HIV-Hepatitis C Co-infection Cohort Study to estimate the association of alcohol abuse with liver fibrosis.

Instrumental variable method for time-to-event data using a pseudo-observation approach. Observational studies are often in peril of unmeasured confounding.

Instrumental variable analysis is a method for controlling for unmeasured confounding. As yet, theory on instrumental variable analysis of censored time-to-event data is scarce. We propose a pseudo-observation approach to instrumental variable analysis of the survival function, the restricted mean, and the cumulative incidence function in competing risks with right-censored data using generalized method of moments estimation. For the purpose of illustrating our proposed method, we study antidepressant exposure in pregnancy and risk of autism spectrum disorder in offspring, and the performance of the method is assessed through simulation studies.

Sample size under the additive hazards model. However, sample size formulas for clinical trials with time to event outcomes are currently based on either the proportional hazards assumption or an assumption of constant hazards. AIMS: The goal is to provide sample size formulas for superiority and non-inferiority trials assuming an additive hazards model but no specific distribution, along with evaluations of the performance of the formulas.

Simulations are conducted to ensure that the formulas attain the desired power. In assessing the efficacy of a time-varying treatment structural nested models SNMs are useful in dealing with confounding by variables affected by earlier treatments. These models often consider treatment allocation and repeated measures at the individual level. We extend SNMMs to clustered observations with time-varying confounding and treatments. We demonstrate how to formulate models with both cluster- and unit-level treatments and show how to derive semiparametric estimators of parameters in such models.

For unit-level treatments, we consider interference, namely the effect of treatment on outcomes in other units of the same cluster. The properties of estimators are evaluated through simulations and compared with the conventional GEE regression method for clustered outcomes. To illustrate our method, we use data from the treatment arm of a glaucoma clinical trial to compare the effectiveness of two commonly used ocular hypertension medications.

Network meta-analysis incorporating randomized controlled trials and non-randomized comparative cohort studies for assessing the safety and effectiveness of medical treatments: challenges and opportunities. Tammyc cadth. Network meta-analysis is increasingly used to allow comparison of multiple treatment alternatives simultaneously, some of which may not have been compared directly in primary research studies. The majority of network meta-analyses published to date have incorporated data from randomized controlled trials RCTs only; however, inclusion of non-randomized studies may sometimes be considered.

Non-randomized studies can complement RCTs or address some of their limitations, such as short follow-up time, small sample size, highly selected population, high cost, and ethical restrictions. In this paper, we discuss the challenges and opportunities of incorporating both RCTs and non-randomized comparative cohort studies into network meta-analysis for assessing the safety and effectiveness of medical treatments. Non-randomized studies with inadequate control of biases such as confounding may threaten the validity of the entire network meta-analysis.

Therefore, identification and inclusion of non-randomized studies must balance their strengths with their limitations. However, thoughtful integration of randomized and non-randomized studies may offer opportunities to provide more timely, comprehensive, and generalizable evidence about the comparative safety and effectiveness of medical treatments.

Meta-Regression | Columbia University Mailman School of Public Health

Estimation of causal effects of binary treatments in unconfounded studies. Gutman R 1 , Rubin DB 1. Recently, Gutman and Rubin proposed a new analysis-phase method for estimating treatment effects when the outcome is binary and there is only one covariate, which viewed causal effect estimation explicitly as a missing data problem. Here, we extend this method to situations with continuous outcomes and multiple covariates and compare it with other commonly used methods such as matching, subclassification, weighting, and covariance adjustment.

In addition, it can estimate finite population average causal effects as well as non-linear causal estimands. This type of analysis also allows the identification of subgroups of units for which the effect appears to be especially beneficial or harmful. Researchers conducting observational studies need to consider 3 types of biases: selection bias, information bias, and confounding bias. A whole arsenal of statistical tools can be used to deal with information and confounding biases.

However, methods for addressing selection bias and unmeasured confounding are less developed. In this paper, we propose general bounding formulas for bias, including selection bias and unmeasured confounding. This should help researchers make more prudent interpretations of their potentially biased results.

Selection bias due to loss to follow up in cohort studies. Over the last fifteen years, stratification-based techniques as well as methods such as inverse probability-of-censoring weighted estimation have been more prominently discussed and offered as a means to correct for selection bias. Evaluation of a weighting approach for performing sensitivity analysis after multiple imputation. A weighting approach based on a selection model has been proposed for performing MNAR analyses to assess the robustness of results obtained under standard MI to departures from MAR.

METHODS: In this article, we use simulation to evaluate the weighting approach as a method for exploring possible departures from MAR, with missingness in a single variable, where the parameters of interest are the marginal mean and probability of a partially observed outcome variable and a measure of association between the outcome and a fully observed exposure. The simulation studies compare the weighting-based MNAR estimates for various numbers of imputations in small and large samples, for moderate to large magnitudes of departure from MAR, where the degree of departure from MAR was assumed known.

Further, we evaluated a proposed graphical method, which uses the dataset with missing data, for obtaining a plausible range of values for the parameter that quantifies the magnitude of departure from MAR. In particular, our findings demonstrate that the weighting approach provides biased parameter estimates, even when a large number of imputations is performed.

In the examples presented, the graphical approach for selecting a range of values for the possible departures from MAR did not capture the true parameter value of departure used in generating the data. Quantifying the impact of time-varying baseline risk adjustment in the self-controlled risk interval design. PURPOSE: The self-controlled risk interval design is commonly used to assess the association between an acute exposure and an adverse event of interest, implicitly adjusting for fixed, non-time-varying covariates.

Explicit adjustment needs to be made for time-varying covariates, for example, age in young children. It can be performed via either a fixed or random adjustment. The random-adjustment approach can provide valid point and interval estimates but requires access to individual-level data for an unexposed baseline sample. The fixed-adjustment approach does not have this requirement and will provide a valid point estimate but may underestimate the variance. We conducted a comprehensive simulation study to evaluate their performance.

The time-varying confounder is age. We considered a variety of design parameters including sample size, relative risk, time-varying baseline risks, and risk interval length. The fixed-adjustment approach can be used as a good alternative when the number of events used to estimate the time-varying baseline risks is at least the number of events used to estimate the relative risk, which is almost always the case. These analyses are often complicated by the number of potential confounders and the possibility of model misspecification.


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METHODS: We conducted a simulation study to compare the performance of logistic regression, propensity score, disease risk score, and stabilized inverse probability weighting methods to adjust for confounding. Model misspecification was induced in the independent derivation dataset. We evaluated performance using relative bias confidence interval coverage of the true effect, among other metrics.

Bias of the disease risk score estimates was at most For the propensity score model, this was 8. At events per coefficient of 1. Bias of misspecified disease risk score models was Despite better performance of disease risk score methods than logistic regression and propensity score models in small events per coefficient settings, bias, and coverage still deviated from nominal. Treatment effect heterogeneity for univariate subgroups in clinical trials: Shrinkage, standardization, or else.

Varadhan R 1, 2 , Wang SJ 3, 4. Treatment effect heterogeneity is a well-recognized phenomenon in randomized controlled clinical trials. In this paper, we discuss subgroup analyses with prespecified subgroups of clinical or biological importance. We explore various alternatives to the naive the traditional univariate subgroup analyses to address the issues of multiplicity and confounding. Specifically, we consider a model-based Bayesian shrinkage Bayes-DS and a nonparametric, empirical Bayes shrinkage approach Emp-Bayes to temper the optimism of traditional univariate subgroup analyses; a standardization approach standardization that accounts for correlation between baseline covariates; and a model-based maximum likelihood estimation MLE approach.

The Bayes-DS and Emp-Bayes methods model the variation in subgroup-specific treatment effect rather than testing the null hypothesis of no difference between subgroups. The standardization approach addresses the issue of confounding in subgroup analyses. The MLE approach is considered only for comparison in simulation studies as the "truth" since the data were generated from the same model.

Using the characteristics of a hypothetical large outcome trial, we perform simulation studies and articulate the utilities and potential limitations of these estimators. Standardization, although it tends to have a larger variance, is suggested when it is important to address the confounding of univariate subgroup effects due to correlation between baseline covariates. KGaA, Weinheim. Can statistical linkage of missing variables reduce bias in treatment effect estimates in comparative effectiveness research studies? AIM: Missing data, particularly missing variables, can create serious analytic challenges in observational comparative effectiveness research studies.

Statistical linkage of datasets is a potential method for incorporating missing variables. Prior studies have focused upon the bias introduced by imperfect linkage. METHODS: This analysis uses a case study of hepatitis C patients to estimate the net effect of statistical linkage on bias, also accounting for the potential reduction in missing variable bias. RESULTS: The results show that statistical linkage can reduce bias while also enabling parameter estimates to be obtained for the formerly missing variables.

New methods for treatment effect calibration, with applications to non-inferiority trials. In comparative effectiveness research, it is often of interest to calibrate treatment effect estimates from a clinical trial to a target population that differs from the study population. One important application is an indirect comparison of a new treatment with a placebo control on the basis of two separate randomized clinical trials: a non-inferiority trial comparing the new treatment with an active control and a historical trial comparing the active control with placebo.

The available methods for treatment effect calibration include an outcome regression OR method based on a regression model for the outcome and a weighting method based on a propensity score PS model. This article proposes new methods for treatment effect calibration: one based on a conditional effect CE model and two doubly robust DR methods. The first DR method involves a PS model and an OR model, is asymptotically valid if either model is correct, and attains the semiparametric information bound if both models are correct.

The second DR method involves a PS model, a CE model, and possibly an OR model, is asymptotically valid under the union of the PS and CE models, and attains the semiparametric information bound if all three models are correct. The various methods are compared in a simulation study and applied to recent clinical trials for treating human immunodeficiency virus infection. These models typically assume that variables are measured without error.

We describe a method to account for differential and nondifferential measurement error in a marginal structural model. METHODS: We illustrate the method estimating the joint effects of antiretroviral therapy initiation and current smoking on all-cause mortality in a United States cohort of 12, patients with HIV followed for up to 5 years between and Smoking status was likely measured with error, but a subset of 3, patients who reported smoking status on separate questionnaires composed an internal validation subgroup.

We compared a standard joint marginal structural model fit using inverse probability weights to a model that also accounted for misclassification of smoking status using multiple imputation. After accounting for misclassification, current smoking without therapy was associated with increased mortality hazard ratio [HR]: 1. The HR for current smoking and therapy [0. When adjusting for comorbidities in statistical models, researchers can include comorbidities individually or through the use of summary measures such as the Charlson Comorbidity Index or Elixhauser score.

We examined the conditions under which individual versus summary measures are most appropriate. We compared the use of the Charlson and Elixhauser scores versus individual comorbidities in prognostic models using a SEER-Medicare data example. We examined the ability of summary comorbidity measures to adjust for confounding using simulations. RESULTS: We devised a mathematical proof that found that the comorbidity summary measures are appropriate prognostic or adjustment mechanisms in survival analyses.

Once one knows the comorbidity score, no other information about the comorbidity variables used to create the score is generally needed. Our data example and simulations largely confirmed this finding. We have provided a theoretical justification that validates the use of such scores under many conditions. Our simulations generally confirm the utility of the summary comorbidity measures as substitutes for use of the individual comorbidity variables in health services research.

One caveat is that a summary measure may only be as good as the variables used to create it. Estimating the effect of treatment on binary outcomes using full matching on the propensity score. Many non-experimental studies use propensity-score methods to estimate causal effects by balancing treatment and control groups on a set of observed baseline covariates. Full matching on the propensity score has emerged as a particularly effective and flexible method for utilizing all available data, and creating well-balanced treatment and comparison groups.

However, full matching has been used infrequently with binary outcomes, and relatively little work has investigated the performance of full matching when estimating effects on binary outcomes. This paper describes methods that can be used for estimating the effect of treatment on binary outcomes when using full matching. It then used Monte Carlo simulations to evaluate the performance of these methods based on full matching with and without a caliper , and compared their performance with that of nearest neighbour matching with and without a caliper and inverse probability of treatment weighting.

The simulations varied the prevalence of the treatment and the strength of association between the covariates and treatment assignment. Results indicated that all of the approaches work well when the strength of confounding is relatively weak. With stronger confounding, the relative performance of the methods varies, with nearest neighbour matching with a caliper showing consistently good performance across a wide range of settings.

We illustrate the approaches using a study estimating the effect of inpatient smoking cessation counselling on survival following hospitalization for a heart attack. Time-dependent prognostic score matching for recurrent event analysis to evaluate a treatment assigned during follow-up. Recurrent events often serve as the outcome in epidemiologic studies. In some observational studies, the goal is to estimate the effect of a new or "experimental" i. The incentive for accepting the new treatment may be that it is more available than the standard treatment.

Given that the patient can choose between the experimental treatment and conventional therapy, it is of clinical importance to compare the treatment of interest versus the setting where the experimental treatment did not exist, in which case patients could only receive no treatment or the standard treatment. Many methods exist for the analysis of recurrent events and for the evaluation of treatment effects.

However, methodology for the intersection of these two areas is sparse. Moreover, care must be taken in setting up the comparison groups in our setting; use of existing methods featuring time-dependent treatment indicators will generally lead to a biased treatment effect since the comparison group construction will not properly account for the timing of treatment initiation. We propose a sequential stratification method featuring time-dependent prognostic score matching to estimate the effect of a time-dependent treatment on the recurrent event rate.

The performance of the method in moderate-sized samples is assessed through simulation. The proposed methods are applied to a prospective clinical study in order to evaluate the effect of living donor liver transplantation on hospitalization rates; in this setting, conventional therapy involves remaining on the wait list or receiving a deceased donor transplant. Shinozaki T 1 , Matsuyama Y. To adjust for confounding variables that contain too many combinations to be fully stratified, two model-based standardization methods exist: regression standardization and use of an inverse probability of exposure weighted-reweighted estimators.

Whereas the former requires an outcome regression model conditional on exposure and confounders, the latter requires a propensity score model. In reconciling among their modeling assumptions, doubly robust estimators, which only require correct specification of either the outcome regression or the propensity score model but do not necessitate both, have been well studied for total populations.

Optimal full matching for survival outcomes: a method that merits more widespread use. Matching on the propensity score is a commonly used analytic method for estimating the effects of treatments on outcomes. Commonly used propensity score matching methods include nearest neighbor matching and nearest neighbor caliper matching. Rosenbaum proposed an optimal full matching approach, in which matched strata are formed consisting of either one treated subject and at least one control subject or one control subject and at least one treated subject.

Full matching has been used rarely in the applied literature. Furthermore, its performance for use with survival outcomes has not been rigorously evaluated. We propose a method to use full matching to estimate the effect of treatment on the hazard of the occurrence of the outcome. An extensive set of Monte Carlo simulations were conducted to examine the performance of optimal full matching with survival analysis.

Its performance was compared with that of nearest neighbor matching, nearest neighbor caliper matching, and inverse probability of treatment weighting using the propensity score. Full matching has superior performance compared with that of the two other matching algorithms and had comparable performance with that of inverse probability of treatment weighting using the propensity score. We illustrate the application of full matching with survival outcomes to estimate the effect of statin prescribing at hospital discharge on the hazard of post-discharge mortality in a large cohort of patients who were discharged from hospital with a diagnosis of acute myocardial infarction.

Optimal full matching merits more widespread adoption in medical and epidemiological research. Moving towards best practice when using inverse probability of treatment weighting IPTW using the propensity score to estimate causal treatment effects in observational studies. Weighting subjects by the inverse probability of treatment received creates a synthetic sample in which treatment assignment is independent of measured baseline covariates. Inverse probability of treatment weighting IPTW using the propensity score allows one to obtain unbiased estimates of average treatment effects.

However, these estimates are only valid if there are no residual systematic differences in observed baseline characteristics between treated and control subjects in the sample weighted by the estimated inverse probability of treatment. We report on a systematic literature review, in which we found that the use of IPTW has increased rapidly in recent years, but that in the most recent year, a majority of studies did not formally examine whether weighting balanced measured covariates between treatment groups. We then proceed to describe a suite of quantitative and qualitative methods that allow one to assess whether measured baseline covariates are balanced between treatment groups in the weighted sample.

The quantitative methods use the weighted standardized difference to compare means, prevalences, higher-order moments, and interactions. The qualitative methods employ graphical methods to compare the distribution of continuous baseline covariates between treated and control subjects in the weighted sample. Finally, we illustrate the application of these methods in an empirical case study. Surrogate markers for time-varying treatments and outcomes.

A good surrogate marker is one where the treatment effect on the surrogate is a strong predictor of the effect of treatment on the outcome. We review the situation when there is one treatment delivered at baseline, one surrogate measured at one later time point, and one ultimate outcome of interest and discuss new issues arising when variables are time-varying. METHODS: Most of the literature on surrogate markers has only considered simple settings with one treatment, one surrogate, and one outcome of interest at a fixed time point.

However, more complicated time-varying settings are common in practice. In this article, we describe the unique challenges in two settings, time-varying treatments and time-varying surrogates, while relating the ideas back to the causal-effects and causal-association paradigms. We hope this article has provided some motivation for future work on estimation and inference in such settings. There are only several papers describing methods that integrate evidence from healthcare databases and SRSs. We propose a methodology that combines ADR signals from these two sources.

The evaluation used a reference standard comprising positive and negative drug-ADR pairs. Future Proofing Adverse Event Monitoring. Propensity score interval matching: using bootstrap confidence intervals for accommodating estimation errors of propensity scores. Propensity score matching, a key component of propensity score methods, normally matches units based on the distance between point estimates of the propensity scores.

The problem with this technique is that it is difficult to establish a sensible criterion to evaluate the closeness of matched units without knowing estimation errors of the propensity scores. METHODS: The present study introduces interval matching using bootstrap confidence intervals for accommodating estimation errors of propensity scores.

In interval matching, if the confidence interval of a unit in the treatment group overlaps with that of one or more units in the comparison group, they are considered as matched units. Imputation approaches for potential outcomes in causal inference. Though often not discussed explicitly in the epidemiological literature, the connections between causal inference and missing data can provide additional intuition.

METHODS: We demonstrate how we can approach causal inference in ways similar to how we address all problems of missing data, using multiple imputation and the parametric g-formula. RESULTS: We explain and demonstrate the use of these methods in example data, and discuss implications for more traditional approaches to causal inference. Comparative validity of methods to select appropriate cutoff weight for probabilistic linkage without unique personal identifiers.

Probabilistic linkage, a method that allows partial match of linkage variables, overcomes disagreements arising from errors and omissions in data entry but also results in false-positive links. The study aimed to assess the validity of probabilistic linkage in the absence of unique personal identifiers UPI and the methods of cutoff weight selection. Histogram inspection suggested an approximate range of optimal cutoffs. With adjusted estimates of the sizes of true matches and searched files, the odds formula method made relatively accurate estimates of cutoff and PPV.

Cutoff selection remains challenging; however, histogram inspection, the duplicate method, and the odds formula method can be used in conjunction when a gold standard is not available. Propensity score matching and persistence correction to reduce bias in comparative effectiveness: the effect of cinacalcet use on all-cause mortality. Observational research can provide complementary findings but is prone to bias.

The effect of treatment crossover was investigated with inverse probability of censoring weighted and lag-censored analyses. Adjusting for non-persistence by 0- and 6-month lag-censoring and by inverse probability of censoring weight, the hazard ratios in AROii 0.

Fall 2014 Biostatistics Colloquia

Persistence-corrected analyses revealed a trend towards reduced ACM in haemodialysis patients receiving cinacalcet therapy. Nonexperimental studies of preventive interventions are often biased because of the healthy-user effect and, in frail populations, because of confounding by functional status.

Bias is evident when estimating influenza vaccine effectiveness, even after adjustment for claims-based indicators of illness. We explored bias reduction methods while estimating vaccine effectiveness in a cohort of adult hemodialysis patients. To improve confounding control, we added frailty indicators to the model, measured time-varying confounders at different time intervals, and restricted the sample in multiple ways. Crude and baseline-adjusted marginal structural models remained strongly biased. Restricting to a healthier population removed some unmeasured confounding; however, this reduced the sample size, resulting in wide confidence intervals.

In this study, the healthy-user bias could not be controlled through statistical adjustment; however, sample restriction reduced much of the bias. Selection and measurement of confounders is critical for successful adjustment in nonrandomized studies. Although the principles behind confounder selection are now well established, variable selection for confounder adjustment remains a difficult problem in practice, particularly in secondary analyses of databases. We present a simulation study that compares the high-dimensional propensity score algorithm for variable selection with approaches that utilize direct adjustment for all potential confounders via regularized regression, including ridge regression and lasso regression.

Simulations were based on 2 previously published pharmacoepidemiologic cohorts and used the plasmode simulation framework to create realistic simulated data sets with thousands of potential confounders. Simulation scenarios varied the true underlying outcome model, treatment effect, prevalence of exposure and outcome, and presence of unmeasured confounding.

Across scenarios, high-dimensional propensity score approaches generally performed better than regularized regression approaches. Do case-only designs yield consistent results across design and different databases? A case study of hip fractures and benzodiazepines. In both, relative risk estimates are obtained within persons, implicitly controlling for time-fixed confounding variables.

Exposure to BZD was divided into non-use, current, recent and past use. However, when we considered separately the day pre-exposure period, the IRR for current period was 1. Matching on the disease risk score in comparative effectiveness research of new treatments. PURPOSE: We use simulations and an empirical example to evaluate the performance of disease risk score DRS matching compared with propensity score PS matching when controlling large numbers of covariates in settings involving newly introduced treatments.

METHODS: We simulated a dichotomous treatment, a dichotomous outcome, and baseline covariates that included both continuous and dichotomous random variables. For the empirical example, we evaluated the comparative effectiveness of dabigatran versus warfarin in preventing combined ischemic stroke and all-cause mortality. Comprehensive Health Insights, Humana Inc. Market St. The purpose of this study was to evaluate two drug safety surveillance CDMs from an ecosystem perspective to better understand how differences in CDMs and analytic tools affect usability and interpretation of results.

Tree-structured data usually contain both topological and geometrical information, and are necessarily considered on manifold instead of Euclidean space for appropriate data parameterization and analysis. In this study, we propose a novel tree-structured data parameterization, called Topology-Attribute matrix T-A matrix , so the data clustering task can be conducted on matrix manifold. We incorporate the structure constraints embedded in data into the negative matrix factorization method to determine meta-trees from the T-A matrix, and the signature vector of each single tree can then be extracted by meta-tree decomposition.

Finally, the T-A matrix based clustering TAMBAC framework is evaluated and compared using both simulated data and real retinal images to illustrate its efficiency and accuracy. Thursday, December 4, Kinesin is a molecular motor that, along with dynein, moves cargo such as organelles and vesicles along microtubules through axons.

Over the last twenty years, these motors have been extensively studied through in vitro experiments of single molecular motors using laser traps and fluorescence techniques. However, an open challenge has been to explain in vivo behavior of these systems when incorporating the data from in vitro experiments into straightforward models.

In this talk, I will discuss recent work with experimental collaborator, Will Hancock Penn State , to understand more subtle behavior of a single kinesin than has previously been studied, such as sliding and detachment and how such behavior can contribute to our understanding of in vivo transport. Data from these experiments include time series taken from fluorescence experiments for kinesin. In particular, we will use novel applications of switching time series models to explain the shifts between different modes of transport.

Thursday, November 13, Measurement of physical activity PA in a free-living setting is essential for several purposes: understanding why some people are more active than others, evaluating the effectiveness of interventions designed to increase PA, performing PA surveillance, and quantifying the relationship between PA dose and health outcomes.

One way to estimate PA is to use an accelerometer a small electronic device that records a time stamped record of acceleration and a statistical model that predicts aspects of PA such as energy expenditure, time walking, time sitting, etc. This talk will describe methods to do this. We will present several calibration studies where acceleration is measured concurrently with objective measurements of PA, describe the statistical models used to relate the two sets of measurements, and examine independent evaluations of the methods.

Thursday, October 30, Keith A. Modern high-throughput biological assays let us ask detailed questions about how diseases operate, and promise to let us personalize therapy. We will present several case studies where simple errors may have put patients at risk. This work has been covered in both the scientific and lay press, and has prompted several journals to revisit the types of information that must accompany publications. We discuss steps we take to avoid such errors, and lessons that can be applied to large data sets more broadly.

Thursday, October 16, Scott R. Unnecessary antibiotic AB use is unsafe, wasteful, and leads to emergence of AB resistance. AB stewardship trials are limited by noninferiority NI design complexities e. RADAR utilizes a superiority design framework evaluating if new strategies are better in totality than current strategies. RADAR has 2 steps: 1 creation of an ordinal overall clinical outcome variable incorporating important patient benefits, harms, and quality of life, and 2 construction of a desirability of outcome ranking DOOR where i patients with better clinical outcomes receive higher ranks than patients with worse outcomes, and ii patients with similar clinical outcomes are ranked by AB exposure, with lower exposure achieving higher ranks.

N is based on a superiority test of ranks. RADAR avoids the complexities associated with NI trials resulting in reduced sample size in many cases , alleviates competing risk problems, provides more informative benefit:risk evaluation, and allows for patient-level interpretation. Researchers should considering using endpoints to analyze patients rather than patients to analyze endpoints in clinical trials. Thursday, October 2, Massive open online classes MOOCs have become an international phenomenon where millions of students are accessing free educational materials from top universities.

At the same time, industry predictions suggest a massive deficit in STEM, and particularly statistics, graduates to meet the growing demand for statistics, machine learning, data analysis, data science and big data expertise. Notably, it features a completely open source educational model. This talk discusses MOOCs, development technology, financial models, and the future of statistics education. Thursday, May 15, Most of the statistical literature on dynamic systems deals with a single time series with equally-spaced observations, focuses on its internal linear structure, and has little to say about causal factors and covariates.

A parameter estimation framework for fitting linear differential equations to data promises to extend and strengthen classical time series analysis in several directions. A variety of examples will serve as illustrations. Thursday, April 10, We introduce a sparse and positive definite estimator of the covariance matrix designed for high-dimensional situations in which the variables have a known ordering.

Our estimator is the solution to a convex optimization problem that involves a hierarchical group lasso penalty. We show how it can be efficiently computed, compare it to other methods such as tapering by a fixed matrix, and develop several theoretical results that demonstrate its strong statistical properties. Finally, we show how using convex banding can improve the performance of high-dimensional procedures such as linear and quadratic discriminant analysis.

Thursday, March 27, While computed tomography and other imaging techniques are measured in absolute units with physical meaning, magnetic resonance images are expressed in arbitrary units that are difficult to interpret and differ between study visits and subjects. Much work in the image processing literature has centered on histogram matching and other histogram mapping techniques, but little focus has been on normalizing images to have biologically interpretable units.

We explore this key goal for statistical analysis and the impact of normalization on cross-sectional and longitudinal segmentation of pathology. Thursday, March 6, Douglas E. We propose methods for estimating the average difference in restricted mean survival time attributable to a time-dependent treatment. In the data structure of interest, the time until treatment is received and the pre-treatment death hazard are both heavily influenced by a longitudinal process.

In addition, subjects may experience periods of treatment ineligibility. The pre-treatment death hazard is modeled using inverse weighted partly conditional methods, while the post-treatment hazard is handled through Cox regression. Subject-specific differences in pre- versus post-treatment survival are estimated, then averaged in order to estimate the average treatment effect among the treated. Asymptotic properties of the proposed estimators are derived and evaluated in finite samples through simulation.

The proposed methods are applied to liver failure data obtained from a national organ transplant registry. This is joint work with Qi Gong. Thursday, January 23, We discuss two statistical learning methods to build effective classification models for predicting at risk subjects and diagnosis of a disease. In the first example, we develop methods to predict whether pre-symptomatic individuals are at risk of a disease based on their marker profiles, which offers an opportunity for early intervention well before receiving definitive clinical diagnosis.

For many diseases, the risk of disease varies with some marker of biological importance such as age, and the markers themselves may be age dependent. To identify effective prediction rules using nonparametric decision functions, standard statistical learning approaches treat markers with clear biological importance e. Therefore, these approaches may be inadequate in singling out and preserving the effects from the biologically important variables, especially in the presence of high-dimensional markers. Using age as an example of a salient marker to receive special care in the analysis, we propose a local smoothing large margin classifier to construct effective age-dependent classification rules.

The method adaptively adjusts for age effect and separately tunes age and other markers to achieve optimal performance. We apply the proposed method to a registry of individuals at risk for Huntington's disease HD and controls to construct age-sensitive predictive scores for the risk of receiving HD diagnosis during study period in premanifest individuals. In the second example, we develop methods for building formal diagnostic criteria sets for a new psychiatric disorder introduced in the recently released fifth edition of the Diagnostic and Statistical Manual of Psychiatric Disorders DSM Thursday, November 7, Non-inferiority equivalence trials are clinical experiments which attempt to show that one intervention is not too much inferior to another on some quantitative scale.

The cutoff value is commonly denoted as Delta. Naturally, a lot of attention is given to choice of Delta. In addition to this, I assert that even more than in superiority clinical trials the scale of Delta in equivalence trials must be carefully chosen. Since null hypotheses in superiority studies generally imply no effect, they are often identical or at least compatible when formulated on different scales.

However, nonzero Deltas on one scale usually conflict with those on another. This can lead to problems in interpretation when the clinically natural scale is not a statistically convenient one. Thursday, October 24, These methods draw on social interaction models, information analytics, and web technologies that have already revolutionized research in other areas. The specific examples include applications in medical informatics to find adverse effects of drugs and chemicals, terrorism analysis, and improvement in the efficiency of communicating individually tailored, policy-relevant information directly to policymakers.

On the surface, traditional information theoretic considerations do not offer solutions. Accordingly, researchers looking for conventional solutions would have difficulty in solving these problems. I will describe how alternative formulations based on statistical underpinnings including Bayesian methods, sequential stopping and combinatorial designs have played a critical role in addressing these challenges.

Tuesday, June 4, Brent A. In post-operative medical care, some drugs are administered intravenously through an infusion pump. For example, in infusion studies following heart surgery, anti-coagulants are delivered intravenously for many hours, even days while a patient recovers. A common primary endpoint of infusion studies is to compare two or more infusion drugs or rates and one can employ standard statistical analyses to address the primary endpoint in an intent-to-treat analysis.

However, the presence of infusion-terminating events can adversely affect the analysis of primary endpoints and complicate statistical analyses of secondary endpoints. In this talk, I will focus on a popular secondary analysis of evaluating infusion lengths. The analysis is complicated due to presence or absence of infusion-terminating events and potential time-dependent confounding in treatment assignment.

I will show how the theory of dynamic treatment regimes lends itself to this problem and offers a principled approach to construct adaptive, personalized infusion length policies. I will present some recent achievements that allow one to construct an improved, doubly-robust estimator for a particular class of nonrandom dynamic treatment regimes.

Thursday, April 18, Medical cost data are often skewed to the right and heteroscedastic, having a nonlinear relation with covariates. To tackle these issues, we consider an extension to generalized linear models by assuming nonlinear covariate effects in the mean function and allowing the variance to be an unknown but smooth function of the mean. We make no further assumption on the distributional form. The unknown functions are described by penalized splines, and the estimation is carried out using nonparametric quasi-likelihood.

Simulation studies show the flexibility and advantages of our approach. We apply the model to the annual medical costs of heart failure patients in the clinical data repository CDR at the University of Virginia Hospital System. We also discuss how to adopt this modeling framework in correlated medical costs data. Thursday, March 28, Patients with a non-curable disease such as many types of cancer usually go through the process of initial treatment, a various number of disease recurrences and salvage treatments. Such multistage treatments are inevitably dynamic.

That is, the choice of the next treatment depends on the patient's response to previous therapies. Dynamic treatment regimes DTRs are routinely used in clinics, but are rarely optimized. A systematic optimization of DTRs is highly desirable, but it poses immense challenges for statisticians given their complex nature. Our approach to address this issue is do optimization by backward induction. That is, we first optimize the treatments for the last stage, conditional on patient treatment and response history.

Again, the optimization of treatments at stage k-1 is done under the assumed AFT model. Repeat this process until the optimization for the first stage is completed. By doing that, the effects of different treatments at each stage on survival can be consistently estimated and fairly compared, and the overall optimal DTR for each patient can be identified. Simulation studies show that the proposed method performs well and is useful in practical situations. The proposed method is applied to a study for acute myeloid leukemia, to identify the optimal treatment strategies for different subgroups of patients.

Potential problems, alternative models, and optimization of the estimation methods are also discussed. Thursday, December 13, The complexity of the human genome makes it challenging to identify genetic markers associated with clinical outcomes. This identification is further complicated by the vast number of available markers, the majority of which are unrelated to outcome.

As a consequence, the standard assessment of individual marginal marker effects on a single outcome is often ineffective. It is thus desirable to borrow information and strength from the large amounts of observed data to develop more powerful testing strategies. In this talk, I will discuss testing procedures that capitalize on various forms of correlation observed in genome-wide association studies. This is joint work with Dr. Thursday, November 29, Richard J. Researchers routinely adopt composite endpoints in multicenter randomized trials designed to evaluate the effect of experimental interventions in cardiovascular disease, diabetes, and cancer.

Despite their widespread use, relatively little attention has been paid to the statistical properties of estimators of treatment effect based on composite endpoints. We consider this here in the context of multivariate models for time to event data in which copula functions link marginal distributions with a proportional hazards structure. We then examine the asymptotic and empirical properties of the estimator of treatment effect arising from a Cox regression model for the time to the first event.

We point out that even when the treatment effect is the same for the component events, the limiting value of the estimator based on the composite endpoint is usually inconsistent for this common value. We find that in this context the limiting value is determined by the degree of association between the events, the stochastic ordering of events, and the censoring distribution.

Within the framework adopted, marginal methods for the analysis of multivariate failure time data yield consistent estimators of treatment effect and are therefore preferred. We illustrate the methods by application to a recent asthma study. This is joint work with Longyang Wu. Thursday, October 25, Growth trajectories play a central role in life course epidemiology, often providing fundamental indicators of prenatal or childhood development, as well as an array of potential determinants of adult health outcomes. Statistical methods for the analysis of growth trajectories have been widely studied, but many challenging problems remain.

Repeated measurements of length, weight and head circumference, for example, may be available on most subjects in a study, but usually only sparse temporal sampling of such variables is feasible. It can thus be challenging to gain a detailed understanding of growth velocity patterns, and smoothing techniques are inevitably needed. Moreover, the problem is exacerbated by the presence of large fluctuations in growth velocity during early infancy, and high variability between subjects.

Existing approaches, however, can be inflexible due to a reliance on parametric models, and require computationally intensive methods that are unsuitable for exploratory analyses. This talk introduces a nonparametric Bayesian inversion approach to such problems, along with an R package that implements the proposed method. Thursday, April 19, Outcome dependent sampling ODS schemes can be cost effective ways to enhance study efficiency. The case-control design has been widely used in epidemiologic studies.

However, when the outcome is measured in continuous scale, dichotomizing the outcome could lead to a loss of efficiency. Recent epidemiologic studies have used ODS sampling schemes where, in addition to an overall random sample, there are also a number of supplemental samples that are collected based on a continuous outcome variable. We consider a semiparametric empirical likelihood inference procedure in which the underlying distribution of covariates is treated as a nuisance parameter and left unspecified.

The proposed estimator has asymptotic normality properties. The likelihood ratio statistic using the semiparametric empirical likelihood function has Wilks type properties in that, under the null, it follows a Chi-square distribution asymptotically and is independent of the nuisance parameters. Simulation results indicate that, for data obtained using an ODS design, the proposed estimator is more efficient than competing estimators with the same size.

Thursday, March 8, Dawn Woodard, PhD Assistant Professor Operations Research and Information Engineering Cornell University We propose a new method for regression using a parsimonious and scientifically interpretable representation of functional predictors. Our approach is designed for data that exhibit features such as spikes, dips, and plateaus whose frequency, location, size, and shape varies stochastically across subjects.

Our method is motivated by the goal of quantifying the association between sleep characteristics and health outcomes, using a large and complex dataset from the Sleep Heart Health Study. We propose Bayesian inference of the joint functional and exposure models, and give a method for efficient computation. We contrast our approach with existing state-of-the-art methods for regression with functional predictors, and show that our method is more effective and efficient for data that include features occurring at varying locations.

A major challenge in making inferences from such observational data is that treatment is not randomly assigned; e. This led to a large class of consistent, asymptotically normal estimators, under the assumption that all confounders are measured. However, estimates and standard errors turn out to depend significantly on the choice of estimators within this class, advocating the study of optimal ones. We will present an explicit solution for the choice of optimal estimators under some extra conditions.

In the absence of those extra conditions, the resulting estimator is still consistent and asymptotically normal, although possibly not optimal. This estimator is also doubly robust: it is consistent and asymptotically normal not only if the model for treatment initiation is correct, but also if a certain outcome-regression model is correct. Delaying the initiation of HAART has the advantage of postponing onset of adverse events or drug resistance, but may also lead to irreversible immune system damage. Application of our methods to observational data on treatment initiation will help provide insight into these tradeoffs.

The current interest in using treatment to control epidemic spread heightens interest in these issues, as early treatment can only be ethically justified if it benefits individual patients, regardless of the potential for community-wide benefits. Thursday, January 19, However, most existing approaches assume a normal distribution for the random effects, and this could affect the bias and efficiency of the fixed-effects estimator. Even in cases where the estimation of the fixed effects is robust with a misspecified distribution of the random effects, the estimation of the random effects could be invalid.

We propose a new approach to estimate fixed and random effects using conditional quadratic inference functions. The new approach does not require the specification of likelihood functions or a normality assumption for random effects. It can also accommodate serial correlation between observations within the same cluster, in addition to mixed-effects modeling.

Other advantages include not requiring the estimation of the unknown variance components associated with the random effects, or the nuisance parameters associated with the working correlations. This is joint work with Peng Wang and Guei-feng Tsai. Alicia L.

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Carriquiry Professor of Statistics Iowa State University The United States government spends billions of dollars each year on food assistance programs, on food safety and food labeling efforts and in general on interventions and other activities with the goal of improving the nutritional status of the population.

To do so, the government relies on large, nationwide food consumption and health surveys that are carried out regularly Of interest to policy makers, researchers and practitioners is the usual intake of a nutrient or other food components. The distribution of usual intakes in population sub-groups is also of interest, as is the association between consumption and health outcomes.

Today we focus on the estimation and interpretation of distributions of nutrient intake and on their use for policy decision-making. From a statistical point of view, estimating the distribution of usual intakes of a nutrient or other food components is challenging. Usual intakes are unobservable in practice and are subject to large measurement error, skewness and other survey-related effects. The problem of estimating usual nutrient intake distributions can therefore be thought of as the problem of estimating the density of a non-normal random variable that is observed with error.

We describe what is now considered to be the standard approach for estimation and will spend some time discussing problems in this area that remain to be addressed. Data from established surveillance networks across the country have provided timely information for intervention, control and prevention. In this talk, I will focus on the study population of drug injection users in Sichuan province over years , and the evaluation of HIV prevalence across regions in this province.

Both simulation studies and real data analysis will be presented. It has been successfully applied to separate source signals of interest from their mixtures. Most existing ICA procedures are carried out by relying solely on the estimation of the marginal density functions. In many applications, correlation structures within each source also play an important role besides the marginal distributions. One important example is functional magnetic resonance imaging fMRI analysis where the brain-function-related signals are temporally correlated.

I shall talk about a novel ICA approach that fully exploits the correlation structures within the source signals. Specifically, we propose to estimate the spectral density functions of the source signals instead of their marginal density functions. Our methodology is described and implemented using spectral density functions from frequently used time series models such as ARMA processes.

The time series parameters and the mixing matrix are estimated via maximizing the Whittle likelihood function. The performance of the proposed method will be illustrated through extensive simulation studies and a real fMRI application. The numerical results indicate that our approach outperforms several popular methods including the most widely used fastICA algorithm. Thursday, May 19, Mary D. Sammel, ScD Department of Biostatistics and Epidemiology University of Pennsylvania School of Medicine In many clinical studies, the disease of interest is multi-faceted and multiple outcomes are needed to adequately characterize the disease or its severity.

In such studies, it is often difficult to determine what constitutes improvement due to the multivariate nature of the response. Identification of population subgroups is of interest as it may enable clinicians to provide targeted treatments or develop accurate prognoses. We propose a multivariate growth curve latent class model that group subjects based on multiple outcomes measured repeatedly over time.

These groups or latent classes are characterized by distinctive longitudinal profiles of a latent variable which is used to summarize the multivariate outcomes at each point in time. The mean growth curve for the latent variable in each class defines the features of the class. We develop this model for any combination of continuous, binary, ordinal or count outcomes within a Bayesian hierarchical framework. Simulation studies are used to validate the estimation procedures. We apply our models to data from a randomized clinical trial evaluating the efficacy of Bacillus Calmette-Guerin in treating symptoms of IC where we are able to identify a class of subjects who were not responsive to treatment, and a class of subjects where treatment was effective in reducing symptoms over time.

Thursday, May 12, Gong Tang, PhD Assistant Professor of Biostatistics University of Pittsburgh Graduate School of Public Health Consider regression analysis on data with nonignorable nonresponse, standard methods require modeling the nonresponse mechanism. Tang, Little and Raghunathan proposed a pseudolikelihood method for analysis of data with a class of nonignorable nonresponse mechanisms without modeling the nonresponse mechanism, and extended it to multivariate monotone data with nonignorable nonresponse.

In the multivariate case, the joint distribution of response variables was factored into the product of conditional distributions and the pseudolikelihood estimates of the conditional distribution parameters were shown asymptotically normal. However, these estimates were based on different subsets of the data, which were dictated by the missing-data pattern, and their joint distribution was unclear. Here we provide a modification of the likelihood functions and derive the asymptotic joint distributions of these estimates.

We also consider an imputation approach for this pseudolikelihood method.

Measurement Error

Usual imputation approaches impute the missing values and summarize via multiple imputations. Without knowing or modeling the nonresponse mechanism in our setting, the missing values cannot be predicted. We propose a novel approach via imputing the necessary sufficient statistics to circumvent this barrier. Thursday, March 10, This then permits the approximation to be put into a form analogous to those given either by Lugananni-Rice or Barndorff-Nielsen. The bubbles are the observed log odds ratios for each study, with bubble sizes proportional to the study weights. The horizontal dashed line represents the null association scenario.

The design, implementation and interpretation of a meta-regression model is a complex process prone to errors and misunderstandings. Some of the most frequent ones are highlighted below:. In spite of the apparent fanciness of the meta-regression methods, there are numerous limitations that can impair the ability of the model to make valid inferences. In first place, sample size is often insufficient to perform a meta-regression. Some estimation methods are based on asymptotical assumptions and can easily be biased when the sample size is small. In addition, published papers may not always measure or appropriately report the information on covariates needed for the model.

These two precedent points may result in the inability to properly adjust for confounding. Even if the information on confounders is present and the number of studies is moderately large, characteristics of the studies tend to be correlated, giving rise to problems of collinearity [9].

Moreover, meta-regression analysis will always be subject to the risk of ecological fallacy, since they attempt to make inferences about individuals using study-level information. Also, it is sometimes difficult to interpret the effect of aggregate variables without the ability to condition on the individual-level analogous measurements e. It has also been noticed that non-differential measurement error at the individual level may be able to bias group-level effects away from the null.

Finally, literature reviews are always susceptible to publication bias, and in particular, quantitative methods are subject to the risk of data dredging and false positive findings. In first place, it is important to realize that meta-regression is not always necessary. Sometimes, a meta-analysis may be sufficient to summarize the published information.

Meta regression may be more useful when there is substantial heterogeneity even if not statistically significant. A rough guide for the interpretation of the amount of heterogeneity is shown below [1]:. Meta-regression should be planned and incorporated in the literature review protocol when it is interesting to explore study-level sources of heterogeneity, especially if there are variables suspected or known to modify the effect of the explored risk factors. Meta-regression may be useful when there is a wide range of values in a continuous moderator variable, but relatively few studies with the exact same value for that moderator.

By contrast, meta-regression may not be feasible when there is too little variability in the observed values of the moderators of interest. Meta-regression serves to appropriately combine and contrast multiple subsets of studies e. Meta-regression can also serve to implement network meta-analysis. In this case, the model will use information about a parameter calculated in individual study groups defined by levels of exposure s.

For instance, instead of using the study specific log OR as an outcome, it is possible to model the log odds in each contrasted group placebo, exposure1, exposure2, etc. Then, the exposure specification can be used in meta-regression as a moderator of the log odds to obtain a log OR. Meta-regression is a powerful tool for exploratory analyses of heterogeneity and for hypothesis generation about cross-level interactions.

Occasionally, it may help to test hypothesis of such effect measure modifications as well as to inform health decision making, as long as the interaction hypothesis are stated a priori and substantiated in scientific theory. Meta-regression may also help to better understand the sources of variability in meta-analysis and to adequately summarize published information in a richer manner than a single number or point estimate does.

Furthermore, it is easy to relate meta-regression to the increasingly popular multilevel modeling methods.

The popularity of the latter may help to make meta-regression results easier to interpret and communicate. However, it needs to be performed with extreme caution, because it is prone to error, poor methodological implementation, and misinterpretations. Investigating heterogeneity. In Cochrane Handbook. Version 5. Cochrane Collab. Provides a clear definition of statistical heterogeneity and its sources in lay terms. Gives an interpretation of the confidence intervals for the pooled estimates from random effects vs. Gives a very brief conceptual summary of what meta-regression is, why random effects meta-regression is preferred, why the number of available studies and the number of moderators is relevant, and why study-level moderators should be specified a priori.

This book provides an extraordinarily clear and intuitive definition and interpretation of statistical heterogeneity, variance components and sources of variability in meta-analysis and the differences and paradoxes of random effects vs. It is a great starting point to initiate an exploration of the topic. Chapter Provides a definition of meta-regression highlighting its analogy with single level regression. It includes a worked example on meta-regression for a BCG vaccine. Presents some useful graphs such as the bubble plot. Clearly states the differences in the hypothesis being tested in random effects vs.

Meta-Analysis with Appl. It states that meta-regression is more useful on the presence of substantial heterogeneity. Provides statistical models for meta-regression in a language that is akin to multilevel models. Provides to estimate the parameters theta, beta, and variance-covariance matrices in random effects meta-regression. It also provides formulas to derive confidence intervals for those parameters. It states the statistical model in matrix notation. Describes some applications of metaregression: explaining heterogeneity, appropriately combining subsets of studies, combining controlled and uncontrolled trials.

Chapter 7. Meta-Analysis with R, , p. Describes meta-regression as an extension of regular weighted multiple regression, describes fixed effects MR as more powerful, but less reliable if between-study variation is significant. Describes statistical model for level 2 variables.

Explicitly states analogy with mixed models. Presents extended examples using R. Lists several methods available to estimate between variance. Compares meta-regression vs. Compares advantages and disadvantages of different options to explore heterogeneity. Provides a slightly different statistical parameterization of the model using a Bayesian perspective on Meta-Regression.

Presents applications to network meta-analysis and regression on baseline risk. Clearly states assumption of equal effect of moderator across groups. Highlights the importance of centering predictors. Discusses appropriateness of tests and information criteria in Bayesian context. Describes the appropriateness of one and two-step approaches to individual patient data meta-analysis. Presents statistical model relating it to multilevel models and presents a conditional notation for the different types of integrative methods fixed effects and random effects meta-analysis, meta-regression.

Control for Confounding in the Presence of Measurment Error in Hierarchical Models Control for Confounding in the Presence of Measurment Error in Hierarchical Models
Control for Confounding in the Presence of Measurment Error in Hierarchical Models Control for Confounding in the Presence of Measurment Error in Hierarchical Models
Control for Confounding in the Presence of Measurment Error in Hierarchical Models Control for Confounding in the Presence of Measurment Error in Hierarchical Models
Control for Confounding in the Presence of Measurment Error in Hierarchical Models Control for Confounding in the Presence of Measurment Error in Hierarchical Models
Control for Confounding in the Presence of Measurment Error in Hierarchical Models Control for Confounding in the Presence of Measurment Error in Hierarchical Models

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