B. Goldstein, E. Polley, F. Briggs, M. J. van der Laan, A. Hubbard
Abstract Comparing the relative fit of competing models can be used to address many different scientific questions. In classical statistics one can, if appropriate, use likelihood ratio tests and information based criterion, whereas clinical medicine has tended to rely on comparisons of fit metrics like C-statistics. However, for many data adaptive modelling procedures such approaches are not suitable. In these cases, statisticians have used cross-validation, which can make inference challenging. In this paper we propose a general approach that focuses on the “conditional” risk difference (conditional on the model fits being fixed) for the improvement in prediction risk. Specifically, we derive a Wald-type test statistic and associated confidence intervals for cross-validated test sets utilizing the independent validation within cross-validation in conjunction with a test for multiple comparisons. We show that this test maintains proper Type I Error under the null fit, and can be used as a general test of relative fit for any semi-parametric model alternative. We apply the test to a candidate gene study to test for the association of a set of genes in a genetic pathway.
{"title":"Testing the Relative Performance of Data Adaptive Prediction Algorithms: A Generalized Test of Conditional Risk Differences","authors":"B. Goldstein, E. Polley, F. Briggs, M. J. van der Laan, A. Hubbard","doi":"10.1515/ijb-2015-0014","DOIUrl":"https://doi.org/10.1515/ijb-2015-0014","url":null,"abstract":"Abstract Comparing the relative fit of competing models can be used to address many different scientific questions. In classical statistics one can, if appropriate, use likelihood ratio tests and information based criterion, whereas clinical medicine has tended to rely on comparisons of fit metrics like C-statistics. However, for many data adaptive modelling procedures such approaches are not suitable. In these cases, statisticians have used cross-validation, which can make inference challenging. In this paper we propose a general approach that focuses on the “conditional” risk difference (conditional on the model fits being fixed) for the improvement in prediction risk. Specifically, we derive a Wald-type test statistic and associated confidence intervals for cross-validated test sets utilizing the independent validation within cross-validation in conjunction with a test for multiple comparisons. We show that this test maintains proper Type I Error under the null fit, and can be used as a general test of relative fit for any semi-parametric model alternative. We apply the test to a candidate gene study to test for the association of a set of genes in a genetic pathway.","PeriodicalId":50333,"journal":{"name":"International Journal of Biostatistics","volume":"12 1","pages":"117 - 129"},"PeriodicalIF":1.2,"publicationDate":"2016-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/ijb-2015-0014","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66987670","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Often the biological and/or clinical experiments result in longitudinal data from multiple related groups. The analysis of such data is quite challenging due to the fact that groups might have shared information on the mean and/or covariance functions. In this article, we consider a Bayesian semiparametric approach of modeling the mean trajectories for longitudinal response coming from multiple related groups. We consider matrix stick-breaking process priors on the group mean parameters which allows information sharing on the mean trajectories across the groups. Simulation studies are performed to demonstrate the effectiveness of the proposed approach compared to the more traditional approaches. We analyze data from a one-year follow-up of nutrition education for hypercholesterolemic children with three different treatments where the children are from different age-groups. Our analysis provides more clinically useful information than the previous analysis of the same dataset. The proposed approach will be a very powerful tool for analyzing data from clinical trials and other medical experiments.
{"title":"A Semiparametric Bayesian Approach for Analyzing Longitudinal Data from Multiple Related Groups","authors":"Kiranmoy Das, Prince Afriyie, Lauren Spirko","doi":"10.1515/ijb-2015-0002","DOIUrl":"https://doi.org/10.1515/ijb-2015-0002","url":null,"abstract":"Abstract Often the biological and/or clinical experiments result in longitudinal data from multiple related groups. The analysis of such data is quite challenging due to the fact that groups might have shared information on the mean and/or covariance functions. In this article, we consider a Bayesian semiparametric approach of modeling the mean trajectories for longitudinal response coming from multiple related groups. We consider matrix stick-breaking process priors on the group mean parameters which allows information sharing on the mean trajectories across the groups. Simulation studies are performed to demonstrate the effectiveness of the proposed approach compared to the more traditional approaches. We analyze data from a one-year follow-up of nutrition education for hypercholesterolemic children with three different treatments where the children are from different age-groups. Our analysis provides more clinically useful information than the previous analysis of the same dataset. The proposed approach will be a very powerful tool for analyzing data from clinical trials and other medical experiments.","PeriodicalId":50333,"journal":{"name":"International Journal of Biostatistics","volume":"30 1","pages":"273 - 284"},"PeriodicalIF":1.2,"publicationDate":"2015-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/ijb-2015-0002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66987590","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We construct an optimal design to simultaneously estimate three common interesting features in a dose-finding trial with possibly different emphasis on each feature. These features are (1) the shape of the dose-response curve, (2) the median effective dose and (3) the minimum effective dose level. A main difficulty of this task is that an optimal design for a single objective may not perform well for other objectives. There are optimal designs for dual objectives in the literature but we were unable to find optimal designs for 3 or more objectives to date with a concrete application. A reason for this is that the approach for finding a dual-objective optimal design does not work well for a 3 or more multiple-objective design problem. We propose a method for finding multiple-objective optimal designs that estimate the three features with user-specified higher efficiencies for the more important objectives. We use the flexible 4-parameter logistic model to illustrate the methodology but our approach is applicable to find multiple-objective optimal designs for other types of objectives and models. We also investigate robustness properties of multiple-objective optimal designs to mis-specification in the nominal parameter values and to a variation in the optimality criterion. We also provide computer code for generating tailor made multiple-objective optimal designs.
{"title":"Multiple-Objective Optimal Designs for Studying the Dose Response Function and Interesting Dose Levels","authors":"Seung Won Hyun, W. Wong","doi":"10.1515/ijb-2015-0044","DOIUrl":"https://doi.org/10.1515/ijb-2015-0044","url":null,"abstract":"Abstract We construct an optimal design to simultaneously estimate three common interesting features in a dose-finding trial with possibly different emphasis on each feature. These features are (1) the shape of the dose-response curve, (2) the median effective dose and (3) the minimum effective dose level. A main difficulty of this task is that an optimal design for a single objective may not perform well for other objectives. There are optimal designs for dual objectives in the literature but we were unable to find optimal designs for 3 or more objectives to date with a concrete application. A reason for this is that the approach for finding a dual-objective optimal design does not work well for a 3 or more multiple-objective design problem. We propose a method for finding multiple-objective optimal designs that estimate the three features with user-specified higher efficiencies for the more important objectives. We use the flexible 4-parameter logistic model to illustrate the methodology but our approach is applicable to find multiple-objective optimal designs for other types of objectives and models. We also investigate robustness properties of multiple-objective optimal designs to mis-specification in the nominal parameter values and to a variation in the optimality criterion. We also provide computer code for generating tailor made multiple-objective optimal designs.","PeriodicalId":50333,"journal":{"name":"International Journal of Biostatistics","volume":"11 1","pages":"253 - 271"},"PeriodicalIF":1.2,"publicationDate":"2015-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/ijb-2015-0044","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66988220","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We study the problem of multiple hypothesis testing for correlated clustered data. As the existing multiple comparison procedures based on maximum likelihood estimation could be computationally intensive, we propose to construct multiple comparison procedures based on composite likelihood method. The new test statistics account for the correlation structure within the clusters and are computationally convenient to compute. Simulation studies show that the composite likelihood based procedures maintain good control of the familywise type I error rate in the presence of intra-cluster correlation, whereas ignoring the correlation leads to erratic performance.
{"title":"Multiple Comparisons Using Composite Likelihood in Clustered Data","authors":"M. Azadbakhsh, Xin Gao, H. Jankowski","doi":"10.1515/ijb-2016-0004","DOIUrl":"https://doi.org/10.1515/ijb-2016-0004","url":null,"abstract":"Abstract We study the problem of multiple hypothesis testing for correlated clustered data. As the existing multiple comparison procedures based on maximum likelihood estimation could be computationally intensive, we propose to construct multiple comparison procedures based on composite likelihood method. The new test statistics account for the correlation structure within the clusters and are computationally convenient to compute. Simulation studies show that the composite likelihood based procedures maintain good control of the familywise type I error rate in the presence of intra-cluster correlation, whereas ignoring the correlation leads to erratic performance.","PeriodicalId":50333,"journal":{"name":"International Journal of Biostatistics","volume":"12 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2014-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/ijb-2016-0004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66988085","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Motivated by modeling and analysis of mass-spectrometry data, a semi- and nonparametric model is proposed that consists of linear parametric components for individual location and scale and a nonparametric regression function for the common shape. A multi-step approach is developed that simultaneously estimates the parametric components and the nonparametric function. Under certain regularity conditions, it is shown that the resulting estimators is consistent and asymptotic normal for the parametric part and achieve the optimal rate of convergence for the nonparametric part when the bandwidth is suitably chosen. Simulation results are presented to demonstrate the effectiveness and finite-sample performance of the method. The method is also applied to a SELDI-TOF mass spectrometry data set from a study of liver cancer patients.
{"title":"Functional and Parametric Estimation in a Semi- and Nonparametric Model with Application to Mass-Spectrometry Data","authors":"Weiping Ma, Yang Feng, Kani Chen, Z. Ying","doi":"10.1515/ijb-2014-0066","DOIUrl":"https://doi.org/10.1515/ijb-2014-0066","url":null,"abstract":"Abstract Motivated by modeling and analysis of mass-spectrometry data, a semi- and nonparametric model is proposed that consists of linear parametric components for individual location and scale and a nonparametric regression function for the common shape. A multi-step approach is developed that simultaneously estimates the parametric components and the nonparametric function. Under certain regularity conditions, it is shown that the resulting estimators is consistent and asymptotic normal for the parametric part and achieve the optimal rate of convergence for the nonparametric part when the bandwidth is suitably chosen. Simulation results are presented to demonstrate the effectiveness and finite-sample performance of the method. The method is also applied to a SELDI-TOF mass spectrometry data set from a study of liver cancer patients.","PeriodicalId":50333,"journal":{"name":"International Journal of Biostatistics","volume":"11 1","pages":"285 - 303"},"PeriodicalIF":1.2,"publicationDate":"2013-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/ijb-2014-0066","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66987582","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Relative risks have become a popular measure of treatment effect for binary outcomes in randomized controlled trials (RCTs). Relative risks can be estimated directly using log binomial regression but the model may fail to converge. Alternative methods are available for estimating relative risks but these have generally only been evaluated for independent data. As some of these methods are now being applied in cluster RCTs, investigation of their performance in this context is needed. We compare log binomial regression and three alternative methods (expanded logistic regression, log Poisson regression and log normal regression) for estimating relative risks in cluster RCTs. Clustering is taken into account using generalized estimating equations (GEEs) with an independence or exchangeable working correlation structure. The results of our large simulation study show that the log binomial GEE generally performs well for clustered data but suffers from convergence problems, as expected. Both the log Poisson GEE and log normal GEE have advantages in certain settings in terms of type I error, bias and coverage. The expanded logistic GEE can perform poorly and is sensitive to the chosen working correlation structure. Conclusions about the effectiveness of treatment often differ depending on the method used, highlighting the need to pre-specify an analysis approach. We recommend pre-specifying that either the log Poisson GEE or log normal GEE will be used in the event that the log binomial GEE fails to converge.
{"title":"Relative Risk Estimation in Cluster Randomized Trials: A Comparison of Generalized Estimating Equation Methods","authors":"L. Yelland, A. Salter, Philip Ryan","doi":"10.2202/1557-4679.1323","DOIUrl":"https://doi.org/10.2202/1557-4679.1323","url":null,"abstract":"Relative risks have become a popular measure of treatment effect for binary outcomes in randomized controlled trials (RCTs). Relative risks can be estimated directly using log binomial regression but the model may fail to converge. Alternative methods are available for estimating relative risks but these have generally only been evaluated for independent data. As some of these methods are now being applied in cluster RCTs, investigation of their performance in this context is needed. We compare log binomial regression and three alternative methods (expanded logistic regression, log Poisson regression and log normal regression) for estimating relative risks in cluster RCTs. Clustering is taken into account using generalized estimating equations (GEEs) with an independence or exchangeable working correlation structure. The results of our large simulation study show that the log binomial GEE generally performs well for clustered data but suffers from convergence problems, as expected. Both the log Poisson GEE and log normal GEE have advantages in certain settings in terms of type I error, bias and coverage. The expanded logistic GEE can perform poorly and is sensitive to the chosen working correlation structure. Conclusions about the effectiveness of treatment often differ depending on the method used, highlighting the need to pre-specify an analysis approach. We recommend pre-specifying that either the log Poisson GEE or log normal GEE will be used in the event that the log binomial GEE fails to converge.","PeriodicalId":50333,"journal":{"name":"International Journal of Biostatistics","volume":"7 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2011-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2202/1557-4679.1323","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68718384","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ruimin Xu, P. McNicholas, A. Desmond, G. Darlington
We investigate a competing risks model, using the specification of the Gompertz distribution for failure times from competing causes and the inverse Gaussian distribution for failure times from the cause of interest. The expectation-maximization algorithm is used for parameter estimation and the model is applied to real data on breast cancer and melanoma. In these applications, our models compare favourably with existing techniques. The proposed method provides a useful technique that may be more broadly applicable than existing alternatives.
{"title":"A First Passage Time Model for Long-Term Survivors with Competing Risks","authors":"Ruimin Xu, P. McNicholas, A. Desmond, G. Darlington","doi":"10.2202/1557-4679.1224","DOIUrl":"https://doi.org/10.2202/1557-4679.1224","url":null,"abstract":"We investigate a competing risks model, using the specification of the Gompertz distribution for failure times from competing causes and the inverse Gaussian distribution for failure times from the cause of interest. The expectation-maximization algorithm is used for parameter estimation and the model is applied to real data on breast cancer and melanoma. In these applications, our models compare favourably with existing techniques. The proposed method provides a useful technique that may be more broadly applicable than existing alternatives.","PeriodicalId":50333,"journal":{"name":"International Journal of Biostatistics","volume":"7 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2011-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2202/1557-4679.1224","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68717157","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work considers the estimation of the size N of a closed population using incomplete lists of its members. Capture histories are constructed by establishing the presence or the absence of each individual in all the lists available. Models for data featuring a heterogeneous catchability and list dependencies are considered. A log-linear model leading to a lower bound for the population size is derived for a known set of list dependencies and a latent catchability variable with an arbitrary distribution. This generalizes Chao’s lower bound to models with interactions. The proposed model can be used to carry out a search for important list interactions. It also provides diagnostic information about the nature of the underlying heterogeneity. Indeed, it is shown that the Poisson maximum likelihood estimator of N under a dichotomous latent class model does not exist for a particular set of LB models. Several distributions for the heterogeneous catchability are considered; they allow to investigate the sensitivity of the population size estimate to the model for the heterogeneous catchability.
{"title":"A Lower Bound Model for Multiple Record Systems Estimation with Heterogeneous Catchability","authors":"L. Rivest","doi":"10.2202/1557-4679.1283","DOIUrl":"https://doi.org/10.2202/1557-4679.1283","url":null,"abstract":"This work considers the estimation of the size N of a closed population using incomplete lists of its members. Capture histories are constructed by establishing the presence or the absence of each individual in all the lists available. Models for data featuring a heterogeneous catchability and list dependencies are considered. A log-linear model leading to a lower bound for the population size is derived for a known set of list dependencies and a latent catchability variable with an arbitrary distribution. This generalizes Chao’s lower bound to models with interactions. The proposed model can be used to carry out a search for important list interactions. It also provides diagnostic information about the nature of the underlying heterogeneity. Indeed, it is shown that the Poisson maximum likelihood estimator of N under a dichotomous latent class model does not exist for a particular set of LB models. Several distributions for the heterogeneous catchability are considered; they allow to investigate the sensitivity of the population size estimate to the model for the heterogeneous catchability.","PeriodicalId":50333,"journal":{"name":"International Journal of Biostatistics","volume":"41 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2011-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2202/1557-4679.1283","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68717594","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This Reader’s Reaction contains some brief remarks regarding Pearl’s concerns regarding the value of principal stratification.
这个读者的反应包含了一些关于珀尔对主要分层的价值的关注的简短评论。
{"title":"Reaction to Pearl's Critique of Principal Stratification","authors":"Arvid Sjolander","doi":"10.2202/1557-4679.1324","DOIUrl":"https://doi.org/10.2202/1557-4679.1324","url":null,"abstract":"This Reader’s Reaction contains some brief remarks regarding Pearl’s concerns regarding the value of principal stratification.","PeriodicalId":50333,"journal":{"name":"International Journal of Biostatistics","volume":"7 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2011-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2202/1557-4679.1324","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68718456","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Variable selection for high dimensional data has recently received a great deal of attention. However, due to the complex structure of the likelihood, only limited developments have been made for time-to-event data where censoring is present. In this paper, we propose a Bayesian variable selection scheme for a Bayesian semiparametric survival model for right censored survival data sets. A special shrinkage prior on the coefficients corresponding to the predictor variables is used to handle cases when the explanatory variables are of very high-dimension. The shrinkage prior is obtained through a scale mixture representation of Normal and Gamma distributions. Our proposed variable selection prior corresponds to the well known lasso penalty. The likelihood function is based on the Cox proportional hazards model framework, where the cumulative baseline hazard function is modeled a priori by a gamma process. We assign a prior on the tuning parameter of the shrinkage prior and adaptively control the sparsity of our model. The primary use of the proposed model is to identify the important covariates relating to the survival curves. To implement our methodology, we have developed a fast Markov chain Monte Carlo algorithm with an adaptive jumping rule. We have successfully applied our method on simulated data sets under two different settings and real microarray data sets which contain right censored survival time. The performance of our Bayesian variable selection model compared with other competing methods is also provided to demonstrate the superiority of our method. A short description of the biological relevance of the selected genes in the real data sets is provided, further strengthening our claims.
{"title":"Bayesian Variable Selection in Semiparametric Proportional Hazards Model for High Dimensional Survival Data","authors":"Kyu Ha Lee, S. Chakraborty, Jianguo Sun","doi":"10.2202/1557-4679.1301","DOIUrl":"https://doi.org/10.2202/1557-4679.1301","url":null,"abstract":"Variable selection for high dimensional data has recently received a great deal of attention. However, due to the complex structure of the likelihood, only limited developments have been made for time-to-event data where censoring is present. In this paper, we propose a Bayesian variable selection scheme for a Bayesian semiparametric survival model for right censored survival data sets. A special shrinkage prior on the coefficients corresponding to the predictor variables is used to handle cases when the explanatory variables are of very high-dimension. The shrinkage prior is obtained through a scale mixture representation of Normal and Gamma distributions. Our proposed variable selection prior corresponds to the well known lasso penalty. The likelihood function is based on the Cox proportional hazards model framework, where the cumulative baseline hazard function is modeled a priori by a gamma process. We assign a prior on the tuning parameter of the shrinkage prior and adaptively control the sparsity of our model. The primary use of the proposed model is to identify the important covariates relating to the survival curves. To implement our methodology, we have developed a fast Markov chain Monte Carlo algorithm with an adaptive jumping rule. We have successfully applied our method on simulated data sets under two different settings and real microarray data sets which contain right censored survival time. The performance of our Bayesian variable selection model compared with other competing methods is also provided to demonstrate the superiority of our method. A short description of the biological relevance of the selected genes in the real data sets is provided, further strengthening our claims.","PeriodicalId":50333,"journal":{"name":"International Journal of Biostatistics","volume":"7 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2011-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2202/1557-4679.1301","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68717639","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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