This paper reviews two main types of prediction interval methods under a parametric framework. First, we describe methods based on an (approximate) pivotal quantity. Examples include the plug-in, pivotal, and calibration methods. Then we describe methods based on a predictive distribution (sometimes derived based on the likelihood). Examples include Bayesian, fiducial, and direct-bootstrap methods. Several examples involving continuous distributions along with simulation studies to evaluate coverage probability properties are provided. We provide specific connections among different prediction interval methods for the (log-)location-scale family of distributions. This paper also discusses general prediction interval methods for discrete data, using the binomial and Poisson distributions as examples. We also overview methods for dependent data, with application to time series, spatial data, and Markov random fields, for example.
{"title":"Methods to Compute Prediction Intervals: A Review and New Results","authors":"Qinglong Tian, D. Nordman, W. Meeker","doi":"10.1214/21-sts842","DOIUrl":"https://doi.org/10.1214/21-sts842","url":null,"abstract":"This paper reviews two main types of prediction interval methods under a parametric framework. First, we describe methods based on an (approximate) pivotal quantity. Examples include the plug-in, pivotal, and calibration methods. Then we describe methods based on a predictive distribution (sometimes derived based on the likelihood). Examples include Bayesian, fiducial, and direct-bootstrap methods. Several examples involving continuous distributions along with simulation studies to evaluate coverage probability properties are provided. We provide specific connections among different prediction interval methods for the (log-)location-scale family of distributions. This paper also discusses general prediction interval methods for discrete data, using the binomial and Poisson distributions as examples. We also overview methods for dependent data, with application to time series, spatial data, and Markov random fields, for example.","PeriodicalId":51172,"journal":{"name":"Statistical Science","volume":" ","pages":""},"PeriodicalIF":5.7,"publicationDate":"2020-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46757725","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A Look at Robustness and Stability of $ell_{1}$-versus $ell_{0}$-Regularization: Discussion of Papers by Bertsimas et al. and Hastie et al.","authors":"Yuansi Chen, Armeen Taeb, P. Bühlmann","doi":"10.1214/20-sts809","DOIUrl":"https://doi.org/10.1214/20-sts809","url":null,"abstract":"","PeriodicalId":51172,"journal":{"name":"Statistical Science","volume":"35 1","pages":"614-622"},"PeriodicalIF":5.7,"publicationDate":"2020-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45899156","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Rejoinder: Best Subset, Forward Stepwise or Lasso? Analysis and Recommendations Based on Extensive Comparisons","authors":"T. Hastie, R. Tibshirani, R. Tibshirani","doi":"10.1214/20-sts733rej","DOIUrl":"https://doi.org/10.1214/20-sts733rej","url":null,"abstract":"","PeriodicalId":51172,"journal":{"name":"Statistical Science","volume":"35 1","pages":"579-592"},"PeriodicalIF":5.7,"publicationDate":"2020-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44684797","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let me begin by congratulating the authors of these two papers, hereafter HTT and BPV, for their superb contributions to the comparisons of methods for variable selection problems in high dimensional regression. The methods considered are truly some of today’s leading contenders for coping with the size and complexity of big data problems of so much current importance. Not surprisingly, there is no clear winner here because the terrain of comparisons is so vast and complex, and no single method can dominate across all situations. The considered setups vary greatly in terms of the number of observations n, the number of predictors p, the number and relative sizes of the underlying nonzero regression coefficients, predictor correlation structures and signal-to-noise ratios (SNRs). And even these only scratch the surface of the infinite possibilities. Further, there is the additional issue as to which performance measure is most important. Is the goal of an analysis exact variable selection or prediction or both? And what about computational speed and scalability? All these considerations would naturally depend on the practical application at hand. The methods compared by HTT and BPV have been unleashed by extraordinary developments in computational speed, and so it is tempting to distinguish them primarily by their novel implementation algorithms. In particular, the recent integer optimization related algorithms for variable selection differ in fundamental ways from the now widely adopted coordinate ascent algorithms for the lasso related methods. Undoubtedly, the impressive improvements in computational speed unleashed by these algorithms are critical for the feasibility of practical applications. However, the more fundamental story behind the performance differences has to do with the differences between the criteria that their algorithms are seeking to optimize. In an important sense, they are being guided by different solutions to the general variable selection problem. Focusing first on the paper of HTT, its main thrust appears to have been kindled by the computational breakthrough of Bertsimas, King and Mazumder (2016) (hereafter BKM), which had proposed a mixed integer opti-
{"title":"Modern Variable Selection in Action: Comment on the Papers by HTT and BPV","authors":"E. George","doi":"10.1214/20-sts808","DOIUrl":"https://doi.org/10.1214/20-sts808","url":null,"abstract":"Let me begin by congratulating the authors of these two papers, hereafter HTT and BPV, for their superb contributions to the comparisons of methods for variable selection problems in high dimensional regression. The methods considered are truly some of today’s leading contenders for coping with the size and complexity of big data problems of so much current importance. Not surprisingly, there is no clear winner here because the terrain of comparisons is so vast and complex, and no single method can dominate across all situations. The considered setups vary greatly in terms of the number of observations n, the number of predictors p, the number and relative sizes of the underlying nonzero regression coefficients, predictor correlation structures and signal-to-noise ratios (SNRs). And even these only scratch the surface of the infinite possibilities. Further, there is the additional issue as to which performance measure is most important. Is the goal of an analysis exact variable selection or prediction or both? And what about computational speed and scalability? All these considerations would naturally depend on the practical application at hand. The methods compared by HTT and BPV have been unleashed by extraordinary developments in computational speed, and so it is tempting to distinguish them primarily by their novel implementation algorithms. In particular, the recent integer optimization related algorithms for variable selection differ in fundamental ways from the now widely adopted coordinate ascent algorithms for the lasso related methods. Undoubtedly, the impressive improvements in computational speed unleashed by these algorithms are critical for the feasibility of practical applications. However, the more fundamental story behind the performance differences has to do with the differences between the criteria that their algorithms are seeking to optimize. In an important sense, they are being guided by different solutions to the general variable selection problem. Focusing first on the paper of HTT, its main thrust appears to have been kindled by the computational breakthrough of Bertsimas, King and Mazumder (2016) (hereafter BKM), which had proposed a mixed integer opti-","PeriodicalId":51172,"journal":{"name":"Statistical Science","volume":"35 1","pages":"609-613"},"PeriodicalIF":5.7,"publicationDate":"2020-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45250262","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
I warmly congratulate the authors Hastie, Tibshirani and Tibshirani (HTT); and Bertsimas, Pauphilet and Van Parys (BPV) for their excellent contributions and important perspectives on sparse regression. Due to space constraints, and my greater familiarity with the content and context of HTT (I have had numerous fruitful discussions with the authors regarding their work), I will focus my discussion on the HTT paper. HTT nicely articulate the relative merits of three canonical estimators in sparse regression: L0, L1 and (forward)stepwise selection. I am humbled that a premise of their work is an article I wrote with Bertsimas and King [4] (BKM). BKM showed that current Mixed Integer Optimization (MIO) algorithms allow us to compute best subsets solutions for problem instances (p ≈ 1000 features) much larger than a previous benchmark (software for best subsets in the R package leaps) that could only handle instances with p ≈ 30. HTT by extending and refining the experiments performed by BKM, have helped clarify and deepen our understanding of L0, L1 and stepwise regression. They raise several intriguing questions that perhaps deserve further attention from the wider statistics and optimization communities. In this commentary, I will focus on some of the key points discussed in HTT, with a bias toward some of the recent work I have been involved in. There is a large and rich body of work in high-dimensional statistics and related optimization techniques that I will not be able to discuss within the limited scope of my commentary.
{"title":"Discussion of “Best Subset, Forward Stepwise or Lasso? Analysis and Recommendations Based on Extensive Comparisons”","authors":"R. Mazumder","doi":"10.1214/20-sts807","DOIUrl":"https://doi.org/10.1214/20-sts807","url":null,"abstract":"I warmly congratulate the authors Hastie, Tibshirani and Tibshirani (HTT); and Bertsimas, Pauphilet and Van Parys (BPV) for their excellent contributions and important perspectives on sparse regression. Due to space constraints, and my greater familiarity with the content and context of HTT (I have had numerous fruitful discussions with the authors regarding their work), I will focus my discussion on the HTT paper. HTT nicely articulate the relative merits of three canonical estimators in sparse regression: L0, L1 and (forward)stepwise selection. I am humbled that a premise of their work is an article I wrote with Bertsimas and King [4] (BKM). BKM showed that current Mixed Integer Optimization (MIO) algorithms allow us to compute best subsets solutions for problem instances (p ≈ 1000 features) much larger than a previous benchmark (software for best subsets in the R package leaps) that could only handle instances with p ≈ 30. HTT by extending and refining the experiments performed by BKM, have helped clarify and deepen our understanding of L0, L1 and stepwise regression. They raise several intriguing questions that perhaps deserve further attention from the wider statistics and optimization communities. In this commentary, I will focus on some of the key points discussed in HTT, with a bias toward some of the recent work I have been involved in. There is a large and rich body of work in high-dimensional statistics and related optimization techniques that I will not be able to discuss within the limited scope of my commentary.","PeriodicalId":51172,"journal":{"name":"Statistical Science","volume":"35 1","pages":"602-608"},"PeriodicalIF":5.7,"publicationDate":"2020-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47846338","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A Conversation with J. Stuart (Stu) Hunter","authors":"R. D. Veaux","doi":"10.1214/19-sts766","DOIUrl":"https://doi.org/10.1214/19-sts766","url":null,"abstract":"","PeriodicalId":51172,"journal":{"name":"Statistical Science","volume":"35 1","pages":"663-671"},"PeriodicalIF":5.7,"publicationDate":"2020-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43654798","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Rejoinder: Sparse Regression: Scalable Algorithms and Empirical Performance","authors":"D. Bertsimas, J. Pauphilet, Bart P. G. Van Parys","doi":"10.1214/20-sts701rej","DOIUrl":"https://doi.org/10.1214/20-sts701rej","url":null,"abstract":"their","PeriodicalId":51172,"journal":{"name":"Statistical Science","volume":" ","pages":""},"PeriodicalIF":5.7,"publicationDate":"2020-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46343056","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Statistical modeling can involve a tension between assumptions and statistical identification. The law of the observable data may not uniquely determine the value of a target parameter without invoking a key assumption, and, while plausible, this assumption may not be obviously true in the scientific context at hand. Moreover, there are many instances of key assumptions which are untestable, hence we cannot rely on the data to resolve the question of whether the target is legitimately identified. Working in the Bayesian paradigm, we consider the grey zone of situations where a key assumption, in the form of a parameter space restriction, is scientifically reasonable but not incontrovertible for the problem being tackled. Specifically, we investigate statistical properties that ensue if we structure a prior distribution to assert that `maybe' or `perhaps' the assumption holds. Technically this simply devolves to using a mixture prior distribution putting just some prior weight on the assumption, or one of several assumptions, holding. However, while the construct is straightforward, there is very little literature discussing situations where Bayesian model averaging is employed across a mix of fully identified and partially identified models.
{"title":"Parameter Restrictions for the Sake of Identification: Is There Utility in Asserting That Perhaps a Restriction Holds?","authors":"P. Gustafson","doi":"10.1214/23-sts885","DOIUrl":"https://doi.org/10.1214/23-sts885","url":null,"abstract":"Statistical modeling can involve a tension between assumptions and statistical identification. The law of the observable data may not uniquely determine the value of a target parameter without invoking a key assumption, and, while plausible, this assumption may not be obviously true in the scientific context at hand. Moreover, there are many instances of key assumptions which are untestable, hence we cannot rely on the data to resolve the question of whether the target is legitimately identified. Working in the Bayesian paradigm, we consider the grey zone of situations where a key assumption, in the form of a parameter space restriction, is scientifically reasonable but not incontrovertible for the problem being tackled. Specifically, we investigate statistical properties that ensue if we structure a prior distribution to assert that `maybe' or `perhaps' the assumption holds. Technically this simply devolves to using a mixture prior distribution putting just some prior weight on the assumption, or one of several assumptions, holding. However, while the construct is straightforward, there is very little literature discussing situations where Bayesian model averaging is employed across a mix of fully identified and partially identified models.","PeriodicalId":51172,"journal":{"name":"Statistical Science","volume":" ","pages":""},"PeriodicalIF":5.7,"publicationDate":"2020-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48381504","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In causal inference, principal stratification is a framework for dealing with a posttreatment intermediate variable between a treatment and an outcome, in which the principal strata are defined by the joint potential values of the intermediate variable. Because the principal strata are not fully observable, the causal effects within them, also known as the principal causal effects, are not identifiable without additional assumptions. Several previous empirical studies leveraged auxiliary variables to improve the inference of principal causal effects. We establish a general theory for identification and estimation of the principal causal effects with auxiliary variables, which provides a solid foundation for statistical inference and more insights for model building in empirical research. In particular, we consider two commonly-used strategies for principal stratification problems: principal ignorability, and the conditional independence between the auxiliary variable and the outcome given principal strata and covariates. For these two strategies, we give non-parametric and semi-parametric identification results without modeling assumptions on the outcome. When the assumptions for neither strategies are plausible, we propose a large class of flexible parametric and semi-parametric models for identifying principal causal effects. Our theory not only ensures formal identification results of several models that have been used in previous empirical studies but also generalizes them to allow for different types of outcomes and intermediate variables.
{"title":"Identification of Causal Effects Within Principal Strata Using Auxiliary Variables","authors":"Zhichao Jiang, Peng Ding","doi":"10.1214/20-sts810","DOIUrl":"https://doi.org/10.1214/20-sts810","url":null,"abstract":"In causal inference, principal stratification is a framework for dealing with a posttreatment intermediate variable between a treatment and an outcome, in which the principal strata are defined by the joint potential values of the intermediate variable. Because the principal strata are not fully observable, the causal effects within them, also known as the principal causal effects, are not identifiable without additional assumptions. Several previous empirical studies leveraged auxiliary variables to improve the inference of principal causal effects. We establish a general theory for identification and estimation of the principal causal effects with auxiliary variables, which provides a solid foundation for statistical inference and more insights for model building in empirical research. In particular, we consider two commonly-used strategies for principal stratification problems: principal ignorability, and the conditional independence between the auxiliary variable and the outcome given principal strata and covariates. For these two strategies, we give non-parametric and semi-parametric identification results without modeling assumptions on the outcome. When the assumptions for neither strategies are plausible, we propose a large class of flexible parametric and semi-parametric models for identifying principal causal effects. Our theory not only ensures formal identification results of several models that have been used in previous empirical studies but also generalizes them to allow for different types of outcomes and intermediate variables.","PeriodicalId":51172,"journal":{"name":"Statistical Science","volume":" ","pages":""},"PeriodicalIF":5.7,"publicationDate":"2020-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48107960","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Hunyong Cho, Joshua P. Zitovsky, Xinyi Li, Minxin Lu, K. Shah, John Sperger, Matthew C. B. Tsilimigras, M. Kosorok
We discuss “Linear mixed models with endogenous covariates: modeling sequential treatment effects with application to a mobile health study” by Qian, Klasnja and Murphy. In this discussion, we study when the linear mixed effects models with endogenous covariates are feasible to use by providing examples and diagnostic tools as well as discussing potential extensions. This includes evaluating feasibility of partial likelihood-based inference, checking the conditional independence assumption, estimation of marginal effects, and kernel extensions of the model.
{"title":"Comment: Diagnostics and Kernel-based Extensions for Linear Mixed Effects Models with Endogenous Covariates","authors":"Hunyong Cho, Joshua P. Zitovsky, Xinyi Li, Minxin Lu, K. Shah, John Sperger, Matthew C. B. Tsilimigras, M. Kosorok","doi":"10.1214/20-sts782","DOIUrl":"https://doi.org/10.1214/20-sts782","url":null,"abstract":"We discuss “Linear mixed models with endogenous covariates: modeling sequential treatment effects with application to a mobile health study” by Qian, Klasnja and Murphy. In this discussion, we study when the linear mixed effects models with endogenous covariates are feasible to use by providing examples and diagnostic tools as well as discussing potential extensions. This includes evaluating feasibility of partial likelihood-based inference, checking the conditional independence assumption, estimation of marginal effects, and kernel extensions of the model.","PeriodicalId":51172,"journal":{"name":"Statistical Science","volume":"35 1","pages":"396-399"},"PeriodicalIF":5.7,"publicationDate":"2020-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47084522","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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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