Suppose one observes a sample of independent and identically distributed observations from a particular data generating distribution. Suppose that one is concerned with estimation of a particular pathwise differentiable Euclidean parameter. A substitution estimator evaluating the parameter of a given likelihood based density estimator is typically too biased and might not even converge at the parametric rate: that is, the density estimator was targeted to be a good estimator of the density and might therefore result in a poor estimator of a particular smooth functional of the density. In this article we propose a one step (and, by iteration, k-th step) targeted maximum likelihood density estimator which involves 1) creating a hardest parametric submodel with parameter epsilon through the given density estimator with score equal to the efficient influence curve of the pathwise differentiable parameter at the density estimator, 2) estimating epsilon with the maximum likelihood estimator, and 3) defining a new density estimator as the corresponding update of the original density estimator. We show that iteration of this algorithm results in a targeted maximum likelihood density estimator which solves the efficient influence curve estimating equation and thereby yields a locally efficient estimator of the parameter of interest, under regularity conditions. In particular, we show that, if the parameter is linear and the model is convex, then the targeted maximum likelihood estimator is often achieved in the first step, and it results in a locally efficient estimator at an arbitrary (e.g., heavily misspecified) starting density.We also show that the targeted maximum likelihood estimators are now in full agreement with the locally efficient estimating function methodology as presented in Robins and Rotnitzky (1992) and van der Laan and Robins (2003), creating, in particular, algebraic equivalence between the double robust locally efficient estimators using the targeted maximum likelihood estimators as an estimate of its nuisance parameters, and targeted maximum likelihood estimators. In addition, it is argued that the targeted MLE has various advantages relative to the current estimating function based approach. We proceed by providing data driven methodologies to select the initial density estimator for the targeted MLE, thereby providing data adaptive targeted maximum likelihood estimation methodology. We illustrate the method with various worked out examples.
假设从一个特定的数据生成分布中观察到一个独立且相同分布的观察样本。假设我们关心的是一个特定的路径可微欧几里得参数的估计。评估给定的基于似然的密度估计器的参数的替代估计器通常过于偏倚，甚至可能不会以参数速率收敛:也就是说，密度估计器的目标是成为密度的良好估计器，因此可能导致密度的特定光滑泛函的差估计器。在本文中，我们提出了一个一步(通过迭代，第k步)目标最大似然密度估计器，它涉及1)通过给定的密度估计器创建参数为epsilon的最难参数子模型，其得分等于密度估计器处路径可微参数的有效影响曲线，2)用最大似然估计器估计epsilon，3)定义一个新的密度估计量作为对原有密度估计量的相应更新。我们证明了该算法的迭代产生了一个目标最大似然密度估计量，它解决了有效的影响曲线估计方程，从而在正则性条件下产生了感兴趣参数的局部有效估计量。特别是，我们表明，如果参数是线性的，模型是凸的，那么目标最大似然估计器通常在第一步就能实现，并且它会在任意(例如，严重错误指定)的起始密度下产生局部有效的估计器。我们还表明，目标最大似然估计量现在与Robins和Rotnitzky(1992)以及van der Laan和Robins(2003)中提出的局部有效估计函数方法完全一致，特别是在使用目标最大似然估计量作为其讨厌参数的估计的双鲁棒局部有效估计量和目标最大似然估计量之间创建了代数等价。此外，本文还认为，相对于目前基于函数的估计方法，目标最大似然算法具有多种优势。我们通过提供数据驱动的方法来选择目标最大似然估计的初始密度估计量，从而提供数据自适应的目标最大似然估计方法。我们用各种算例来说明该方法。
{"title":"Targeted Maximum Likelihood Learning","authors":"M. J. van der Laan, D. Rubin","doi":"10.2202/1557-4679.1043","DOIUrl":"https://doi.org/10.2202/1557-4679.1043","url":null,"abstract":"Suppose one observes a sample of independent and identically distributed observations from a particular data generating distribution. Suppose that one is concerned with estimation of a particular pathwise differentiable Euclidean parameter. A substitution estimator evaluating the parameter of a given likelihood based density estimator is typically too biased and might not even converge at the parametric rate: that is, the density estimator was targeted to be a good estimator of the density and might therefore result in a poor estimator of a particular smooth functional of the density. In this article we propose a one step (and, by iteration, k-th step) targeted maximum likelihood density estimator which involves 1) creating a hardest parametric submodel with parameter epsilon through the given density estimator with score equal to the efficient influence curve of the pathwise differentiable parameter at the density estimator, 2) estimating epsilon with the maximum likelihood estimator, and 3) defining a new density estimator as the corresponding update of the original density estimator. We show that iteration of this algorithm results in a targeted maximum likelihood density estimator which solves the efficient influence curve estimating equation and thereby yields a locally efficient estimator of the parameter of interest, under regularity conditions. In particular, we show that, if the parameter is linear and the model is convex, then the targeted maximum likelihood estimator is often achieved in the first step, and it results in a locally efficient estimator at an arbitrary (e.g., heavily misspecified) starting density.We also show that the targeted maximum likelihood estimators are now in full agreement with the locally efficient estimating function methodology as presented in Robins and Rotnitzky (1992) and van der Laan and Robins (2003), creating, in particular, algebraic equivalence between the double robust locally efficient estimators using the targeted maximum likelihood estimators as an estimate of its nuisance parameters, and targeted maximum likelihood estimators. In addition, it is argued that the targeted MLE has various advantages relative to the current estimating function based approach. We proceed by providing data driven methodologies to select the initial density estimator for the targeted MLE, thereby providing data adaptive targeted maximum likelihood estimation methodology. We illustrate the method with various worked out examples.","PeriodicalId":50333,"journal":{"name":"International Journal of Biostatistics","volume":"2 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2006-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2202/1557-4679.1043","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68715717","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}
Optimal designs of dose levels in order to estimate parameters from a model for binary response data have a long and rich history. These designs are based on parametric models. Here we consider fully nonparametric models with interest focused on estimation of smooth functionals using plug-in estimators based on the nonparametric maximum likelihood estimator. An important application of the results is the derivation of the optimal choice of the monitoring time distribution function for current status observation of a survival distribution. The optimal choice depends in a simple way on the dose-response function and the form of the functional. The results can be extended to allow dependence of the monitoring mechanism on covariates.
{"title":"Choice of Monitoring Mechanism for Optimal Nonparametric Functional Estimation for Binary Data","authors":"N. Jewell, M. J. van der Laan, S. Shiboski","doi":"10.2202/1557-4679.1031","DOIUrl":"https://doi.org/10.2202/1557-4679.1031","url":null,"abstract":"Optimal designs of dose levels in order to estimate parameters from a model for binary response data have a long and rich history. These designs are based on parametric models. Here we consider fully nonparametric models with interest focused on estimation of smooth functionals using plug-in estimators based on the nonparametric maximum likelihood estimator. An important application of the results is the derivation of the optimal choice of the monitoring time distribution function for current status observation of a survival distribution. The optimal choice depends in a simple way on the dose-response function and the form of the functional. The results can be extended to allow dependence of the monitoring mechanism on covariates.","PeriodicalId":50333,"journal":{"name":"International Journal of Biostatistics","volume":"2 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2006-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68715343","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}
Van der Laan (2005) proposed a targeted method used to construct variable importance measures coupled with respective statistical inference. This technique involves determining the importance of a variable in predicting an outcome. This method can be applied as inverse probability of treatment weighted (IPTW) or double robust inverse probability of treatment weighted (DR-IPTW) estimators. The variance and respective p-value of the estimate are calculated by estimating the influence curve. This article applies the Van der Laan (2005) variable importance measures and corresponding inference to HIV-1 sequence data. In this application, the method is targeted at every codon position. In this data application, protease and reverse transcriptase codon positions on the HIV-1 strand are assessed to determine their respective variable importance, with respect to an outcome of viral replication capacity. We estimate the DR-IPTW W-adjusted variable importance measure for a specified set of potential effect modifiers W. In addition, simulations were performed on two separate datasets to examine the DR-IPTW estimator.
Van der Laan(2005)提出了一种有针对性的方法，用于构建变量重要性度量，并结合各自的统计推断。这项技术包括确定变量在预测结果中的重要性。该方法可应用于加权处理逆概率估计(IPTW)或双鲁棒加权处理逆概率估计(DR-IPTW)。通过估计影响曲线来计算估计的方差和各自的p值。本文将Van der Laan(2005)变量重要性度量和相应的推断应用于HIV-1序列数据。在本应用中，该方法针对每个密码子位置。在此数据应用中，评估了HIV-1链上蛋白酶和逆转录酶密码子的位置，以确定它们各自的变量重要性，以及病毒复制能力的结果。我们估计了DR-IPTW w调整后的变量重要性测量值对一组特定的潜在效应修饰因子w的影响。此外，在两个独立的数据集上进行了模拟，以检验DR-IPTW估计器。
{"title":"Application of a Variable Importance Measure Method","authors":"M. Birkner, M. J. van der Laan","doi":"10.2202/1557-4679.1013","DOIUrl":"https://doi.org/10.2202/1557-4679.1013","url":null,"abstract":"Van der Laan (2005) proposed a targeted method used to construct variable importance measures coupled with respective statistical inference. This technique involves determining the importance of a variable in predicting an outcome. This method can be applied as inverse probability of treatment weighted (IPTW) or double robust inverse probability of treatment weighted (DR-IPTW) estimators. The variance and respective p-value of the estimate are calculated by estimating the influence curve. This article applies the Van der Laan (2005) variable importance measures and corresponding inference to HIV-1 sequence data. In this application, the method is targeted at every codon position. In this data application, protease and reverse transcriptase codon positions on the HIV-1 strand are assessed to determine their respective variable importance, with respect to an outcome of viral replication capacity. We estimate the DR-IPTW W-adjusted variable importance measure for a specified set of potential effect modifiers W. In addition, simulations were performed on two separate datasets to examine the DR-IPTW estimator.","PeriodicalId":50333,"journal":{"name":"International Journal of Biostatistics","volume":"2 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2006-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2202/1557-4679.1013","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68715116","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}
Comparing two large multivariate distributions is potentially complicated at least for the following reasons. First, some variable/level combinations may have a redundant difference in prevalence between groups in the sense that the difference can be completely explained in terms of lower-order combinations. Second, the total number of variable/level combinations to compare between groups is very large, and likely computationally prohibitive. In this paper, for both the paired and independent sample case, an approximate comparison method is proposed, along with a computationally efficient algorithm, that estimates the set of variable/level combinations that have a non-redundant different prevalence between two populations. The probability that the estimate contains one or more false or redundant differences is asymptotically bounded above by any pre-specified level for arbitrary data-generating distributions. The method is shown to perform well for finite samples in a simulation study, and is used to investigate HIV-1 genotype evolution in a recent AIDS clinical trial.
{"title":"The Two Sample Problem for Multiple Categorical Variables","authors":"A. DiRienzo","doi":"10.2202/1557-4679.1019","DOIUrl":"https://doi.org/10.2202/1557-4679.1019","url":null,"abstract":"Comparing two large multivariate distributions is potentially complicated at least for the following reasons. First, some variable/level combinations may have a redundant difference in prevalence between groups in the sense that the difference can be completely explained in terms of lower-order combinations. Second, the total number of variable/level combinations to compare between groups is very large, and likely computationally prohibitive. In this paper, for both the paired and independent sample case, an approximate comparison method is proposed, along with a computationally efficient algorithm, that estimates the set of variable/level combinations that have a non-redundant different prevalence between two populations. The probability that the estimate contains one or more false or redundant differences is asymptotically bounded above by any pre-specified level for arbitrary data-generating distributions. The method is shown to perform well for finite samples in a simulation study, and is used to investigate HIV-1 genotype evolution in a recent AIDS clinical trial.","PeriodicalId":50333,"journal":{"name":"International Journal of Biostatistics","volume":"2 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2006-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2202/1557-4679.1019","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68715010","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 paper develops empirical likelihood based simultaneous confidence bands for differences and ratios of two distribution functions from independent samples of right-censored survival data. The proposed confidence bands provide a flexible way of comparing treatments in biomedical settings, and bring empirical likelihood methods to bear on important target functions for which only Wald-type confidence bands have been available in the literature. The approach is illustrated with a real data example.
{"title":"Comparing Distribution Functions via Empirical Likelihood","authors":"I. McKeague, Yichuan Zhao","doi":"10.2202/1557-4679.1007","DOIUrl":"https://doi.org/10.2202/1557-4679.1007","url":null,"abstract":"This paper develops empirical likelihood based simultaneous confidence bands for differences and ratios of two distribution functions from independent samples of right-censored survival data. The proposed confidence bands provide a flexible way of comparing treatments in biomedical settings, and bring empirical likelihood methods to bear on important target functions for which only Wald-type confidence bands have been available in the literature. The approach is illustrated with a real data example.","PeriodicalId":50333,"journal":{"name":"International Journal of Biostatistics","volume":"1 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2006-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2202/1557-4679.1007","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68714433","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}
A natural choice of time scale for analyzing recurrent event data is the ``gap" (or soujourn) time between successive events. In many situations it is reasonable to assume correlation exists between the successive events experienced by a given subject. This paper looks at the problem of extending the accelerated failure time (AFT) model to the case of dependent recurrent event data via intensity modeling. Specifically, the accelerated gap times model of Strawderman (2005), a semiparametric intensity model for independent gap time data, is extended to the case of multiplicative gamma frailty. As argued in Aalen & Husebye (1991), incorporating frailty captures the heterogeneity between subjects and the ``hazard" portion of the intensity model captures gap time variation within a subject. Estimators are motivated using semiparametric efficiency theory and lead to useful generalizations of the rank statistics considered in Strawderman (2005). Several interesting distinctions arise in comparison to the Cox-Andersen-Gill frailty model (e.g., Nielsen et al, 1992; Klein, 1992). The proposed methodology is illustrated by simulation and data analysis.
分析重复事件数据的时间尺度的自然选择是连续事件之间的“间隙”(或逗留)时间。在许多情况下，假设给定主体所经历的连续事件之间存在相关性是合理的。本文研究了通过强度建模将加速失效时间(AFT)模型扩展到相关循环事件数据的问题。具体而言，Strawderman(2005)的加速间隙时间模型(独立间隙时间数据的半参数强度模型)被扩展到乘法伽马脆弱的情况。正如Aalen & Husebye(1991)所指出的那样，纳入脆弱性捕获了受试者之间的异质性，而强度模型的“危险”部分捕获了受试者内部的间隙时间变化。估计器使用半参数效率理论进行激励，并导致了Strawderman(2005)中考虑的秩统计的有用推广。与Cox-Andersen-Gill脆弱性模型相比，出现了几个有趣的区别(例如，Nielsen et al, 1992;克莱恩,1992)。通过仿真和数据分析说明了所提出的方法。
{"title":"A Regression Model for Dependent Gap Times","authors":"R. Strawderman","doi":"10.2202/1557-4679.1005","DOIUrl":"https://doi.org/10.2202/1557-4679.1005","url":null,"abstract":"A natural choice of time scale for analyzing recurrent event data is the ``gap\" (or soujourn) time between successive events. In many situations it is reasonable to assume correlation exists between the successive events experienced by a given subject. This paper looks at the problem of extending the accelerated failure time (AFT) model to the case of dependent recurrent event data via intensity modeling. Specifically, the accelerated gap times model of Strawderman (2005), a semiparametric intensity model for independent gap time data, is extended to the case of multiplicative gamma frailty. As argued in Aalen & Husebye (1991), incorporating frailty captures the heterogeneity between subjects and the ``hazard\" portion of the intensity model captures gap time variation within a subject. Estimators are motivated using semiparametric efficiency theory and lead to useful generalizations of the rank statistics considered in Strawderman (2005). Several interesting distinctions arise in comparison to the Cox-Andersen-Gill frailty model (e.g., Nielsen et al, 1992; Klein, 1992). The proposed methodology is illustrated by simulation and data analysis.","PeriodicalId":50333,"journal":{"name":"International Journal of Biostatistics","volume":"2 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2006-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68714794","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}
Marginal structural models (MSM) provide a powerful tool for estimating the causal effect of a treatment. These models, introduced by Robins, model the marginal distributions of treatment-specific counterfactual outcomes, possibly conditional on a subset of the baseline covariates. Marginal structural models are particularly useful in the context of longitudinal data structures, in which each subject's treatment and covariate history are measured over time, and an outcome is recorded at a final time point. However, the utility of these models for some applications has been limited by their inability to incorporate modification of the causal effect of treatment by time-varying covariates. Particularly in the context of clinical decision making, such time-varying effect modifiers are often of considerable or even primary interest, as they are used in practice to guide treatment decisions for an individual. In this article we propose a generalization of marginal structural models, which we call history-adjusted marginal structural models (HA-MSM). These models allow estimation of adjusted causal effects of treatment, given the observed past, and are therefore more suitable for making treatment decisions at the individual level and for identification of time-dependent effect modifiers. Specifically, a HA-MSM models the conditional distribution of treatment-specific counterfactual outcomes, conditional on the whole or a subset of the observed past up till a time-point, simultaneously for all time-points. Double robust inverse probability of treatment weighted estimators have been developed and studied in detail for standard MSM. We extend these results by proposing a class of double robust inverse probability of treatment weighted estimators for the unknown parameters of the HA-MSM. In addition, we show that HA-MSM provide a natural approach to identifying the dynamic treatment regimen which follows, at each time-point, the history-adjusted (up till the most recent time point) optimal static treatment regimen. We illustrate our results using an example drawn from the treatment of HIV infection.
{"title":"History-Adjusted Marginal Structural Models and Statically-Optimal Dynamic Treatment Regimens","authors":"M. J. van der Laan, M. Petersen, M. Joffe","doi":"10.2202/1557-4679.1003","DOIUrl":"https://doi.org/10.2202/1557-4679.1003","url":null,"abstract":"Marginal structural models (MSM) provide a powerful tool for estimating the causal effect of a treatment. These models, introduced by Robins, model the marginal distributions of treatment-specific counterfactual outcomes, possibly conditional on a subset of the baseline covariates. Marginal structural models are particularly useful in the context of longitudinal data structures, in which each subject's treatment and covariate history are measured over time, and an outcome is recorded at a final time point. However, the utility of these models for some applications has been limited by their inability to incorporate modification of the causal effect of treatment by time-varying covariates. Particularly in the context of clinical decision making, such time-varying effect modifiers are often of considerable or even primary interest, as they are used in practice to guide treatment decisions for an individual. In this article we propose a generalization of marginal structural models, which we call history-adjusted marginal structural models (HA-MSM). These models allow estimation of adjusted causal effects of treatment, given the observed past, and are therefore more suitable for making treatment decisions at the individual level and for identification of time-dependent effect modifiers. Specifically, a HA-MSM models the conditional distribution of treatment-specific counterfactual outcomes, conditional on the whole or a subset of the observed past up till a time-point, simultaneously for all time-points. Double robust inverse probability of treatment weighted estimators have been developed and studied in detail for standard MSM. We extend these results by proposing a class of double robust inverse probability of treatment weighted estimators for the unknown parameters of the HA-MSM. In addition, we show that HA-MSM provide a natural approach to identifying the dynamic treatment regimen which follows, at each time-point, the history-adjusted (up till the most recent time point) optimal static treatment regimen. We illustrate our results using an example drawn from the treatment of HIV infection.","PeriodicalId":50333,"journal":{"name":"International Journal of Biostatistics","volume":"1 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2005-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2202/1557-4679.1003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68714712","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}
In this paper we introduce three natural ``score statistics" for testing the hypothesis that F(t_0)takes on a fixed value in the context of nonparametric inference with current status data. These three new test statistics have natural interpretations in terms of certain (weighted) L_2 distances, and are also connected to natural ``one-sided" scores. We compare these new test statistics with the analogue of the classical Wald statistic and the likelihood ratio statistic introduced in Banerjee and Wellner (2001) for the same testing problem. Under classical ``regular" statistical problems the likelihood ratio, score, and Wald statistics all have the same chi-squared limiting distribution under the null hypothesis. In sharp contrast, in this non-regular problem all three statistics have different limiting distributions under the null hypothesis. Thus we begin by establishing the limit distribution theory of the statistics under the null hypothesis, and discuss calculation of the relevant critical points for the test statistics. Once the null distribution theory is known, the immediate question becomes that of power. We establish the limiting behavior of the three types of statistics under local alternatives. We have also compared the power of these five different statistics via a limited Monte-Carlo study. Our conclusions are: (a) the Wald statistic is less powerful than the likelihood ratio and score statistics; and (b) one of the score statistics may have more power than the likelihood ratio statistic for some alternatives.
{"title":"Score Statistics for Current Status Data: Comparisons with Likelihood Ratio and Wald Statistics","authors":"M. Banerjee, J. Wellner","doi":"10.2202/1557-4679.1001","DOIUrl":"https://doi.org/10.2202/1557-4679.1001","url":null,"abstract":"In this paper we introduce three natural ``score statistics\" for testing the hypothesis that F(t_0)takes on a fixed value in the context of nonparametric inference with current status data. These three new test statistics have natural interpretations in terms of certain (weighted) L_2 distances, and are also connected to natural ``one-sided\" scores. We compare these new test statistics with the analogue of the classical Wald statistic and the likelihood ratio statistic introduced in Banerjee and Wellner (2001) for the same testing problem. Under classical ``regular\" statistical problems the likelihood ratio, score, and Wald statistics all have the same chi-squared limiting distribution under the null hypothesis. In sharp contrast, in this non-regular problem all three statistics have different limiting distributions under the null hypothesis. Thus we begin by establishing the limit distribution theory of the statistics under the null hypothesis, and discuss calculation of the relevant critical points for the test statistics. Once the null distribution theory is known, the immediate question becomes that of power. We establish the limiting behavior of the three types of statistics under local alternatives. We have also compared the power of these five different statistics via a limited Monte-Carlo study. Our conclusions are: (a) the Wald statistic is less powerful than the likelihood ratio and score statistics; and (b) one of the score statistics may have more power than the likelihood ratio statistic for some alternatives.","PeriodicalId":50333,"journal":{"name":"International Journal of Biostatistics","volume":"1 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2005-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2202/1557-4679.1001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68714657","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}
Backcalculation is a technique that was originally developed for the study of HIV incidence. Here we introduce some variants of the estimation technique that allow for (i) correlation of the unobserved disease incidence counts, and (ii) the use of a smoothing step as part of the maximizing step in the EM algorithm to reduce instability due to small diagnosis counts. Both of these issues can be important in the analysis of small "epidemics." In addition, identification of correlation between diagnosis counts provides indirect evidence of correlation among unobserved incidence counts, hinting at the possibility of an infectious agent. We illustrate the ideas by reconstructing an incidence intensity function for the onset of multiple sclerosis, using data from the Faroe Islands. Previously, this data had been examined statistically, by Joseph, Wolfson & Wolfson (1990), to address the issue of infectiousness of multiple sclerosis. We argue that the incidence function cannot directly shed light on the enigmatic origin of multiple sclerosis in the Faroe Islands during World War II, and, in particular, cannot discriminate between hypotheses of an infectious or environmental agent.
{"title":"Some Variants of the Backcalculation Method for Estimation of Disease Incidence: An Application to Multiple Sclerosis Data from the Faroe Islands","authors":"N. Jewell, B. Lu","doi":"10.2202/1557-4679.1002","DOIUrl":"https://doi.org/10.2202/1557-4679.1002","url":null,"abstract":"Backcalculation is a technique that was originally developed for the study of HIV incidence. Here we introduce some variants of the estimation technique that allow for (i) correlation of the unobserved disease incidence counts, and (ii) the use of a smoothing step as part of the maximizing step in the EM algorithm to reduce instability due to small diagnosis counts. Both of these issues can be important in the analysis of small \"epidemics.\" In addition, identification of correlation between diagnosis counts provides indirect evidence of correlation among unobserved incidence counts, hinting at the possibility of an infectious agent. We illustrate the ideas by reconstructing an incidence intensity function for the onset of multiple sclerosis, using data from the Faroe Islands. Previously, this data had been examined statistically, by Joseph, Wolfson & Wolfson (1990), to address the issue of infectiousness of multiple sclerosis. We argue that the incidence function cannot directly shed light on the enigmatic origin of multiple sclerosis in the Faroe Islands during World War II, and, in particular, cannot discriminate between hypotheses of an infectious or environmental agent.","PeriodicalId":50333,"journal":{"name":"International Journal of Biostatistics","volume":"1 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2005-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2202/1557-4679.1002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68714703","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}
In many clinical trials related to diseases such as cancers and HIV, patients are treated by different combinations of therapies. This leads to two-stage designs, where patients are initially randomized to a primary therapy and then depending on disease remission and patients' consent, a maintenance therapy will be randomly assigned. In such designs, the effects of different treatment policies, i.e., combinations of primary and maintenance therapy are of great interest. In this paper, we propose an estimator for the survival distribution for each treatment policy in such two-stage studies with right-censoring using the method of weighted estimation equations within risk sets. We also derive the large-sample properties. The method is demonstrated and compared with other estimators through simulations and applied to analyze a two-stage randomized study with leukemia patients.
{"title":"A Weighted Risk Set Estimator for Survival Distributions in Two-Stage Randomization Designs with Censored Survival Data","authors":"Xiang Guo, A. Tsiatis","doi":"10.2202/1557-4679.1000","DOIUrl":"https://doi.org/10.2202/1557-4679.1000","url":null,"abstract":"In many clinical trials related to diseases such as cancers and HIV, patients are treated by different combinations of therapies. This leads to two-stage designs, where patients are initially randomized to a primary therapy and then depending on disease remission and patients' consent, a maintenance therapy will be randomly assigned. In such designs, the effects of different treatment policies, i.e., combinations of primary and maintenance therapy are of great interest. In this paper, we propose an estimator for the survival distribution for each treatment policy in such two-stage studies with right-censoring using the method of weighted estimation equations within risk sets. We also derive the large-sample properties. The method is demonstrated and compared with other estimators through simulations and applied to analyze a two-stage randomized study with leukemia patients.","PeriodicalId":50333,"journal":{"name":"International Journal of Biostatistics","volume":"1 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2005-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2202/1557-4679.1000","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68714613","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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