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}
In the case of incomplete data we give general relationships between the first and second derivatives of the loglikelihood relative to the full and the incomplete observation set-ups. In the case where these quantities are easy to compute for the full observation set-up we propose to compute their analogue for the incomplete observation set-up using the above mentioned relationships: this involves numerical integrations. Once we are able to compute these quantities, Newton-Raphson type algorithms can be applied to find the maximum likelihood estimators, together with estimates of their variances. We detail the application of this approach to parametric multiplicative frailty models and we show that the method works well in practice using both a real data and a simulated example. The proposed algorithm outperforms a Newton-Raphson type algorithm using numerical derivatives.
{"title":"Relationship between Derivatives of the Observed and Full Loglikelihoods and Application to Newton-Raphson Algorithm","authors":"D. Commenges, V. Rondeau","doi":"10.2202/1557-4679.1010","DOIUrl":"https://doi.org/10.2202/1557-4679.1010","url":null,"abstract":"In the case of incomplete data we give general relationships between the first and second derivatives of the loglikelihood relative to the full and the incomplete observation set-ups. In the case where these quantities are easy to compute for the full observation set-up we propose to compute their analogue for the incomplete observation set-up using the above mentioned relationships: this involves numerical integrations. Once we are able to compute these quantities, Newton-Raphson type algorithms can be applied to find the maximum likelihood estimators, together with estimates of their variances. We detail the application of this approach to parametric multiplicative frailty models and we show that the method works well in practice using both a real data and a simulated example. The proposed algorithm outperforms a Newton-Raphson type algorithm using numerical derivatives.","PeriodicalId":50333,"journal":{"name":"International Journal of Biostatistics","volume":"2 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68714578","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}
We provide a method for calculating the sample size required to attain a given average power (the ratio of rejected hypotheses to the number of false hypotheses) and a given false discovery rate (the number of incorrect rejections divided by the number of rejections) in adaptive versions of the Benjamini-Hochberg method of multiple testing. The method works in an asymptotic sense as the number of hypotheses grows to infinity and under quite general conditions, and it requires data from a pilot study. The consistency of the method follows from several results in classical areas of nonparametric statistics developed in a new context of "weak" dependence.
{"title":"Approximate Power and Sample Size Calculations with the Benjamini-Hochberg Method","authors":"J. A. Ferreira, A. Zwinderman","doi":"10.2202/1557-4679.1018","DOIUrl":"https://doi.org/10.2202/1557-4679.1018","url":null,"abstract":"We provide a method for calculating the sample size required to attain a given average power (the ratio of rejected hypotheses to the number of false hypotheses) and a given false discovery rate (the number of incorrect rejections divided by the number of rejections) in adaptive versions of the Benjamini-Hochberg method of multiple testing. The method works in an asymptotic sense as the number of hypotheses grows to infinity and under quite general conditions, and it requires data from a pilot study. The consistency of the method follows from several results in classical areas of nonparametric statistics developed in a new context of \"weak\" dependence.","PeriodicalId":50333,"journal":{"name":"International Journal of Biostatistics","volume":"2 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2202/1557-4679.1018","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68714885","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}
P. Sottas, N. Robinson, S. Giraud, F. Taroni, M. Kamber, P. Mangin, M. Saugy
Blood doping has been challenging the scientific community since the early 1970's, where it was demonstrated that blood transfusion significantly improves physical performance. Here, we present through 3 applications how statistical classification techniques can assist the implementation of effective tests to deter blood doping in elite sports. In particular, we developed a new indirect and universal test of blood doping, called Abnormal Blood Profile Score (ABPS), based on the statistical classification of indirect biomarkers of altered erythropoiesis. Up to 601 hematological profiles have been compiled in a reference database. Twenty-one of them were obtained from blood samples withdrawn from professional athletes convicted of blood doping by other direct tests. Discriminative training algorithms were used jointly with cross-validation techniques to map these labeled reference profiles to target outputs. The strict cross-validation procedure facilitates the adherence to medico-legal standards mandated by the World Anti Doping Agency (WADA). The test has a sensitivity to recombinant erythropoietin (rhEPO) abuse up to 3 times better than current generative models, independently whether the athlete is currently taking rhEPO or has stopped the treatment. The test is also sensitive to any form of blood transfusion, autologous transfusion included. We finally conclude why a probabilistic approach should be encouraged for the evaluation of evidence in anti-doping area of investigation.
{"title":"Statistical Classification of Abnormal Blood Profiles in Athletes","authors":"P. Sottas, N. Robinson, S. Giraud, F. Taroni, M. Kamber, P. Mangin, M. Saugy","doi":"10.2202/1557-4679.1011","DOIUrl":"https://doi.org/10.2202/1557-4679.1011","url":null,"abstract":"Blood doping has been challenging the scientific community since the early 1970's, where it was demonstrated that blood transfusion significantly improves physical performance. Here, we present through 3 applications how statistical classification techniques can assist the implementation of effective tests to deter blood doping in elite sports. In particular, we developed a new indirect and universal test of blood doping, called Abnormal Blood Profile Score (ABPS), based on the statistical classification of indirect biomarkers of altered erythropoiesis. Up to 601 hematological profiles have been compiled in a reference database. Twenty-one of them were obtained from blood samples withdrawn from professional athletes convicted of blood doping by other direct tests. Discriminative training algorithms were used jointly with cross-validation techniques to map these labeled reference profiles to target outputs. The strict cross-validation procedure facilitates the adherence to medico-legal standards mandated by the World Anti Doping Agency (WADA). The test has a sensitivity to recombinant erythropoietin (rhEPO) abuse up to 3 times better than current generative models, independently whether the athlete is currently taking rhEPO or has stopped the treatment. The test is also sensitive to any form of blood transfusion, autologous transfusion included. We finally conclude why a probabilistic approach should be encouraged for the evaluation of evidence in anti-doping area of investigation.","PeriodicalId":50333,"journal":{"name":"International Journal of Biostatistics","volume":"2 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2202/1557-4679.1011","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68714923","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}
We consider the inverse problem of estimating a survival distribution when the survival times are only observed to be in one of the intervals of a random bisection of the time axis. We are particularly interested in the case that high-dimensional and/or time-dependent covariates are available, and/or the survival events and censoring times are only conditionally independent given the covariate process. The method of estimation consists of regularizing the survival distribution by taking the primitive function or smoothing, estimating the regularized parameter by using estimating equations, and finally recovering an estimator for the parameter of interest.
{"title":"Estimating a Survival Distribution with Current Status Data and High-dimensional Covariates","authors":"A. van der Vaart, M. J. van der Laan","doi":"10.2202/1557-4679.1014","DOIUrl":"https://doi.org/10.2202/1557-4679.1014","url":null,"abstract":"We consider the inverse problem of estimating a survival distribution when the survival times are only observed to be in one of the intervals of a random bisection of the time axis. We are particularly interested in the case that high-dimensional and/or time-dependent covariates are available, and/or the survival events and censoring times are only conditionally independent given the covariate process. The method of estimation consists of regularizing the survival distribution by taking the primitive function or smoothing, estimating the regularized parameter by using estimating equations, and finally recovering an estimator for the parameter of interest.","PeriodicalId":50333,"journal":{"name":"International Journal of Biostatistics","volume":"2 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2202/1557-4679.1014","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68715195","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}
Many statistical problems involve the learning of an importance/effect of a variable for predicting an outcome of interest based on observing a sample of $n$ independent and identically distributed observations on a list of input variables and an outcome. For example, though prediction/machine learning is, in principle, concerned with learning the optimal unknown mapping from input variables to an outcome from the data, the typical reported output is a list of importance measures for each input variable. The approach in prediction has been to learn the unknown optimal predictor from the data and derive, for each of the input variables, the variable importance from the obtained fit. In this article we propose a new approach which involves for each variable separately 1) defining variable importance as a real valued parameter, 2) deriving the efficient influence curve and thereby optimal estimating function for this parameter in the assumed (possibly nonparametric) model, and 3) develop a corresponding double robust locally efficient estimator of this variable importance, obtained by substituting for the nuisance parameters in the optimal estimating function data adaptive estimators. We illustrate this methodology in the context of prediction, and obtain in this manner double robust locally optimal estimators of marginal variable importance, accompanied with p-values and confidence intervals. In addition, we present a model based and machine learning approach to estimate covariate-adjusted variable importance. Finally, we generalize this methodology to variable importance parameters for time-dependent variables.
{"title":"Statistical Inference for Variable Importance","authors":"M. J. van der Laan","doi":"10.2202/1557-4679.1008","DOIUrl":"https://doi.org/10.2202/1557-4679.1008","url":null,"abstract":"Many statistical problems involve the learning of an importance/effect of a variable for predicting an outcome of interest based on observing a sample of $n$ independent and identically distributed observations on a list of input variables and an outcome. For example, though prediction/machine learning is, in principle, concerned with learning the optimal unknown mapping from input variables to an outcome from the data, the typical reported output is a list of importance measures for each input variable. The approach in prediction has been to learn the unknown optimal predictor from the data and derive, for each of the input variables, the variable importance from the obtained fit. In this article we propose a new approach which involves for each variable separately 1) defining variable importance as a real valued parameter, 2) deriving the efficient influence curve and thereby optimal estimating function for this parameter in the assumed (possibly nonparametric) model, and 3) develop a corresponding double robust locally efficient estimator of this variable importance, obtained by substituting for the nuisance parameters in the optimal estimating function data adaptive estimators. We illustrate this methodology in the context of prediction, and obtain in this manner double robust locally optimal estimators of marginal variable importance, accompanied with p-values and confidence intervals. In addition, we present a model based and machine learning approach to estimate covariate-adjusted variable importance. Finally, we generalize this methodology to variable importance parameters for time-dependent variables.","PeriodicalId":50333,"journal":{"name":"International Journal of Biostatistics","volume":"2 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2202/1557-4679.1008","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68714461","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}
Several measures have been proposed to summarize the Receiver Operating Characteristic (ROC) curve, including the Projected Length of the Curve (PLC) and the Area Swept out by the Curve (ASC). These indices were first proposed by Lee (Epidemiology 1996; 7:605-611) to avoid certain deficiencies of the traditional Area Under the Curve (AUC) summary measure. More recently meta-analysis methods for assessing diagnostic test accuracy have been developed and the Summary Receiver Operating Characteristic (SROC) curve has been recommended to represent the performance of a diagnostic test. Some properties of the SROC curve were discussed by Walter (Statist. Med. 2002; 21:1237-1256). Here we extend that work to focus on properties of PLC and ASC in the context of SROC curve. Mathematical expressions for these two indices and their variances are derived in terms of the overall diagnostic odds ratio and the magnitude of inter-study heterogeneity in the odds ratio. Expressions for PLC and ASC and their variances are easily computed in homogeneous studies, and their values provide good approximations to the corresponding values for heterogeneous studies in most practical situations. General variances of PLC and ASC are derived by using delta methods, and are found to be smaller if the odds ratio is large. The methods are illustrated using data from two studies, the first being a meta-analysis on the detection of metastases in cervical cancer patients, and the second being a single study of HPV infection and pre-invasive cervical lesions.
{"title":"Properties of the Projected Length of the Curve (PLC) and Area Swept out by the Curve (ASC) Indices for the Receiver Operating Characteristic (SROC) Curve","authors":"Xuan Zhang, S. Walter, R. Agnihotram","doi":"10.2202/1557-4679.1096","DOIUrl":"https://doi.org/10.2202/1557-4679.1096","url":null,"abstract":"Several measures have been proposed to summarize the Receiver Operating Characteristic (ROC) curve, including the Projected Length of the Curve (PLC) and the Area Swept out by the Curve (ASC). These indices were first proposed by Lee (Epidemiology 1996; 7:605-611) to avoid certain deficiencies of the traditional Area Under the Curve (AUC) summary measure. More recently meta-analysis methods for assessing diagnostic test accuracy have been developed and the Summary Receiver Operating Characteristic (SROC) curve has been recommended to represent the performance of a diagnostic test. Some properties of the SROC curve were discussed by Walter (Statist. Med. 2002; 21:1237-1256). Here we extend that work to focus on properties of PLC and ASC in the context of SROC curve. Mathematical expressions for these two indices and their variances are derived in terms of the overall diagnostic odds ratio and the magnitude of inter-study heterogeneity in the odds ratio. Expressions for PLC and ASC and their variances are easily computed in homogeneous studies, and their values provide good approximations to the corresponding values for heterogeneous studies in most practical situations. General variances of PLC and ASC are derived by using delta methods, and are found to be smaller if the odds ratio is large. The methods are illustrated using data from two studies, the first being a meta-analysis on the detection of metastases in cervical cancer patients, and the second being a single study of HPV infection and pre-invasive cervical lesions.","PeriodicalId":50333,"journal":{"name":"International Journal of Biostatistics","volume":"5 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2202/1557-4679.1096","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68715363","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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