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Journal of Biopharmaceutical Statistics Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/lbps20
Odds Ratio for 2 × 2 Tables: Mantel–Haenszel Estimator, Profile Likelihood, and Presence of Surrogate Responses a
a
Buddhananda Banerjee & Atanu Biswas a
Applied Statistics Unit, Indian Statistical Institute, Kolkata, India Accepted author version posted online: 03 Apr 2014.Published online: 15 Apr 2014.
Click for updates To cite this article: Buddhananda Banerjee & Atanu Biswas (2014) Odds Ratio for 2 × 2 Tables: Mantel–Haenszel Estimator, Profile Likelihood, and Presence of Surrogate Responses, Journal of Biopharmaceutical Statistics, 24:3, 649-659, DOI: 10.1080/10543406.2014.888568 To link to this article: http://dx.doi.org/10.1080/10543406.2014.888568
PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &
Downloaded by [ECU Libraries] at 03:26 26 April 2015
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Journal of Biopharmaceutical Statistics, 24: 649–659, 2014 Copyright © Taylor & Francis Group, LLC ISSN: 1054-3406 print/1520-5711 online DOI: 10.1080/10543406.2014.888568
ODDS RATIO FOR 2 × 2 TABLES: MANTEL–HAENSZEL ESTIMATOR, PROFILE LIKELIHOOD, AND PRESENCE OF SURROGATE RESPONSES Buddhananda Banerjee and Atanu Biswas
Downloaded by [ECU Libraries] at 03:26 26 April 2015
Applied Statistics Unit, Indian Statistical Institute, Kolkata, India Use of surrogate outcome to improve the inference in biomedical problems is an area of growing interest. Here, we consider a setup where both the true and surrogate endpoints are binary and we observe all the surrogate endpoints along with a few true endpoints. In a two-treatment setup we study the surrogate-augmented Mantel–Haenszel estimator based on observations from different groups when the group effect is present. We compare the Mantel– Haenszel estimator with the one obtained by maximizing profile likelihood in a surrogate augmented setup. We observe that the performances of these estimators are very close. Key Words: Log-odds ratio; Mantel–Haenszel estimator; Profile likelihood; Surrogate endpoint; True endpoint.
1. INTRODUCTION Many clinical outcomes are such that the response variables are often difficult or highly expensive to measure, or the responses are delayed where short-term measures are needed for inferential and administrative purposes. Therefore, to evaluate the effects of treatments or exposures on the true endpoint in medical studies, a closely related variable is used as a surrogate response. For example, damage to the heart muscle due to myocardial infraction can be accurately assessed by arterioscintography. As this is an expensive procedure, peak cardiac enzyme level in the blood stream, which is more easily obtainable, is used as surrogate measure of heart vascular damage (Wittes et al., 1989). Often an observed value of the response variable in the middle of an experiment is considered as a surrogate endpoint. High-dose interferon-α is used for patients with age-related macular degeneration (ARMD) who progressively loose their vision. Observations are taken after 6 months and 1 year for surrogate and true endpoints, respectively (Buyse and Molenberghs, 1998). CD4 cell count is used as a surrogate for HIV patients where the survival time is the true endpoint. A statistical definition and validation criteria for surrogate endpoints were first introduced by Prentice (1989). Begg and Leung (2000) proposed that a measure of concordance can be a possible validation criterion for discrete random variables. In the present article, we are interested in inferential problems using properly validated surrogate endpoints where we assume that the validation is done a priori. Day and Duffy (1996) Received January 31, 2012; Accepted May 4, 2013 Address correspondence to Professor Atanu Biswas, Applied Statistics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700 108, India; E-mail: [email protected]
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focused on the use of surrogate for the true endpoint in a clinical or preservation trial to shorten the duration of the trial and to increase the power. Freedman et al. (1992) supplemented these criteria with the so-called proportion explained by an intermediate endpoint for binary outcome variable, which has been extended by Lin et al. (1997) to failure time endpoints. Pepe (1992) considered a semiparametric model to estimate regression coefficients of covariates when surrogate data are present and showed its consistency. Chen (2000) developed a robust imputation method in the same setup. Chen et al. (2007) proposed two notions for a consistent surrogate and a strictly consistent surrogate to avoid the surrogate paradox. Molenberghs et al. (2001) introduced the situation where the surrogate is binary and the true endpoint is continuous, or vice versa. In addition, they considered the case of ordinal endpoints. In a recent study Banerjee and Biswas (2011) studied the estimation of the difference in success probabilities in the two-treatment setup in the presence of surrogate responses. In this article we consider the two-treatment problem. Suppose a multicenter Phase III clinical trial is being carried out for two competing treatments. If the responses are binary, a 2 × 2 table is generated as the outcome in each center. The Mantel–Haenszel (MH) estimator is widely used to get a pooled estimate odds ratio from multiple tables. A meta-analysis consisting of data on different studies is another example of applicability of the MH estimator. For example, Ergin and Ergin (2005) described a meta-analysis of 11 studies resulting eleven 2 × 2 tables of thrombolytic therapy versus placebo for the treatment of acute ischemic stroke patients. Assume that the response variable is binary for true as well as surrogate endpoints. Surrogate responses are more quickly and easily available than the true ones. There are two completely different problems studied in literature regarding surrogate endpoints: (i) the validation problem and (ii) the inferential problem. The validation problem deals with the derangement of appropriateness of a particular surrogate. Begg and Leung (2000) concluded that absolute validity of surrogate endpoint in the context of real clinical trial is unattainable. In the present article the surrogate endpoint is assumed to be validated (maybe from past experiments). However, it is preferable to use surrogate having high association with true endpoints (Begg and Leung, 2000; Prentice, 1989; Molenberghs et al., 2001). In this article we do not consider the validation problem; rather, we are interested in some inferential problems by optimal use of surrogate endpoints. Our approach developed in this article does not suffer from the surrogate paradox (Chen et al., 2007). The data where true and surrogate both are available are used for model building, and that is used where only surrogate endpoints are available for inferential purpose. Here we compare treatment effects in terms of log-odds ratio when large numbers of surrogate endpoints are available along with moderate numbers of true and surrogate paired observations when observations are available from multiple tables. Estimating the common odds ratio for multiple tables is a well studied problem (Breslow, 1981; Anscombe, 1956; Hauck, 1979; Breslow and Liang, 1982; Robins et al., 1986). We propose a methodology to obtain Mantel–Haenszel (MH) (Böhning et al., 2008; Mantel and Haenszel, 1959) and profile likelihood (PMLE) (Murphy and van der Vaart, 2000) based estimators in the surrogate augmented framework. We established asymptotic normality of the estimators. Our extensive simulation study also shows remarkable closeness in the behavior of the two surrogate augmented estimators: MH and PMLE. The rest of the article is as follows: Section 2 describes the data structure of multiple tables in the presence of surrogate endpoints. In section 3 we discuss the MH estimator and the PMLE without surrogate. Imputation with surrogate data is described in section 4. Section 5 studies the MH estimator and the PMLE in the presence of surrogate data.
MANTEL–HAENSZEL ESTIMATOR AND SURROGATE RESPONSES
651
Simulation results are reported in section 6 to show the closeness of the estimators. A simulated data example in a real situation is discussed in section 7, and section 8 concludes.
Downloaded by [ECU Libraries] at 03:26 26 April 2015
2. DATA STRUCTURE We consider a setup of two treatment binary endpoints with binary surrogates. Begg and Leung (2000) pointed out that for binary endpoints the probability of concordance is an indicator of association between true and surrogate endpoints. Suppose nA and nB patients are allotted to treatments A and B, respectively but we get only mA and mB true endpoints along with all surrogate endpoints within the stipulated time frame or cost limit, where mt
Journal of Biopharmaceutical Statistics Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/lbps20
Odds Ratio for 2 × 2 Tables: Mantel–Haenszel Estimator, Profile Likelihood, and Presence of Surrogate Responses a
a
Buddhananda Banerjee & Atanu Biswas a
Applied Statistics Unit, Indian Statistical Institute, Kolkata, India Accepted author version posted online: 03 Apr 2014.Published online: 15 Apr 2014.
Click for updates To cite this article: Buddhananda Banerjee & Atanu Biswas (2014) Odds Ratio for 2 × 2 Tables: Mantel–Haenszel Estimator, Profile Likelihood, and Presence of Surrogate Responses, Journal of Biopharmaceutical Statistics, 24:3, 649-659, DOI: 10.1080/10543406.2014.888568 To link to this article: http://dx.doi.org/10.1080/10543406.2014.888568
PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &
Downloaded by [ECU Libraries] at 03:26 26 April 2015
Conditions of access and use can be found at http://www.tandfonline.com/page/termsand-conditions
Journal of Biopharmaceutical Statistics, 24: 649–659, 2014 Copyright © Taylor & Francis Group, LLC ISSN: 1054-3406 print/1520-5711 online DOI: 10.1080/10543406.2014.888568
ODDS RATIO FOR 2 × 2 TABLES: MANTEL–HAENSZEL ESTIMATOR, PROFILE LIKELIHOOD, AND PRESENCE OF SURROGATE RESPONSES Buddhananda Banerjee and Atanu Biswas
Downloaded by [ECU Libraries] at 03:26 26 April 2015
Applied Statistics Unit, Indian Statistical Institute, Kolkata, India Use of surrogate outcome to improve the inference in biomedical problems is an area of growing interest. Here, we consider a setup where both the true and surrogate endpoints are binary and we observe all the surrogate endpoints along with a few true endpoints. In a two-treatment setup we study the surrogate-augmented Mantel–Haenszel estimator based on observations from different groups when the group effect is present. We compare the Mantel– Haenszel estimator with the one obtained by maximizing profile likelihood in a surrogate augmented setup. We observe that the performances of these estimators are very close. Key Words: Log-odds ratio; Mantel–Haenszel estimator; Profile likelihood; Surrogate endpoint; True endpoint.
1. INTRODUCTION Many clinical outcomes are such that the response variables are often difficult or highly expensive to measure, or the responses are delayed where short-term measures are needed for inferential and administrative purposes. Therefore, to evaluate the effects of treatments or exposures on the true endpoint in medical studies, a closely related variable is used as a surrogate response. For example, damage to the heart muscle due to myocardial infraction can be accurately assessed by arterioscintography. As this is an expensive procedure, peak cardiac enzyme level in the blood stream, which is more easily obtainable, is used as surrogate measure of heart vascular damage (Wittes et al., 1989). Often an observed value of the response variable in the middle of an experiment is considered as a surrogate endpoint. High-dose interferon-α is used for patients with age-related macular degeneration (ARMD) who progressively loose their vision. Observations are taken after 6 months and 1 year for surrogate and true endpoints, respectively (Buyse and Molenberghs, 1998). CD4 cell count is used as a surrogate for HIV patients where the survival time is the true endpoint. A statistical definition and validation criteria for surrogate endpoints were first introduced by Prentice (1989). Begg and Leung (2000) proposed that a measure of concordance can be a possible validation criterion for discrete random variables. In the present article, we are interested in inferential problems using properly validated surrogate endpoints where we assume that the validation is done a priori. Day and Duffy (1996) Received January 31, 2012; Accepted May 4, 2013 Address correspondence to Professor Atanu Biswas, Applied Statistics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700 108, India; E-mail: [email protected]
649
Downloaded by [ECU Libraries] at 03:26 26 April 2015
650
BANERJEE AND BISWAS
focused on the use of surrogate for the true endpoint in a clinical or preservation trial to shorten the duration of the trial and to increase the power. Freedman et al. (1992) supplemented these criteria with the so-called proportion explained by an intermediate endpoint for binary outcome variable, which has been extended by Lin et al. (1997) to failure time endpoints. Pepe (1992) considered a semiparametric model to estimate regression coefficients of covariates when surrogate data are present and showed its consistency. Chen (2000) developed a robust imputation method in the same setup. Chen et al. (2007) proposed two notions for a consistent surrogate and a strictly consistent surrogate to avoid the surrogate paradox. Molenberghs et al. (2001) introduced the situation where the surrogate is binary and the true endpoint is continuous, or vice versa. In addition, they considered the case of ordinal endpoints. In a recent study Banerjee and Biswas (2011) studied the estimation of the difference in success probabilities in the two-treatment setup in the presence of surrogate responses. In this article we consider the two-treatment problem. Suppose a multicenter Phase III clinical trial is being carried out for two competing treatments. If the responses are binary, a 2 × 2 table is generated as the outcome in each center. The Mantel–Haenszel (MH) estimator is widely used to get a pooled estimate odds ratio from multiple tables. A meta-analysis consisting of data on different studies is another example of applicability of the MH estimator. For example, Ergin and Ergin (2005) described a meta-analysis of 11 studies resulting eleven 2 × 2 tables of thrombolytic therapy versus placebo for the treatment of acute ischemic stroke patients. Assume that the response variable is binary for true as well as surrogate endpoints. Surrogate responses are more quickly and easily available than the true ones. There are two completely different problems studied in literature regarding surrogate endpoints: (i) the validation problem and (ii) the inferential problem. The validation problem deals with the derangement of appropriateness of a particular surrogate. Begg and Leung (2000) concluded that absolute validity of surrogate endpoint in the context of real clinical trial is unattainable. In the present article the surrogate endpoint is assumed to be validated (maybe from past experiments). However, it is preferable to use surrogate having high association with true endpoints (Begg and Leung, 2000; Prentice, 1989; Molenberghs et al., 2001). In this article we do not consider the validation problem; rather, we are interested in some inferential problems by optimal use of surrogate endpoints. Our approach developed in this article does not suffer from the surrogate paradox (Chen et al., 2007). The data where true and surrogate both are available are used for model building, and that is used where only surrogate endpoints are available for inferential purpose. Here we compare treatment effects in terms of log-odds ratio when large numbers of surrogate endpoints are available along with moderate numbers of true and surrogate paired observations when observations are available from multiple tables. Estimating the common odds ratio for multiple tables is a well studied problem (Breslow, 1981; Anscombe, 1956; Hauck, 1979; Breslow and Liang, 1982; Robins et al., 1986). We propose a methodology to obtain Mantel–Haenszel (MH) (Böhning et al., 2008; Mantel and Haenszel, 1959) and profile likelihood (PMLE) (Murphy and van der Vaart, 2000) based estimators in the surrogate augmented framework. We established asymptotic normality of the estimators. Our extensive simulation study also shows remarkable closeness in the behavior of the two surrogate augmented estimators: MH and PMLE. The rest of the article is as follows: Section 2 describes the data structure of multiple tables in the presence of surrogate endpoints. In section 3 we discuss the MH estimator and the PMLE without surrogate. Imputation with surrogate data is described in section 4. Section 5 studies the MH estimator and the PMLE in the presence of surrogate data.
MANTEL–HAENSZEL ESTIMATOR AND SURROGATE RESPONSES
651
Simulation results are reported in section 6 to show the closeness of the estimators. A simulated data example in a real situation is discussed in section 7, and section 8 concludes.
Downloaded by [ECU Libraries] at 03:26 26 April 2015
2. DATA STRUCTURE We consider a setup of two treatment binary endpoints with binary surrogates. Begg and Leung (2000) pointed out that for binary endpoints the probability of concordance is an indicator of association between true and surrogate endpoints. Suppose nA and nB patients are allotted to treatments A and B, respectively but we get only mA and mB true endpoints along with all surrogate endpoints within the stipulated time frame or cost limit, where mt
