Journal of Biopharmaceutical Statistics
ISSN: 1054-3406 (Print) 1520-5711 (Online) Journal homepage: http://www.tandfonline.com/loi/lbps20
Sample size calculation based on efficient unconditional tests for clinical trials with historical controls Guogen Shan, Sheniz Moonie & Jay Shen To cite this article: Guogen Shan, Sheniz Moonie & Jay Shen (2014): Sample size calculation based on efficient unconditional tests for clinical trials with historical controls, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2014.1000545 To link to this article: http://dx.doi.org/10.1080/10543406.2014.1000545
Accepted author version posted online: 31 Dec 2014. Published online: 31 Dec 2014. Submit your article to this journal
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Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=lbps20 Download by: [Deakin University Library]
Date: 06 November 2015, At: 08:15
JOURNAL OF BIOPHARMACEUTICAL STATISTICS http://dx.doi.org/10.1080/10543406.2014.1000545
Sample size calculation based on efficient unconditional tests for clinical trials with historical controls Guogen Shana, Sheniz Mooniea and Jay Shenb
Downloaded by [Deakin University Library] at 08:15 06 November 2015
a Department of Environmental and Occupational Health, Epidemiology and Biostatistics Program, School of Community Health Sciences, University of Nevada Las Vegas, Las Vegas, Nevada, USA; bDepartment of Health Care Administration and Policy, School of Community Health Sciences, University of Nevada Las Vegas, Las Vegas, Nevada, USA
ABSTRACT
ARTICLE HISTORY
In historical clinical trials, the sample size and the number of success in the control group are often considered as given. The traditional method for sample size calculation is based on an asymptotic approach developed by Makuch and Simon (1980). Exact unconditional approaches may be considered as alternative to control for the type I error rate where the asymptotic approach may fail to do so. We provide the sample size calculation using an efficient exact unconditional testing procedure based on estimation and maximization. The sample size using the exact unconditional approach based on estimation and maximization is generally smaller than those based on the other approaches.
Received 8 August 2013 Accepted 17 October 2014 KEYWORDS
Exact test; E+M approach; historical clinical trial; sample size; unconditional test
1. Introduction An important goal of phase II clinical trials is to determine the effectiveness of a new treatment. The effectiveness is often seen as an improvement in the response rate, and it may be confirmed by conducting a randomized clinical trial by comparing the new treatment to one or more established golden standards if they exist. In randomized clinical trials (Simon et al. 1985), subjects are generally randomized into two or more treatment groups, and the same treatment is applied within each group. It is often true that one of the groups is a control group or a group receiving the standard treatment. In some cases, a randomized clinical trial is not applicable due to some practical issues. For example, it may not be easy to find a control group in rare cancer trials. A historical control trial may be then considered as an alternative to determine the effectiveness of a new treatment by comparing the new treatment to the historical data. The historical data could be obtained from a very recent clinical trial or a combination of historical clinical trials. The historical control trial is preferred over the randomized clinical trial for sample size savings, ethical considerations, and budget limitations (Makuch and Simon, 1980; Fleming, 1987; Chang et al., 2004). The widely used method to calculate the sample size for clinical trials with historical controls was proposed by Makuch and Simon (1980) for binary outcomes (referred to as the MS approach). The angular transformation of estimated response rates is used to improve the variance-stabilizing properties and has good normal approximation. The sample size for a new treatment group is then determined by solving Equation (5) in Makuch and Simon (1980) iteratively. The MS approach has been extensively investigated by many researchers (Fleming, 1987; Wolff et al., 1987; Chow et al., 2007; Hutson et al., 2009; Hunsberger et al., 2009). Subsequentily, Chang et al. (1999) proposed a two-stage design for phase II clinical trials with binary outcomes based on normal approximation of the angular transformation of response rates to save the sample size. CONTACT Guogen Shan [email protected] Department of Environmental and Occupational Health, Epidemiology and Biostatistics Program, School of Community Health Sciences, University of Nevada Las Vegas, Las Vegas, NV 89154, USA. © 2015 Taylor & Francis
Downloaded by [Deakin University Library] at 08:15 06 November 2015
2
G. SHAN ET AL.
An alternative to the sample size calculation based on asymptotic approaches is an exact approach. Exact approaches for analyzing independent data have been studied for many years and significant progress were documented by Fisher (1970), Barnard (1945), Agresti (2002), Hilton and Mehta (1993), Mehta and Hilton (1993), and Shan et al. (2012). Chang et al. (2004) were among the first researchers to apply the exact unconditional approach for the sample size calculation for phase II clinical trials with historical controls. The exact unconditional method used in their manuscript was proposed by Berger and Boos (1994). Chang et al. (2004) found that the sample sizes produced by the exact unconditional approach and the asymptotic based MS approach are usually fairly similar. Recently, another efficient unconditional testing approach based on estimation and maximization (referred to as the E+M approach) has been proposed by Lloyd (2008) to further reduce the conservativeness of traditional unconditional exact test based on maximization. It has been shown to gain higher power for parallel-arm comparison for proportions. The p-value of the estimated approach is calculated by replacing the unknown nuisance parameter in the null tail probability with the estimator under the null hypothesis. The p-value of the E+M approach is then obtained by maximizing the null tail probability using the estimated p-value as a test statistic. The E+M approach has been successfully applied to the difference between two proportions (Lloyd and Moldovan, 2008), trends among K binomial populations (Shan et al., 2012; Shan et al., 2013), and the cluster data (Shan, 2013; Shan and Ma, 2014). It has been recommended for use in practice due to its efficiency compared to other testing procedures. We provide sample size calculation for a clinical trial with historical control based on this exact unconditional E+M approach. We then compare sample size calculation based on the asymptotic MS approach, the exact unconditional approach in Chang et al. (2004), and the new exact unconditional E+M approach. The remainder of this article is organized as follows. Section 2 will briefly review existing methods for sample size calculation for historical clinical trials, and develop the efficient unconditional procedure. Section 3 will provide the sample size calculation criterion and algorithm, and compare the sample size of the competing tests. Section 4 will provide some concluding remarks.
2. Methods Let nc and ne be the total number of subjects in the control group and the experimental group, respectively. The total number of subjects in the study is n ¼ ne þ nc . With the number of response Xc and Xe in the control group and the experimental group, associated estimated response rates are ^pc ¼ Xc =nc and ^pe ¼ Xe =ne , respectively. Investigators are often interested to test the hypotheses: H0: pe pc against the alternative Ha: pe
ISSN: 1054-3406 (Print) 1520-5711 (Online) Journal homepage: http://www.tandfonline.com/loi/lbps20
Sample size calculation based on efficient unconditional tests for clinical trials with historical controls Guogen Shan, Sheniz Moonie & Jay Shen To cite this article: Guogen Shan, Sheniz Moonie & Jay Shen (2014): Sample size calculation based on efficient unconditional tests for clinical trials with historical controls, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2014.1000545 To link to this article: http://dx.doi.org/10.1080/10543406.2014.1000545
Accepted author version posted online: 31 Dec 2014. Published online: 31 Dec 2014. Submit your article to this journal
Article views: 52
View related articles
View Crossmark data
Citing articles: 3 View citing articles
Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=lbps20 Download by: [Deakin University Library]
Date: 06 November 2015, At: 08:15
JOURNAL OF BIOPHARMACEUTICAL STATISTICS http://dx.doi.org/10.1080/10543406.2014.1000545
Sample size calculation based on efficient unconditional tests for clinical trials with historical controls Guogen Shana, Sheniz Mooniea and Jay Shenb
Downloaded by [Deakin University Library] at 08:15 06 November 2015
a Department of Environmental and Occupational Health, Epidemiology and Biostatistics Program, School of Community Health Sciences, University of Nevada Las Vegas, Las Vegas, Nevada, USA; bDepartment of Health Care Administration and Policy, School of Community Health Sciences, University of Nevada Las Vegas, Las Vegas, Nevada, USA
ABSTRACT
ARTICLE HISTORY
In historical clinical trials, the sample size and the number of success in the control group are often considered as given. The traditional method for sample size calculation is based on an asymptotic approach developed by Makuch and Simon (1980). Exact unconditional approaches may be considered as alternative to control for the type I error rate where the asymptotic approach may fail to do so. We provide the sample size calculation using an efficient exact unconditional testing procedure based on estimation and maximization. The sample size using the exact unconditional approach based on estimation and maximization is generally smaller than those based on the other approaches.
Received 8 August 2013 Accepted 17 October 2014 KEYWORDS
Exact test; E+M approach; historical clinical trial; sample size; unconditional test
1. Introduction An important goal of phase II clinical trials is to determine the effectiveness of a new treatment. The effectiveness is often seen as an improvement in the response rate, and it may be confirmed by conducting a randomized clinical trial by comparing the new treatment to one or more established golden standards if they exist. In randomized clinical trials (Simon et al. 1985), subjects are generally randomized into two or more treatment groups, and the same treatment is applied within each group. It is often true that one of the groups is a control group or a group receiving the standard treatment. In some cases, a randomized clinical trial is not applicable due to some practical issues. For example, it may not be easy to find a control group in rare cancer trials. A historical control trial may be then considered as an alternative to determine the effectiveness of a new treatment by comparing the new treatment to the historical data. The historical data could be obtained from a very recent clinical trial or a combination of historical clinical trials. The historical control trial is preferred over the randomized clinical trial for sample size savings, ethical considerations, and budget limitations (Makuch and Simon, 1980; Fleming, 1987; Chang et al., 2004). The widely used method to calculate the sample size for clinical trials with historical controls was proposed by Makuch and Simon (1980) for binary outcomes (referred to as the MS approach). The angular transformation of estimated response rates is used to improve the variance-stabilizing properties and has good normal approximation. The sample size for a new treatment group is then determined by solving Equation (5) in Makuch and Simon (1980) iteratively. The MS approach has been extensively investigated by many researchers (Fleming, 1987; Wolff et al., 1987; Chow et al., 2007; Hutson et al., 2009; Hunsberger et al., 2009). Subsequentily, Chang et al. (1999) proposed a two-stage design for phase II clinical trials with binary outcomes based on normal approximation of the angular transformation of response rates to save the sample size. CONTACT Guogen Shan [email protected] Department of Environmental and Occupational Health, Epidemiology and Biostatistics Program, School of Community Health Sciences, University of Nevada Las Vegas, Las Vegas, NV 89154, USA. © 2015 Taylor & Francis
Downloaded by [Deakin University Library] at 08:15 06 November 2015
2
G. SHAN ET AL.
An alternative to the sample size calculation based on asymptotic approaches is an exact approach. Exact approaches for analyzing independent data have been studied for many years and significant progress were documented by Fisher (1970), Barnard (1945), Agresti (2002), Hilton and Mehta (1993), Mehta and Hilton (1993), and Shan et al. (2012). Chang et al. (2004) were among the first researchers to apply the exact unconditional approach for the sample size calculation for phase II clinical trials with historical controls. The exact unconditional method used in their manuscript was proposed by Berger and Boos (1994). Chang et al. (2004) found that the sample sizes produced by the exact unconditional approach and the asymptotic based MS approach are usually fairly similar. Recently, another efficient unconditional testing approach based on estimation and maximization (referred to as the E+M approach) has been proposed by Lloyd (2008) to further reduce the conservativeness of traditional unconditional exact test based on maximization. It has been shown to gain higher power for parallel-arm comparison for proportions. The p-value of the estimated approach is calculated by replacing the unknown nuisance parameter in the null tail probability with the estimator under the null hypothesis. The p-value of the E+M approach is then obtained by maximizing the null tail probability using the estimated p-value as a test statistic. The E+M approach has been successfully applied to the difference between two proportions (Lloyd and Moldovan, 2008), trends among K binomial populations (Shan et al., 2012; Shan et al., 2013), and the cluster data (Shan, 2013; Shan and Ma, 2014). It has been recommended for use in practice due to its efficiency compared to other testing procedures. We provide sample size calculation for a clinical trial with historical control based on this exact unconditional E+M approach. We then compare sample size calculation based on the asymptotic MS approach, the exact unconditional approach in Chang et al. (2004), and the new exact unconditional E+M approach. The remainder of this article is organized as follows. Section 2 will briefly review existing methods for sample size calculation for historical clinical trials, and develop the efficient unconditional procedure. Section 3 will provide the sample size calculation criterion and algorithm, and compare the sample size of the competing tests. Section 4 will provide some concluding remarks.
2. Methods Let nc and ne be the total number of subjects in the control group and the experimental group, respectively. The total number of subjects in the study is n ¼ ne þ nc . With the number of response Xc and Xe in the control group and the experimental group, associated estimated response rates are ^pc ¼ Xc =nc and ^pe ¼ Xe =ne , respectively. Investigators are often interested to test the hypotheses: H0: pe pc against the alternative Ha: pe
