ࡱ> FHCDE5@ Ebjbj22 .XXE@ddddd@e^fffffg g$#Ru$5nffnn$ffYrrrnffrnr*rssfe a&d\qss4o0s2rsd@ss 'g%ilrj$k!'g'g'g$$@@]dr @@d HYPERLINK "http://www.comcen.com.au/%7Ejournals/Pharkin.htm" BACK TO MAIN PAGE  1996-2005 All Rights Reserved. Online Journal of Pharmacokinetics. You may not store these pages in any form except for your own personal use. All other usage or distribution is illegal under international copyright treaties. Permission to use any of these pages in any other way besides the before mentioned must be gained in writing from the publisher. This article is exclusively copyrighted in its entirety to OJK publications. This article may be copied once but may not be, reproduced or re-transmitted without the express permission of the editors.  OJPKTM Online Journal of Pharmacokinetics Volume 3 : 1-12, 2004.  Comparison of Schuirmanns Two One-sided Tests With Nonparametric Two One-sided Tests for Non-normal Data in Clinical Pharmacokinetic Drug-Drug Interaction Studies Jihao Zhou, PhD1, Yulei He, MS 2, Ying Yuan, MA, MS1 1Department of Clinical Biostatistics, Pfizer Global Research and Development, La Jolla Laboratories, 11085 Torreyana Road, San Diego, California, 92121, USA. 2Department of Biostatistics, The University of Michigan, Ann Arbor, Michigan, 48109 USA. ;Fax: 01(858) 678-8248, e-mail:  HYPERLINK "mailto:jihao.zhou@pfizer.com" jihao.zhou@pfizer.com;  HYPERLINK "mailto:yuleih@umich.edu" yuleih@umich.edu;  HYPERLINK "mailto:yuany@umich.edu" yuany@umich.edu. Corresponding author  Abstract Zhou J, He Y, Yuan Y, Comparison of Schuirmanns Two One-sided Tests With Nonparametric Two One-sided Tests for Non-normal Data in Clinical Pharmacokinetic Drug-Drug Interaction Studies. Online Journal of Pharmacokinetics. 3 : 1-12, 2004. Schuirmanns two one-sided tests (TOST) approach is widely used in clinical drug-drug interaction studies. However, it requires normality assumption, which may not hold in practice. The objective of this paper was to investigate the statistical performance of Schuirmanns TOST procedure for non-normal data, and then to compare it with nonparametric TOSTs. Monte Carlo simulations were used to generate non-normal data with different skewness and kurtosis. The statistical performances of Schuirmanns TOST and nonparametric method-based TOSTs were compared in terms of empirical power, size and coverage probability. The nonparametric TOST approaches were based on Wilcoxon signed-rank test, Jackknife method, and Bootstrap methods. Our simulations show that Schuirmanns TOST is fairly robust to slightly skewed data, but may have poor performance when data are heavily skewed. In contrast, Bootstrap-T approach consistently yields the best coverage probabilities and reasonable empirical sizes. KEY WORDS drug-drug interaction, two-one sided tests, nonparametric, clinical trial.  Introduction Clinical drug-drug interaction is a key issue in clinical practice, especially for HIV/AIDS teatment. Seventy percent of clinical drug-drug interaction studies were conducted as a fixed-sequence design (sometimes called a crossover like design) (Huang, et al, 1999). Schuirmanns two one-sided tests (TOST) approach (Schuirmann, 1987), the FDA preferred method for clinical drug-drug interaction studies, is based on normality assumption of pharmacokinetic data such as logeAUC and logeCmax. This assumption may not be always satisfied in reality. In addition, the ability to detect the non-normality is low given the commonly used small sample sizes. Based on a recent FDA survey, the median sample size for clicical drug-drug interaction studes submitted to US FDA was 12 (Huang, et al, 1999). Nonparametric method-based TOSTs, such as TOST based on Wilcoxon signed-rank test and Bootstrap methods, have been proposed to address this problem (Chow and Liu, 2000). In this paper, we first investigated the statistical peroformance of Schuirmanns TOST approach for non-normal data in terms of empirical power, size and coverage probability, and then, compared it with nonparametric method-based TOSTs under non-normality. The nonparametric methods included Wilcoxon signed-rank test, Jackknife method, and Bootstrap methods.  Methods Normal and Non-normal Random Data Generation: Skewness and kurtosis for our simulations are defined as  and . For normal data, both skewness and kurtosis are zero. We use SAS function RANNOR() to generate normally distributed data. For non-normal data generation, we modifed SAS macro (Fan, et al, 2002) programs and adopt Fleishman s power transformation method (Fleishman, 1978), which uses a cubic polynomial transformation to transform a normally distributed variable to a variable with specified degrees of skewness and kurtosis. Schuirmann s TOST: In a clinical drug-drug interaction study using the fixed-sequence design, the hypotheses of testing can usually be formulated as follows:  where md is the mean difference of a pharmacokinetic parameter such as logeAUC between a two-drug combination treatment and one-drug reference while qL and qU are certain critically meaningful limits such as (80%, 125%), which is the FDA default decision criteria for claiming pharmaceutical bioequivalence. If 100(1-2a) confidence interval (CI) of the mean md is entirely within (qL, qU), we reject the null hypothesis H0 and accept the alternative hypothesis Ha at a significance level a and conclude no drug-drug interaction; otherwise we fail to reject H0. Often times, natural logarithmic transformation (logemd) is used and qL and qU are 0.2231 (loge(80%), loge(125%)), since the pharmacokinetic parameters such as AUC and Cmax are generally believed to follow log-normal distribution (Chow and Liu, 2000). Nonparametric TOST: Wilcoxon signed rank test (WC):It is a nonparametric test for the median of the difference to be zero consisting of sorting the difference values from smallest to largest, assigning ranks to the absolute values and then finding the sum of the ranks of the positive differences (Rosner, 1995). If H0 is true, the sum of the ranks of the positive differences should be about the same as the sum of the ranks of the negative differences. The CI of the median difference is constructed from the distribution of these ranks under H0. Jackknife confidence interval approach ( JK): Jackknife is a resampling approach by drawing samples that leave out one observation at a time. Assuming a normal sampling distribution, the Jackknife CI is constructed from mean estimate and standard error estimate which are calculated from the Jackknife samples (Efron and Tibshirani, 1993). Bootstrap methods: Bootstrap is a resampling approach by drawing samples of the same size as the original using sampling with replacement from the observed data and statistics are calculated from the bootstrap samples (Efron and Tibshirani, 1993). There are many variations of bootstrap to get CI. Briefly, the following methods were considered in our simulations. Bootstrap-Normal CI (BN): assuming a normal sampling distribution, CI is constructed based on bootstrap mean estimate and bootstrap standard error estimate . Bootstrap-T CI (BT): CI is constructed based on bootstrap mean estimate , bootstrap standard error estimate and bootstrap t-statistic estimate . Bootstrap-T CI (BT): CI is constructed based on bootstrap mean estimate , bootstrap standard error estimate and bootstrap t-statistic estimate . Bootstrap-Percentile CI (BP): is constructed from (100a%, 100(1-a)%) percentile of the bootstrap distribution of the estimated mean . Bootstrap-Hybrid CI (BH): the reverse of Bootstrap-Percentile method. Bootstrap-BC CI (BBC): a Bootstrap-Percentile CI corrected for bias. Bootstrap-BCa CI (BBCa): a Bootstrap-Percentile CI corrected for both bias and skewness. We adopted the SAS Macro programs provided by The SAS Institute Inc. (SAS, 2003) to implement the above nonparametric TOST methods. Monte Carlo Simulation: Monte Carlo simulation relates to or involves the use of random sampling techniques and often the use of computer simulation to obtain approximate solutions to mathematical or physical problems especially in terms of a range of values each of which has a calculated probability of being the solution (Fan, et al, 2002). In a nutshell, Monte Carlo simulation simulates the sampling process from a defined population repeatedly by using a computer instead of actually drawing multiple samples to estimate the sampling distributions of the events of the interest. In our simulation, we generated 5000 non-normal sample replicates of size 12 for each scenario of skewness and kurtosis, and then applied different TOSTs methods to these replicates obtain the performance evaluation statistics, which include empirical coverage probability, empirircal size and empirical power. Specifically, empirical coverage probability is determined as the estimated proportion of the replicates whose CI includes the true population mean. Empirircal size is determined as the estimated proportion of the repliates in which we reject H0 given H0 is true. Empirical power is determined as the estimated proportion of the repliates in which we reject H0 given H0 is true.  Results Tables 1 and 2 show the empirical coverage probability, empirical size and power for non-normal data with different scenarios of skewness and kurtosis. The coverage probability of Schuirmanns TOST is close to the nominal value (90%) when skewness is less than one. When skewness is further greater than one, the coverage probability decreases gradually. The empirical size of Schuirmanns TOST at the positive boundary gradually deviates from the nominal value (5%) while the empirical size at the negative boundary decreases, as data is getting more positively skewed. The power curve on non-normal data is asymmetric and positively skewed. Table 1 Empirical coverage probability (%) for non-normal data skewnesskurtosisProbskewnesskurtosisProbskewnesskurtosisProb0.00.490.30.61.289.31.42.688.30.00.890.30.80.489.31.43.088.20.01.290.00.80.889.41.63.085.80.20.489.80.81.289.51.63.487.60.20.890.11.00.888.71.63.886.80.21.289.61.01.289.11.84.286.10.40.490.31.01.689.41.84.687.20.40.890.91.21.488.31.85.087.00.41.289.91.21.888.51.85.685.60.60.490.01.22.287.61.86.086.20.60.889.61.42.288.11.86.486.8Note: Simulations are based on md = 0.2231, s = 0.24, sample size = 12, and the number of replicates = 5000. Table 2 Empirical size and power (%) for non-normal data mdmdskewnesskurtosis0.22310-0.2231skewnesskurtosis0.22310-0.22310.00.44.782.84.81.00.88.785.92.30.00.85.182.35.31.01.28.384.12.80.01.25.181.85.11.01.68.184.42.40.20.46.281.24.21.21.49.484.72.30.20.85.582.14.41.21.88.784.42.10.21.25.682.54.31.22.29.884.92.30.40.45.782.94.11.42.29.985.02.00.40.85.882.03.71.42.69.783.72.10.41.26.181.94.21.43.09.783.81.90.60.46.882.13.61.63.012.385.21.30.60.86.583.53.91.63.410.584.81.50.61.26.981.93.61.63.811.584.81.60.80.47.285.03.11.84.212.083.01.20.80.87.584.12.91.84.611.284.81.10.81.27.083.72.71.85.011.184.51.3Note: Simulations are based on md = 0.2231/0/-0.2231, s = 0.24, sample size = 12, and the number of replicates = 5000. Tables 3 and 4 present the coverage probability and empirical size of both Schuirmanns and nonparametric TOSTs. Bootstrap-T method appears to be the most robust approach compared to other TOST approaches in terms of coverage and empirical size. It consistently gives reasonable coverage probability and empirical size under different skewness and kurtosis in our simulation. Wilcoxon signed-rank test has higher coverage probability and lower empirical size than those of Schuirmanns TOST. It gives proper coverage probability only when skewness is low, but its empirical size is deviated from nominal value. Other nonparametric TOSTs methods generally fail to yield satisfactory coverage probability and empirical size for non-normal data. Table 3 Coverage probability (%) of parametric and nonparametric TOSTs parametric*NonparametricSkewnesskurtosisWCJKBNBTBHBPBBCBBCa0089.990.687.085.490.085.485.385.485.20.00.490.190.487.085.589.485.585.185.084.90.21.290.392.087.485.689.085.585.084.383.30.61.289.592.386.784.989.184.984.684.484.11.01.688.892.186.084.489.484.184.384.684.51.43.088.591.385.984.389.984.184.484.784.91.85.086.788.084.282.189.081.482.783.083.02.06.486.587.784.182.389.381.582.582.983.3Note: Simulations are based on md = 0.2231, s = 0.24, sample size = 12, and the number of replicates = 5000. Bootstrap samples of 2000 are used. *: Schuirmann s two one-sided tests; for the abbreviations of nonparametric methods see Methods section. Table 4 Empirical size (%) of parametric and nonparametric TOSTs parametric*NonparametricskewnesskurtosisWCJKBNBTBHBPBBCBBCamd = 0.2231004.92.16.57.54.97.47.37.47.40.00.44.92.16.47.35.17.27.57.57.60.21.25.82.77.48.46.28.58.58.79.00.61.27.03.68.89.96.410.19.79.69.31.01.68.34.89.910.76.611.310.49.89.21.43.09.57.211.111.86.412.611.210.59.41.85.011.811.013.314.77.615.813.512.611.32.06.412.111.313.915.07.916.114.113.011.6md = -0.2231004.96.66.17.04.97.07.17.07.10.00.45.36.86.67.45.77.37.77.88.00.21.24.66.55.86.75.46.67.27.67.20.61.23.84.54.85.74.45.46.06.36.71.01.63.12.54.24.93.94.55.35.66.21.43.01.91.73.0?@QRSTUu ݘ~v^P?: hJo( hJ5CJOJQJ\^JaJhJCJOJQJ^JaJ/hJ56B* CJ8OJQJ\]^JaJ8phhJCJaJ#hJ56B* CJ4\]aJ4phjh UhJCJOJQJaJhJB* CJOJQJaJphhJCJOJQJaJhJjh U hJ0J5CJOJQJ\aJhJ5CJOJQJ\aJ%jhJ5CJOJQJU\aJSU q r s&,, ,,},?,,,,,PS,.,,,PS ,PS,PS,PS,PS ,PS,edN\? $a$edN\? [$\$edN\? $a$edN\? $a$edN\? edN\? edJ#D$a$edN\? 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BMC154dComparison of Schuirmann s Two One-sided Tests With Nonparametric Two One-sided Tests for Non-normal Data in Clinical Pharmacokinetic Drug-Drug Interaction StudiesjizhouVincent GuerriniOh+'0$LXht      Comparison of Schuirmanns Two One-sided Tests With Nonparametric Two One-sided Tests for Non-normal Data in Clinical Pharmacokinetic Drug-Drug Interaction StudiesompjizhousizhizhBMC154dVincent Guerriniuir2ncMicrosoft Word 10.0@F#@!@ $@ $e C;՜.+,D՜.+, hp   Life Science Communications Ltd~#EA Comparison of Schuirmanns Two One-sided Tests With Nonparametric Two One-sided Tests for Non-normal Data in Clinical Pharmacokinetic Drug-Drug Interaction Studies Title 8@ _PID_HLINKSAl^ 1http://www.comcen.com.au/%7Ejournals/Pharkin.htmUg mailto:yuany@umich.edu#mailto:yuleih@umich.eduL:mailto:jihao.zhou@pfizer.com^1http://www.comcen.com.au/%7Ejournals/Pharkin.htm  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdeghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./012456789:<=>?@ABGRoot Entry F0'IData fE1TableWordDocument.SummaryInformation(3DocumentSummaryInformation8;CompObjj  FMicrosoft Word Document MSWordDocWord.Document.89q