Comparison of Schuirmann’s Two One-sided Tests With ...



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©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.

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OJPKTM

Online Journal of Pharmacokinetics ©

Volume 3 : 1-12, 2004.

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Comparison of Schuirmann’s 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: jihao.zhou@; yuleih@umich.edu; yuany@umich.edu.  §Corresponding author

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Abstract

Zhou J, He Y,  Yuan Y, Comparison of Schuirmann’s 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. Schuirmann’s 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 Schuirmann’s 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 Schuirmann’s 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 Schuirmann’s 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.

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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).   Schuirmann’s 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 Schuirmann’s 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.

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Methods

Normal and Non-normal Random Data Generation:

Skewness and kurtosis for our simulations are defined as

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and

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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:

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where μd is the mean difference of a pharmacokinetic parameter such as logeAUC between a two-drug combination treatment and one-drug reference while θL and θU are certain critically meaningful limits such as (80%, 125%), which is the FDA default decision criteria for claiming pharmaceutical bioequivalence. If 100(1-2α) confidence interval (CI) [pic]of the mean μd is entirely within (θL, θU), we reject the null hypothesis H0 and accept the alternative hypothesis Ha at a significance level α and conclude no drug-drug interaction; otherwise we fail to reject H0. Often times, natural logarithmic transformation (logeμd) is used and θL and θU 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 [pic] is constructed from mean estimate [pic]and standard error estimate [pic] 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 [pic] is constructed based on bootstrap mean estimate [pic] and bootstrap standard error estimate [pic].

Bootstrap-T CI (BT): CI [pic] is constructed based on bootstrap mean estimate [pic], bootstrap standard error estimate [pic] and bootstrap t-statistic estimate [pic].

Bootstrap-T CI (BT): CI [pic] is constructed based on bootstrap mean estimate [pic], bootstrap standard error estimate [pic] and bootstrap t-statistic estimate [pic].

 Bootstrap-Percentile CI (BP): [pic]is constructed from (100α%, 100(1-α)%) percentile of the bootstrap distribution of the estimated mean [pic].

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.

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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 Schuirmann’s 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 Schuirmann’s 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

|skewness |kurtosis |Prob |skewness |kurtosis |Prob |

|skewness |kurtosis |0.2231 |0 |

|Skewness |kurtosis | |WC |

|skewness |

|0 |

0 |0 |4.9 |6.6 |6.1 |7.0 |4.9 |7.0 |7.1 |7.0 |7.1 | |0.0 |0.4 |5.3 |6.8 |6.6 |7.4 |5.7 |7.3 |7.7 |7.8 |8.0 | |0.2 |1.2 |4.6 |6.5 |5.8 |6.7 |5.4 |6.6 |7.2 |7.6 |7.2 | |0.6 |1.2 |3.8 |4.5 |4.8 |5.7 |4.4 |5.4 |6.0 |6.3 |6.7 | |1.0 |1.6 |3.1 |2.5 |4.2 |4.9 |3.9 |4.5 |5.3 |5.6 |6.2 | |1.4 |3.0 |1.9 |1.7 |3.0 |3.8 |2.9 |3.5 |4.1 |4.5 |5.4 | |1.8 |5.0 |1.4 |0.9 |2.5 |3.2 |2.5 |2.8 |3.6 |4.2 |4.8 | |2.0 |6.4 |1.6 |0.9 |2.3 |2.9 |2.6 |2.6 |3.5 |3.9 |4.6 | |

Note: Simulations are based on μd = 0.2231/- 0.2231, σ = 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.

  

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Discussion

The parametric methods assume that data come from some underlying distributions and statistical inferences are based on these assumptions. If these assumptions are violated, the inferences may be misleading. As expected, Schuirmann’s TOST, a parametric approach with normality assumption, has poor statistical performance when data are heavily skewed. However, it is quite robust to slightly skewed data based on our simulations.

In contrast, nonparametric approaches make few assumptions about the underlying distribution from which the data are sampled. They are robust to the misspecification of underlying distribution, and can be used when the underlying distribution is unknown. Our simulations suggest that Bootstrap-T method is robust to non-normality. Surprisingly, other nonparametric approaches have poor statistical performances. However, since most nonparametric methods are based on large sample theory and our simulation sample size is only 12, which is the most widely used sample size in clinical drug-drug interaction studies (Huang, et al, 1999), thus they may fail to yield reasonable statistical performance under such small sample size. If data is highly skewed, we recommend the Bootstrap-T method over other nonparametric approaches considered here. Schuirmann’s TOST is still a very good choice even when data is slightly skewed.

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Conclusions

Our simulations have shown that Schuirmann’s TOST is fairly robust to slightly skewed data, but this approach may have poor statistical performance when data are heavily skewed. In contrast, Bootstrap-T approach consistently yields the best coverage probabilities and reasonable empirical sizes. 

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Acknowledgements

The authors would like to thank Dr. Eric Yan for valuable suggestions.

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References

Huang S.-M., Lesko LJ, and Wiiliams, RL: Assessment of the quality and quantity of dug-drug interaction studies in recent NDA submissions: Study design and data analysis issues.  Journal of Clin Pharmacology, 39: 1006-1014, 1999.

Schuirmann DJ: A Comparison of the Two One-Sides Tests Procedure and the Power Approach for Assessing the Equivalence of Average Bioavailablity.  Journal of Pharamcokinetics and Biopharmaceutics, 15(6): 657-680, 1987.

Chow S-C and Liu J-P.  Design and analysis of bioavailability and bioequivalence stuides. Marcel Dekker, Inc, 2000.

Fan X,  Felsovalyi A., Sivo SA, and Keenan SC: SAS® for Monte Carlo Studies: A guide for Quantitative Researchers. Cary, NC: SAS Institute Inc. pp 66-71, 2002.

Fleishman, AI.  A Method of Simulating Non-Normal Distributions. Psychometrika 1978, 43: 521-531, 1978.

Rosner B: Fundamentals of Biostatistics. Harrisonburg, VA: Duxbury press, pp 560. 1995

Efron B. and Tibshirani RJ: An Introduction to the Bootstrap. New York, NY: Chapman and Hall, Inc. 1993.

SAS: SAS Macros JACKBOOT. The SAS Institute, Inc. Cary, NC, 2003.

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This study was supported by Pfizer Inc. Yulei He was a recipient of a Pfizer internship (2003) and Ying Yuan was a recipient of a Pfizer internship (2004).

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©1996-2006 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.

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