An Efficient Permutation Approach for Classical and Bioequivalence Hypothesis Testing of Biomedical Shape Study Chunxiao Zhou 1, 2, 5 , Yongmei Michelle Wang 1, 3, 4, 5 \ Departments of 1 Statistics, 2 Electrical&Computer Engineering, 3 Psychology, 4 Bioengineering, 5 Beckman Institute, University of Illinois at Urbana-Champaign, IL, USA czhou4@uiuc.edu, ymw@uiuc.edu Abstract A new statistical permutation analysis method is presented in this paper to efficiently and accurately localize regionally specific shape differences between groups of 3D biomedical images. It can improve the system’s efficiency by approximating the permutation distribution of the test statistic with Pearson distribution series. This procedure involves the calculation of the first four moments of the permutation distribution, which are derived theoretically and analytically without any permutation. Furthermore, bioequivalence testing aims for practical significances between the two groups that are statistically significant with the shape differences larger than a desired threshold. Experimental results based on both classical and bioequivalence hypothesis tests using simulated data and real biomedical images are presented to demonstrate the advantages of the proposed approach. 1. Introduction In biomedical image analysis, the distribution of the data is usually unknown and sample size is quite small. When hypothesis testing involved in the analysis, the parametric approaches may not be optimal despite their simplicity. Permutation tests are among the most powerful nonparametric tests that can be applied when parametric tests do not work. They obtain p-values from permutation distributions of a test statistic, rather than from parametric distributions. In addition, permutation tests require few assumptions concerning statistical distributions but exchangeability. They belong to the nonparametric “distribution-free” category of hypothesis testing and are thus flexible, and have been used successfully in biomedical MR image analysis [6]. There are three major approaches to construct the permutation distribution [5]. First, exact permutation enumerates all possible arrangements. The second approach is an approximate permutation distribution based on random sampling from all possible permutations. Third, permutation distribution approximation uses the analytical moments of the exact permutation distribution under the null hypothesis. The computational cost is the main disadvantage of the exact permutation, due to the factorial increase in the number of permutations with the increasing number of subjects. The second technique has the problem of replication and causes more type I errors. When a large number of repeated tests are needed, the random permutation strategy is also computationally expensive to achieve satisfied p-value accuracy. Sometimes, the moments of the exact permutation distribution do not actually exist. In addition, if they ever exist, it is difficult to obtain them. These two factors are two main limitations of the third approach. In this paper, we propose to use a novel hybrid strategy to take advantage of nonparametric permutation tests and parametric Pearson distribution approximation for both efficiency and accuracy/flexibility. Specifically, we employ a general theoretical method to derive moments of permutation distribution for any linear test statistics. Here, the term “linear test statistic” refers to a linear function of test statistic coefficients, instead of that of data. The key idea is to separate the moments of permutation distribution into two parts, permutation of test statistic coefficients and function of the data. We can then obtain the moments without any permutation since the permutation of test statistic coefficients can be derived theoretically. Given the first four moments, the permutation distribution can be well fitted by Pearson distribution series. The p-values are then estimated without any real permutation. For multiple comparison of two-group difference, given the sample size n 1 = 21 and n 2 = 21, the number of tests is m = 2000. In this case, m×(n 1 +n 2 )!/ n 1 !/ n 2 ! ≈ 1.1×10 15 permutations are needed for an exact permutation test. Even for 20,000 random permutations per test, 4×10 7 permutations are still required. Alternatively, our hybrid permutation method using Pearson distribution approximation only involves the calculation of analytically derived first four moments of exact permutation distributions while 2008 International Conference on BioMedical Engineering and Informatics 978-0-7695-3118-2/08 $25.00 © 2008 IEEE DOI 10.1109/BMEI.2008.192 1661 2008 International Conference on BioMedical Engineering and Informatics 978-0-7695-3118-2/08 $25.00 © 2008 IEEE DOI 10.1109/BMEI.2008.192 737