1. Multi-Response Permutation Procedure Matt Vanlandeghem Natascha Israel Grant Sorensen

2. MRPP - Introduction • Nonparametric procedure • Tests null hypothesis of no difference between two or more groups of entities • Groups determined a priori • Procedure based on distances among data points within groups

3. How MRPP works • Scatter diagram • Measure distance among all points within each group in diagram • Take averages of distances for each group • Calculate observed delta (δobs) • Weighted average • Determine if δobs is unusual • Calculate number of possible permutations (M) • Calculate p value

4. MRPP - Benefits • Avoids making distributional assumptions • No need for normality or equality of variance (or variance-covariance matrices) • Robust to outliers • If sample size (n) < number of variables (p) • Non-parametric approach to MANOVA

5. MRPP - Example • Freshwater Ecology

6. MRPP - Downfalls • No direct procedure in SAS • Most use Euclidean distance • Computationally strenuous for large data sets • Number of total combinations • N!/(n1!n2!)

7. If M is too big (M>106) • Approximate p values using Pearson type III distribution • Calculate mean (μ δobs), variance (σ2δobs), and skewness for delta under null hypothesis • Standardized test statistic (T) under null hypothesis • T= (δobs - μ δobs)/ σ2δobs • Calculate p value

8. MRPP - Caveats • Not always the best alternatives to MANOVA • Test stat can be un-reliable for large data sets (n >20) • Monte Carlo exercises • standard parametric tests out performed MRPP when data satisfies standard assumptions • REF

9. MRPP – Alt Examples • Social Sciences • Multi-response data that tends to violate normality • Data from only a small group of individuals • Medical sciences • 20 year study on cancer patients • Low n but high p • Data prone to extreme outliers

10. References • Biondini, M.E., MielkeJr., P.W., Berry, K.J. 1988. Data-dependent permutation techniques for the analysis of ecological data. Vegetatio75: 161-168 (1988). • Cai, L. 2004. Multi-response Permutation Procedure as An Alternative to the Analysis of Variance: an SPSS Implementation. Department of Psychology, University of North Carolina. • Mielke, P.W. & Berry, K.J. 1994. Permuation tests for common locations among samples with unequal variances. Journal of Educational and Behavioral Statistics 19:217-236. • Mielke, P.W. & Berry, K.J. 2001. Permuation methods: A distance function approach. New York: Springer-Verlag. • Mielke, P.W., Berry K.J., Johnson, E.S. 1976. Multi-response permuation procedure for a priori classifications. Communications in Statistics – Theory and Methods5: 1409-1424.