Multi-Response Permutation
Procedure
Matt Vanlandeghem
Natascha Israel
Grant Sorensen
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
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
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
MRPP - Example
• Freshwater Ecology
MRPP - Downfalls
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No direct procedure in SAS
Most use Euclidean distance
Computationally strenuous for large data sets
Number of total combinations
– N!/(n1!n2!)
• If M is too big (M>106)
– Approximate p values using Pearson type III
distribution
– Calculate mean (μ δobs), variance (σ2 δobs), and
skewness for delta under null hypothesis
– Standardized test statistic (T) under null
hypothesis
• T= (δobs - μ δobs)/ σ2 δobs
– Calculate p value
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
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
References
• Biondini, M.E., Mielke Jr., P.W., Berry, K.J. 1988. Data-dependent
permutation techniques for the analysis of ecological data.
Vegetatio 75: 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 Methods 5: 1409-1424.