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1 This article was downloaded by: [Northwestern University] On: 04 February 2015, At: 07:24 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Communications in Statistics - Simulation and Computation Publication details, including instructions for authors and subscription information: Computation of exact probability values for multiresponse permutation procedures (MRPP) Kenneth J. Berry a & Paul W. Mielke a a Colorado State University, Fort Collins, Colorado, Published online: 27 Jun To cite this article: Kenneth J. Berry & Paul W. Mielke (1984) Computation of exact probability values for multiresponse permutation procedures (MRPP), Communications in Statistics - Simulation and Computation, 13:3, , DOI: / To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the Content ) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at

2 COMMUN. STATIST.-SIMULA. COMPUTA., 13(3), (1984) COMPUTATION OF EXACT PROBABILITY VALUES FOR MULTI-RESPONSE PERMUTATION PROCEDURES (MRPP) Kenneth J. Berry and Paul W. Mielke, Jr. Colorado State University Fort Collins, Colorado ABSTRACT An algorithm is presented for computing the exact probability value associated with a recently-developed class of statistical inference techniques termed multi-response permutation procedures (?IRPP). 1. INTRODUCTION Multi-response permutation procedures (MRPP) are used to compare a. puc~ti classified groups of objects where r observations have been obtained from each of N objects (Mielke, 1979; Mielke & d., 1976, 1981a). The units of analysis are the distance functions (e.g., Euclidean distances) between the r-dimensional points associated with the r observations on each object. Discus- sions of specific applications of the MRPP and their relation to well-horn techniques such as the Kruskal-Wallis test and the permutation version of one-way analysis of variance are given by Mielke ~t d. (1981a, 1981b, 1982). Employing a moment approximation technique developed by Mielke Cd d. (1976) and by Mielke (1979), Berry and Mielke (1983) 1984 by Marcel Dekker, Inc.

3 418 BERRY AND MIELKE have presented an algor ithm for computing the finite population parameters and approximate MRPP probability value which is appropriate for moderate and large samples. An algorithm for computing exact MRPP probability values for small samples is given in the next sect ion. 2. DESCRIPTION OF ALGORITHM Subroutine MREP computes the number of paired comparisons to be made, determines the proportional weight of each group, and calculates the distance functions. Acting as a driver program, subroutine MREP then calls subroutine PERY which generates all possible permutations of the N objects with group sizes fixed, passing the permutations, one at a time, to subroutine DVAL. routine DVAL calculates the test statistics anci counts those less than or equal to the realized value. The count is returned to subroutine MREP where the probability value is computed. Following the notation of Mielke ~t d. (1976, 1981a) and Sub- Mielke (l979), let R = {ul,..., us! designate a finite population of N objects, let xi - [xli,...' x 1 denote r commensurate obr I servations on object oi (I = 1,..., N), and let Sl,..., Sg+l represent an exhaustive partitioning of the N objects comprising Cl into g+l disjoint groups, where ni 2 2 is the number of (clas- sified) objects in group Si (i = 1,..., g), K = zg n an? i=l i' n = N - K is the number of remaining (unclassified) objects in g+l the excess group S g+l cations). Let (the excess group is empty in many appl.i- designate the distance function (measure of separation) of r ob- servations between objects ot and o with v = 1 for Euclidean 3' distance, v = 2 for squared distance, etc. Subroutine NXEP N calculates the (2) A values and stores them for later retrieval. I 7 J n. -1 Let F = (21) I AI, J$i(~I)$i(~J) be the average betveen-object i I<J

4 EXACT PROBABILITY VALUES 419 (i = 1,..., g), distance function for all objects within group S i where C is the sumover all Iand J such that 151 < JzNand I<J $J~(W~) is unity if o belongs to S and zero otherwise for all I i I = 1,..., N. Then the test statistic of interest is the weighted average of the 5 values (i = 1,..., g) and is given by i 6 = Zg C 5 where C = n./k > 0 (i = 1,..., g) and ztz1ci = 1. i=1 i i i 1 An exact MRPP probability value requires the complete enumeration of all permutations of the N objects considered nl,..., n at a time, where Zg'ln. = N and nl,..., ngtl are fixed for g+l 1=11 all permutations. Utilizing a sequence of linked combination generators (Berry, 1982), each of the M = N!/(n~;f:n~:) required permutat ion structures is generated by subroutine PEW. Subroutine DVAL utilizes the permutation structures from sub- routine PERM to calculate the M possible 6 values. Because, under the null hypothesis, each of the M permutations has an equal probability (1/M) of occurrence, the exact MRPP probability value is the proportion of 6 values less than or equal to the realized (i.e., initial) 6 value. 3. RELATED ALGORITHES A version of algorithm AS 179 (Berry, 1982) is used in sub- routine PERM to compute the required permutation structures. 4. PROGRAE! CONSTATJTS -12 PREC = a rounding factor (1.0 x 10 ) applied to the initial (realized) 6 value to insure that subseque.nt 6 values which are actually identical to the initial value will not exceed the initial value simply due to machine rounding error. This work has been supported by the National Science Founda- tion, under Grant ATM Berry, K. J. (1982). AS 179: Enumeration of all permutations of multi-sets with fixed repetition numbers. Appl. Stati-~t

5 420 BERRY AND MIELKE Berry, K. J. and Mielke, P. W. (1983). Computation of finite population parameters and approximate probability values for mult i-response permutation procedures (MRP1). Ccwtmun. SAatL5R. - SimuLa. Compi~fa. 12, Mielke, P. W. (1979). On asymptotic non-normality of null distributions of MRPP statistics. Ccmmun. Strfin%. - Tizec~. Me-th. A8, Errata: A10, 1795; All, 847. Mielke, P. W., Berry, K. J. and Brier, G. W. (l98la). Application of multi-response permutation procedures for examining seasonal changes in monthly mean sea-level pressure patterns. ML. Wea. Rev. 109, Nielke, P. W., Berry, K. J., Brockwell, P. J. and Williams, J. S. (1981b). A class of nonparametric tests based on multiresponse permutat ion procedures. &cm&kci-,a 68, Mielke, P. W., Berry, K. J. and Johnson, E. S. (1976). Yultiresponse permutation procedures for I? t3tr: XL classifications. Cctnmur~. St*rf~bt. - Tim?. \ie,t/i. A5, Mielke, P. W., Berry, K. J. and Medina, J. G. (1982). Climax I and 11: distortion resistant residual analyses. J. At2)sP. u&&~t. 21, Received by Editorial Board member July, 1983; Revised December Recommended by Wm. J. Kennedy, Iowas State University, Ames, IA Refereed by Kathleen Shelley, Iowa State University, Ames, IA

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