Dario Mazzaro, Fortunato Pesarin, Luigi Salmaso (University of Padova, Italy)

Repeated measures designs: a permutation approach and closed testing

We deal with permutation testing for multiresponse repeated measures designs and we consider a replicated unbalanced homoscedastic factorial design with fixed effects (Milliken, 1984) as the basic experimental plan. The design responses are measured in L time occasions. The usual linear model for single responses is: ; j = 1, 2; i = 1, 2; l = 1, ..., L; r = 1, ..., ; , where are the experimental responses; is the population mean for the l-th measure; is the effect of the j-th level of factor A in the l-th measure; is the effect of the i-th level of factor B in the l-th measure; is the interaction effect between levels j and i of factors A and B in the l-th measure; are exchangeable experimental errors in the l measure from an unknown distribution P with zero mean and variance ; finally, is the number of observations for each factor's levels combination. Thus, the total sample size is . The overall system of hypotheses is , against the alternative , where the three partial hypotheses for each measure are vs , vs , vs , so that, the null hypothesis is true if all three partial sub hypotheses are true. Let us consider the three partial tests for effects in every measure. For example, the l-th permutation test for the effect of factor A is constructed by a linear combination of the two following statistics: , , where the weights are defined as: , , , , we jointly consider the L measures, then the permutation solution is based on the nonparametric combination methodology (Pesarin, 1999). It is worth noting that the new permutation approach, presented here, is highly robust, with respect to departures from normality of error terms in the linear model for responses, since it is conditioned to the sufficient statistic represented by the data matrix. A comparative simulation study has been performed in order to evaluate the power of such exact tests. Multiple comparisons for the above tests are also discussed (Westfall and Young, 1993).


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  2. Pesarin F. (1999). Permutation testing of multidimensional hypotheses by nonparametric combination of dependent tests. CLEUP, Padua.
  3. Pesarin F., Salmaso L. (1999). Exact permutation testing on effects in replicated 2k factorial designs. Working Paper series, Department of Statistics, University of Padua. Submitted for publication.
  4. Westfall P.H., Young S.S. (1993) Resampling-based Multiple Testing. Wiley, New York.