A Permutation F-Test

[Pages:14]9.3 A Permutation F-Test

? The data setup is the same as Friedman's Test. That is, we have k treatments in either b blocks from a RCBD or b subjects from a SRMD.

? Assume ? is the overall mean, i is the ith treatment effect, j is the jth block (subject)

effect, and ij is the random error of the observation. The linear model for a RCBD or

SRMD is

yij = ? + i + j + ij and ij IIDN (0, 2).

(20)

? In the traditional analysis of variance (ANOVA) F -test, we are testing the null hypothesis of equality (no differences) in treatment effects

H0 : 1 = 2 = ? ? ? = k

against the alternative hypothesis

H1 : i = j for some i = j.

That is, if H1 is true, then all treatment effects are not equal. ? To compare k 3 treatment means, the test statistic is

F

=

SStrt/(k - 1) SSE/(N - k)

=

M Strt MSE

where N = kb the total number of observations in the data set.

? The RCBD or SRMD total sum of squares (SStotal) is partitioned into 3 components:

SST otal = SST rt + SSBlock + SSE

? Formulas to calculate SST otal, SST rt and SSBlock are

k

SST otal =

b

yi2j

-

y?2? kb

i=1 j=1

SST rt =

k

yi2? - y?2? b kb

i=1

SSBlock =

b

y?2j - y?2? k kb

j=1

SSE = SST otal - SST rt - SSBlock

where

y?2? is the correction factor. kb

Dot notation: y?? = the sum of all of the responses, yi? = sum of responses for treatment i, and y?j = sum of responses for block j.

? If the residual errors are approximately normally distributed with equal variances, then the test statistic F has an F -distribution with k - 1 degrees of freedom for the numerator and (k - 1)(b - 1) degrees of freedom for the denominator.

? In this case, the experimenter compares the F -statistic to the F [(k - 1)(b - 1)] distribution to determine a p-value for the test.

? However, if the assumptions are violated, (that is, the residual errors are not normally distributed with constant variance), then a permutation F -test may be appropriate.

212

The Steps in the Permutation F -Test (Monte-Carlo Approach)

? Calculate the F -statistic from the original data. Call this Fobs.

? Generate a large number Prep of permutations where observations are permuted within each block. That is, we are randomly permuting the treatments to the observations within blocks .

? For each permutation, calculate the F -statistic.

? Find the proportion of this set of Prep permutation F -statistics that are Fobs. This is the p-value for the Permutation F -test.

R code for Permutation F-Test for RCBD

# RCBD data from Table 4.4.3 (Higgins, page 130)

library(lmPerm)

# Enter vector of responses y ................
................

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