page.test.RdGiven N replications of k different treatments/conditions,
tests whether the median ordinal ranks \(m_i\) of the treatments
are identical $$m_1 = m_2 = \ldots = m_k$$ against the alternative
hypothesis $$m_1 \leq m_2 \leq \ldots \leq m_k$$ where at least
one of the inequalities is a strict inequality (Siegel and Castellan
1988, p.184). Given that even a single point change in the distribution of
ranks across conditions represents evidence against the null hypothesis,
the Page test is simply a test for some ordered differences in
ranks, but not a 'trend test' in any meaningful way (see also the
Page test tutorial).
page.test(data, verbose = TRUE) page.L(data, verbose = TRUE, ties.method = "average") page.compute.exact(k, N, L)
| data | a matrix with the different conditions along its |
|---|---|
| verbose | whether to print the final rankings based on which the L statistic is computed |
| ties.method | how to resolve tied ranks. Passed on to
|
| k | number of conditions/generations |
| N | number of replications/chains |
| L | value of the Page L statistic |
page.test returns a list of class pagetest (and
htest) containing the following elements:
statisticvalue of the L statistic for the data set
parametera named vector specifying the number of conditions (k) and replications (N) of the data (which is the number of columns and rows of the data set, respectively)
p.valuesignificance level
p.typewhether the computed p-value is "exact" or
"approximate"
Tests the given matrix for monotonically increasing ranks across k
linearly ordered conditions (along columns) based on N replications
(along rows). To test for monotonically decreasing ranks, either reverse
the order of columns, or simply invert the rank ordering by calling - on
the entire dataset.
Exact p-values are computed for k up to 22, using the pre-computed null
distributions from the
pspearman package. For
larger k, p-values are computed based on a Normal distribution
approximation (Siegel and Castellan, 1988).
page.test: See above.
page.L: Calculate Page's L statistic for the given dataset.
page.compute.exact: Calculate exact significance levels of the Page L
statistic. Returns a single numeric indicating the null probability of
the Page statistic with the given k, N being greater or equal than the
given L.
Siegel, S., and N. J. Castellan, Jr. (1988). Nonparametric Statistics for the Behavioral Sciences. McGraw-Hill.
# exact p value computation for N=4, k=4 page.test(t(replicate(4, sample(4))))#> Ranking used: #> [,1] [,2] [,3] [,4] #> N=1 2 4 3 1 #> N=2 3 4 2 1 #> N=3 1 4 3 2 #> N=4 4 2 1 3#> #> Page test of monotonicity of ranks #> #> data: t(replicate(4, sample(4))) #> L = 93, k = 4, N = 4, p-value = 0.8992 #> alternative hypothesis: at least one difference in ranks #># exact p value computation for N=4, k=10 page.test(t(replicate(4, sample(10))))#> Ranking used: #> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] #> N=1 6 2 1 3 10 7 5 8 4 9 #> N=2 3 7 8 4 2 10 9 6 1 5 #> N=3 2 7 1 4 5 6 3 8 9 10 #> N=4 9 10 8 2 3 1 5 6 7 4#> #> Page test of monotonicity of ranks #> #> data: t(replicate(4, sample(10))) #> L = 1269, k = 10, N = 4, p-value = 0.146 #> alternative hypothesis: at least one difference in ranks #># approximate p value computation for N=4, k=23 result <- page.test(t(replicate(4, sample(23))), verbose = FALSE)#>print(result)#> #> Page test of monotonicity of ranks #> #> data: t(replicate(4, sample(23))) #> L = 12581, k = 23, N = 4, p-value = 0.9389 #> alternative hypothesis: at least one difference in ranks #># raw calculation of the significance levels page.compute.exact(6, 4, 322)#> [1] 0.03932265