All-Pairs Comparisons for Simply Ordered Mean Ranksums
Performs Nashimoto and Wright's all-pairs comparison procedure for simply ordered mean ranksums (NPT'-test and NPY'-test).
According to the authors, the procedure shall only be
applied after Chacko's test (see chackoTest) indicates
global significance.
chaAllPairsNashimotoTest(x, ...)
## Default S3 method:
chaAllPairsNashimotoTest(
x,
g,
p.adjust.method = c(p.adjust.methods),
alternative = c("greater", "less"),
dist = c("Normal", "h"),
...
)
## S3 method for class 'formula'
chaAllPairsNashimotoTest(
formula,
data,
subset,
na.action,
p.adjust.method = c(p.adjust.methods),
alternative = c("greater", "less"),
dist = c("Normal", "h"),
...
)x |
a numeric vector of data values, or a list of numeric data vectors. |
... |
further arguments to be passed to or from methods. |
g |
a vector or factor object giving the group for the
corresponding elements of |
p.adjust.method |
method for adjusting p values. Ignored if |
alternative |
the alternative hypothesis. Defaults to |
dist |
the test distribution. Defaults to |
formula |
a formula of the form |
data |
an optional matrix or data frame (or similar: see
|
subset |
an optional vector specifying a subset of observations to be used. |
na.action |
a function which indicates what should happen when
the data contain |
The modified procedure uses the property of a simple order, θ_m' - θ_m ≤ θ_j - θ_i ≤ θ_l' - θ_l \qquad (l ≤ i ≤ m~\mathrm{and}~ m' ≤ j ≤ l'). The null hypothesis H_{ij}: θ_i = θ_j is tested against the alternative A_{ij}: θ_i < θ_j for any 1 ≤ i < j ≤ k.
Let R_{ij} be the rank of X_{ij}, where X_{ij} is jointly ranked from ≤ft\{1, 2, …, N \right\}, ~~ N = ∑_{i=1}^k n_i, then the test statistics for all-pairs comparisons and a balanced design is calculated as
SEE PDF
with n = n_i; ~ N = ∑_i^k n_i ~~ (1 ≤ i ≤ k), \bar{R}_i the mean rank for the ith group, and the expected variance (without ties) σ_a^2 = N ≤ft(N + 1 \right) / 12.
For the NPY'-test (dist = "h"), if T_{ij} > h_{k-1,α,∞}.
For the unbalanced case with moderate imbalance the test statistic is
SEE PDF
For the NPY'-test (dist="h") the null hypothesis is rejected in an unbalanced design,
if \hat{T}_{ij} > h_{k,α,∞} / √{2}.
In case of a NPY'-test, the function does not return p-values. Instead the critical h-values
as given in the tables of Hayter (1990) for α = 0.05 (one-sided)
are looked up according to the number of groups (k-1) and
the degree of freedoms (v = ∞).
For the NPT'-test (dist = "Normal"), the null hypothesis is rejected, if
T_{ij} > √{2} t_{α,∞} = √{2} z_α. Although Nashimoto and Wright (2005) originally did not use any p-adjustment,
any method as available by p.adjust.methods can
be selected for the adjustment of p-values estimated from
the standard normal distribution.
Either a list of class "osrt" if dist = "h" or a list
of class "PMCMR" if dist = "Normal".
a character string indicating what type of test was performed.
a character string giving the name(s) of the data.
the estimated statistic(s)
critical values for α = 0.05.
a character string describing the alternative hypothesis.
the parameter(s) of the test distribution.
a string that denotes the test distribution.
There are print and summary methods available.
A list with class "PMCMR" containing the following components:
a character string indicating what type of test was performed.
a character string giving the name(s) of the data.
lower-triangle matrix of the estimated quantiles of the pairwise test statistics.
lower-triangle matrix of the p-values for the pairwise tests.
a character string describing the alternative hypothesis.
a character string describing the method for p-value adjustment.
a data frame of the input data.
a string that denotes the test distribution.
The function will give a warning for the unbalanced case and returns the critical value h_{k-1,α,∞} / √{2} if applicable.
Hayter, A. J.(1990) A One-Sided Studentised Range Test for Testing Against a Simple Ordered Alternative, J Amer Stat Assoc 85, 778–785.
Nashimoto, K., Wright, F.T. (2007) Nonparametric Multiple-Comparison Methods for Simply Ordered Medians. Comput Stat Data Anal 51, 5068–5076.
## Example from Shirley (1977) ## Reaction times of mice to stimuli to their tails. x <- c(2.4, 3, 3, 2.2, 2.2, 2.2, 2.2, 2.8, 2, 3, 2.8, 2.2, 3.8, 9.4, 8.4, 3, 3.2, 4.4, 3.2, 7.4, 9.8, 3.2, 5.8, 7.8, 2.6, 2.2, 6.2, 9.4, 7.8, 3.4, 7, 9.8, 9.4, 8.8, 8.8, 3.4, 9, 8.4, 2.4, 7.8) g <- gl(4, 10) ## Shirley's test ## one-sided test using look-up table shirleyWilliamsTest(x ~ g, alternative = "greater") ## Chacko's global hypothesis test for 'greater' chackoTest(x , g) ## post-hoc test, default is standard normal distribution (NPT'-test) summary(chaAllPairsNashimotoTest(x, g, p.adjust.method = "none")) ## same but h-distribution (NPY'-test) chaAllPairsNashimotoTest(x, g, dist = "h") ## NPM-test NPMTest(x, g) ## Hayter-Stone test hayterStoneTest(x, g) ## all-pairs comparisons hsAllPairsTest(x, g)
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