chaAllPairsNashimotoTest: All-Pairs Comparisons for Simply Ordered Mean Ranksums

Description

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.

Usage

chaAllPairsNashimotoTest(x, ...)
# S3 method for default
chaAllPairsNashimotoTest(
  x,
  g,
  p.adjust.method = c(p.adjust.methods),
  alternative = c("greater", "less"),
  dist = c("Normal", "h"),
  ...
)
# S3 method for formula
chaAllPairsNashimotoTest(
  formula,
  data,
  subset,
  na.action,
  p.adjust.method = c(p.adjust.methods),
  alternative = c("greater", "less"),
  dist = c("Normal", "h"),
  ...
)

Arguments

a numeric vector of data values, or a list of numeric data vectors.

…

further arguments to be passed to or from methods.

a vector or factor object giving the group for the corresponding elements of "x". Ignored with a warning if "x" is a list.

p.adjust.method

method for adjusting p values. Ignored if dist = "h".

alternative

the alternative hypothesis. Defaults to greater.

dist

the test distribution. Defaults to Normal.

formula

a formula of the form response ~ group where response gives the data values and group a vector or factor of the corresponding groups.

data

an optional matrix or data frame (or similar: see model.frame) containing the variables in the formula formula. By default the variables are taken from environment(formula).

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 NAs. Defaults to getOption("na.action").

Value

Either a list of class "osrt" if dist = "h" or a list of class "PMCMR" if dist = "Normal".

method: a character string indicating what type of test was performed.
data.name: a character string giving the name(s) of the data.
statistic: the estimated statistic(s)
crit.value: critical values for $\alpha = 0.05$.
alternative: a character string describing the alternative hypothesis.
parameter: the parameter(s) of the test distribution.
dist: a string that denotes the test distribution.

There are print and summary methods available.

A list with class "PMCMR" containing the following components:

method: a character string indicating what type of test was performed.
data.name: a character string giving the name(s) of the data.
statistic: lower-triangle matrix of the estimated quantiles of the pairwise test statistics.
p.value: lower-triangle matrix of the p-values for the pairwise tests.
alternative: a character string describing the alternative hypothesis.
p.adjust.method: a character string describing the method for p-value adjustment.
model: a data frame of the input data.
dist: a string that denotes the test distribution.

Details

The modified procedure uses the property of a simple order, $\theta_m' - \theta_m \le \theta_j - \theta_i \le \theta_l' - \theta_l \qquad (l \le i \le m~\mathrm{and}~ m' \le j \le l')$. The null hypothesis H$_{ij}: \theta_i = \theta_j$ is tested against the alternative A$_{ij}: \theta_i < \theta_j$ for any $1 \le i < j \le k$.

Let $R_{ij}$ be the rank of $X_{ij}$, where $X_{ij}$ is jointly ranked from $\left\{1, 2, \ldots, N \right\}, ~~ N = \sum_{i=1}^k n_i$, then the test statistics for all-pairs comparisons and a balanced design is calculated as $$ \hat{T}_{ij} = \max_{i \le m < m' \le j} \frac{\left(\bar{R}_{m'} - \bar{R}_m \right)} {\sigma_a / \sqrt{n}}, $$

with $n = n_i; ~ N = \sum_i^k n_i ~~ (1 \le i \le k)$, $\bar{R}_i$ the mean rank for the $i$th group, and the expected variance (without ties) $\sigma_a^2 = N \left(N + 1 \right) / 12$.

For the NPY'-test (dist = "h"), if $T_{ij} > h_{k-1,\alpha,\infty}$.

For the unbalanced case with moderate imbalance the test statistic is $$ \hat{T}_{ij} = \max_{i \le m < m' \le j} \frac{\left(\bar{R}_{m'} - \bar{R}_m \right)} {\sigma_a \left(1/n_m + 1/n_{m'}\right)^{1/2}}, $$

For the NPY'-test (dist="h") the null hypothesis is rejected in an unbalanced design, if $\hat{T}_{ij} > h_{k,\alpha,\infty} / \sqrt{2}$. 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 $\alpha = 0.05$ (one-sided) are looked up according to the number of groups ($k-1$) and the degree of freedoms ($v = \infty$).

For the NPT'-test (dist = "Normal"), the null hypothesis is rejected, if $T_{ij} > \sqrt{2} t_{\alpha,\infty} = \sqrt{2} z_\alpha$. 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.

References

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.

Examples

Run this code

# NOT RUN {
## 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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