NPMTest: All-Pairs Comparisons for Simply Ordered Mean Ranksums
In PMCMRplus: Calculate Pairwise Multiple Comparisons of Mean Rank Sums Extended
NPMTest R Documentation
All-Pairs Comparisons for Simply Ordered Mean Ranksums
Description
Performs Nashimoto and Wright's all-pairs comparison procedure
for simply ordered mean ranksums.
Usage
NPMTest(x, ...)
## Default S3 method:
NPMTest(
x,
g,
alternative = c("greater", "less"),
method = c("look-up", "boot", "asympt"),
nperm = 10000,
...
)
## S3 method for class 'formula'
NPMTest(
formula,
data,
subset,
na.action,
alternative = c("greater", "less"),
method = c("look-up", "boot", "asympt"),
nperm = 10000,
...
)
Arguments
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 "x".
Ignored with a warning if "x" is a list.
alternative
the alternative hypothesis. Defaults to greater.
method
a character string specifying the test statistic to use.
Defaults to "look-up" that uses published Table values of Williams (1972).
nperm
number of permutations for the asymptotic permutation test.
Defaults to 1000. Ignored, if method = "look-up".
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").
Details
The 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.
The all-pairs comparisons test statistics for a balanced design are
\hat{h}_{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 ith group,
and \sigma_a = \sqrt{N \left(N + 1 \right) / 12}. The null hypothesis is rejected,
if h_{ij} > h_{k,\alpha,\infty}.
For the unbalanced case with moderate imbalance the test statistic is
\hat{h}_{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}},
The null hypothesis is rejected, if \hat{h}_{ij} > h_{k,\alpha,\infty} / \sqrt{2}.
If method = "look-up" the function will 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) and
the degree of freedoms (v = \infty).
If method = "boot" an asymetric permutation test
is conducted and p-values is returned.
If method = "asympt" is selected the asymptotic
p-value is estimated as implemented in the
function pHayStonLSA of the package NSM3.
Value
- 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.
Either a list of class "PMCMR" or a
list with class "osrt" that contains 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
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.
Note
The function will give a warning for the unbalanced case and returns the
critical value h_{k,\alpha,\infty} / \sqrt{2}.
Source
If method = "asympt" is selected, this function calls
an internal probability function pHS. The GPL-2 code for
this function was taken from pHayStonLSA of the
the package NSM3:
Grant Schneider, Eric Chicken and Rachel Becvarik (2020) NSM3:
Functions and Datasets to Accompany Hollander, Wolfe, and
Chicken - Nonparametric Statistical Methods, Third Edition. R
package version 1.15. https://CRAN.R-project.org/package=NSM3
References
Hayter, A. J.(1990) A One-Sided Studentised Range
Test for Testing Against a Simple Ordered Alternative,
Journal of the American Statistical Association
85, 778–785.
Nashimoto, K., Wright, F.T. (2007)
Nonparametric Multiple-Comparison Methods for Simply
Ordered Medians.
Comput Stat Data Anal 51, 5068–5076.
See Also
MTest
Examples
## 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)
PMCMRplus documentation built on Nov. 27, 2023, 1:08 a.m.
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| NPMTest | R Documentation |
All-Pairs Comparisons for Simply Ordered Mean Ranksums
Description
Performs Nashimoto and Wright's all-pairs comparison procedure for simply ordered mean ranksums.
Usage
NPMTest(x, ...)
## Default S3 method:
NPMTest(
x,
g,
alternative = c("greater", "less"),
method = c("look-up", "boot", "asympt"),
nperm = 10000,
...
)
## S3 method for class 'formula'
NPMTest(
formula,
data,
subset,
na.action,
alternative = c("greater", "less"),
method = c("look-up", "boot", "asympt"),
nperm = 10000,
...
)
Arguments
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 |
alternative |
the alternative hypothesis. Defaults to |
method |
a character string specifying the test statistic to use.
Defaults to |
nperm |
number of permutations for the asymptotic permutation test.
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 |
Details
The 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.
The all-pairs comparisons test statistics for a balanced design are
\hat{h}_{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 ith group,
and \sigma_a = \sqrt{N \left(N + 1 \right) / 12}. The null hypothesis is rejected,
if h_{ij} > h_{k,\alpha,\infty}.
For the unbalanced case with moderate imbalance the test statistic is
\hat{h}_{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}},
The null hypothesis is rejected, if \hat{h}_{ij} > h_{k,\alpha,\infty} / \sqrt{2}.
If method = "look-up" the function will 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) and
the degree of freedoms (v = \infty).
If method = "boot" an asymetric permutation test
is conducted and p-values is returned.
If method = "asympt" is selected the asymptotic
p-value is estimated as implemented in the
function pHayStonLSA of the package NSM3.
Value
- 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.
Either a list of class "PMCMR" or a
list with class "osrt" that contains 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
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.
Note
The function will give a warning for the unbalanced case and returns the
critical value h_{k,\alpha,\infty} / \sqrt{2}.
Source
If method = "asympt" is selected, this function calls
an internal probability function pHS. The GPL-2 code for
this function was taken from pHayStonLSA of the
the package NSM3:
Grant Schneider, Eric Chicken and Rachel Becvarik (2020) NSM3: Functions and Datasets to Accompany Hollander, Wolfe, and Chicken - Nonparametric Statistical Methods, Third Edition. R package version 1.15. https://CRAN.R-project.org/package=NSM3
References
Hayter, A. J.(1990) A One-Sided Studentised Range Test for Testing Against a Simple Ordered Alternative, Journal of the American Statistical Association 85, 778–785.
Nashimoto, K., Wright, F.T. (2007) Nonparametric Multiple-Comparison Methods for Simply Ordered Medians. Comput Stat Data Anal 51, 5068–5076.
See Also
MTest
Examples
## 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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