flignerWolfeTest {PMCMRplus}

Testing Several Treatments With One Control

Description

Performs Fligner-Wolfe non-parametric test for simultaneous testing of several locations of treatment groups against the location of the control group.

Usage

flignerWolfeTest(x, ...)

## Default S3 method:
flignerWolfeTest(
  x,
  g,
  alternative = c("greater", "less"),
  dist = c("Wilcoxon", "Normal"),
  ...
)

## S3 method for class 'formula'
flignerWolfeTest(
  formula,
  data,
  subset,
  na.action,
  alternative = c("greater", "less"),
  dist = c("Wilcoxon", "Normal"),
  ...
)

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".

dist

the test distribution. Defaults to "Wilcoxon".

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

For a one-factorial layout with non-normally distributed residuals the Fligner-Wolfe test can be used.

Let there be k-1-treatment groups and one control group, then the null hypothesis, H_0: \theta_i - \theta_c = 0 ~ (1 \le i \le k-1) is tested against the alternative (greater), A_1: \theta_i - \theta_c > 0 ~ (1 \le i \le k-1), with at least one inequality being strict.

Let n_c denote the sample size of the control group, N^t = \sum_{i=1}^{k-1} n_i the sum of all treatment sample sizes and N = N^t + n_c. The test statistic without taken ties into account is

W = \sum_{j=1}^{k-1} \sum_{i=1}^{n_i} r_{ij} - \frac{N^t \left(N^t + 1 \right) }{2}

with r_{ij} the rank of variable x_{ij}. The null hypothesis is rejected, if W > W_{\alpha,m,n} with m = N^t and n = n_c.

In the presence of ties, the statistic is

\hat{z} = \frac{W - n_c N^t / 2}{s_W},

where

s_W = \frac{n_c N^t}{12 N \left(N - 1 \right)} \sum_{j=1}^g t_j \left(t_j^2 - 1\right),

with g the number of tied groups and t_j the number of tied values in the jth group. The null hypothesis is rejected, if \hat{z} > z_\alpha (as cited in EPA 2006).

If dist = Wilcoxon, then the p-values are estimated from the Wilcoxon distribution, else the Normal distribution is used. The latter can be used, if ties are present.

Value

A list with class "htest" 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

the estimated quantile of the test statistic.

p.value

the p-value for the test.

parameter

the parameters of the test statistic, if any.

alternative

a character string describing the alternative hypothesis.

estimates

the estimates, if any.

null.value

the estimate under the null hypothesis, if any.

Note

Factor labels for g must be assigned in such a way, that they can be increasingly ordered from zero-dose control to the highest dose level, e.g. integers {0, 1, 2, ..., k} or letters {a, b, c, ...}. Otherwise the function may not select the correct values for intended zero-dose control.

It is safer, to i) label the factor levels as given above, and to ii) sort the data according to increasing dose-levels prior to call the function (see order, factor).

References

EPA (2006) Data Quality Assessment: Statistical Methods for Practitioners (Guideline No. EPA QA/G-9S), US-EPA.

Fligner, M.A., Wolfe, D.A. (1982) Distribution-free tests for comparing several treatments with a control. Stat Neerl 36, 119–127.

See Also

kruskalTest and shirleyWilliamsTest of the package PMCMRplus, kruskal.test of the library stats.

Examples

## Example from Sachs (1997, p. 402)
x <- c(106, 114, 116, 127, 145,
       110, 125, 143, 148, 151,
       136, 139, 149, 160, 174)
g <- gl(3,5)
levels(g) <- c("A", "B", "C")

## Chacko's test
chackoTest(x, g)

## Cuzick's test
cuzickTest(x, g)

## Johnson-Mehrotra test
johnsonTest(x, g)

## Jonckheere-Terpstra test
jonckheereTest(x, g)

## Le's test
leTest(x, g)

## Spearman type test
spearmanTest(x, g)

## Murakami's BWS trend test
bwsTrendTest(x, g)

## Fligner-Wolfe test
flignerWolfeTest(x, g)

## Shan-Young-Kang test
shanTest(x, g)