flignerWolfeTest() R function from [PMCMRplus]

Testing Several Treatments With One Control

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


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").

Returns

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.

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}%SEE PDF

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},%SEE PDF

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),%SEE PDF

with $g$ the number of tied groups and $t_j$

the number of tied values in the $j$ th 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.

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).

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)

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.

flignerWolfeTest function