View source: R/flignerWolfeTest.R
flignerWolfeTest | R Documentation |
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"),
...
)
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 |
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 |
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 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.
A list with class "htest"
containing the following components:
a character string indicating what type of test was performed.
a character string giving the name(s) of the data.
the estimated quantile of the test statistic.
the p-value for the test.
the parameters of the test statistic, if any.
a character string describing the alternative hypothesis.
the estimates, if any.
the estimate under the null hypothesis, if any.
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
).
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.
kruskalTest
and shirleyWilliamsTest
of the package PMCMRplus,
kruskal.test
of the library stats.
## 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)
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