| shanTest {PMCMRplus} | R Documentation |
Testing against Ordered Alternatives (Shan-Young-Kang Test)
Description
Performs the Shan-Young-Kang test for testing against ordered alternatives.
Usage
shanTest(x, ...)
## Default S3 method:
shanTest(x, g, alternative = c("greater", "less"), ...)
## S3 method for class 'formula'
shanTest(
formula,
data,
subset,
na.action,
alternative = c("greater", "less"),
...
)
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 |
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 null hypothesis, H_0: \theta_1 = \theta_2 = \ldots = \theta_k
is tested against a simple order hypothesis,
H_\mathrm{A}: \theta_1 \le \theta_2 \le \ldots \le
\theta_k,~\theta_1 < \theta_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,
the the test statistic is
S = \sum_{i = 1}^{k-1} \sum_{j = i + 1}^k D_{ij},
with
D_{ij} = \sum_{l = 1}^{n_i} \sum_{m=1}^{n_j} \left(R_{jm} - R_{il} \right)~ \mathrm{I}\left(X_{jm} > X_{il} \right),
where
\mathrm{I}(u) = \left\{
\begin{array}{c}
1, \qquad \forall~ u > 0 \\
0, \qquad \forall~ u \le 0
\end{array}
\right.
.
The test statistic is asymptotically normal distributed:
z = \frac{S - \mu_{\mathrm{S}}}{\sqrt{s^2_{\mathrm{S}}}}
The p-values are estimated from the standard normal distribution.
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
The variance estimation (see Theorem 2.1, Shan et al. 2014)
can become negative for certain combinations of N,~n_i,~k
\qquad (1 \le i \le k). In these cases the function will return
a warning and the returned p-value will be NaN.
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
Shan, G., Young, D., Kang, L. (2014) A New Powerful Nonparametric Rank Test for Ordered Alternative Problem. PLOS ONE 9, e112924. https://doi.org/10.1371/journal.pone.0112924
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)