| chenTest {PMCMRplus} | R Documentation |
Chen's Many-to-One Comparisons Test
Description
Performs Chen's nonparametric test for contrasting increasing (decreasing) dose levels of a treatment.
Usage
chenTest(x, ...)
## Default S3 method:
chenTest(
x,
g,
alternative = c("greater", "less"),
p.adjust.method = c("SD1", p.adjust.methods),
...
)
## S3 method for class 'formula'
chenTest(
formula,
data,
subset,
na.action,
alternative = c("greater", "less"),
p.adjust.method = c("SD1", p.adjust.methods),
...
)
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 |
p.adjust.method |
method for adjusting p values
(see |
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
Chen's test is a non-parametric step-down trend test for
testing several treatment levels with a zero control.
Let X_{0j} denote a variable with the j-th
realization of the control group (1 \le j \le n_0)
and X_{ij} the j-the realization
in the i-th treatment group (1 \le i \le k).
The variables are i.i.d. of a least ordinal scale with
F(x) = F(x_0) = F(x_i), ~ (1 \le i \le k).
A total of m = k hypotheses can be tested:
\begin{array}{ll}
\mathrm{H}_{m}: \theta_0 = \theta_1 = \ldots = \theta_m, & \mathrm{A}_{m} = \theta_0 \le \theta_1 \le \ldots \theta_m, \theta_0 < \theta_m \\
\mathrm{H}_{m-1}: \theta_0 = \theta_1 = \ldots = \theta_{m-1}, & \mathrm{A}_{m-1} = \theta_0 \le \theta_1 \le \ldots \theta_{m-1}, \theta_0 < \theta_{m-1} \\
\vdots & \vdots \\
\mathrm{H}_{1}: \theta_0 = \theta_1, & \mathrm{A}_{1} = \theta_0 < \theta_1\\
\end{array}
The statistics T_i are based on a Wilcoxon-type ranking:
T_i = \sum_{j=0}^{i=1} \sum_{u=1}^{n_i} \sum_{v=1}^{n_j} I(x_{iu} - x_{jv}), \qquad (1 \leq i \leq k),
where the indicator function returns I(a) = 1, ~ \mathrm{if}~ a > 0, 0.5 ~ \mathrm{if} a = 0
otherwise 0.
The expected ith mean is
\mu(T_i) = n_i N_{i-1} / 2,
with N_j = \sum_{j =0}^i n_j and the ith variance:
\sigma^2(T_i) = n_i N_{i-1} / 12 ~ \left\{N_i + 1 -
\sum_{j=1}^g t_j \left(t_j^2 - 1 \right) /
\left[N_i \left( N_i - 1 \right)\right]\right\}.
The test statistic T_i^* is asymptotically standard normal
T_i^* = \frac{T_i - \mu(T_i)}
{\sqrt{\sigma^2(T_i)}}, \qquad (1 \leq i \leq k).
The p-values are calculated from the standard normal distribution.
The p-values can be adjusted with any method as available
by p.adjust or by the step-down procedure as proposed
by Chen (1999), if p.adjust.method = "SD1".
Value
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
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
Chen, Y.-I., 1999, Nonparametric Identification of the Minimum Effective Dose. Biometrics 55, 1236–1240. doi:10.1111/j.0006-341X.1999.01236.x
See Also
Examples
## Chen, 1999, p. 1237,
## Minimum effective dose (MED)
## is at 2nd dose level
df <- data.frame(x = c(23, 22, 14,
27, 23, 21,
28, 37, 35,
41, 37, 43,
28, 21, 30,
16, 19, 13),
g = gl(6, 3))
levels(df$g) <- 0:5
ans <- chenTest(x ~ g, data = df, alternative = "greater",
p.adjust.method = "SD1")
summary(ans)