chenTest {PMCMRplus} | R Documentation |
Performs Chen's nonparametric test for contrasting increasing (decreasing) dose levels of a treatment.
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), ... )
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 |
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 ≤ j ≤ n_0) and X_{ij} the j-the realization in the i-th treatment group (1 ≤ i ≤ k). The variables are i.i.d. of a least ordinal scale with F(x) = F(x_0) = F(x_i), ~ (1 ≤ i ≤ k). A total of m = k hypotheses can be tested:
\begin{array}{ll} \mathrm{H}_{m}: θ_0 = θ_1 = … = θ_m, & \mathrm{A}_{m} = θ_0 ≤ θ_1 ≤ … θ_m, θ_0 < θ_m \\ \mathrm{H}_{m-1}: θ_0 = θ_1 = … = θ_{m-1}, & \mathrm{A}_{m-1} = θ_0 ≤ θ_1 ≤ … θ_{m-1}, θ_0 < θ_{m-1} \\ \vdots & \vdots \\ \mathrm{H}_{1}: θ_0 = θ_1, & \mathrm{A}_{1} = θ_0 < θ_1\\ \end{array}
The statistics T_i are based on a Wilcoxon-type ranking:
T_i = ∑_{j=0}^{i=1} ∑_{u=1}^{n_i} ∑_{v=1}^{n_j} I(x_{iu} - x_{jv}), \qquad (1 ≤q i ≤q 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
μ(T_i) = n_i N_{i-1} / 2,
with N_j = ∑_{j =0}^i n_j and the ith variance:
σ^2(T_i) = n_i N_{i-1} / 12 ~ ≤ft\{N_i + 1 - ∑_{j=1}^g t_j ≤ft(t_j^2 - 1 \right) / ≤ft[N_i ≤ft( N_i - 1 \right)\right]\right\}.
The test statistic T_i^* is asymptotically standard normal
T_i^* = \frac{T_i - μ(T_i)} {√{σ^2(T_i)}}, \qquad (1 ≤q i ≤q 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"
.
A list with class "PMCMR"
containing the following components:
a character string indicating what type of test was performed.
a character string giving the name(s) of the data.
lower-triangle matrix of the estimated quantiles of the pairwise test statistics.
lower-triangle matrix of the p-values for the pairwise tests.
a character string describing the alternative hypothesis.
a character string describing the method for p-value adjustment.
a data frame of the input data.
a string that denotes the test distribution.
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
).
Chen, Y.-I., 1999, Nonparametric Identification of the Minimum Effective Dose. Biometrics 55, 1236–1240. doi: 10.1111/j.0006-341X.1999.01236.x
## 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)