chenJanTest | R Documentation |
Performs Chen and Jan nonparametric test for contrasting increasing (decreasing) dose levels of a treatment in a randomized block design.
chenJanTest(y, ...)
## Default S3 method:
chenJanTest(
y,
groups,
blocks,
alternative = c("greater", "less"),
p.adjust.method = c("single-step", "SD1", p.adjust.methods),
...
)
y |
a numeric vector of data values, or a list of numeric data vectors. |
groups |
a vector or factor object giving the group for the
corresponding elements of |
blocks |
a vector or factor object giving the block for the
corresponding elements of |
alternative |
the alternative hypothesis. Defaults to |
p.adjust.method |
method for adjusting p values
(see |
... |
further arguments to be passed to or from methods. |
Chen's test is a non-parametric step-down trend test for
testing several treatment levels with a zero control. Let
there be k
groups including the control and let
the zero dose level be indicated with i = 0
and the highest
dose level with i = m
, then the following m = k - 1
hypotheses are 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}
Let Y_{ij1}, Y_{ij2}, \ldots, Y_{ijn_{ij}}
(i = 1, 2, \dots, b, j = 0, 1, \ldots, k ~ \mathrm{and} ~ n_{ij} \geq 1)
be
a i.i.d. random variable of at least ordinal scale. Further,the zero dose
control is indicated with j = 0
.
The Mann-Whittney statistic is
T_{ij} = \sum_{u=0}^{j-1} \sum_{s=1}^{n_{ij}}
\sum_{r=1}^{n_{iu}} I(Y_{ijs} - Y_{iur}),
\qquad i = 1, 2, \ldots, b, ~ j = 1, 2, \ldots, k,
where where the indicator function returns I(a) = 1, ~ \mathrm{if}~ a > 0, 0.5 ~ \mathrm{if} a = 0
otherwise 0
.
Let
N_{ij} = \sum_{s=0}^j n_{is} \qquad i = 1, 2, \ldots, b, ~ j = 1, 2, \ldots, k,
and
T_j = \sum_{i=1}^b T_{ij} \qquad j = 1, 2, \ldots, k.
The mean and variance of T_j
are
\mu(T_j) = \sum_{i=1}^b n_{ij} ~ N_{ij-1} / 2 \qquad \mathrm{and}
\sigma(T_j) = \sum_{i=1}^b n_{ij} ~ N_{ij-1} \left[
\left(N_{ij} + 1\right) - \sum_{u=1}^{g_i}
\left(t_u^3 - t_u \right) /
\left\{N_{ij} \left(N_{ij} - 1\right) \right\} \right]/ 2,
with g_i
the number of ties in the i
th block and
t_u
the size of the tied group u
.
The test statistic T_j^*
is asymptotically multivariate normal
distributed.
T_j^* = \frac{T_j - \mu(T_j)}{\sigma(T_j)}
If p.adjust.method = "single-step"
than the p-values
are calculated with the probability function of the multivariate
normal distribution with \Sigma = I_k
. Otherwise
the standard normal distribution is used to calculate
p-values and any method as available
by p.adjust
or by the step-down procedure as proposed
by Chen (1999), if p.adjust.method = "SD1"
can be used
to account for \alpha
-error inflation.
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.
Chen, Y.I., Jan, S.L., 2002. Nonparametric Identification of the Minimum Effective Dose for Randomized Block Designs. Commun Stat-Simul Comput 31, 301–312.
Normal
pmvnorm
## Example from Chen and Jan (2002, p. 306)
## MED is at dose level 2 (0.5 ppm SO2)
y <- c(0.2, 6.2, 0.3, 0.3, 4.9, 1.8, 3.9, 2, 0.3, 2.5, 5.4, 2.3, 12.7,
-0.2, 2.1, 6, 1.8, 3.9, 1.1, 3.8, 2.5, 1.3, -0.8, 13.1, 1.1,
12.8, 18.2, 3.4, 13.5, 4.4, 6.1, 2.8, 4, 10.6, 9, 4.2, 6.7, 35,
9, 12.9, 2, 7.1, 1.5, 10.6)
groups <- gl(4,11, labels = c("0", "0.25", "0.5", "1.0"))
blocks <- structure(rep(1:11, 4), class = "factor",
levels = c("1", "2", "3", "4", "5", "6", "7", "8", "9", "10", "11"))
summary(chenJanTest(y, groups, blocks, alternative = "greater"))
summary(chenJanTest(y, groups, blocks, alternative = "greater", p.adjust = "SD1"))
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