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.