chenTest() R function from [PMCMRplus]

Chen's Many-to-One Comparisons Test

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),
  ...
)

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 "x". Ignored with a warning if "x" is a list.
alternative: the alternative hypothesis. Defaults to two.sided.
p.adjust.method: method for adjusting p values (see p.adjust)
formula: a formula of the form response ~ group where response gives the data values and group a vector or factor of the corresponding groups.
data: an optional matrix or data frame (or similar: see model.frame) containing the variables in the formula formula. By default the variables are taken from environment(formula).
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 NAs. Defaults to getOption("na.action").

Returns

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.

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 $i$ th mean is

\mu(T_i) = n_i N_{i-1} / 2,

with $N_j = \sum_{j =0}^i n_j$ and the $i$ th 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".

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).

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)

References

Chen, Y.-I., 1999, Nonparametric Identification of the Minimum Effective Dose. Biometrics 55 , 1236--1240. tools:::Rd_expr_doi("10.1111/j.0006-341X.1999.01236.x")

chenTest function