spearmanTest: Testing against Ordered Alternatives (Spearman Test)

Description

Performs a Spearman type test for testing against ordered alternatives.

Usage

spearmanTest(x, ...)
# S3 method for default
spearmanTest(x, g, alternative = c("two.sided", "greater", "less"), ...)
# S3 method for formula
spearmanTest(
  formula,
  data,
  subset,
  na.action,
  alternative = c("two.sided", "greater", "less"),
  ...
)

Value

A list with class "htest" 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: the estimated quantile of the test statistic.
p.value: the p-value for the test.
parameter: the parameters of the test statistic, if any.
alternative: a character string describing the alternative hypothesis.
estimates: the estimates, if any.
null.value: the estimate under the null hypothesis, if any.

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".
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").

Details

A one factorial design for dose finding comprises an ordered factor, .e. treatment with increasing treatment levels. The basic idea is to correlate the ranks $R_{ij}$ with the increasing order number $1 \le i \le k$ of the treatment levels (Kloke and McKean 2015). More precisely, $R_{ij}$ is correlated with the expected mid-value ranks under the assumption of strictly increasing median responses. Let the expected mid-value rank of the first group denote $E_1 = \left(n_1 + 1\right)/2$. The following expected mid-value ranks are $E_j = n_{j-1} + \left(n_j + 1 \right)/2$ for $2 \le j \le k$. The corresponding number of tied values for the $i$th group is $n_i$. # The sum of squared residuals is $D^2 = \sum_{i=1}^k \sum_{j=1}^{n_i} \left(R_{ij} - E_i \right)^2$. Consequently, Spearman's rank correlation coefficient can be calculated as:

$$ r_\mathrm{S} = \frac{6 D^2} {\left(N^3 - N\right)- C}, $$

with $$ C = 1/2 - \sum_{c=1}^r \left(t_c^3 - t_c\right) + 1/2 - \sum_{i=1}^k \left(n_i^3 - n_i \right) $$ and $t_c$ the number of ties of the $c$th group of ties. Spearman's rank correlation coefficient can be tested for significance with a $t$-test. For a one-tailed test the null hypothesis of $r_\mathrm{S} \le 0$ is rejected and the alternative $r_\mathrm{S} > 0$ is accepted if

$$ r_\mathrm{S} \sqrt{\frac{\left(n-2\right)}{\left(1 - r_\mathrm{S}\right)}} > t_{v,1-\alpha}, $$

with $v = n - 2$ degree of freedom.

References

Kloke, J., McKean, J. W. (2015) Nonparametric statistical methods using R. Boca Raton, FL: Chapman & Hall/CRC.

Examples

Run this code

## Example from Sachs (1997, p. 402)
x <- c(106, 114, 116, 127, 145,
       110, 125, 143, 148, 151,
       136, 139, 149, 160, 174)
g <- gl(3,5)
levels(g) <- c("A", "B", "C")

## Chacko's test
chackoTest(x, g)

## Cuzick's test
cuzickTest(x, g)

## Johnson-Mehrotra test
johnsonTest(x, g)

## Jonckheere-Terpstra test
jonckheereTest(x, g)

## Le's test
leTest(x, g)

## Spearman type test
spearmanTest(x, g)

## Murakami's BWS trend test
bwsTrendTest(x, g)

## Fligner-Wolfe test
flignerWolfeTest(x, g)

## Shan-Young-Kang test
shanTest(x, g)

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