PMCMRplus (version 1.9.12)

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 $E_1=(n_1+1)/2$. The following expected mid-value ranks are $E_j=n_{j-1}+(n_j+1)/2$ 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 $D^2=\sum _{i=1}^k\sum _{j=1}^{n_i}{(R_{ij}-E_i)}^2$. Consequently, Spearman's rank correlation coefficient can be calculated as:

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

with $$C=1/2-\sum _{c=1}^r(t_c^3-t_c)+1/2-\sum _{i=1}^k(n_i^3-n_i)$$ 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

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

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.

See Also

kruskalTest and shirleyWilliamsTest of the package PMCMRplus, kruskal.test of the library stats.

Examples

## 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)