Performs Shirley's nonparametric equivalent of William's test for contrasting increasing dose levels of a treatment.
shirleyWilliamsTest(x, ...)# S3 method for default
shirleyWilliamsTest(x, g, alternative = c("two.sided",
"greater", "less"), method = c("look-up", "boot"), nperm = 10000, ...)
# S3 method for formula
shirleyWilliamsTest(formula, data, subset, na.action,
alternative = c("two.sided", "greater", "less"), method = c("look-up",
"boot"), nperm = 10000, ...)
a numeric vector of data values, or a list of numeric data vectors.
further arguments to be passed to or from methods.
a vector or factor object giving the group for the
corresponding elements of "x".
Ignored with a warning if "x" is a list.
the alternative hypothesis. Defaults to two.sided
a character string specifying the test statistic to use.
Defaults to "look-up" that uses published Table values of Williams (1972).
number of permutations for the assymptotic permutation test.
Defaults to 1000. Ignored, if method = "look-up".
a formula of the form response ~ group where
response gives the data values and group a vector or
factor of the corresponding groups.
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).
an optional vector specifying a subset of observations to be used.
a function which indicates what should happen when
the data contain NAs. Defaults to getOption("na.action").
Either a list with class "williamsTest" or al list with class "PMCMR".
The list with class "williamsTest".
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 critical t\'-values for \(\alpha = 0.05\).
the degree of freedom
a character string describing the alternative hypothesis.
a data frame of the input data.
a string that denotes the test distribution.
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.
The Shirley-William 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:
$$ \begin{array}{ll} \mathrm{H}_{m}: \theta_0 = \theta_1 = \ldots = \theta_m, & \mathrm{A}_{m} = \theta_0 \le \theta_1 \le \ldots \theta_m, \theta_0 < \theta_m \\ \mathrm{H}_{m-1}: \theta_0 = \theta_1 = \ldots = \theta_{m-1}, & \mathrm{A}_{m-1} = \theta_0 \le \theta_1 \le \ldots \theta_{m-1}, \theta_0 < \theta_{m-1} \\ \vdots & \vdots \\ \mathrm{H}_{1}: \theta_0 = \theta_1, & \mathrm{A}_{1} = \theta_0 < \theta_1\\ \end{array} $$
The procedure starts from the highest dose level (\(m\)) to the the lowest dose level (\(1\)) and stops at the first non-significant test. The consequent lowest effect dose is the treatment level of the previous test number. This function has included the modifications as recommended by Williams (1986).
If method = "look-up" is selected, the function does not return p-values. Instead the critical t-values
as given in the tables of Williams (1972) for \(\alpha = 0.05\) (one-sided)
are looked up according to the degree of freedoms (\(v = \infty\)) and the order number of the
dose level (\(i\)) and (potentially) modified according to the given extrapolation
coefficient \(\beta\).
Non tabulated values are linearly interpolated with the function
approx.
For the comparison of the first dose level (i = 1) with the control, the critical
z-value from the standard normal distribution is used (Normal).
If method = "boot", the p-values are estimated through an assymptotic
boot-strap method. The p-values for H\(_1\)
are calculated from the t distribution with infinite degree of freedom.
Shirley, E., (1977) Nonparametric Equivalent of Williams Test for Contrasting Increasing Dose Levels of a Treatment, Biometrics 33, 386--389.
Williams, D. A. (1986) Note on Shirley's nonparametric test for comparing several dose levels with a zero-dose control, Biometrics 42, 183--186.
# NOT RUN {
## Example from Shirley (1977)
## Reaction times of mice to stimuli to their tails.
y <- c(2.4, 3, 3, 2.2, 2.2, 2.2, 2.2, 2.8, 2, 3,
2.8, 2.2, 3.8, 9.4, 8.4, 3, 3.2, 4.4, 3.2, 7.4, 9.8, 3.2, 5.8,
7.8, 2.6, 2.2, 6.2, 9.4, 7.8, 3.4, 7, 9.8, 9.4, 8.8, 8.8, 3.4,
9, 8.4, 2.4, 7.8)
g <- gl(4, 10)
# }
# NOT RUN {
## two.sided test
summary(shirleyWilliamsTest(y ~ g, method = "boot", alternative = "two.sided"))
# }
# NOT RUN {
## one-sided test using look-up table
shirleyWilliamsTest(y ~ g, alternative = "greater")
# }
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