This function uses pairwise Wilcoxon tests, comparing a common control sample with each of
several treatment samples, in a multiple comparison fashion. The experiment wise
significance probabity is calculated, estimated, or approximated, when
testing
the hypothesis that all independent samples arise
from a common unspecified distribution, or that treatments have no effect when assigned
randomly to the given subjects.
Usage
Steel.test(..., data = NULL,
method = c("asymptotic", "simulated", "exact"),
alternative = c("greater","less","two-sided"),
dist = FALSE, Nsim = 10000)
Arguments
...
Either several sample vectors, say
x_1, …, x_k,
with x_i containing n_i sample values.
n_i > 4 is recommended for reasonable asymptotic
P-value calculation. The pooled sample size is denoted
by N=n_1+…+n_k. The first vector serves as control sample,
the others as treatment samples.
or a list of such sample vectors.
or a formula y ~ g, where y contains the pooled sample values
and g (same length as y) is a factor with levels identifying
the samples to which the elements of y belong. The lowest factor level
corresponds to the control sample, the other levels to treatment samples.
data
= an optional data frame providing the variables in formula y ~ g or y, g input
method
= c("asymptotic","simulated","exact"), where
"asymptotic" uses only an asymptotic normal approximation
to approximate the P-value, This calculation is always done.
"simulated" uses Nsim simulated standardized
Steel statistics based on random splits of the
pooled samples into samples of sizes
n_1, …, n_k, to estimate the P-value.
"exact" uses full enumeration of all sample splits with resulting
standardized Steel statistics to obtain the exact P-value.
It is used only when Nsim is at least as large as the number
N!/(n_1!… n_k!)
of full enumerations. Otherwise, method reverts to
"simulated" using the given Nsim. It also reverts
to "simulated" when ncomb > 1e8 and dist = TRUE.
alternative
= c("greater","less","two-sided"), where for "greater" the
maximum of the pairwise standardized Wilcoxon test statistics is used and
a large maximum value is judged significant.
For "less" the minimum of the pairwise standardized Wilcoxon test
statistics is used and a low minimum value is judged significant.
For "two-sided" the maximum of the absolute pairwise standardized Wilcoxon test
statistics is used and a large maximum value is judged significant.
dist
= FALSE (default) or TRUE. If TRUE, the
simulated or fully enumerated null distribution vector null.dist
is returned for the Steel test statistic, as chosen via alternative.
Otherwise, NULL is returned. When dist = TRUE then
Nsim <- min(Nsim, 1e8), to limit object size.
Nsim
= 10000 (default), number of simulation sample splits to use.
It is only used when method = "simulated",
or when method = "exact" reverts to method = "simulated", as previously explained.
Details
The Steel criterion uses the Wilcoxon test statistic in the pairwise comparisons of the
common control sample with each of the treatment samples. These statistics are used in
standardized form, using the means and standard deviations as they apply conditionally
given the tie pattern in the pooled data, see Scholz (2016). This conditional treatment allows for
correct usage in the presence of ties and is appropriate either when the samples are independent
and come from the same distribution (continuous or not) or when treatments are assigned
randomly among the total of N subjects. However, in the case of ties the significance probability
has to be viewed conditionally given the tie pattern.
The Steel statistic is used to test the hypothesis that the samples all come
from the same but unspecified distribution function F(x), or, under random treatment
assigment, that the treatments have no effect. The significance probability is the probability
of obtaining test results as extreme or more extreme than the observed test statistic,
when testing for the possibility of a treatment effect under any of the treatments.
For small sample sizes exact (conditional) null distribution
calculations are possible (with or without ties), based on a recursively extended
version of Algorithm C (Chase's sequence) in Knuth (2011), which allows the
enumeration of all possible splits of the pooled data into samples of
sizes of n_1, …, n_k, as appropriate under treatment randomization. This
is done in C, as is the simulation of such splits.
NA values are removed and the user is alerted with the total NA count.
It is up to the user to judge whether the removal of NA's is appropriate.
Value
A list of class kSamples with components
test.name
"Steel"
alternative
"greater", "less", or "two-sided"
k
number of samples being compared, including the control sample as the first one
ns
vector (n_1,…,n_k) of the k sample sizes
N
size of the pooled sample = n_1+…+n_k
n.ties
number of ties in the pooled sample
st
2 (or 3) vector containing the observed standardized Steel statistic,
its asymptotic P-value,
(its simulated or exact P-value)
warning
logical indicator, warning = TRUE when at least one
n_i < 5
null.dist
simulated or enumerated null distribution vector
of the test statistic. It is NULL when dist = FALSE or when
method = "asymptotic".
method
the method used.
Nsim
the number of simulations used.
W
vector
(W_{12},…, W_{1k})
of Mann-Whitney statistics comparing each observation under treatment i (> 1)
against each observation of the control sample.
mu
mean vector (n_1n_2/2,…,n_1n_k/2) of W.
tau
vector of standard deviations of W.
sig0
standard deviation used in calculating the significance probability
of the maximum (minimum) of (absolute) standardized Mann-Whitney statistics, see Scholz (2016).
sig
vector
(σ_1,…, σ_k)
of standard deviations used in calculating the significance probability
of the maximum (minimum) of (absolute) standardized Mann-Whitney statistics, see Scholz (2016).
warning
method = "exact" should only be used with caution.
Computation time is proportional to the number of enumerations.
Experiment with system.time and trial values for
Nsim to get a sense of the required computing time.
In most cases
dist = TRUE should not be used, i.e.,
when the returned distribution objects
become too large for R's work space.
References
Knuth, D.E. (2011), The Art of Computer Programming, Volume 4A
Combinatorial Algorithms Part 1, Addison-Wesley
Lehmann, E.L. (2006),
Nonparametrics, Statistical Methods Based on Ranks, Revised First Edition,
Springer Verlag.
Scholz, F.W. (2016), "On Steel's Test with Ties", submitted to Journal of Nonparametric Statistics.
Examples
z1 <- c(103, 111, 136, 106, 122, 114)
z2 <- c(119, 100, 97, 89, 112, 86)
z3 <- c( 89, 132, 86, 114, 114, 125)
z4 <- c( 92, 114, 86, 119, 131, 94)
y <- c(z1, z2, z3, z4)
g <- as.factor(c(rep(1, 6), rep(2, 6), rep(3, 6), rep(4, 6)))
set.seed(2627)
Steel.test(list(z1, z2, z3, z4), method = "simulated",
alternative = "less", Nsim = 1000)
# or with same seed
# Steel.test(z1, z2, z3, z4,method = "simulated",
# alternative = "less", Nsim = 1000)
# or with same seed
# Steel.test(y ~ g, method = "simulated",
# alternative = "less", Nsim=1000)