Steel.test {kSamples}

Steel's Multiple Comparison Wilcoxon Tests

Description

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, \ldots, 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+\ldots+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, \ldots, 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

ncomb = \frac{N!}{n_1!\ldots 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, \ldots, 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,\ldots,n_k) of the k sample sizes

N

size of the pooled sample = n_1+\ldots+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},\ldots, 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,\ldots,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 (\sigma_1,\ldots, \sigma_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. (2023), "On Steel's Test with Ties", https://arxiv.org/abs/2308.05873

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)