A nonparametric multiple-comparison procedure for comparing several treatment groups with a single control group based on Wilcoxon rank-sum statistics.
Arguments
- x
a numeric vector of observations, or a list of numeric vectors.
- ...
further arguments passed to methods.
- formula
a formula of the form
response ~ group.- data
an optional data frame containing the variables in
formula.- subset
an optional expression specifying a subset of observations to be used.
- na.action
a function specifying how missing values should be handled.
- g
a grouping factor corresponding to
x. Ignored ifxis a list.- control
the level of the control group against which all treatment groups are compared. Defaults to the first group.
- alternative
character string specifying the alternative hypothesis. One of
"two.sided","greater"or"less".- output
character string specifying the output format. One of
"list"(default) or"matrix".- alpha
the significance level used to compile the groups flagged as significantly different in the label attribute of the p-value matrix (default is
0.05)
Value
An object of class "rankTest" containing:
- res
comparison results. For
output="list"a matrix with columnsW,zandpval; foroutput="matrix"a many-to-one matrix of adjusted p-values.- pmat
matrix of adjusted p-values.
- statistic
observed Steel test statistic.
- p.value
asymptotic p-value for the global Steel test.
Additional information is stored in attributes:
method, alternative, output,
main, and data.name.
Details
Performs Steel's many-to-one rank test for independent samples. The procedure is the nonparametric analogue of Dunnett's test and controls the family-wise error rate when comparing multiple treatment groups against a common control.
The test statistic is based on pairwise Wilcoxon rank-sum statistics comparing each treatment group with the control group. Ties are handled using the asymptotic variance and covariance corrections described by Scholz (2023). Adjusted p-values are obtained from the asymptotic multivariate normal distribution of the standardized statistics.
If x is a list, its elements are taken as the samples to be
compared and hence have to be numeric data vectors. In this case,
g is ignored and one can simply use steelTest(x).
Otherwise, x must be a numeric vector and g a grouping
factor (or vector coercible to a factor) of the same length.
Steel's test is the nonparametric analogue of Dunnett's test. Unlike Dunn's or Conover's procedures, only comparisons against the designated control group are performed.
The asymptotic covariance structure follows Scholz (2023), including tie corrections for the Wilcoxon rank-sum statistics. P-values are obtained from the multivariate normal distribution corresponding to the joint asymptotic distribution of the standardized Steel statistics.
References
Steel, R. G. D. (1959). A multiple comparison rank sum test: Treatments versus control. Biometrics, 15, 560–572.
Scholz, F. W. (2023). Improved tie correction methods for Steel-type rank tests. Journal of Nonparametric Statistics, 35, 541–563.
See also
Other test.posthoc:
conoverTest(),
dscfTest(),
dunnTest(),
dunnettTest(),
gamesHowellTest(),
nemenyiTest(),
plot.PostHocTest(),
postHoc,
scheffeTest(),
signifDiff()
Examples
## Hollander & Wolfe example
x <- c(2.9, 3.0, 2.5, 2.6, 3.2)
y <- c(3.8, 2.7, 4.0, 2.4)
z <- c(2.8, 3.4, 3.7, 2.2, 2.0)
steelTest(list(x, y, z))
#>
#> Steel test for multiple comparisons with a control
#>
#> W z pval
#> 2-1 12 0.4898979 0.8464
#> 3-1 12 -0.1044466 0.9924
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
x <- c(x, y, z)
g <- factor(rep(1:3, c(5, 4, 5)))
steelTest(x, g)
#>
#> Steel test for multiple comparisons with a control
#>
#> W z pval
#> 2-1 12 0.4898979 0.8464
#> 3-1 12 -0.1044466 0.9924
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
## Specify control group
steelTest(x, g, control = "1")
#>
#> Steel test for multiple comparisons with a control
#>
#> W z pval
#> 2-1 12 0.4898979 0.8464
#> 3-1 12 -0.1044466 0.9924
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
## Formula interface
steelTest(Ozone ~ factor(Month), data = airquality)
#>
#> Steel test for multiple comparisons with a control
#>
#> W z pval
#> 6-5 152.0 1.321795 0.50052
#> 7-5 566.5 4.183685 0.00012 ***
#> 8-5 548.5 3.854117 0.00043 ***
#> 9-5 470.0 1.568481 0.34258
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>