Steel's Many-to-One Rank Test
Performs Steel's non-parametric many-to-one comparison test for Wilcox-type ranked data.
steelTest(x, ...) ## Default S3 method: steelTest(x, g, alternative = c("greater", "less"), ...) ## S3 method for class 'formula' steelTest( formula, data, subset, na.action, alternative = c("greater", "less"), ... )
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 |
alternative |
the alternative hypothesis. Defaults to |
formula |
a formula of the form |
data |
an optional matrix or data frame (or similar: see
|
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 |
For many-to-one comparisons (pairwise comparisons with one control) in an one-factorial balanced layout with non-normally distributed residuals Steels's non-parametric single-step test can be performed. Let there be k treatment levels (excluding the control), then k pairwise comparisons can be performed between the i-th treatment level and the control. H_i: θ_0 = θ_i is tested in the one-tailed case (less) against A_i: θ_0 > θ_i, ~~ (1 ≤ i ≤ k).
For each control - treatment level the data are ranked in increasing order.
The ranksum R_i for the i-th treatment level is compared
to a critical R value and is significantly(p = 0.05) less,
if R_i ≤ R. For the alternative = "greater"
the sign is changed.
The function does not return p-values. Instead the critical R-values as given in the tables of USEPA (2002) for α = 0.05 (one-sided, less) are looked up according to the balanced sample sizes (n) and the order number of the dose level (i).
A list with class "osrt"
that contains the following components:
a character string indicating what type of test was performed.
a character string giving the name(s) of the data.
the estimated statistic(s)
critical values for α = 0.05.
a character string describing the alternative hypothesis.
the parameter(s) of the test distribution.
a string that denotes the test distribution.
There are print and summary methods available.
The critical rank sum values were taken from Table E.5 of USEPA (2002).
USEPA (2002) Short-term Methods for Estimating the Chronic Toxicity of Effluents and Receiving Waters to Freshwater Organisms, 4th edition, EPA-821-R-02-013.
Steel's Many-to-One Rank test is only applicable for balanced designs and directional hypotheses. An error message will occur, if the design is unbalanced. In the current implementation, only one-sided tests on the level of α = 0.05 can be performed.
Steel, R. G. D. (1959) A multiple comparison rank sum test: treatments versus control, Biometrics 15, 560–572.
## 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("0", "I", "II") ## Steel's Test steelTest(x ~ g) ## Example from USEPA (2002): ## Reproduction data from a Ceriodaphnia dubia ## 7-day chronic test to several concentrations ## of effluent. Dose level 50% is excluded. x <- c(20, 26, 26, 23, 24, 27, 26, 23, 27, 24, 13, 15, 14, 13, 23, 26, 0, 25, 26, 27, 18, 22, 13, 13, 23, 22, 20, 22, 23, 22, 14, 22, 20, 23, 20, 23, 25, 24, 25, 21, 9, 0, 9, 7, 6, 10, 12, 14, 9, 13, rep(0,10)) g <- gl(6, 10) levels(g) <- c("Control", "3%", "6%", "12%", "25%", "50%") ## NOEC at 3%, LOEC at 6% steelTest(x ~ g, subset = g != "50%", alternative = "less")
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