Siegel-Tukey Test for Comparing Dispersion Between Two Independent Groups

The Siegel-Tukey test examines the null hypothesis that the variability (scale) of x and y is equal. Rejection indicates that the two groups differ in spread. The test is distribution-free and does not assume normality, but it does assume equal medians under the null hypothesis of equal scale. If the medians differ, the test may detect that difference rather than a difference in scale; use adjustMedian = TRUE to remove median differences before testing.

Ranks are assigned to the combined sorted sample in the pattern 1, 4, 5, 8, 9, ... from the extremes inward, and 2, 3, 6, 7, ... from the second-lowest upward. If the combined sample size is odd, the median observation is dropped before ranking (it is taken from the larger group when group sizes differ).

Ties receive average ranks. The p-value is computed exactly (via pwilcox()) when there are no ties and both samples are smaller than 50 observations; otherwise a normal approximation with tie-corrected variance is used. This behaviour can be overridden with exact.

Note: The Siegel-Tukey test has relatively low power compared to alternatives such as ansari.test() or mood.test(), and may indicate significance due to median differences rather than scale differences when adjustMedian = FALSE.

Usage

siegelTukeyTest(x, ...)

# S3 method for class 'formula'
siegelTukeyTest(formula, data, subset, na.action = na.pass, ...)

# Default S3 method
siegelTukeyTest(
  x,
  y,
  alternative = c("two.sided", "less", "greater"),
  mu = 0,
  adjustMedian = FALSE,
  exact = NA,
  correct = TRUE,
  ...
)

Arguments

x, y: numeric vectors of data values.
...: further arguments passed to or from methods.
formula: a formula of the form response ~ group, where response is a numeric vector and group a factor or vector with exactly two levels.
data: an optional data frame (or matrix, coerced to data frame) containing the variables in formula. If not supplied, variables are taken from environment(formula).
subset: an optional vector specifying a subset of observations to use.
na.action: a function indicating how to handle NAs in the formula interface. Defaults to na.pass; NAs in x or y are silently dropped in the default method.
alternative: a character string specifying the alternative hypothesis: "two.sided" (default), "greater", or "less". Partial matching is allowed.
mu: a single number specifying the location parameter under the null hypothesis. Default is 0.
adjustMedian: logical; if TRUE, the median of x is shifted to equal the median of y before ranking, to prevent median differences from inflating the test statistic. Default is FALSE.
exact: logical; if TRUE, an exact p-value is computed via pwilcox(). Exact computation is not possible in the presence of ties; a warning is issued and the normal approximation is used instead. If NA (default), exact computation is used when both samples have fewer than 50 observations and there are no ties.
correct: logical; if TRUE (default), a continuity correction is applied in the normal approximation. Ignored when exact = TRUE or when ties are present (continuity correction is not appropriate with tie-corrected variance).

Value

An object of class "htest" with the following components:

statistic: the Wilcoxon rank-sum statistic \(W\) computed on the Siegel-Tukey ranks.
p.value: the p-value of the test.
null.value: the location parameter mu under the null hypothesis.
alternative: a character string describing the alternative hypothesis.
method: a character string identifying the test.
data.name: a character string giving the names of the data.
exact: logical indicating whether the exact distribution was used.
ties: logical indicating whether ties were present in the Siegel-Tukey ranks.

Details

A nonparametric test for differences in scale (variability) between two independent groups. Ranks are assigned by alternating between the extremes and the center of the combined sorted sample, and a Wilcoxon-Mann-Whitney statistic is then computed on these ranks.

Note

Originally based on a blog post by Tal Galili:
https://www.r-statistics.com/2010/02/siegel-tukey-a-non-parametric-test-for-equality-in-variability-r-code/

References

Siegel, S. and Tukey, J. W. (1960): A nonparametric sum of ranks procedure for relative spread in unpaired samples. Journal of the American Statistical Association, 55(291), 429–445.

Sheskin, D. J. (2004): Handbook of Parametric and Nonparametric Statistical Procedures, 3rd ed. Chapman & Hall/CRC, Boca Raton, FL.

Examples

# Duller, S. 183
x <- c(12, 13, 29, 30)
y <- c(15, 17, 18, 24, 25, 26)
siegelTukeyTest(x, y)
#> 
#> 	Siegel-Tukey test for scale differences
#> 
#> data:  x and y
#> W = 45, p-value = 0.009524
#> alternative hypothesis: true mu is not equal to 0
#> 
siegelTukeyTest(x, y, alternative = "greater")
#> 
#> 	Siegel-Tukey test for scale differences
#> 
#> data:  x and y
#> W = 45, p-value = 0.004762
#> alternative hypothesis: true mu is greater than 0
#> 

# Duller, S. 323
old <- c(870, 930, 935, 1045, 1050, 1052, 1055)
new <- c(932, 970, 980, 1001, 1009, 1030, 1032, 1040, 1046)
siegelTukeyTest(old, new, alternative = "greater")
#> 
#> 	Siegel-Tukey test for scale differences
#> 
#> data:  old and new
#> W = 102, p-value = 0.002622
#> alternative hypothesis: true mu is greater than 0
#> 
# recommended alternatives:
mood.test(old, new, alternative = "greater")
#> 
#> 	Mood two-sample test of scale
#> 
#> data:  old and new
#> Z = 2.8666, p-value = 0.002075
#> alternative hypothesis: greater
#> 
ansari.test(old, new, alternative = "greater")
#> 
#> 	Ansari-Bradley test
#> 
#> data:  old and new
#> AB = 18, p-value = 0.001573
#> alternative hypothesis: true ratio of scales is greater than 1
#> 

# Bortz, S. 250 -- formula interface
x  <- c(26.3, 26.5, 26.8, 27.0, 27.0, 27.2, 27.3,
        27.3, 27.4, 27.5, 27.6, 27.8, 27.9)
id <- c(2, 2, 2, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2) - 1
siegelTukeyTest(x ~ factor(id))
#> 
#> 	Siegel-Tukey test for scale differences
#> 
#> data:  groups[[1L]] and groups[[2L]]
#> W = 53.5, p-value = 0.8649
#> alternative hypothesis: true mu is not equal to 0
#> 

# Sachs (2007), S. 314
A <- c(10.1, 7.3, 12.6, 2.4, 6.1, 8.5, 8.8, 9.4, 10.1, 9.8)
B <- c(15.3, 3.6, 16.5, 2.9, 3.3, 4.2, 4.9, 7.3, 11.7, 13.1)
siegelTukeyTest(A, B)
#> 
#> 	Siegel-Tukey test for scale differences
#> 
#> data:  A and B
#> W = 75.5, p-value = 0.02825
#> alternative hypothesis: true mu is not equal to 0
#> 

# equal medians, different spread
x <- c(4, 4, 5, 5, 6, 6)
y <- c(0, 0, 1, 9, 10, 10)
siegelTukeyTest(x, y)
#> 
#> 	Siegel-Tukey test for scale differences
#> 
#> data:  x and y
#> W = 21, p-value = 0.003601
#> alternative hypothesis: true mu is not equal to 0
#> 

# unequal group sizes
x <- c(4, 4, 5, 5, 6, 6)
y <- c(0, 0, 1, 9, 10)
siegelTukeyTest(x, y)
#> 
#> 	Siegel-Tukey test for scale differences
#> 
#> data:  x and y
#> W = 15, p-value = 0.01141
#> alternative hypothesis: true mu is not equal to 0
#> 

# median adjustment
x <- c(177, 200, 227, 230, 232, 268, 272, 297)
y <- c(47, 105, 126, 142, 158, 172, 197, 220, 225, 230, 262, 270)
siegelTukeyTest(x, y, adjustMedian = TRUE)
#> 
#> 	Siegel-Tukey test for scale differences
#> 
#> data:  x and y
#> W = 104, p-value = 0.09788
#> alternative hypothesis: true mu is not equal to 0
#> 

# floating-point robustness (previously affected by merge precision bug)
y2 <- c(-1, 2, 2.1, 3)
x2 <- c(-5, -9, 13, 12, 90, 100)
siegelTukeyTest(x2, y2, adjustMedian = TRUE)  
#> 
#> 	Siegel-Tukey test for scale differences
#> 
#> data:  x2 and y2
#> W = 30, p-value = 0.1143
#> alternative hypothesis: true mu is not equal to 0
#> 
# p ~ 0.1143