# SiegelTukeyTest: Siegel-Tukey Test For Equality In Variability In DescTools: Tools for Descriptive Statistics

## Description

Non-parametric Siegel-Tukey test for equality in variability. The null hypothesis is that the variability of x is equal between two groups. A rejection of the null hypothesis indicates that variability differs between the two groups. `SiegelTukeyRank` returns the ranks, calculated after Siegel Tukey logic.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13``` ```SiegelTukeyTest(x, ...) ## Default S3 method: SiegelTukeyTest(x, y, adjust.median = FALSE, alternative = c("two.sided", "less", "greater"), mu = 0, exact = NULL, correct = TRUE, conf.int = FALSE, conf.level = 0.95, ...) ## S3 method for class 'formula' SiegelTukeyTest(formula, data, subset, na.action, ...) SiegelTukeyRank(x, g, drop.median = TRUE) ```

## Arguments

 `x, y` numeric vector of data values. Non-finite (e.g. infinite or missing) values will be omitted. `g` a vector or factor object giving the group for the corresponding elements of x. `adjust.median` Should between-group differences in medians be leveled before performing the test? In certain cases, the Siegel-Tukey test is susceptible to median differences and may indicate significant differences in variability that, in reality, stem from differences in medians. Default is `FALSE`. `alternative` a character string specifying the alternative hypothesis, must be one of `"two.sided"` (default), `"greater"` or `"less"`. You can specify just the initial letter. `mu` a number specifying an optional parameter used to form the null hypothesis. See Details. `exact` a logical indicating whether an exact p-value should be computed. This is passed directly to `wilcox.test`. `correct` a logical indicating whether to apply continuity correction in the normal approximation for the p-value. `conf.int` a logical indicating whether a confidence interval should be computed. `conf.level` confidence level of the interval. `formula` a formula of the form `lhs ~ rhs` where `lhs` gives the data values and rhs the corresponding groups. `data` an optional matrix or data frame (or similar: see `model.frame`) containing the variables in the formula `formula`. By default the variables are taken from `environment(formula)`. `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 NAs. Defaults to `getOption("na.action")`. `drop.median` logical, defining whether the median of the combined samples should be left out, ensuring that there's an even number of elements (which is a requirement of the Siegel-Tukey test). Defaults to `TRUE`. `...` further arguments to be passed to or from methods.

## Details

The Siegel-Tukey test has relatively low power and may, under certain conditions, indicate significance due to differences in medians rather than differences in variabilities (consider using the argument `adjust.median`). Consider also using `mood.test` or `ansari.test`.

## Value

A list of class `htest`, containing the following components:

 `statistic` Siegel-Tukey test (Wilcoxon test on tie-adjusted Siegel-Tukey ranks, after the median adjustment if specified). `p.value` the p-value for the test `null.value` is the value of the median specified by the null hypothesis. This equals the input argument `mu`. `alternative` a character string describing the alternative hypothesis. `method` the type of test applied `data.name` a character string giving the names of the data.

## Author(s)

Daniel Malter, Tal Galili <tal.galili@gmail.com>, Andri Signorell <andri@signorell.net>
published on: https://www.r-statistics.com/2010/02/siegel-tukey-a-non-parametric-test-for-equality-in-variability-r-code/

## References

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

Sheskin, D. J. (2004): Handbook of parametric and nonparametric statistical procedures 3rd edition. Chapman and Hall/CRC. Boca Raton, FL.

`mood.test`, `ansari.test`, `wilcox.test`, `LeveneTest`

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62``` ```# Duller, S. 183 x <- c(12, 13, 29, 30) y <- c(15, 17, 18, 24, 25, 26) SiegelTukeyTest(x, y) SiegelTukeyTest(x, y, alternative="greater") # 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") # compare to the recommended alternatives mood.test(old, new, alternative="greater") ansari.test(old, new, alternative="greater") # Bortz, S. 250 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 ~ id) # Sachs, Angewandte Statistik, 12. Auflage, 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) ### 1 x <- c(4,4,5,5,6,6) y <- c(0,0,1,9,10,10) SiegelTukeyTest(x, y) ### 2 # example for a non equal number of cases: x <- c(4,4,5,5,6,6) y <- c(0,0,1,9,10) SiegelTukeyTest(x, y) ### 3 x <- c(33, 62, 84, 85, 88, 93, 97, 4, 16, 48, 51, 66, 98) id <- c(0,0,0,0,0,0,0,1,1,1,1,1,1) SiegelTukeyTest(x ~ id) ### 4 x <- c(177,200,227,230,232,268,272,297,47,105,126,142,158,172,197,220,225,230,262,270) id <- c(rep(0,8),rep(1,12)) SiegelTukeyTest(x ~ id, adjust.median=TRUE) ### 5 x <- c(33,62,84,85,88,93,97) y <- c(4,16,48,51,66,98) SiegelTukeyTest(x, y) ### 6 x <- c(0,0,1,4,4,5,5,6,6,9,10,10) id <- c(0,0,0,1,1,1,1,1,1,0,0,0) SiegelTukeyTest(x ~ id) ### 7 x <- c(85,106,96, 105, 104, 108, 86) id <- c(0,0,1,1,1,1,1) SiegelTukeyTest(x ~ id) ```

### Example output

```	Siegel-Tukey-test for equal variability

data:  x and y
ST = 10, p-value = 0.009524
alternative hypothesis: true ratio of scales is not equal to 1

Siegel-Tukey-test for equal variability

data:  x and y
ST = 10, p-value = 0.004762
alternative hypothesis: true ratio of scales is greater than 1

Siegel-Tukey-test for equal variability

data:  old and new
ST = 34, p-value = 0.002622
alternative hypothesis: true ratio of scales is greater than 1

Mood two-sample test of scale

data:  old and new
Z = 2.8666, p-value = 0.002075
alternative hypothesis: greater

data:  old and new
AB = 18, p-value = 0.001573
alternative hypothesis: true ratio of scales is greater than 1

Siegel-Tukey-test for equal variability

data:  x by id
ST = 35.5, p-value = 0.6842
alternative hypothesis: true ratio of scales is not equal to 1

Warning message:
In wilcox.test.default(ranks0, ranks1, alternative = alternative,  :
cannot compute exact p-value with ties

Siegel-Tukey-test for equal variability

data:  A and B
ST = 134.5, p-value = 0.02324
alternative hypothesis: true ratio of scales is not equal to 1

Warning message:
In wilcox.test.default(ranks0, ranks1, alternative = alternative,  :
cannot compute exact p-value with ties

Siegel-Tukey-test for equal variability

data:  x and y
ST = 57, p-value = 0.003601
alternative hypothesis: true ratio of scales is not equal to 1

Warning message:
In wilcox.test.default(ranks0, ranks1, alternative = alternative,  :
cannot compute exact p-value with ties

Siegel-Tukey-test for equal variability

data:  x and y
ST = 15, p-value = 0.01141
alternative hypothesis: true ratio of scales is not equal to 1

Warning message:
In wilcox.test.default(ranks0, ranks1, alternative = alternative,  :
cannot compute exact p-value with ties

Siegel-Tukey-test for equal variability

data:  x by id
ST = 24, p-value = 0.202
alternative hypothesis: true ratio of scales is not equal to 1

Siegel-Tukey-test for equal variability

data:  x by id
ST = 106, p-value = 0.09788
alternative hypothesis: true ratio of scales is not equal to 1

Siegel-Tukey-test for equal variability

data:  x and y
ST = 24, p-value = 0.202
alternative hypothesis: true ratio of scales is not equal to 1

Siegel-Tukey-test for equal variability

data:  x by id
ST = 21, p-value = 0.003601
alternative hypothesis: true ratio of scales is not equal to 1

Warning message:
In wilcox.test.default(ranks0, ranks1, alternative = alternative,  :
cannot compute exact p-value with ties

Siegel-Tukey-test for equal variability

data:  x by id
ST = 4, p-value = 0.2667
alternative hypothesis: true ratio of scales is not equal to 1
```

DescTools documentation built on June 17, 2021, 5:12 p.m.