# ScaleTests

##### Two- and \(K\)-Sample Scale Tests

Testing the equality of the distributions of a numeric response variable in two or more independent groups against scale alternatives.

- Keywords
- htest

##### Usage

```
# S3 method for formula
taha_test(formula, data, subset = NULL, weights = NULL, …)
# S3 method for IndependenceProblem
taha_test(object, conf.int = FALSE, conf.level = 0.95, …)
```# S3 method for formula
klotz_test(formula, data, subset = NULL, weights = NULL, …)
# S3 method for IndependenceProblem
klotz_test(object, ties.method = c("mid-ranks", "average-scores"),
conf.int = FALSE, conf.level = 0.95, …)

# S3 method for formula
mood_test(formula, data, subset = NULL, weights = NULL, …)
# S3 method for IndependenceProblem
mood_test(object, ties.method = c("mid-ranks", "average-scores"),
conf.int = FALSE, conf.level = 0.95, …)

# S3 method for formula
ansari_test(formula, data, subset = NULL, weights = NULL, …)
# S3 method for IndependenceProblem
ansari_test(object, ties.method = c("mid-ranks", "average-scores"),
conf.int = FALSE, conf.level = 0.95, …)

# S3 method for formula
fligner_test(formula, data, subset = NULL, weights = NULL, …)
# S3 method for IndependenceProblem
fligner_test(object, ties.method = c("mid-ranks", "average-scores"),
conf.int = FALSE, conf.level = 0.95, …)

# S3 method for formula
conover_test(formula, data, subset = NULL, weights = NULL, …)
# S3 method for IndependenceProblem
conover_test(object, conf.int = FALSE, conf.level = 0.95, …)

##### Arguments

- formula
a formula of the form

`y ~ x | block`

where`y`

is a numeric variable,`x`

is a factor and`block`

is an optional factor for stratification.- data
an optional data frame containing the variables in the model formula.

- subset
an optional vector specifying a subset of observations to be used. Defaults to

`NULL`

.- weights
an optional formula of the form

`~ w`

defining integer valued case weights for each observation. Defaults to`NULL`

, implying equal weight for all observations.- object
- conf.int
a logical indicating whether a confidence interval for the ratio of scales should be computed. Defaults to

`FALSE`

.- conf.level
a numeric, confidence level of the interval. Defaults to

`0.95`

.- ties.method
a character, the method used to handle ties: the score generating function either uses mid-ranks (

`"mid-ranks"`

, default) or averages the scores of randomly broken ties (`"average-scores"`

).- …
further arguments to be passed to

`independence_test`

.

##### Details

`taha_test`

, `klotz_test`

, `mood_test`

, `ansari_test`

,
`fligner_test`

and `conover_test`

provide the Taha test, the Klotz
test, the Mood test, the Ansari-Bradley test, the Fligner-Killeen test and the
Conover-Iman test. A general description of these methods is given by
Hollander and Wolfe (1999). For the adjustment of scores for tied
values see H<U+00E1>jek, <U+0160>id<U+00E1>k and Sen (1999,
pp. 133--135).

The null hypothesis of equality, or conditional equality given `block`

,
of the distribution of `y`

in the groups defined by `x`

is tested
against scale alternatives. In the two-sample case, the two-sided null
hypothesis is \(H_0\!: V(Y_1) / V(Y_2) = 1\),
where \(V(Y_s)\) is the variance of the responses in the \(s\)th sample.
In case `alternative = "less"`

, the null hypothesis is \(H_0\!: V(Y_1)
/ V(Y_2) \ge 1\). When
`alternative = "greater"`

, the null hypothesis is \(H_0\!: V(Y_1) /
V(Y_2) \le 1\). Confidence intervals for the
ratio of scales are available and computed according to Bauer (1972).

The Fligner-Killeen test uses median centering in each of the samples, as suggested by Conover, Johnson and Johnson (1981), whereas the Conover-Iman test, following Conover and Iman (1978), uses mean centering in each of the samples.

The conditional null distribution of the test statistic is used to obtain
\(p\)-values and an asymptotic approximation of the exact distribution is
used by default (`distribution = "asymptotic"`

). Alternatively, the
distribution can be approximated via Monte Carlo resampling or computed
exactly for univariate two-sample problems by setting `distribution`

to
`"approximate"`

or `"exact"`

respectively. See
`asymptotic`

, `approximate`

and `exact`

for details.

##### Value

An object inheriting from class `"'>IndependenceTest"`

.
Confidence intervals can be extracted by confint.

##### Note

In the two-sample case, a *large* value of the Ansari-Bradley
statistic indicates that sample 1 is *less* variable than sample
2, whereas a *large* value of the statistics due to Taha, Klotz,
Mood, Fligner-Killeen, and Conover-Iman indicate that sample 1 is
*more* variable than sample 2.

##### References

Bauer, D. F. (1972). Constructing confidence sets using rank statistics.
*Journal of the American Statistical Association* **67**(339),
687--690. 10.1080/01621459.1972.10481279

Conover, W. J. and Iman, R. L. (1978). Some exact tables for the squared
ranks test. *Communications in Statistics -- Simulation and Computation*
**7**(5), 491--513. 10.1080/03610917808812093

Conover, W. J., Johnson, M. E. and Johnson, M. M. (1981). A comparative
study of tests for homogeneity of variances, with applications to the outer
continental shelf bidding data. *Technometrics* **23**(4), 351--361.
10.1080/00401706.1981.10487680

H<U+00E1>jek, J., <U+0160>id<U+00E1>k, Z. and Sen, P. K. (1999).
*Theory of Rank Tests*, Second Edition. San Diego: Academic Press.

Hollander, M. and Wolfe, D. A. (1999). *Nonparametric Statistical
Methods*, Second Edition. York: John Wiley & Sons.

##### Examples

```
# NOT RUN {
## Serum Iron Determination Using Hyland Control Sera
## Hollander and Wolfe (1999, p. 147, Tab 5.1)
sid <- data.frame(
serum = c(111, 107, 100, 99, 102, 106, 109, 108, 104, 99,
101, 96, 97, 102, 107, 113, 116, 113, 110, 98,
107, 108, 106, 98, 105, 103, 110, 105, 104,
100, 96, 108, 103, 104, 114, 114, 113, 108, 106, 99),
method = gl(2, 20, labels = c("Ramsay", "Jung-Parekh"))
)
## Asymptotic Ansari-Bradley test
ansari_test(serum ~ method, data = sid)
## Exact Ansari-Bradley test
pvalue(ansari_test(serum ~ method, data = sid,
distribution = "exact"))
## Platelet Counts of Newborn Infants
## Hollander and Wolfe (1999, p. 171, Tab. 5.4)
platelet <- data.frame(
counts = c(120, 124, 215, 90, 67, 95, 190, 180, 135, 399,
12, 20, 112, 32, 60, 40),
treatment = factor(rep(c("Prednisone", "Control"), c(10, 6)))
)
## Approximative (Monte Carlo) Lepage test
## Hollander and Wolfe (1999, p. 172)
lepage_trafo <- function(y)
cbind("Location" = rank_trafo(y), "Scale" = ansari_trafo(y))
independence_test(counts ~ treatment, data = platelet,
distribution = approximate(nresample = 10000),
ytrafo = function(data)
trafo(data, numeric_trafo = lepage_trafo),
teststat = "quadratic")
## Why was the null hypothesis rejected?
## Note: maximum statistic instead of quadratic form
ltm <- independence_test(counts ~ treatment, data = platelet,
distribution = approximate(nresample = 10000),
ytrafo = function(data)
trafo(data, numeric_trafo = lepage_trafo))
## Step-down adjustment suggests a difference in location
pvalue(ltm, method = "step-down")
## The same results are obtained from the simple Sidak-Holm procedure since the
## correlation between Wilcoxon and Ansari-Bradley test statistics is zero
cov2cor(covariance(ltm))
pvalue(ltm, method = "step-down", distribution = "marginal", type = "Sidak")
# }
```

*Documentation reproduced from package coin, version 1.3-1, License: GPL-2*