ScaleTests | R Documentation |

`K`

-Sample Scale TestsTesting the equality of the distributions of a numeric response variable in two or more independent groups against scale alternatives.

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

`formula` |
a formula of the form |

`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 |

`weights` |
an optional formula of the form |

`object` |
an object inheriting from class |

`conf.int` |
a logical indicating whether a confidence interval for the ratio of scales
should be computed. Defaults to |

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

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

`...` |
further arguments to be passed to |

`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 Hájek, Šidák 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.

An object inheriting from class `"IndependenceTest"`

.
Confidence intervals can be extracted by `confint()`

.

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.

Bauer, D. F. (1972). Constructing confidence sets using rank statistics.
*Journal of the American Statistical Association* **67**(339),
687–690. \Sexpr[results=rd]{tools:::Rd_expr_doi("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. \Sexpr[results=rd]{tools:::Rd_expr_doi("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.
\Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/00401706.1981.10487680")}

Hájek, J., Šidák, 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.

```
## 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")
```

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