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
wherey
is a numeric variable,x
is a factor andblock
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 toNULL
, 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")
# }