# ScaleTests

0th

Percentile

##### Independent Two- and K-Sample Scale Tests

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

Keywords
htest
##### Usage
## S3 method for class 'formula':
ansari_test(formula, data, subset = NULL, weights = NULL, \dots)
## S3 method for class 'IndependenceProblem':
ansari_test(object,
alternative = c("two.sided", "less", "greater"),
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, \dots)
## S3 method for class 'IndependenceProblem':
fligner_test(object,
ties.method = c("mid-ranks", "average-scores"),
distribution = c("asymptotic", "approximate"),
...)
##### Details

The null hypothesis of the equality of the distribution of y in the groups given by x is tested. In particular, the methods documented here are designed to detect scale alternatives. For a general description of the test procedures documented here we refer to Hollander & Wolfe (1999).

The asymptotic null distribution is computed by default for both procedures. Exact p-values may be computed for the Ansari-Bradley test can be approximated via Monte-Carlo for the Fligner-Killeen procedure. Exact p-values are computed either by the shift algorithm (Streitberg & R"ohmel, 1986, 1987) or by the split-up algorithm (van de Wiel, 2001).

The Ansari-Bradley test can be used to test the two-sided hypothesis $var(Y_1) / var(Y_2) = 1$, where $var(Y_i)$ is the variance of the responses in the ith group. Confidence intervals for the ratio of scales are available for the Ansari-Bradley test and are computed according to Bauer (1972). In case alternative = "less", the null hypothesis $var(Y_1) / var(Y_2) \ge 1$ is tested and alternative = "greater" corresponds to $var(Y_1) / var(Y_2) \le 1$.

For the adjustment of scores for tied values see Hajek, Sidak and Sen (1999), page 131ff.

##### Value

• An object inheriting from class IndependenceTest-class with methods show, statistic, expectation, covariance and pvalue. The null distribution can be inspected by pperm, dperm, qperm and support methods. Confidence intervals can be extracted by confint.

##### References

Myles Hollander & Douglas A. Wolfe (1999). Nonparametric Statistical Methods, 2nd Edition. New York: John Wiley & Sons.

Bernd Streitberg & Joachim R"ohmel (1986). Exact distributions for permutations and rank tests: An introduction to some recently published algorithms. Statistical Software Newsletter 12(1), 10--17.

Bernd Streitberg & Joachim R"ohmel (1987). Exakte Verteilungen f"ur Rang- und Randomisierungstests im allgemeinen $c$-Stichprobenfall. EDV in Medizin und Biologie 18(1), 12--19.

Mark A. van de Wiel (2001). The split-up algorithm: a fast symbolic method for computing p-values of rank statistics. Computational Statistics 16, 519--538.

David F. Bauer (1972). Constructing confidence sets using rank statistics. Journal of the American Statistical Association 67, 687--690.

Jaroslav Hajek, Zbynek Sidak & Pranab K. Sen (1999). Theory of Rank Tests. San Diego, London: Academic Press.

##### Examples
### Serum Iron Determination Using Hyland Control Sera
### Hollander & Wolfe (1999), page 147
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 = factor(gl(2, 20), labels = c("Ramsay", "Jung-Parekh")))

ansari_test(serum ~ method, data = sid)

### exact p-value
ansari_test(serum ~ method, data = sid, distribution = "exact")

### Platelet Counts of Newborn Infants
### Hollander & Wolfe (1999), Table 5.4, page 171
platalet_counts <- data.frame(
counts = c(120, 124, 215, 90, 67, 95, 190, 180, 135, 399,
12, 20, 112, 32, 60, 40),
treatment = factor(c(rep("Prednisone", 10), rep("Control", 6))))

### Lepage test, Hollander & Wolfe (1999), page 172
lt <- independence_test(counts ~ treatment, data = platalet_counts,
ytrafo = function(data) trafo(data, numeric_trafo = function(x)
cbind(rank(x), ansari_trafo(x))),
teststat = "quad", distribution = approximate(B = 9999))

lt

### where did the rejection come from? Use maximum statistic
ltmax <- independence_test(counts ~ treatment, data = platalet_counts,
ytrafo = function(data) trafo(data, numeric_trafo = function(x)
matrix(c(rank(x), ansari_trafo(x)), ncol = 2,
dimnames = list(1:length(x), c("Location", "Scale")))),
teststat = "max")

### points to a difference in location
pvalue(ltmax, method = "single-step")

### Funny: We could have used a simple Bonferroni procedure
### since the correlation between the Wilcoxon and Ansari-Bradley
### test statistics is zero
covariance(ltmax)
Documentation reproduced from package coin, version 1.0-7, License: GPL-2

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