LocationTests: Independent Two- and K-Sample Location Tests

Description

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

Usage

## S3 method for class 'formula':
oneway_test(formula, data, subset = NULL, weights = NULL, \dots)
## S3 method for class 'IndependenceProblem':
oneway_test(object, 
    alternative = c("two.sided", "less", "greater"),
    distribution = c("asymptotic", "approximate", "exact"), ...)
## S3 method for class 'formula':
wilcox_test(formula, data, subset = NULL, weights = NULL, \dots)
## S3 method for class 'IndependenceProblem':
wilcox_test(object, 
    alternative = c("two.sided", "less", "greater"),
    distribution = c("asymptotic", "approximate", "exact"),
    conf.int = FALSE, conf.level = 0.95, ...)
## S3 method for class 'formula':
normal_test(formula, data, subset = NULL, weights = NULL, \dots)
## S3 method for class 'IndependenceProblem':
normal_test(object, 
    alternative = c("two.sided", "less", "greater"),
    distribution = c("asymptotic", "approximate", "exact"),
    ties.method = c("mid-ranks", "average-scores"),
    conf.int = FALSE, conf.level = 0.95, ...)
## S3 method for class 'formula':
median_test(formula, data, subset = NULL, weights = NULL, \dots)
## S3 method for class 'IndependenceProblem':
median_test(object, 
    alternative = c("two.sided", "less", "greater"),
    distribution = c("asymptotic", "approximate", "exact"),
    conf.int = FALSE, conf.level = 0.95, ...)
## S3 method for class 'formula':
kruskal_test(formula, data, subset = NULL, weights = NULL, \dots)
## S3 method for class 'IndependenceProblem':
kruskal_test(object, 
    distribution = c("asymptotic", "approximate"), ...)

Arguments

formula

a formula of the form y ~ x | block where y is a numeric variable giving the data values and x a factor with two or more levels giving the corresponding groups. block is an optional factor fo

data

an optional data frame containing the variables in the model formula.

subset

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

weights

an optional formula of the form ~ w defining integer valued weights for the observations.

object

an object of class IndependenceProblem.

alternative

a character, the alternative hypothesis must be one of "two.sided" (default), "greater" or "less". You can specify just the initial letter.

distribution

a character, the null distribution of the test statistic can be computed exactly or can be approximated by its asymptotic distribution (asymptotic) or via Monte-Carlo resampling (approximate). Alterna

ties.method

a character, two methods are available to adjust scores for ties, either the score generating function is applied to mid-ranks or the scores computed based on random ranks are averaged for all tied values (average-scor

conf.int

a logical indicating whether a confidence interval for the difference in location should be computed.

conf.level

confidence level of the interval.

...

further arguments to be passed to or from methods.

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.

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 shift alternatives. For a general description of the test procedures documented here we refer to Hollander & Wolfe (1999).

The test procedures apply a rank transformation to the response values y, except of oneway_test which computes a test statistic using the untransformed response values.

The asymptotic null distribution is computed by default for all procedures. Exact p-values may be computed for the two-sample problems and can be approximated via Monte-Carlo resampling for all procedures. 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 linear rank tests for two samples (wilcox_test, normal_test and median_test) can be used to test the two-sided hypothesis $H_0: Y_1 - Y_2 = 0$, where $Y_i$ is the median of the responses in the ith group. Confidence intervals for the difference in location are available for the rank-based procedures and are computed according to Bauer (1972). In case alternative = "less", the null hypothesis $H_0: Y_1 - Y_2 \ge 0$ is tested and alternative = "greater" corresponds to a null hypothesis $H_0: Y_1 - Y_2 \le 0$.

In case x is an ordered factor, kruskal_test computes the linear-by-linear association test for ordered alternatives.

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

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

Run this code

### Tritiated Water Diffusion Across Human Chorioamnion
### Hollander & Wolfe (1999), Table 4.1, page 110
water_transfer <- data.frame(
    pd = c(0.80, 0.83, 1.89, 1.04, 1.45, 1.38, 1.91, 1.64, 0.73, 1.46,
           1.15, 0.88, 0.90, 0.74, 1.21),
    age = factor(c(rep("At term", 10), rep("12-26 Weeks", 5))))

### Wilcoxon-Mann-Whitney test, cf. Hollander & Wolfe (1999), page 111
### exact p-value and confidence interval for the difference in location
### (At term - 12-26 Weeks)
wt <- wilcox_test(pd ~ age, data = water_transfer, 
            distribution = "exact", conf.int = TRUE)
print(wt)

### extract observed Wilcoxon statistic, i.e, the sum of the
### ranks for age = "12-26 Weeks"
statistic(wt, "linear")

### its expectation
expectation(wt)

### and variance
covariance(wt)

### and, finally, the exact two-sided p-value
pvalue(wt)

### Confidence interval for difference (12-26 Weeks - At term)
wilcox_test(pd ~ age, data = water_transfer, 
    xtrafo = function(data) 
        trafo(data, factor_trafo = function(x) as.numeric(x == levels(x)[2])),
        distribution = "exact", conf.int = TRUE)


### Permutation test, asymptotic p-value
oneway_test(pd ~ age, data = water_transfer)

### approximate p-value (with 99\% confidence interval)
pvalue(oneway_test(pd ~ age, data = water_transfer, 
                 distribution = approximate(B = 9999)))
### exact p-value
pt <- oneway_test(pd ~ age, data = water_transfer, distribution = "exact")
pvalue(pt)

### plot density and distribution of the standardised 
### test statistic
layout(matrix(1:2, nrow = 2))
s <- support(pt)
d <- sapply(s, function(x) dperm(pt, x))
p <- sapply(s, function(x) pperm(pt, x))
plot(s, d, type = "S", xlab = "Teststatistic", ylab = "Density")
plot(s, p, type = "S", xlab = "Teststatistic", ylab = "Cumm. Probability")


### Length of YOY Gizzard Shad from Kokosing Lake, Ohio,
### sampled in Summer 1984, Hollander & Wolfe (1999), Table 6.3, page 200
YOY <- data.frame(length = c(46, 28, 46, 37, 32, 41, 42, 45, 38, 44, 
                             42, 60, 32, 42, 45, 58, 27, 51, 42, 52, 
                             38, 33, 26, 25, 28, 28, 26, 27, 27, 27, 
                             31, 30, 27, 29, 30, 25, 25, 24, 27, 30),
                  site = factor(c(rep("I", 10), rep("II", 10),
                                  rep("III", 10), rep("IV", 10))))

### Kruskal-Wallis test, approximate exact p-value
kw <- kruskal_test(length ~ site, data = YOY, 
             distribution = approximate(B = 9999))
kw
pvalue(kw)

### Nemenyi-Damico-Wolfe-Dunn test (joint ranking)
### Hollander & Wolfe (1999), page 244 
### (where Steel-Dwass results are given)
if (require("multcomp")) {

    NDWD <- oneway_test(length ~ site, data = YOY,
        ytrafo = function(data) trafo(data, numeric_trafo = rank),
        xtrafo = function(data) trafo(data, factor_trafo = function(x)
            model.matrix(~x - 1) %*% t(contrMat(table(x), "Tukey"))),
        teststat = "maxtype", distribution = approximate(B = 90000))

    ### global p-value
    print(pvalue(NDWD))

    ### sites (I = II) != (III = IV) at alpha = 0.01 (page 244)
    print(pvalue(NDWD, method = "single-step"))
}

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