mWilcoxonTest {DepthProc}

Multivariate Wilcoxon test for equality of dispersion.

Description

Depth based multivariate Wilcoxon test for a scale difference.

Usage

mWilcoxonTest(x, y, alternative = "two.sided", depth_params = list())

Arguments

x

data matrix

y

data matrix

alternative

a character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater" or "less".

depth_params

list of parameters for function depth (method, threads, ndir, la, lb, pdim, mean, cov, exact).

Details

Having two samples {X} ^ {n} and {Y} ^ {m} using any depth function, we can compute depth values in a combined sample {Z} ^ {n + m} = {X} ^ {n} \cup {Y} ^ {m} , assuming the empirical distribution calculated basing on all observations, or only on observations belonging to one of the samples {X} ^ {n} or {Y} ^ {m} .

For example if we observe {X}_{l}'s depths are more likely to cluster tightly around the center of the combined sample, while {Y}_{l}'s depths are more likely to scatter outlying positions, then we conclude {Y} ^ {m} was drawn from a distribution with larger scale.

Properties of the DD plot based statistics in the i.i.d setting were studied by Li & Liu (2004). Authors proposed several DD-plot based statistics and presented bootstrap arguments for their consistency and good effectiveness in comparison to Hotelling T ^ 2 and multivariate analogues of Ansari-Bradley and Tukey-Siegel statistics. Asymptotic distributions of depth based multivariate Wilcoxon rank-sum test statistic under the null and general alternative hypotheses were obtained by Zuo & He (2006). Several properties of the depth based rang test involving its unbiasedness was critically discussed by Jureckova & Kalina (2012). Basing on DD-plot object, which is available within the DepthProc it is possible to define several multivariate generalizations of one-dimensional rank and order statistics in an easy way. These generalizations cover well known Wilcoxon rang-sum statistic.

The depth based multivariate Wilcoxon rang sum test is especially useful for the multivariate scale changes detection and was introduced among other by Liu & Singh (2003) and intensively studied by Jureckowa & Kalina (2012) and Zuo & He (2006) in the i.i.d. setting.

For the samples {{{X}} ^ {m}} = \{{{{X}}_{1}}, ..., {{{X}}_{m}}\} , {{{Y}} ^ {n}} = \{{{{Y}}_{1}}, ..., {{{Y}}_{n}}\} , their d_{1} ^ {X}, ..., d_{m} ^ {X} , d_{1} ^ {Y}, ..., d_{n} ^ {Y} , depths w.r.t. a combined sample {{Z}} = {{{X}} ^ {n}} \cup {{{Y}} ^ {m}} the Wilcoxon statistic is defined as S = \sum\limits_{i = 1} ^ {m}{{{R}_{i}}} , where {R}_{i} denotes the rang of the i-th observation, i = 1, ..., m in the combined sample R({{{y}}_{l}}) = \sharp\left\{ {{{z}}_{j}} \in {{{Z}}}:D({{{z}}_{j}}, {{Z}}) \le D({{{y}}_{l}}, {{Z}}) \right\}, l = 1, ..., m .

The distribution of S is symmetric about E(S) = \frac{ 1 }{ 2 }m(m + n + 1) , its variance is {{D} ^ {2}}(S) = \frac{ 1 }{ 12 }mn(m + n + 1) .

References

Jureckova J, Kalina J (2012). Nonparametric multivariate rank tests and their unbiasedness. Bernoulli, 18(1), 229–251. Li J, Liu RY (2004). New nonparametric tests of multivariate locations and scales using data depth. Statistical Science, 19(4), 686–696. Liu RY, Singh K (1995). A quality index based on data depth and multivariate rank tests. Journal of American Statistical Association, 88, 252–260. Zuo Y, He X (2006). On the limiting distributions of multivariate depth-based rank sum statistics and related tests. The Annals of Statistics, 34, 2879–2896.

Examples

# EXAMPLE 1
x <- mvrnorm(100, c(0, 0), diag(2))
y <- mvrnorm(100, c(0, 0), diag(2) * 1.4)
mWilcoxonTest(x, y)
mWilcoxonTest(x, y, depth_params = list(method = "LP"))

# EXAMPLE 2
data(under5.mort)
data(inf.mort)
data(maesles.imm)
data2011 <- na.omit(cbind(under5.mort[, 22], inf.mort[, 22],
                          maesles.imm[, 22]))
data1990 <- na.omit(cbind(under5.mort[, 1], inf.mort[, 1], maesles.imm[, 1]))
mWilcoxonTest(data2011, data1990)