Depth based multivariate Wilcoxon test for a scale difference.

Description

Depth based multivariate Wilcoxon test for a scale difference.

Usage

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

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".

...

arguments passed to depth function(e.g. method)

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=∑\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}})= \#≤ft\{ {{{z}}_{j}}\in {{{Z}}}:D({{{z}}_{j}},{{Z}})≤ D({{{y}}_{l}},{{Z}}) \right\}, l=1,...,m.

The distribution of S is symmetric about E(S)=1/2m(m{+}n{+1)} , its variance is {{D}^{2}}(S)={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

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x = mvrnorm(100, c(0,0), diag(2))
y = mvrnorm(100, c(0,0), diag(2)*1.4)
mWilcoxonTest(x,y)
mWilcoxonTest(x,y, 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)

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