DepthProc (version 1.0.1)

mWilcoxonTest: Depth based multivariate Wilcoxon test for a scale difference.

Description

Depth based multivariate Wilcoxon test for a scale difference.

Usage

mWilcoxonTest(x, y, alternative = "greater")

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".

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 $R({{{y}}_{l}})= \#\left{ {{{z}}_{j}}\in {{{Z}}}:D({{{z}}_{j}},{{Z}})\le D({{{y}}_{l}},{{Z}}) \right}, l=1,...,m.$

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

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

#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)