Depth based multivariate Wilcoxon test for a scale difference.

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

## Arguments

x data matrix data matrix a character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater" or "less". 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)#>
#> 	Multivariate Wilcoxon test for equality of dispersion
#>
#> data:  dep_x and dep_y
#> W = 5618, p-value = 0.1314
#> alternative hypothesis: true dispersion ratio is not equal to 1
#> mWilcoxonTest(x, y, depth_params = list(method = "LP"))#>
#> 	Multivariate Wilcoxon test for equality of dispersion
#>
#> data:  dep_x and dep_y
#> W = 5608, p-value = 0.1377
#> alternative hypothesis: true dispersion ratio is not equal to 1
#>
# 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)#>
#> 	Multivariate Wilcoxon test for equality of dispersion
#>
#> data:  dep_x and dep_y
#> W = 20182, p-value = 6.228e-06
#> alternative hypothesis: true dispersion ratio is not equal to 1
#>