mWilcoxonTest function

Multivariate Wilcoxon test for equality of dispersion.

Depth based multivariate Wilcoxon test for a scale difference.

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 ${{D} ^ {2}}(S) = \frac{ 1 }{ 12 }mn(m + n + 1)$.

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)

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.