Package 'GraphRankTest'

Rank-based Two-Sample Tests on Similarity / Graph-Induced Rank Matrices

Description

RISE constructs a nonnegative, symmetric rank/graph matrix $R$ from two samples $X$ and $Y$ (or from a pre-computed similarity matrix $S$), then computes a Hotelling-type quadratic statistic with an asymptotic chi-square p-value. Optionally, a permutation p-value is returned.

Usage

RISE(
  X = NULL,
  Y = NULL,
  S = NULL,
  sample1ID = NULL,
  sample2ID = NULL,
  k = 10,
  rank.type = "RgNN",
  perm = 0
)

Arguments

X	Numeric matrix of size $m \times p$ (first sample). Optional if S is supplied.
Y	Numeric matrix of size $n \times p$ (second sample). Optional if S is supplied.
S	Numeric similarity matrix of size $N \times N$ with $N=m+n$ (larger values indicate greater similarity). If X and Y are provided, S is constructed internally as -dist(rbind(X, Y)).
sample1ID	Integer indices (length $m$) for sample $X$ in S. Ignored if X and Y are given.
sample2ID	Integer indices (length $n$) for sample $Y$ in S. Ignored if X and Y are given.
k	Positive integer tuning parameter. For "RgNN"/"RoNN", it is the neighborhood size in the k-nearest-neighbor graph (k-NNG); for "RgMST"/"RoMST", it controls the number of minimum-spanning-tree layers (k-MST); for "RoMDP", it specifies the number of rounds of minimum-distance non-bipartite matching (k-MDP).
rank.type	Character, one of c("RgNN","RoNN","RgMST","RoMST","RoMDP"). Prefix "Rg" denotes graph-induced ranks; prefix "Ro" denotes overall ranks obtained by ordering all selected edges. See the references for precise definitions.
perm	Integer, number of permutations for a permutation p-value (default 0).

Details

From $S$ (or from $X$, $Y$), the procedure constructs a symmetric matrix $R$ with zero diagonal using one of the supported graph/ranking schemes. It then forms the within-group edge sums $U_x = \sum_{i,j \in X} R_{ij}$ and $U_y = \sum_{i,j \in Y} R_{ij}$. The expectation vector and covariance matrix of $(U_x, U_y)$ are derived under the permutation null distribution. The test statistic is

$T = ( U_x - \mu_x, U_y - \mu_y ) \Sigma^{-1} \begin{pmatrix} U_x - \mu_x \\ U_y - \mu_y \end{pmatrix},$

where $\mu_x, \mu_y$ are the expected values and $\Sigma$ is the covariance matrix. Under the null hypothesis, $T$ is asymptotically chi-square distributed with 2 degrees of freedom.

Value

A list with components:

• test.statistic: quadratic form $T_R$.

• pval.approx: asymptotic p-value (chi-square, df = 2).

• pval.perm: permutation p-value (present only if perm > 0).

References

Zhou, D. and Chen, H. (2023). A new ranking scheme for modern data and its application to two-sample hypothesis testing. In Proceedings of the 36th Annual Conference on Learning Theory (COLT 2023), PMLR, pp. 3615–3668.

See Also

rTests.base, Cov.asy

Examples

set.seed(1)
X <- matrix(rnorm(50*100, mean = 0), nrow=50)
Y <- matrix(rnorm(50*100, mean = 0.3), nrow=50)
# RgNN: graph-induced ranks from the k-nearest-neighbor graph
out.RgNN <- RISE(X = X, Y = Y, k = 10, rank.type = "RgNN", perm = 1000)
out.RgNN

# RoNN: overall ranks obtained by ordering edges from the k-NN graph
out.RoNN <- RISE(X = X, Y = Y, k = 10, rank.type = "RoNN", perm = 1000)
out.RoNN

# RgMST: graph-induced ranks from layered minimum spanning trees

out.RgMST <- RISE(X = X, Y = Y, k = 10, rank.type = "RgMST", perm = 1000)
out.RgMST


# RoMST: overall ranks obtained by ordering edges in the MST

out.RoMST <- RISE(X = X, Y = Y, k = 10, rank.type = "RoMST", perm = 1000)
out.RoMST


# RoMDP: overall ranks obtained by ordering edges from minimum-distance pairings

out.RoMDP <- RISE(X = X, Y = Y, k = 10, rank.type = "RoMDP", perm = 1000)
out.RoMDP

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