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Selective Hotelling T-squared test after clustering

Source: R/test.R
selective_hotelling_test.Rd

Clusters X (k-means or hierarchical clustering, K clusters) and tests whether the discovered clusters k1 and k2 have the same mean, with Hotelling's \(T^2\) and an unknown covariance. The p-value is the Besag–Clifford test selective_hotelling_bc() with the random Givens kernel on the event that the clustering returns the tested pair (Algorithm 1 of the paper). It is exactly uniform under the null for any M and L.

Usage

selective_hotelling_test(
  X,
  k1,
  k2,
  K = 3,
  clustering = c("kmeans", "hclust"),
  hclust_method = "average",
  hclust_square = TRUE,
  M = 5000L,
  L = 2L,
  kernel = "givens",
  ridge = 1e-08,
  verbose = FALSE
)

Arguments

X

A numeric data matrix with observations in rows.

k1, k2

The two cluster labels (in 1:K) to compare.

K

Number of clusters.

clustering

"kmeans" (default, Lloyd's algorithm from fixed initial centres) or "hclust" (agglomerative clustering cut at K).

hclust_method

Agglomeration method for clustering = "hclust", passed to hclust_fun() / stats::hclust(). Ignored for clustering = "kmeans".

hclust_square

If TRUE (default) the tree is built on squared Euclidean distances (the clusterpval convention). Ignored for clustering = "kmeans".

M

Path length (the path carries M + 1 points). The p-value cannot be smaller than \(1/(M+1)\).

L

Number of kernel steps between consecutive path points.

kernel

Kernel passed to selective_hotelling_bc(): "givens" (default), "cayley", or "givens_slice".

ridge

Ridge added to the pooled covariance in hotelling_T2().

verbose

Whether to print progress messages.

Value

An S3 object of class "selectiveHotelling"; see selective_hotelling_bc() for its elements. It also contains cluster, the cluster labels of the clustering that was tested, so that k1 and k2 can be matched to observations.

Details

For k-means, the initial centres are drawn once (from the current random seed) and then held fixed, so that re-running the clustering on perturbed data is a deterministic function of the data. For the one-dimensional test, use selective_truncated_F().

See also

selective_hotelling_bc(), selective_truncated_F(), kmeans_fun(), hclust_fun()

Examples

set.seed(2)
Xb <- rbind(
  matrix(rnorm(20), 10, 2),
  matrix(rnorm(20) + 6, 10, 2),
  matrix(rnorm(20) - 6, 10, 2)
)
res <- selective_hotelling_test(Xb, 1, 2, K = 3, M = 200)
res$pval
#> [1] 0.02457418
print(res)
#> Selective Hotelling T^2 test (unknown covariance)
#>   Clusters tested : 1 vs 2 (of K = 3)
#>   Sampler         : besag-clifford
#>   Conditioning    : pair-preserved
#>   Observed T^2    : 213.2
#>   Selective p-val : 0.02457 (MC s.e. ~ 0.011)
res_h <- selective_hotelling_test(Xb, 1, 2, K = 3, clustering = "hclust", M = 200)
res_h$pval
#> [1] 0.006160104