One-dimensional selective Hotelling test (unknown covariance)

The one-dimensional test \(p_{\mathrm{polar,1d}}\) of the paper: it conditions on every coordinate of the polar factor \(Q\) except the Hotelling angle \(\theta\), so that the selective null law of \(\theta\) is a truncated Beta (equivalently, a truncated F for the scaled \(T^2\)).

Usage

selective_truncated_F(
  X,
  k1,
  k2,
  clustering = c("kmeans", "hclust", "custom"),
  method = c("bc", "exact", "is"),
  K = 3L,
  init_centers_idx = NULL,
  hclust_method = "average",
  hclust_square = TRUE,
  hclust_event = "trace",
  cl_fun = NULL,
  cl_obs = NULL,
  M = 5000L,
  L = 2L,
  onedim_kernel = c("givens", "slice"),
  n_is = 2000L,
  is_mix = 0.5,
  is_scale = NULL,
  ridge = 1e-08,
  verbose = FALSE
)

Arguments

X

A numeric \(n \times p\) data matrix with observations in rows.

k1, k2

The two tested cluster labels. Each cluster must have at least three members, and \(r - 1 > p\).

clustering

"kmeans", "hclust", or "custom" (use cl_fun).

method

"bc" (default), "exact" (closed form; "kmeans" and "hclust" only), or "is".

K

Number of clusters (built-in "kmeans"/"hclust").

init_centers_idx

Fixed initial-centre row indices for "kmeans"; drawn via kmeans_fun() if NULL.

hclust_method, hclust_square

Linkage and squared-distance flag for "hclust" (defaults match hclust_fun()).

hclust_event

Kept for compatibility; the only value is "trace", the merge order used by method = "exact".

cl_fun

A clustering routine function(Xmat) -> labels (a pure function of its data) for clustering = "custom"; optional otherwise.

cl_obs

Optional precomputed observed clustering (saves one call).

M, L

Path length and number of kernel steps between path points for method = "bc".

onedim_kernel

For method = "bc": "givens" (default; random step on the angle, step size drawn log-uniformly on [1e-3, 1] per block of L steps) or "slice" (slice sampling with shrinkage).

n_is, is_mix, is_scale

Number of draws, weight of the null law in the defensive mixture, and the standard deviation of its normal part, for method = "is".

ridge

Small non-negative ridge for the reported observed \(T^2\).

verbose

If TRUE, print a one-line diagnostic.

Value

An S3 object of class "selectiveHotelling" with pval, T_obs, k1, k2, K, condition_on, sampler, p_uncond (the classical F p-value), theta_obs, F_obs, df1, df2, and, for method = "exact", the feasible arcs intervals.

Details

With \(\cos^2\theta = u(Q)\), \(T^2 = (r-2)\cot^2\theta\), and under the null \(\cos^2\theta \sim \mathrm{Beta}(p/2, (r-1-p)/2)\) independently of the other coordinates of \(Q\). Given those coordinates, the clustering restricts \(\theta\) to a set \(S_\theta\), and the p-value is the Beta mass of \(S_\theta \cap (0, \theta_{obs}]\) divided by the Beta mass of \(S_\theta\).

method = "bc" (default) computes it for any clustering, on the event that the clustering returns the tested pair, with the Besag–Clifford construction on the angle (Algorithm 2 of the paper); it is exact for any M. method = "exact" gives a closed form, with no Monte Carlo, for two stronger events: the whole Lloyd trajectory (clustering = "kmeans") and the merge order up to the cut (clustering = "hclust", average or single linkage). These events condition on more and cost power. method = "is" uses importance sampling (consistent, not exact).

See also

selective_hotelling_bc(), selective_hotelling_test(), hotelling_T2()

Examples

set.seed(1)
X <- rbind(matrix(rnorm(40), 20, 2), matrix(rnorm(40) + 5, 20, 2),
           matrix(rnorm(40) - 5, 20, 2))
cl_fun <- function(x) hclust_fun(x, k = 3)$cluster
res <- selective_truncated_F(X, 1, 2, clustering = "custom", cl_fun = cl_fun, M = 200)
res$pval
#> [1] 0.01469148