Equality test of K unknown component distributions

Equality test of the unknown component distributions coming from K (K > 1) admixture models, based on the Inversion - Best Matching (IBM) approach. Recall that we have K populations following admixture models, each one with probability density functions (pdf) $$ \ell_k = p_k f_k + (1 - p_k) g_k, \quad k=1,...,K, $$ where \(g_k\) is the known pdf and \(l_k\) corresponds to the observed sample. In such a context, perform the following hypothesis test: $$ H_0: \, f_1 = f_K \quad \mbox{against} \quad H_1: \, \exists \mbox{ at least } i \neq j \mbox{ such that } f_i \neq f_j, \; i \neq j, \, i,j \in \{1,...,K\}. $$

Usage

IBM_k_samples_test(
  samples,
  admixMod,
  conf_level = 0.95,
  sim_U = NULL,
  tune_penalty = TRUE,
  n_sim_tab = 100,
  parallel = FALSE,
  n_cpu = 2
)

Arguments

samples

A list of the K samples to be studied, all following admixture distributions.

admixMod

A list of objects of class admix_model, containing information about known distributions and known associated parameters.

conf_level

The confidence level of the K-sample test.

sim_U

(default to NULL) Random draws of the inner convergence part of the contrast as defined in the IBM approach (see references below).

tune_penalty

(default to TRUE) A boolean that allows to choose between a classical penalty term or an optimized penalty (embedding some tuning parameters, automatically optimized). Optimized penalty is particularly useful for low or unbalanced sample sizes to detect alternatives to the null hypothesis (H0). It is recommended to set it to TRUE.

n_sim_tab

(default to 100) Number of simulated Gaussian processes when tabulating the inner convergence distribution in the 'icv' testing method using the IBM estimation approach.

parallel

(default to FALSE) Boolean to indicate whether parallel computations are performed (speed-up the tabulation).

n_cpu

(default to 2) Number of cores used when paralleling computations.

Value

An object of class 'IBM_test', containing 17 attributes: 1) the number of samples for the test; 2) the sizes of each sample; 3) the information about component distributions for each sample; 4) the reject decision of the test; 5) the confidence level of the test (1-alpha, where alpha refers to the first-type error); 6) the test p-value; 7) the 95th-percentile of the contrast tabulated distribution; 8) the test statistic value; 9) the selected rank (number of terms involved in the test statistic); 10) the terms (pairwise contrasts) involved in the test statistic; 11) A boolean indicating whether the penalization corresponds to the null hypothesis has been considered; 12) the value of penalized test statistics; 13) the selected optimal 'gamma' parameter (see reference below); 14) the selected optimal constant involved in the penalization process (see also the reference); 15) the tabulated distribution of the contrast; 16) the estimated mixing proportions (not implemented yet, since that makes sense only in case of equal unknown component distributions); 17) the matrix of pairwise contrasts (distance between two samples); 18) the matrix of the ranks of pairwise contrasts; and 19) the matrix of identifiers of pairwise contrasts.

References

Milhaud X, Pommeret D, Salhi Y, Vandekerkhove P (2024). “Contamination-source based K-sample clustering.” Journal of Machine Learning Research, 25(287), 1–32. https://jmlr.org/papers/v25/23-0914.html.

See also

get_tabulated_dist() to access the tabulated distribution under the null hypothesis, which defines the quantile against which the test statistics is tested; get_discrepancy_rank(), get_discrepancy_matrix() to access the measure of discrepancy between pairs of samples; get_statistic_components() in k-sample test;

Author

Xavier Milhaud xavier.milhaud.research@gmail.com

Examples

if (FALSE) { # \dontrun{
####### Under the null hypothesis H0 (with K=3 populations):
## Simulate mixture data:
mixt1 <- twoComp_mixt(n = 450, weight = 0.4,
                      comp.dist = list("norm", "norm"),
                      comp.param = list(list("mean" = -2, "sd" = 0.5),
                                        list("mean" = 0, "sd" = 1)))
mixt2 <- twoComp_mixt(n = 380, weight = 0.7,
                      comp.dist = list("norm", "norm"),
                      comp.param = list(list("mean" = -2, "sd" = 0.5),
                                        list("mean" = 1, "sd" = 1)))
mixt3 <- twoComp_mixt(n = 400, weight = 0.8,
                      comp.dist = list("norm", "exp"),
                      comp.param = list(list("mean" = -2, "sd" = 0.5),
                                        list("rate" = 1)))
data1 <- get_mixture_data(mixt1)
data2 <- get_mixture_data(mixt2)
data3 <- get_mixture_data(mixt3)
## Define the admixture models:
admixMod1 <- admix_model(knownComp_dist = mixt1$comp.dist[[2]],
                         knownComp_param = mixt1$comp.param[[2]])
admixMod2 <- admix_model(knownComp_dist = mixt2$comp.dist[[2]],
                         knownComp_param = mixt2$comp.param[[2]])
admixMod3 <- admix_model(knownComp_dist = mixt3$comp.dist[[2]],
                         knownComp_param = mixt3$comp.param[[2]])
## Perform the 3-samples test:
IBM_k_samples_test(samples = list(data1, data2, data3),
                   admixMod = list(admixMod1, admixMod2, admixMod3),
                   conf_level = 0.95, sim_U = NULL, n_sim_tab = 5,
                   tune_penalty = FALSE, parallel = FALSE)
} # }