ARHT {ARHT}

An adaptable generalized Hotelling's T^2 test for high dimensional data

Description

This function performs the adaptable regularized Hotelling's T^2 test (ARHT) (Li et al., (2016) <arXiv:1609.08725>) for the one-sample and two-sample test problem, where we're interested in detecting the mean vector in the one-sample problem or the difference between mean vectors in the two-sample problem in a high dimensional regime.

Usage

ARHT(X, Y = NULL, mu_0 = NULL, prob_alt_prior = list(c(1, 0, 0), c(0, 1,
  0), c(0, 0, 1)), Type1error_calib = c("cube_root", "sqrt", "chi_sq",
  "none"), lambda_range = NULL, nlambda = 2000, bs_size = 1e+05)

Arguments

X

the n1-by-p observation matrix with numeric column variables.

Y

an optional n2-by-p observation matrix; if NULL, a one-sample test is conducted on X; otherwise, a two-sample test is conducted on X and Y.

mu_0

the null hypothesis vector to be tested; if NULL, the default value is the 0 vector of length p.

prob_alt_prior

a non-empty list; Each field is a numeric vector with sum 1. The default value is the "canonical weights" list(c(1,0,0), c(0,1,0), c(0,0,1)); Each field represents a probabilistic prior model specified by weights of I_p, \Sigma, \Sigma^2, etc, where \Sigma is the population covariance matrix of the observations.

Type1error_calib

the method to calibrate Type 1 error rate of ARHT. Choose its first element when more than one are specified. Four values are allowed:

• cube_root The default value; cube-root transformation;

• sqrt Square-root transformation;

• chi_sq Chi-square approximation, not available when more than three models are specified in prob_alt_prior;

• none No calibration.

lambda_range

optional user-supplied lambda range; If NULL, ARHT chooses its own range.

nlambda

optional user-supplied number of lambda's in grid search; default to be 2000; the grid is progressively coarser.

bs_size

positive numeric with default value 1e5; only effective when more than one prior models are specified in prob_alt_prior; control the size of the bootstrap sample used to approximate the ARHT p-value.

Details

The method incorporates ridge-regularization in the classic Hotelling's T^2 test with the regularization parameter chosen such that the asymptotic power under a class of probabilistic alternative prior models is maximized. ARHT combines different prior models by taking the maximum of statistics under all models. ARHT is distributed as the maximum of a correlated multivariate normal random vector. We estimate its covariance matrix and bootstrap its distribution. The returned p-value is a Monte Carlo approximation to its true value using the bootstrap sample, therefore not deterministic. Various methods are available to calibrate the slightly inflated Type 1 error rate of ARHT, including Cube-root transformation, square-root transformation and chi-square approximation.

Value

• ARHT_pvalue: The p-value of ARHT test.

  • If length(prob_alt_prior)==1, it is identical to RHT_pvalue.

  • If length(prob_alt_prior)>1, it is the p-value after combining results from all prior models. The value is bootstrapped, therefore not deterministic.

• RHT_opt_lambda: The optimal lambda's chosen under each of the prior models in prob_alt_prior. It has the same length and order as prob_alt_prior.

• RHT_pvalue: The p-value of RHT tests with the lambda's in RHT_opt_lambda.

• RHT_std: The standardized RHT statistics with the lambda's in RHT_opt_lambda. Take its maximum to get the statistic of ARHT test.

• Theta1: As defined in Li et al. (2016) <arXiv:1609.08725>, the estimated asymptotic means of RHT statistics with the lambda's in RHT_opt_lambda.

• Theta2: As defined in Li et al. (2016) <arXiv:1609.08725>, 2*Theta2 are the estimated asymptotic variances of RHT statistics the lambda's in RHT_opt_lambda.

• Corr_RHT: The estimated correlation matrix of the statistics in RHT_std.

References

Li, H. Aue, A., Paul, D. Peng, J., & Wang, P. (2016). An adaptable generalization of Hotelling's T^2 test in high dimension. <arXiv:1609:08725>.

Chen, L., Paul, D., Prentice, R., & Wang, P. (2011). A regularized Hotelling's T^2 test for pathway analysis in proteomic studies. Journal of the American Statistical Association, 106(496), 1345-1360.

Examples

set.seed(10086)
# One-sample test
n1 = 300; p =500
dataX = matrix(rnorm(n1 * p), nrow = n1, ncol = p)
res1 = ARHT(dataX)

# Two-sample test
n2= 400
dataY = matrix(rnorm(n2 * p), nrow = n2, ncol = p )
res2 = ARHT(dataX, dataY, mu_0 = rep(0.01,p))

# Specify probabilistic alternative priors model
res3 = ARHT(dataX, dataY, mu_0 = rep(0.01,p),
     prob_alt_prior = list(c(1/3, 1/3, 1/3), c(0,1,0)))

# Change Type 1 error calibration method
res4 = ARHT(dataX, dataY, mu_0 = rep(0.01,p),
     Type1error_calib = "sqrt")

RejectOrNot = res4$ARHT_pvalue < 0.05