ARHT {ARHT}  R Documentation 
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 onesample
and twosample test problem, where we're interested in detecting the mean vector in the onesample problem or the difference
between mean vectors in the twosample 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 n1byp observation matrix with numeric column variables. 
Y 
an optional n2byp observation matrix; if 
mu_0 
the null hypothesis vector to be tested; if 
prob_alt_prior 
a nonempty list; Each field is a numeric vector with sum 1. The default value is the "canonical weights"

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:

lambda_range 
optional usersupplied lambda range; If 
nlambda 
optional usersupplied number of lambda's in grid search; default to be 
bs_size 
positive numeric with default value 
Details
The method incorporates ridgeregularization 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 pvalue 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 Cuberoot transformation,
squareroot transformation and chisquare approximation.
Value
ARHT_pvalue
: The pvalue of ARHT test.If
length(prob_alt_prior)==1
, it is identical toRHT_pvalue
.If
length(prob_alt_prior)>1
, it is the pvalue 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 inprob_alt_prior
. It has the same length and order asprob_alt_prior
.RHT_pvalue
: The pvalue of RHT tests with the lambda's inRHT_opt_lambda
.RHT_std
: The standardized RHT statistics with the lambda's inRHT_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 inRHT_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 inRHT_opt_lambda
.Corr_RHT
: The estimated correlation matrix of the statistics inRHT_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), 13451360.
Examples
set.seed(10086)
# Onesample test
n1 = 300; p =500
dataX = matrix(rnorm(n1 * p), nrow = n1, ncol = p)
res1 = ARHT(dataX)
# Twosample 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