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 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, Σ, Σ^2, etc, where Σ 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

```

[Package ARHT version 0.1.0 Index]