fPASS (version 1.0.0)

Sim_HotellingT_unequal_var: Samples from the non-null distribution of the Hotelling-\(T^2\) statistic under unequal covariance.

Description

The function Sim_HotellingT_unequal_var() generates samples from the (non-null) distribution of the two-sample Hotelling-\(T^2\) statistic under the assuming of unequal covariance of the multivariate response between the two groups. This function is used to compute the power function of Two-Sample (TS) Projection-based test (Wang 2021, EJS.) for sparsely observed univariate functional data.

Usage

Sim_HotellingT_unequal_var(
  total_sample_size,
  mean_diff,
  sig1,
  sig2,
  alloc.ratio = c(1, 1),
  nsim = 10000
)

Value

A named list with two elements.

1. samples - a vector of length nsim, containing The samples from the distribution of the Hotelling T statistic under unequal variance.

2. denom.df - The denominator degrees of freedom of the chi-square statistic obtained by approximation of the sum of two Wishart distribution under unequal variance.

Arguments

total_sample_size

Target sample size, must be a positive integer.

mean_diff

The difference in the mean vector between the two groups, must be a vector.

sig1

The true (or estimate) of covariance matrix for the first group. Must be symmetric (is.symmetric(sig1) == TRUE) and positive definite (chol(sig1) without an error!).

sig2

The true (or estimate) of covariance matrix for the second group. Must be symmetric (is.symmetric(sig2) == TRUE) and positive definite (chol(sig2) without an error!).

alloc.ratio

Allocation of total sample size into the two groups. Must set as a vector of two positive numbers. For equal allocation it should be put as c(1,1), for non-equal allocation one can put c(2,1) or c(3,1) etc.

nsim

The number of samples to be generated from the alternate distribution.

Author

Salil Koner
Maintainer: Salil Koner

Details

Under the assumption of the equal variance, we know that the alternative distribution of the Hotelling-\(T^2\) statistic has an F distribution with the non-centrality depending on the difference between the true mean vectors and the (common) covariance of the response. However, when the true covariance of the true groups of responses differ, the alternate distribution becomes non-trivial. Koner and Luo (2023) proved that the alternate distribution of the test-statistic approximately follows a ratio of the linear combination of the K (dimension of the response) non-central chi-squared random variables (where the non-centrality parameter depends on the mean difference) and a chi-squared distribution whose degrees of freedom depends on a complicated functions of sample size in the two groups. See Koner and Luo (2023) for more details on the formula of the non-null distribution.

References

Wang, Qiyao (2021) Two-sample inference for sparse functional data, Electronic Journal of Statistics, Vol. 15, 1395-1423
tools:::Rd_expr_doi("https://doi.org/10.1214/21-EJS1802").

See Also

Hotelling::hotelling.test(), Hotelling::hotelling.stat() to generate empirical samples from the Hotelling T-statistic from empirical data.

Examples

# Case 1: Null hypothesis is true. True mean difference is zero, and the true
# covariance of the two groups are same.
k <- 5
mu1  <- rep(0,k); del  <- 0; mu2 <- mu1 + rep(del, k);
sig1 <- diag(k); sig2 <- sig1 + del*toeplitz(c(1,rep(0.5, k-1))); n <- 200;
null.dist.samples <- Sim_HotellingT_unequal_var(total_sample_size=n, mean_diff=mu1-mu2,
                     sig1=sig1, sig2=sig2, alloc.ratio=c(1,1), nsim=1e3)
# The following Kolmogorov Smirnov test confirms that under null hypothesis
# and when the covariances are same, the distribution is exactly a
# central F distribution with \eqn{k} and \eqn{n-k}  degrees of freedom.
ks.test(null.dist.samples$samples, {{(n - 2) * k}/(n - k -1)} * {rf(n=1e3, k, n-k-1)} )


# Case 2: Alternate hypothesis is true. The mean difference is non-zero,
# and the covariances of the two groups are same:
k <- 6
mu1  <- rep(0,k); del  <- 0.15; mu2 <- mu1 + rep(del, k);
sig1 <- diag(k); sig2 <- sig1;
n1 <- 100; n2 <- 100;
alt.dist.samples <- Sim_HotellingT_unequal_var(total_sample_size=n1+n2, mean_diff=mu1-mu2,
                                               sig1=sig1, sig2=sig2, alloc.ratio=c(1,1), nsim=1e3)
ks.test(alt.dist.samples$samples,
        {(n1+n2 - 2) * k /(n1+n2 - k -1)}*rf(n=1e3, k, n1+n2-k-1,
          ncp = {(n1*n2)/(n1+n2)}*as.vector(crossprod(mu1-mu2, solve(sig1, mu1-mu2))) ) )


# Case 3: Alternate hypothesis is true. The mean difference is non-zero,
# and the covariances of the two groups are different
k <- 5
mu1  <- rep(0,k); del  <- 0.25; mu2 <- mu1 + rep(del, k);
sig1 <- diag(k); sig2 <- sig1 + del*toeplitz(c(1,rep(0.5, k-1)))
alt.dist.samples <- Sim_HotellingT_unequal_var(total_sample_size=200, mean_diff=mu1-mu2,
sig1=sig1, sig2=sig2, alloc.ratio=c(1,1), nsim=1e3)

# Generate samples with unequal allocation ratio:
k <- 8
mu1  <- rep(0,k); del  <- 0.4; mu2 <- mu1 + rep(del, k);
sig1 <- diag(k); sig2 <- sig1 + del*toeplitz(c(1,rep(0.5, k-1)))
alt.dist.samples <- Sim_HotellingT_unequal_var(total_sample_size=150, mean_diff=mu1-mu2,
sig1=sig1, sig2=sig2, alloc.ratio=c(2,1), nsim=1e3)