fPASS (version 1.0.0)

Power_Proj_Test_ufDA: Power of the Two-sample Projection-based test for functional data with known (or estimated) eigencomponents.

Description

The function Power_Proj_Test_ufDA() computes the power of of the two-sample projection-based test for functional response data setting, when the group difference, the eigenfunctions of the covariance of the data are specified at dense grid of time points, along with the (estimated) covariance of the shrinkage scores.

Usage

Power_Proj_Test_ufDA(
  total_sample_size,
  argvals,
  mean_vector,
  eigen_matrix,
  scores_var1,
  scores_var2,
  weights,
  sig.level = 0.05,
  alloc.ratio = c(1, 1),
  npc_to_pick = ncol(eigen_matrix),
  nsim = 10000
)

Value

Power of the projection-based test for specified difference in the mean function and the eigencomponents of the covariance of the functional data.

Arguments

total_sample_size

Total sample size combing the two groups, must be a positive integer.

argvals

The working grid of timepoints to evaluate the eigenfunctions and the mean functions. It is preferred to take the working grid as dense grid so that \(\int [\mu_1(t) - \mu_2(t)]\phi_k(t) \,dt\) can be calculated with a required precision.

mean_vector

The difference in the mean function evaluated at argvals, must be a numeric vector of length same as that that of argavls.

eigen_matrix

The matrix of eigenfunctions evaluated at argvals, must be a length(argvals) by K matrix, where K is the number of eigenfunctions.

scores_var1

The true (or estimate) of covariance matrix of the shrinkage scores for the first group. Must be symmetric (is.symmetric(scores_var1) == TRUE) and positive definite (chol(scores_var1) without an error!).

scores_var2

The true (or estimate) of covariance matrix of the shrinkage scores for the second group. Must be symmetric (is.symmetric(scores_var2) == TRUE) and positive definite (chol(scores_var2) without an error!).

weights

The weights to put to compute the projection \(\int [\mu_1(t) - \mu_2(t)]\phi_k(t) \,dt\), for each \(k=1,\dots, K\). The integral is numerically approximated as sum(mean_diff(argvals)*eigen_matrix[,k]*weights).

sig.level

Significance level of the test, default set at 0.05, must be less than 0.2.

alloc.ratio

The allocation ratio of samples in the each group. Note that the eigenfunctions will still be estimated based on the total sample_size, however, the variance of the shrinkage scores (which is required to compute the power function) will be estimated based on the allocation of the samples in each group. Must be given as vector of length 2. Default value is set at c(1, 1), indicating equal sample size.

npc_to_pick

Number of eigenfunction to be used to compute the power. Typically this is becomes handy when the user want to discard few of the last eigenfunctions, typically with a very small eigenvalues.

nsim

The number of samples to be generated from the alternate distribution of Hotelling T statistic. Default value is 10000.

Author

Salil Koner
Maintainer: Salil Koner

Details

The projection-based test first extracts K eigenfunctions from the data, and then project the mean difference function onto each of the eigenfunctions to obtain a K-dimensional projection vector that reflects the group difference. Wang (2021) pointed that under the null hypothesis the covariance of K-dimensional functional principal component analysis (fPCA) scores are the same, and thus a Hotelling \(T^2\) test with assuming equal variance of the shrinkage scores is a valid test. However, Koner and Luo (2023) pointed out that under the alternate hypothesis, when the difference is mean is significant, the covariance of the shrinkage scores also differ between the groups. Therefore, while computing the power of test, we must have to derive the distribution of the Hotelling \(T^2\) statistic under the assumption of unequal variance. The alogrithm for the power of multivariate Hotelling \(T^2\) under unequal variance is coded in pHotellingT() function. This particular function is a wrapper around that function, which inputs the mean difference as a function, and the eigenfunctions and the scores, and subsequently call the pHotellingT() function to compute the power under unequal variance. 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

See pHotellingT() and Sim_HotellingT_unequal_var() for samples from Hotelling T distribution.

Examples

ngrid          <- 101
interval       <- c(-1,1)
gauss.quad.pts <- gss::gauss.quad(ngrid,interval) # evaluation points
working.grid   <- gauss.quad.pts$pt
mean_fn        <- function(t) {0.4*sin(2*pi*t)}
mean_vector    <- mean_fn(working.grid)
eigen_fn       <- function(t, k){ sqrt(2)*{(k==2)*sin(2*pi*t) + (k==1)*cos(2*pi*t)} }
eigen_matrix   <- cbind(eigen_fn(working.grid,1), eigen_fn(working.grid,2))
mean_proj      <- sapply(1:2, function(r) integrate(function(x)
eigen_fn(x,r)*mean_fn(x), interval[1], interval[2])$value)
sig1           <- diag(2)
sig2           <- 2*diag(2)
alp            <- 0.05
n              <- 100
k              <- ncol(eigen_matrix)
cutoff         <- {(n - 2)*k/(n - k -1)}*qf(1-alp, k, n-k-1)
func_power     <- Power_Proj_Test_ufDA(total_sample_size=n,
argvals=working.grid,
mean_vector = mean_vector, eigen_matrix = eigen_matrix,
scores_var1 = sig1, scores_var2= sig2, weights = gauss.quad.pts$wt,
sig.level=alp, alloc.ratio = c(1,1), npc_to_pick=ncol(eigen_matrix),
nsim = 5e3)