parametric_power: Power calculation for the Hotelling's parametric test
In astamm/fdahotelling: Inference for Functional Data Analysis in R

Description Usage Arguments Details Value See Also Examples

parametric_power computes an estimate of the statistical power of Hotelling's T^2 parametric test on the mean function (or on the difference between the mean functions) using Monte-Carlo simulations.

1 2	parametric_power(mu1 = 0, Sigma = diag(length(mu1)), n1 = 10L, n2 = NULL, MC = 1000L, alpha = 0.05, paired = FALSE, step_size = 0)

`mu1`	True mean function or difference between mean functions (default: 0).
`Sigma`	True covariance matrix Σ (default: 1).
`n1`	Sample size of data pertaining to the 1st population (default: 10).
`n2`	Sample size of data pertaining to the 2nd population (default: `NULL`).
`MC`	Number of Monte-Carlo runs to estimate statistical power (default: 1000).
`alpha`	Significance level (default: 0.05).
`paired`	Is the input data paired? (default: `FALSE`).
`step_size`	The step size used to perform integral approximation via the method of rectangles (default: `0`). When set to `0`, it assumes that we are dealing with multivariate data rather than functional data and thus no integration is necessary.

This function computes the statistical power of Hotelling's T^2 parametric test based on a specific generative model. It is assumed that data is generated from a Gaussian distribution. The user-defined inputs are

mu1: a numeric vector containing the actual mean function of the distribution (or difference between mean functions) evaluated in a pointwise fashion on a uniform grid.
Sigma: a numeric matrix containing the actual covariance kernel of the distribution (or pooled covariance kernel) evaluated in a pointwise fashion on a uniform grid.

An estimate of the statistical power of the test.

The underlying statistical test is described in details in Secchi, P., Stamm, A., & Vantini, S. (2013). Inference for the mean of large p small n data: A finite-sample high-dimensional generalization of Hotelling theorem. Electronic Journal of Statistics, 7, pp. 2005-2031. doi:10.1214/13-EJS833, available online at http://projecteuclid.org/euclid.ejs/1375708877.

# Set the sample sizes for a two-sample test
n1 <- 10
n2 <- 10
# Set the dimensionality for curve approximation
p <- 100
# Set the actual covariance kernel
Sigma <- diag(1, p)
# The following lines of code computes the actual significance level of the
# test.
mu1 <- rep(0, p)
parametric_power(mu1, Sigma, n1, n2, MC = 1000)
# We can use more complex covariance matrices and compute an estimate of the
# statistical power of the test for a given non-null mean difference.
mu1 <- rep(4, p)
s <- seq(-1, 1, length.out = 100)
Sigma <- outer(s, s, function(t, s) exp(-abs(t - s)))
parametric_power(mu1, Sigma, n1, n2, MC = 1000)