Likelihood ratio test for homogeneity across groups (Gaussian AD data)

Tests the null hypothesis that G groups have the same AD(p) covariance structure against the alternative that they have different AD(p) structures. This implements Theorem 6.6 of Zimmerman & Núñez-Antón (2009).

Usage

test_homogeneity_gau(y, blocks, p = 1L, use_modified = TRUE)

Arguments

y: Numeric matrix with n_subjects rows and n_time columns.
blocks: Integer vector of length n_subjects indicating group membership.
p: Antedependence order. This is the same order parameter named order in fit_gau.
use_modified: Logical. If TRUE (default), use modified test statistic for better small-sample approximation.

Value

A list with class gau_homogeneity_test containing:

method: Inference method used ("lrt").
statistic: Test statistic value
statistic_modified: Modified test statistic (if use_modified = TRUE)
df: Degrees of freedom
p_value: P-value from chi-square distribution
p_value_modified: P-value from modified test
G: Number of groups
group_sizes: Sample sizes for each group
order: Antedependence order

Details

The test compares:

H0: All G groups have the same AD(p) covariance matrix Sigma(theta)
H1: Groups have different AD(p) covariance matrices Sigma(theta_g)

The likelihood ratio test statistic (Theorem 6.6) involves comparing pooled and within-group RSS values. The degrees of freedom are (G-1)(2n - p)(p + 1)/2.

This test is useful for determining whether a common covariance structure can be assumed across treatment groups before performing mean comparisons.

References

Zimmerman, D.L. and Núñez-Antón, V. (2009). Antedependence Models for Longitudinal Data. Chapman & Hall/CRC. Section 6.4.

Kenward, M.G. (1987). A method for comparing profiles of repeated measurements. Applied Statistics, 36, 296-308.

Examples

if (FALSE) { # \dontrun{
# Simulate data from two groups with same covariance
n1 <- 30
n2 <- 35
y1 <- simulate_gau(n1, n_time = 6, order = 1, phi = 0.5, sigma = 1)
y2 <- simulate_gau(n2, n_time = 6, order = 1, phi = 0.5, sigma = 1)
y <- rbind(y1, y2)
blocks <- c(rep(1, n1), rep(2, n2))

# Test homogeneity
test <- test_homogeneity_gau(y, blocks, p = 1)
print(test)
} # }