Test linear hypotheses on the mean under antedependence

Tests the null hypothesis C * mu = c for a specified contrast matrix C and vector c, under an AD(p) covariance structure. This implements Theorem 7.2 of Zimmerman & Núñez-Antón (2009).

Usage

test_contrast_gau(y, C, c = NULL, p = 1L)

Arguments

y: Numeric matrix with n_subjects rows and n_time columns.
C: Contrast matrix with c rows and n_time columns, where c is the number of contrasts being tested. Rows must be linearly independent.
c: Right-hand side vector of length equal to nrow(C). Default is a vector of zeros.
p: Antedependence order of the covariance structure. This is the same order parameter named order in fit_gau.

Value

A list with class gau_contrast_test containing:

method: Inference method used ("wald").
C: Contrast matrix
c: Right-hand side vector
mu_hat: Estimated mean vector
contrast_est: Estimated value of C * mu
statistic: Wald test statistic
df: Degrees of freedom (number of contrasts)
p_value: P-value from chi-square distribution

Details

The Wald test statistic (Theorem 7.2) is: $$(C\bar{Y} - c)^T (C \hat{\Sigma} C^T)^{-1} (C\bar{Y} - c)$$

where $\hat{\Sigma}$ is the REML estimator of the covariance matrix under the AD(p) model.

Common examples include:

Testing if mean is constant: C is the first-difference matrix
Testing for linear trend: C tests deviations from linearity

References

Zimmerman, D.L. and Núñez-Antón, V. (2009). Antedependence Models for Longitudinal Data. Chapman & Hall/CRC. Chapter 7.

Examples

if (FALSE) { # \dontrun{
y <- simulate_gau(n_subjects = 50, n_time = 5, order = 1)

# Test if mean is constant (all differences = 0)
# C is 4x5 matrix of first differences
C <- matrix(0, nrow = 4, ncol = 5)
for (i in 1:4) {
  C[i, i] <- 1
  C[i, i+1] <- -1
}
test <- test_contrast_gau(y, C = C, p = 1)
print(test)
} # }