Likelihood ratio test for antedependence order (Gaussian AD data)

Tests the null hypothesis that the data follow an AD(p) model against the alternative that they follow an AD(p+q) model, using the likelihood ratio test described in Theorem 6.4 and 6.5 of Zimmerman & Núñez-Antón (2009).

Usage

test_order_gau(
  y,
  p = 0L,
  q = 1L,
  mu = NULL,
  use_modified = TRUE,
  order_null = NULL,
  order_alt = NULL
)

Arguments

y

Numeric matrix with n_subjects rows and n_time columns.

p

Order under the null hypothesis (default 0). This is the same antedependence order parameter named order in fit_gau.

q

Order increment under the alternative (default 1, so alternative is AD(p+q)).

mu

Optional mean vector. If NULL, the saturated mean (sample means) is used.

use_modified

Logical. If TRUE (default), use the modified test statistic (formula 6.9) which has better small-sample properties.

order_null

Optional alias for p (null order).

order_alt

Optional absolute order under the alternative. When supplied, q is computed as order_alt - p.

Value

A list with class gau_order_test containing:

method

Inference method used ("lrt").

p

Order under null hypothesis

q

Order increment

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 (if use_modified = TRUE)

n_subjects

Number of subjects

n_time

Number of time points

Details

The test is based on the intervenor-adjusted sample partial correlations. Under the null hypothesis AD(p), the partial correlations r_(i,i-k|(i-k+1:i-1)) should be zero for k > p.

The likelihood ratio test statistic (Theorem 6.4) is: $$-N \sum_{j=1}^{q} \sum_{i=p+j+1}^{n} \log(1 - r^2_{i,i-p-j\cdot(i-p-j+1:i-1)})$$

which is asymptotically chi-square with (2n - 2p - q - 1)(q/2) degrees of freedom.

The modified test (formula 6.9) adjusts for small-sample bias using Kenward's (1987) correction.

References

Zimmerman, D.L. and Núñez-Antón, V. (2009). Antedependence Models for Longitudinal Data. Chapman & Hall/CRC. Chapter 6.

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

See also

test_one_sample_gau, test_homogeneity_gau

Examples

if (FALSE) { # \dontrun{
# Simulate AD(1) data
y <- simulate_gau(n_subjects = 50, n_time = 6, order = 1, phi = 0.5)

# Test AD(0) vs AD(1)
test01 <- test_order_gau(y, p = 0, q = 1)
print(test01)

# Test AD(1) vs AD(2)
test12 <- test_order_gau(y, p = 1, q = 1)
print(test12)
} # }