Estimate Reliability Coefficients (Section 3.1)

Estimates the reliability coefficients \(\lambda_i\) for each indicator in a univariate latent factor model, using pairwise covariances and a quasi-Poisson GLM with log link.

Usage

estimate_reliability(X, na.rm = TRUE)

Arguments

X: numeric matrix (n x d) of indicator variables, d >= 3.
na.rm: logical; if TRUE, rows with any NA are removed.

Value

A list with components:

lambda: numeric vector of length d with estimated reliabilities.
glm_fit: the fitted glm object.
V_k: matrix (n x d) of per-subject estimating function contributions for the reliability parameters (used internally by test_t0).
U_pairs: matrix (n x choose(d,2)) of pairwise estimating function contributions (used internally by test_t0).
pairs: matrix (choose(d,2) x 2) of indicator index pairs.
Xc: matrix (n x d) of mean-centred indicators.

Details

Under the structural model \(X_i = \mu_i + \lambda_i \eta + \varepsilon_i\), we have \(\mathrm{Cov}(X_i, X_j) = \lambda_i \lambda_j\). The log of each pairwise covariance is modelled as a linear combination of \(\log(\lambda_i)\), yielding a quasi-Poisson GLM.

References

VanderWeele, T. J., & Vansteelandt, S. (2022). A statistical test to reject the structural interpretation of a latent factor model. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 84(5), 2032–2054. doi:10.1111/rssb.12555

Examples

# Simulate data from a one-factor model
set.seed(12345)
n <- 1000
eta <- rnorm(n)
lambda_true <- c(1.0, 0.8, 0.6)
X <- cbind(
  lambda_true[1] * eta + rnorm(n, sd = 0.5),
  lambda_true[2] * eta + rnorm(n, sd = 0.5),
  lambda_true[3] * eta + rnorm(n, sd = 0.5)
)

rel <- estimate_reliability(X)
rel$lambda
#> [1] 1.0258103 0.7859982 0.5831385

# Ratios lambda_i / lambda_1 should be close to truth
rel$lambda / rel$lambda[1]
#> [1] 1.0000000 0.7662218 0.5684662