T1 Test: Reliability-Independent Test of Structural Interpretation (Section 3.3)

Tests whether the structural interpretation of a univariate latent factor model can be rejected, without requiring estimation of reliability coefficients.

Usage

test_t1(X, z, na.rm = TRUE, max_iter = 1000L, tol = 1e-25, verbose = FALSE)

Arguments

X: numeric matrix (n x d) of indicator variables, d >= 2.
z: numeric vector or factor of length n encoding a discrete auxiliary variable. Must have at least 3 distinct levels. If z is continuous, discretise it (e.g., into quantile groups) before use.
na.rm: logical; if TRUE, rows with any NA are removed.
max_iter: integer; maximum iterations for nlm.
tol: numeric; convergence tolerance for the alternating LS initializer.
verbose: logical; if TRUE, print progress information.

Value

An object of class c("structest_t1", "structest", "htest") containing:

statistic: the T1 test statistic.
parameter: degrees of freedom, $(d-1)(p-2)$.
p.value: p-value from chi-squared distribution.
method: description of the test.
data.name: name of the data objects.
estimates: list with gamma ($\gamma_i$), alpha ($\alpha_i$, with $\alpha_1 = 1$), and beta ($\beta_j$, with $\beta_1 = 0$).
n_obs: number of observations used.
d: number of indicators.
p: number of Z-levels.
convergence: convergence code from nlm.
optim_details: full output from nlm.

Details

Under the structural model $X_i = \mu_i + \lambda_i \eta + \varepsilon_i$, the conditional expectations satisfy $$E(X_i \mid Z = z_j) = \gamma_i + \alpha_i \beta_j$$ where $\gamma_i$ are intercepts absorbing the reference-level means, $\alpha_i$ are parameters (with $\alpha_1 = 1$ for identification), and $\beta_j$ are parameters (with $\beta_1 = 0$ for the reference level).

This gives $d \times p$ moment conditions: $$E[I(Z = z_j)(X_i - \gamma_i - \alpha_i \beta_j)] = 0$$ with $2d + p - 2$ free parameters ($d$ intercepts, $d - 1$ alphas, $p - 1$ betas), yielding $dp - (2d + p - 2) = (d-1)(p-2)$ degrees of freedom.

The test checks whether the mean-difference matrix $\Delta_{ij} = E(X_i \mid Z = z_j) - E(X_i \mid Z = z_1)$ has rank $\le 1$, which is the testable implication of the structural model (Theorem 2 in the paper).

Consistent generalised methods of moments estimators (Newey & McFadden, 1994) are obtained by minimising a distance metric statistic. The procedure uses two-step estimation to obtain stable initial estimates, then minimises the criterion with the weight matrix recomputed at each parameter value. The test statistic is asymptotically $\chi^2_{(d-1)(p-2)}$ under the null.

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

set.seed(12345)
n <- 1000
z <- sample(0:2, n, replace = TRUE)
eta <- 1 + 0.3 * (z == 1) + 0.7 * (z == 2) + rnorm(n)
lambda <- c(1.0, 0.8, 0.6)
X <- cbind(
  2 + lambda[1] * eta + rnorm(n, sd = 0.5),
  3 + lambda[2] * eta + rnorm(n, sd = 0.5),
  1 + lambda[3] * eta + rnorm(n, sd = 0.5)
)

result <- test_t1(X, z)
result
#> 
#> 	 T1: Reliability-independent test of structural interpretation (VanderWeele & Vansteelandt, 2022) 
#> 
#> data:   X and z 
#> statistic = 0.4784, df = 2, p-value = 0.7872
#> n = 1000, d = 3 indicators, p = 3 Z-levels
#>