T0 Test: Reliability-Dependent Test of Structural Interpretation (Section 3.2)

Tests whether the structural interpretation of a univariate latent factor model can be rejected, using estimated reliability coefficients. The variance of the estimating equations is adjusted for uncertainty in the reliability estimates (see paper, p. 2041).

Usage

test_t0(X, z, na.rm = TRUE, max_iter = 1000L, verbose = FALSE)

Arguments

X

numeric matrix (n x d) of indicator variables, d >= 3.

z

numeric vector or factor of length n encoding a discrete auxiliary variable. Must have at least 2 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.

verbose

logical; if TRUE, print progress information.

Value

An object of class c("structest_t0", "structest", "htest") containing:

statistic

the T0 test statistic.

parameter

degrees of freedom, \((d-1)(p-1)\).

p.value

p-value from chi-squared distribution.

method

description of the test.

data.name

name of the data objects.

estimates

list with gamma (intercepts), beta (Z effects), and lambda (reliabilities).

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, \(E[X_i | Z = z_j] = \gamma_i + (\lambda_i / \lambda_1) \beta_j\). This gives \(d \times p\) moment conditions with \(d + p - 1\) free parameters (\(\gamma_1, \ldots, \gamma_d\) and \(\beta_2, \ldots, \beta_p\) with \(\beta_1 = 0\)), yielding \((d-1)(p-1)\) degrees of freedom.

The variance of the estimating equations is adjusted for the uncertainty in the reliability estimates \(\lambda_i\) (see paper, p. 2041). Consistent generalised methods of moments estimators (Newey & McFadden, 1994) are obtained by minimising the resulting distance metric statistic.

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

See also

test_t1 for the reliability-independent test; fit_structural for parameter estimates with standard errors; estimate_reliability for the reliability estimation step.

Examples

set.seed(12345)
n <- 1000
z <- rbinom(n, 1, 0.5)
eta <- 1 + 0.5 * z + 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_t0(X, z)
result
#> 
#> 	 T0: Reliability-dependent test of structural interpretation (VanderWeele & Vansteelandt, 2022) 
#> 
#> data:   X and z 
#> statistic = 6.452, df = 2, p-value = 0.03971
#> n = 1000, d = 3 indicators, p = 2 Z-levels
#>