Residual correlations for local dependence (Yen's Q3) — residual

The pairwise correlations of the standardised response residuals are Yen's (1984) Q3 statistics. Under unidimensionality and local independence the off-diagonal values sit near -1/(L-1); large positive values flag local dependence between item pairs. Following Christensen, Makransky and Horton (2017), each Q3 is also reported relative to the average off-diagonal value (q3_star). There is no universal adjusted-Q3 critical value: it depends on sample size, test length, category structure, and missingness. The default therefore reports the statistics without a binary flag. A user-supplied flag is an explicitly heuristic screening threshold, not a calibrated significance test.

Usage

residual_correlations(fit, flag = NULL)

Arguments

fit: A fitted object from rasch.
flag: Optional heuristic excess above the average off-diagonal Q3 at which a pair is flagged. The default NULL withholds binary flags.

Value

A list with the Q3 matrix, the adjusted-Q3 star_matrix (each Q3 less the average off-diagonal value, diagonal empty), the average off-diagonal value, pairs (every item pair with q3, q3_star and a flagged indicator, sorted by q3), and the subset of flagged pairs.

References

Yen, W. M. (1984). Effects of local item dependence on the fit and equating performance of the three-parameter logistic model. Applied Psychological Measurement, 8(2), 125-145.

Christensen, K. B., Makransky, G., & Horton, M. (2017). Critical values for Yen's Q3: identification of local dependence in the Rasch model using residual correlations. Applied Psychological Measurement, 41(3), 178-194.

Examples

set.seed(1)
d <- seq(-2, 2, length.out = 8)
X <- matrix(rbinom(300 * 8, 1, plogis(outer(rnorm(300), d, "-"))), 300, 8)
colnames(X) <- paste0("I", 1:8)
residual_correlations(rasch(X))$average
#> [1] -0.1297187