Estimate the magnitude of response dependence between two items

Quantifies how strongly a dependent item's response follows an independent item's response, in logits, by the resolution method of Andrich and Kreiner (2010; polytomous generalisation Andrich, Humphry and Marais 2012), by resolution of the dependent item. The dependent item is resolved into one item per category of the independent item (each carrying the responses of the persons who gave that category), both original items are removed, and the model refitted. Under dependence of magnitude \(d\), threshold \(k\) of the resolved item for category \(x_i\) is shifted by \(-d\) when \(k \le x_i\) and \(+d\) otherwise, so each threshold yields \(\hat d_k = (\hat\delta_{ji(k)}(x_i = k-1) - \hat\delta_{ji(k)}(x_i = k))/2\) and \(\hat d\) is their mean (eq. 24.7 of Andrich and Marais 2019). The resolved threshold estimates share the calibration of the remaining items and are therefore correlated. The standard error of \(\hat d\) is calculated from their full sandwich covariance. If that covariance is unavailable or not positive semidefinite, the point estimate is retained as a descriptive magnitude and Wald inference is withheld. When repeated person identifiers contribute to the refit, probabilities use a t reference with the number of independent person clusters minus one degree of freedom. Independent response rows retain the asymptotic normal reference, represented by infinite degrees of freedom.

Usage

dependence_magnitude(fit, dependent, independent)

Arguments

fit

A fitted object from rasch.

dependent, independent

Item names or indices: the item hypothesised to depend, and the item it depends on. Both must share the same maximum score (the formalisation requires it).

Value

A list of class "rasch_dependence": the estimate d, its se, t, reference df and p for the hypothesis \(d = 0\), the per-threshold table thresholds (columns k, delta_lo, delta_hi, d_k, se_k), and the resolved refit.

Details

Polytomous resolution requires an unconstrained partial credit model so that each resolved threshold can move independently. A rating scale or principal-component threshold constraint is therefore refused. The refit otherwise retains the original fit grouping, keyed scoring, anchors on items that remain, and optimisation controls. Both calibrations must converge before the magnitude and its standard error are reported. MFRM virtual items are resolved through this unconstrained PCM. EFRM virtual frames are mutually exclusive and must first be reduced to an observable frame or linked design block. For an explanatory fit, the remaining items retain their explanatory restrictions and the resolved copies receive free fixed departures. Resolution is refused if any retained item or resolved copy loses its original score categories during calibration.

References

Andrich, D. and Kreiner, S. (2010). Quantifying response dependence between two dichotomous items using the Rasch model. Applied Psychological Measurement, 34, 181-192. Andrich, D., Humphry, S. M. and Marais, I. (2012). Quantifying local, response dependence between two polytomous items using the Rasch model. Applied Psychological Measurement, 36, 309-324.

Examples

set.seed(1); N <- 700
d0 <- seq(-1.5, 1.5, length.out = 8)
X <- matrix(rbinom(N * 8, 1, plogis(outer(rnorm(N), d0, "-"))), N, 8)
X[, 5] <- ifelse(runif(N) < 0.75, X[, 4], X[, 5])   # I5 follows I4
colnames(X) <- paste0("I", 1:8)
dependence_magnitude(rasch(X), dependent = "I5", independent = "I4")
#> Response dependence of I5 on I4 (Andrich & Kreiner resolution)
#>   d = 1.992 logits (se 0.142), z = 14.01, p = < 0.001