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). Because the resolved items are answered by disjoint persons, the estimates are independent, and the standard error pools the threshold variances: \(\hat\sigma^2_k = (\hat\sigma^2_{(k)(k-1)} + \hat\sigma^2_{(k)(k)})/4\), with \(V[\hat d] = \bar{\hat\sigma^2_k}/m\) (eqs. 24.9-24.11).

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, z 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.

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 0.126), z = 15.81, p = < 0.001