Magnitude of multidimensionality from a subtest analysis — dimensionality

Estimates how strongly two or more hypothesised subscales measure distinct traits, by Andrich's (2016) comparison of two reliability calculations: one treating all items as independent (which inflates reliability under multidimensionality) and one on the subtest analysis in which each subscale is combined into a single polytomous super-item (which absorbs the unique subscale variance). Under the bifactor formalisation $\beta_{ns} = \beta_n + c\,\beta'_{ns}$ (Marais and Andrich 2008), with $S$ subscales of $K$ items, $$c^2 = S\,(r_1/r_2 - 1) \frac{SK - 1}{S(K - 1)},$$ the latent correlation between subscales is $\rho = 1/(1 + c^2)$, and $A = S/(S + c^2)$ is the proportion of common (non-unique, non-error) variance. Both the person separation index and coefficient alpha versions are reported (Andrich and Marais 2019, ch. 24). Both the original and subtest calibrations must converge.

Usage

dimensionality_magnitude(fit, subtests)

Arguments

fit: A fitted object from rasch.
subtests: A list of character vectors assigning every item of the fit to one subscale (at least two subscales of two or more items). The published magnitude formula requires equal subscale sizes.

Value

A list of class "rasch_dim_magnitude": the comparison table (rows PSI and alpha; columns run1, subtest, c2, c, rho, A, n, n_excluded), the subtest refit, and the design constants S and K.

Details

Both reliability calculations use rows with responses to every item. PSI further requires a finite person estimate and standard error in both fits. The calibrations are retained; only the reliability sample changes. With missing responses, the result describes this complete-response sample, not necessarily the full population. The table reports rows used and excluded.

References

Andrich, D. (2016). Components of variance of scales with a bifactor subscale structure from two calculations of alpha. Educational Measurement: Issues and Practice, 35(4), 25-30.

Examples

set.seed(1); N <- 500
common <- rnorm(N); u1 <- rnorm(N); u2 <- rnorm(N)
d <- rep(seq(-1, 1, length.out = 5), 2)
X <- sapply(1:10, function(i) rbinom(N, 1,
  plogis(common + 0.8 * (if (i <= 5) u1 else u2) - d[i])))
colnames(X) <- paste0("I", 1:10)
fit <- rasch(X)
dimensionality_magnitude(fit,
  list(paste0("I", 1:5), paste0("I", 6:10)))$table
#>  index  run1 subtest    c2     c   rho     A   n n_excluded
#>    PSI 0.577   0.369 1.268 1.126 0.441 0.612 500          0
#>  alpha 0.662   0.497 0.747 0.864 0.572 0.728 500          0