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).

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). Unequal subscale sizes use their mean as $K$.

Value

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

References

Andrich, D. (2016). Components of variance of scales with a bifactor structure from two calculations of coefficient 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
#> 1   PSI 0.5770681 0.3690656 1.2680828 1.1260918 0.4409010 0.6119796
#> 2 alpha 0.6618793 0.4969475 0.7467516 0.8641479 0.5724912 0.7281328