The classical companion table conventionally reported alongside a Rasch analysis (Andrich and Marais 2019, chs. 3-5), on complete cases by default: per item the facility (mean score over maximum), the item-total and corrected item-rest correlations, the discrimination index DI = PRU - PRL (mean proportion-of-maximum in the upper third of total scores minus the lower third). Equal total scores remain in the same third, so the group sizes can differ and DI is withheld when three distinct score groups cannot be formed. The table also gives alpha if the item is deleted; the summary gives coefficient alpha, the raw-score mean, SD, and the classical standard error of measurement \(s\sqrt{1 - \alpha}\), which unlike the Rasch SE is one value for all persons. The SEM is withheld when alpha is negative, since a negative coefficient is not a usable reliability estimate. With missing responses, available-case mode also withholds SEM: its pairwise alpha and complete-case score SD describe different samples. Use complete-case mode to estimate SEM on a consistent sample. Alpha if deleted is checked separately for each retained item set; it can be available even when the full-scale covariance matrix is incomplete.
Usage
ctt_table(fit, missing = c("complete", "available"))Arguments
- fit
A fitted object from
raschwhose columns form one administered item set. Expanded EFRM and MFRM response-cell matrices are not accepted; one-cell-per-item reductions are.- missing
"complete"(default) computes the classical table on respondents who answered every item, matching the textbook total-score definitions."available"retains itemwise and pairwise available cases as an explicitly exploratory summary; respondents answering different item sets need not be comparable.
Value
A list of class "rasch_ctt": the per-item table
(item, n, min, max, facility,
item_total, item_rest, di, alpha_drop), and
the scalars
alpha, n (complete cases), mean, sd, and
sem.
References
Cronbach, L. J. (1951). Coefficient alpha and the internal structure of tests. Psychometrika, 16, 297–334.
Examples
set.seed(1)
d <- seq(-2, 2, length.out = 8)
X <- matrix(rbinom(400 * 8, 1, plogis(outer(rnorm(400), d, "-"))), 400, 8)
colnames(X) <- paste0("I", 1:8)
ctt_table(rasch(X))
#> Traditional statistics (complete cases; item n 400-400; 400 complete)
#> Raw score mean 4.08, SD 1.71 (complete responders); alpha 0.529; SEM 1.17
#> item n min max facility item_total item_rest di alpha_drop
#> I1 400 0 1 0.848 0.370 0.169 0.257 0.519
#> I2 400 0 1 0.787 0.443 0.221 0.364 0.504
#> I3 400 0 1 0.672 0.479 0.226 0.458 0.504
#> I4 400 0 1 0.575 0.526 0.267 0.539 0.488
#> I5 400 0 1 0.432 0.579 0.333 0.613 0.460
#> I6 400 0 1 0.338 0.495 0.243 0.486 0.497
#> I7 400 0 1 0.250 0.533 0.313 0.466 0.472
#> I8 400 0 1 0.175 0.407 0.198 0.293 0.511