Andrich's (1985) least-upper-bound screen: the spread component
\(\lambda\) of a polytomous item (half the distance between successive
thresholds in the principal-components parameterisation, estimated here
by pcml_pc) cannot fall below the value implied by the
binomial distribution when the item is a subtest of equally difficult,
independent dichotomous items; different difficulties only raise it.
Sampling uncertainty matters when the fitted spread lies near the bound.
The function therefore reports whether the point estimate is below the
bound separately from a one-sided test of \(H_0: \lambda \geq \lambda_0\)
against \(H_1: \lambda < \lambda_0\). Evidence of dependence requires the
adjusted one-sided probability to be below alpha; a point estimate
below the bound alone is not treated as a verdict. The tabulated bounds
are exact for subtests of two and three items. For larger subtests the
binomial thresholds are no longer equally spaced, and the spread that
pcml_pc recovers from independent, equally difficult
components sits above the tabulated value (about 0.45, 0.41, 0.34, and
0.27 for four, five, six, and eight items against 0.41, 0.35, 0.29, and
0.22), so the screen is conservative there: a subtest of four or more
items needs a spread well below the tabulated bound before the test
reports dependence.
Applied to the superitems recorded by combine_items. The
binomial bound applies only when every component was dichotomous; a
composite containing a polytomous item is shown but its bound and verdict
are withheld. When person identifiers repeat, the spread refit's sandwich
covariance clusters the score contributions by person and probabilities use
a t reference with the number of independent person clusters minus one
degree of freedom. Independent rows retain the asymptotic normal reference.
The input calibration and the principal-components refit must both converge.
Arguments
- fit
A fitted object from
rasch.- maxit, tol
Passed to the
pcml_pcrefit.- alpha
Significance level for the one-sided dependence screen.
- p_adjust
Multiplicity adjustment across the eligible superitems; one of
stats::p.adjust.methods. An eligible superitem remains in the family if its probability is unavailable.
Value
A data frame with one row per recorded superitem: item,
m, whether the binomial bound is eligible, the
spread estimate and its se, the bound lub
(available for dichotomous-component subtests with maximum scores 2 to
8), t = (spread - lub)/se, reference df, the one-sided
p and adjusted p_adj, below_bound for the
point-estimate comparison, and
dependent for adjusted evidence at alpha.
Items not formed by combine_items() are omitted.
The result retains alpha and p_adjust as attributes.
References
Andrich, D. (1985). An elaboration of Guttman scaling with Rasch models for measurement. In N. B. Tuma (Ed.), Sociological Methodology 1985 (pp. 33–80). Jossey-Bass.
Andrich, D. and Marais, I. (2019). A Course in Rasch Measurement Theory: Measuring in the Educational, Social and Health Sciences. Springer.
Examples
set.seed(1); N <- 600
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) < .85, X[, 4], 1 - X[, 4])
X[, 6] <- ifelse(runif(N) < .75, X[, 4], 1 - X[, 4]) # a dependent triple
colnames(X) <- paste0("I", 1:8)
fit2 <- combine_items(rasch(X), list(c("I4", "I5", "I6"), c("I1", "I2", "I3")))
spread_test(fit2)
#> Spread-parameter screen (Andrich 1985): one-sided evidence below the binomial bound (holm adjustment; alpha 0.050)
#> item m eligible spread se bound t df p p_adj below_bound
#> I4+I5+I6 3 * -0.118 0.051 0.550 -13.110 Inf < 0.001 < 0.001 *
#> I1+I2+I3 3 * 0.501 0.081 0.550 -0.610 Inf 0.271 0.271 *
#> dependent
#> *
#>