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.
A spread estimate below the bound therefore indicates response
dependence among the members (Andrich and Marais 2019, Table 24.1).
Typically applied after combine_items, whose super-items
are exactly such subtests.
Value
A data frame with one row per polytomous item: item,
m, the spread estimate and its se, the bound
lub (available for maximum scores 2 to 8), z =
(spread - lub)/se, and dependent = spread below the bound.
Dichotomous items carry no spread and are omitted.
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] <- X[, 4]; X[, 6] <- 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): spread below the binomial bound indicates dependence
#> item m spread se bound z dependent
#> I1+I2+I3 3 0.653 0.114 0.550 0.899