The per-class-interval breakdown behind an item's item-trait chi-square, as dissected in Andrich and Marais (2019, ch. 13): for every class interval the size, the maximum and mean person location, the standardised residual between observed and expected interval means, its squared chi-square component, the observed and expected means (OM, EV), the sample-size-free effect size ES = (OM - EV)/sqrt(mean V), and per response category the observed proportion (OBS.P), the mean model probability (EST.P), and the observed conditional threshold proportion (OBS.T), the proportion scoring k among those scoring k - 1 or k.
Arguments
- fit
A fitted object from
rasch.- item
Item name or index.
Value
A list with item, location, the intervals
data frame, the categories data frame, the whole-sample observed
mean ave, and the item's total chisq, df, and
p. Intervals with fewer than 2 responders are shown but carry no
chi-square contribution (used = FALSE), matching the item-trait
computation. The same applies when the model variance for an interval
is unavailable or zero. The probability is NA when person IDs
repeat because the asymptotic reference counts response rows rather than
independent persons; the interval summaries and chi-square remain
descriptive.
See also
fit_bootstrap, which refers the item's total, and
every other item fit statistic, to a bootstrap null rather than to its
asymptotic distribution.
Examples
set.seed(1)
d <- seq(-1.5, 1.5, length.out = 6)
X <- matrix(rbinom(400 * 6, 1, plogis(outer(rnorm(400), d, "-"))), 400, 6)
colnames(X) <- paste0("I", 1:6)
chisq_detail(rasch(X), "I3")$intervals
#> interval n theta_max theta_mean obs_mean exp_value residual
#> 1 1 52 -1.61901055 -1.61901055 0.07692308 0.1962929 -2.1671803
#> 2 2 67 -0.73551099 -0.73551099 0.38805970 0.3714206 0.2818740
#> 3 3 118 0.01710555 0.01710555 0.54237288 0.5563815 -0.3062983
#> 4 4 78 0.75817394 0.75817394 0.78205128 0.7246324 1.1352375
#> 5 5 43 1.61003743 1.61003743 0.93023256 0.8604966 1.3198477
#> chisq es used
#> 1 4.69667044 -0.30053383 TRUE
#> 2 0.07945293 0.03443639 TRUE
#> 3 0.09381862 -0.02819704 TRUE
#> 4 1.28876421 0.12854034 TRUE
#> 5 1.74199785 0.20127488 TRUE