Computation of item fit statistics for ltm, rasch and tpm models.
item.fit(object, G = 10, FUN = median,
simulate.p.value = FALSE, B = 100)a model object inheriting either from class ltm, class rasch or class tpm.
either a number or a numeric vector. If a number, then it denotes the number of categories sample units are grouped according to their ability estimates.
a function to summarize the ability estimate with each group (e.g., median, mean, etc.).
logical; if TRUE, then the Monte Carlo procedure described in the Details
section is used to approximate the the distribution of the item-fit statistic under the null hypothesis.
the number of replications in the Monte Carlo procedure.
An object of class itemFit is a list with components,
a numeric vector with item-fit statistics.
a numeric vector with the corresponding \(p\)-values.
the value of the G argument.
the value of the simulate.p.value argument.
the value of the B argument.
a copy of the matched call of object.
The item-fit statistic computed by item.fit() has the form: $$\sum \limits_{j = 1}^G \frac{N_j
(O_{ij} - E_{ij})^2}{E_{ij} (1 - E_{ij})},$$
where \(i\) is the item, \(j\) is the interval created by grouping sample units on the basis of their ability
estimates, \(G\) is the number of sample units groupings (i.e., G argument), \(N_j\) is the number of
sample units with ability estimates falling in a given interval \(j\), \(O_{ij}\) is the observed proportion of
keyed responses on item \(i\) for interval \(j\), and \(E_{ij}\) is the expected proportion of keyed responses
on item \(i\) for interval \(j\) based on the IRT model (i.e., object) evaluated at the ability estimate
\(z^*\) within the interval, with \(z^*\) denoting the result of FUN applied to the ability estimates in
group \(j\).
If simulate.p.value = FALSE, then the \(p\)-values are computed assuming a chi-squared distribution with
degrees of freedom equal to the number of groups G minus the number of estimated parameters. If
simulate.p.value = TRUE, a Monte Carlo procedure is used to approximate the distribution of the item-fit
statistic under the null hypothesis. In particular, the following steps are replicated B times:
Simulate a new data-set of dichotomous responses under the assumed IRT model, using the maximum
likelihood estimates \(\hat{\theta}\) in the original data-set, extracted from object.
Fit the model to the simulated data-set, extract the maximum likelihood estimates \(\theta^*\) and compute the ability estimates \(z^*\) for each response pattern.
For the new data-set, and using \(z^*\) and \(\theta^*\), compute the value of the item-fit statistic.
Denote by \(T_{obs}\) the value of the item-fit statistic for the original data-set. Then the \(p\)-value is approximated according to the formula $$\left(1 + \sum_{b = 1}^B I(T_b \geq T_{obs})\right) / (1 + B),$$ where \(I(.)\) denotes the indicator function, and \(T_b\) denotes the value of the item-fit statistic in the \(b\)th simulated data-set.
Reise, S. (1990) A comparison of item- and person-fit methods of assessing model-data fit in IRT. Applied Psychological Measurement, 14, 127--137.
Yen, W. (1981) Using simulation results to choose a latent trait model. Applied Psychological Measurement, 5, 245--262.
# NOT RUN {
# item-fit statistics for the Rasch model
# for the Abortion data-set
item.fit(rasch(Abortion))
# Yen's Q1 item-fit statistic (i.e., 10 latent ability groups; the
# mean ability in each group is used to compute fitted proportions)
# for the two-parameter logistic model for the LSAT data-set
item.fit(ltm(LSAT ~ z1), FUN = mean)
# }
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