ltm, rasch and tpm models.item.fit(object, G = 10, FUN = median,
simulate.p.value = FALSE, B = 100)ltm, class rasch or class tpm.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.itemFit is a list with components,G argument.simulate.p.value argument.B argument.object.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 the 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 give 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:
[object Object],[object Object],[object Object]
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.person.fit,
margins,
GoF.rasch,# 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)Run the code above in your browser using DataLab