item.fit | R Documentation |

Computation of item fit statistics for `ltm`

, `rasch`

and `tpm`

models.

item.fit(object, G = 10, FUN = median, simulate.p.value = FALSE, B = 100)

`object` |
a model object inheriting either from class |

`G` |
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. |

`FUN` |
a function to summarize the ability estimate with each group (e.g., median, mean, etc.). |

`simulate.p.value` |
logical; if |

`B` |
the number of replications in the Monte Carlo procedure. |

The item-fit statistic computed by `item.fit()`

has the form:

*∑_{j = 1}^g 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:

- Step 1:
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`

.- Step 2:
Fit the model to the simulated data-set, extract the maximum likelihood estimates

*theta^**and compute the ability estimates*z^**for each response pattern.- Step 3:
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

*(1 +
sum_{b = 1}^B I(T_b >= T_{obs})) / (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.

An object of class `itemFit`

is a list with components,

`Tobs` |
a numeric vector with item-fit statistics. |

`p.values` |
a numeric vector with the corresponding |

`G` |
the value of the |

`simulate.p.value` |
the value of the |

`B` |
the value of the |

`call` |
a copy of the matched call of |

Dimitris Rizopoulos d.rizopoulos@erasmusmc.nl

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.

`person.fit`

,
`margins`

,
`GoF.gpcm`

,
`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)

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