Description Usage Arguments Details Value Author(s) References See Also Examples

Returns a statistic of item fit together with its degrees of freedom and p-value. Optionally produces a plot.

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`resp` |
A matrix of responses: persons as rows, items as columns, entries are either 0 or 1, no missing data |

`ip` |
Item parameters: the object returned by |

`item` |
A single number pointing to the item (column of |

`stat` |
The statistic to be computed, either |

`theta` |
A vector containing some viable estimate of the latent variable
for the same persons whose responses are given in |

`standardize` |
Standardize the distribution of ability estimates? |

`mu` |
Mean of the standardized distribution of ability estimates |

`sigma` |
Standard deviation of the standardized distribution of ability estimates |

`bins` |
Desired number of bins (default is 9) |

`breaks` |
A vector of cutpoints. Overrides |

`equal` |
Either |

`type` |
The points at which |

`do.plot` |
Whether to do a plot |

`main` |
The title of the plot if one is desired |

Given a long test, say 20 items or more, a large-test statistic of item fit could be constructed by dividing examinees into groups of similar ability, and comparing the observed proportion of correct answers in each group with the expected proportion under the proposed model. Different statistics have been proposed for this purpose.

The chi-squared statistic

*X^2=∑_g(N_g\frac{(p_g-π_g)^2}{π_g(1-π_g)},*

where *N_g*
is the number of examinees in group *g*, *p_g=r_g/N_g*, *r_g* is
the number of correct responses to the item in group *g*, and
*π_g* is the IRF of the proposed model for the median ability in group
*g*, is attributed by Embretson & Reise to R. D. Bock, although the
article they cite does not actually mention it. The statistic is the sum of
the squares of quantities that are often called "Pearson residuals" in the
literature on categorical data analysis.

BILOG uses the likelihood-ratio statistic

*X^2=2∑_g≤ft[r_g\log\frac{p_g}{π_g} +
(N_g-r_g)\log\frac{(1-p_g)}{(1-π_g)}\right],*

where *π_g* is now the
IRF for the mean ability in group *g*, and all other symbols are as
above.

Both statistics are assumed to follow the chi-squared distribution with
degrees of freedom equal to the number of groups minus the number of
parameters of the model (eg 2 in the case of the 2PL model). The first
statistic is obtained in `itf`

with `stat="chi"`

, and the second
with `stat="lr"`

(or not specifying `stat`

at all).

In the real world we can only work with estimates of ability, not with
ability itself. `irtoys`

allows use of any suitable ability measure
via the argument `theta`

. If `theta`

is not specified, `itf`

will compute EAP estimates of ability, group them in 9 groups having
approximately the same number of cases, and use the means of the ability
eatimates in each group. This is the approximate behaviour of BILOG.

If the test has less than 20 items, `itf`

will issue a warning.
For tests of 10 items or less, BILOG has a special statistic of fit, which
can be found in the BILOG output. Also of interest is the fit in 2- and
3-way marginal tables in package `ltm`

.

A vector of three numbers:

`Statistic` |
The value of the statistic of item fit |

`DF` |
The degrees of freedom |

`P-value` |
The p-value |

Ivailo Partchev

S. E. Embretson and S. P. Reise (2000), Item Response Theory for Psychologists, Lawrence Erlbaum Associates, Mahwah, NJ

M. F. Zimowski, E. Muraki, R. J. Mislevy and R. D. Bock (1996), BILOG–MG. Multiple-Group IRT Analysis and Test Maintenance for Binary Items, SSI Scientific Software International, Chicago, IL

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