itf(resp, ip, item, stat = "lr", theta,
standardize = TRUE, mu = 0, sigma = 1, bins = 9,
breaks = NULL, equal = "count", type = "means",
do.plot = TRUE, main = "Item fit")resp, row of ip), for which fit is to be
tested"chi" or "lr". Default is "lr". See
details below.resp. If not given (and group
is also missing), EAP estimates will be computed from
resp and <bins if present."width" for bins of equal
width, or "count" for bins with roughly counts of
observations. Default is "quant"itf will evaluate
the IRF. One of "mids" (the mid-point of each
bin), "meds" (the median of the values in the
bin), or "means" (the mean of the values in the
bin). Default is The chi-squared statistic $$X^2=\sum_g(N_g\frac{(p_g-\pi_g)^2}{\pi_g(1-\pi_g)},$$ 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 $\pi_g$ 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 $$X^2=2\sum_g\left[r_g\log\frac{p_g}{\pi_g} + (N_g-r_g)\log\frac{(1-p_g)}{(1-\pi_g)}\right],$$ where $\pi_g$ 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.
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
eap, qrsfit <- itf(resp=Scored, ip=Scored2pl$est, item=7)Run the code above in your browser using DataLab