itf(resp, ip, item, stat = "lr", theta, groups, standardize = TRUE, mu = 0, sigma = 1,
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 <grp.
If not given, grp will be applied on theta
with its default values.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, so the approach is a
bit circular in defining the groups. I have tried to
offer some extra flexibility with the arguments
theta nor group:
thetanorgroupis specified,item.testwill compute EAP estimates of ability
for the proposed model, group them, and use medians for"chi"or means for"lr". This is the
approximate behaviour of BILOG (assumingstat="lr").qrsand
passing them toitem.testastheta."chi", means for"lr") can be overriden by preparing the groups
withgrpand passing them toitem.testasgroup. In that case,thetais not needed. If the test has less than 20 items, item.test 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
grp, eap, qrsp.2pl <- est(Scored, model = "2PL", engine = "ltm")
fit <- itf(resp = Scored, ip = p.2pl, item = 7)Run the code above in your browser using DataLab