Returns a statistic of item fit together with its degrees of freedom and p-value. Optionally produces a plot.
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")A matrix of responses: persons as rows, items as columns, entries are either 0 or 1, no missing data
Item parameters: the object returned by \(est\), or the equivalent of its first part.
A single number pointing to the item (column of resp, row
of ip), for which fit is to be tested
The statistic to be computed, either "chi" or
"lr". Default is "lr". See details below.
A vector containing some viable estimate of the latent variable
for the same persons whose responses are given in resp. If not given
(and group is also missing), WLE estimates will be computed from
resp and ip.
Standardize the distribution of ability estimates?
Mean of the standardized distribution of ability estimates
Standard deviation of the standardized distribution of ability estimates
Desired number of bins (default is 9)
A vector of cutpoints. Overrides bins if present.
Either "width" for bins of equal width, or
"count" for bins with roughly counts of observations. Default is
"quant"
The points at which 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 "means".
Whether to do a plot
The title of the plot if one is desired
A vector of three numbers:
The value of the statistic of item fit
The degrees of freedom
The p-value
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 $$X^2=\sum_g(N_g\frac{(p_g-\pi_g)^2}{\pi_g(1-\pi_g)},$$ 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.
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
# NOT RUN {
fit <- itf(resp=Scored, ip=Scored2pl, item=7)
# }
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