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.
1 2 3 4 |
resp |
A matrix of responses: persons as rows, items as columns, entries are either 0 or 1, no missing data |
ip |
Item parameters: a matrix with one row per item, and three columns: [,1] item discrimination a, [,2] item difficulty b, and [,3] asymptote c. |
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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