mirt fits an unconditional maximum likelihood
factor analysis model to dichotomous and polychotomous
data under the item response theory paradigm.
Pseudo-guessing parameters may be included but must be
declared as constant.mirt(data, nfact, guess = 0, upper = 1, SE = FALSE,
rotate = 'varimax', Target = NULL, prev.cor = NULL,
par.prior = FALSE, startvalues = NULL, quadpts = NULL,
verbose = FALSE, debug = FALSE, technical = list(),
...) ## S3 method for class 'mirt':
summary(object, rotate = '', suppress =
0, digits = 3, print = FALSE, ...)
## S3 method for class 'mirt':
coef(object, rotate = '', digits = 3,
...)
## S3 method for class 'mirt':
anova(object, object2, ...)
## S3 method for class 'mirt':
fitted(object, digits = 3, ...)
## S3 method for class 'mirt':
plot(x, type = 'info', npts = 50, rot =
list(x = -70, y = 30, z = 10), ...)
## S3 method for class 'mirt':
residuals(object, restype = 'LD', digits
= 3, printvalue = NULL, ...)
matrix or data.frame that
consists of only 0, 1, and NA values to be factor
analyzed. If scores have been recorded by the response
pattern then they can be recoded to dichotomous format
using the NULLslope and int to
specify the coefficients beta prior for the slopes and
normal prior for the interceptssummary; default is 'varimax'. See
below for list of possible rotations. If rotate !=
'' in the summaryres='exp' option. Only prints patterns
that have standardized residuals greater than
abs(printvalue). The default (NULL) prints all
response patternsmirt to be plotted or
printedmirt of class
mirtClassmirt
of class mirtClass with more estimated parameters
than object'curve'
for the total test score as a function of two dimensions,
or 'info' to show the test information function
for two dimensions'LD' for a local dependence matrix (Chen &
Thissen, 1997) or 'exp' for the expected values
for the frequencies of every response patternmirt follows the item factor analysis strategy by
marginal maximum likelihood estimation (MML) outlined in
Bock and Aiken (1981), Bock, Gibbons and Muraki (1988),
and Muraki and Carlson (1995). Nested models may be
compared via the approximate chi-squared difference test
or by a reduction in AIC/BIC values (comparison via
anova). The general equation used for
multidimensional item response theory is a logistic form
with a scaling correction of 1.702. This correction is
applied to allow comparison to mainstream programs such
as TESTFACT (2003) and POLYFACT. The general IRT equation
for dichotomous items is$$P(X | \theta; \bold{a}_j; d_j; g_j, u_j) = g_j + (u_j - g_j) / (1 + exp(-1.702(\bold{a}_j' \theta + d_j)))$$
where j is the item index, $\bold{a}_j$ is the vector of discrimination parameters (i.e., slopes), $$\theta$$ is the vector of factor scores, $d_j$ is the intercept, $g_j$ is the pseudo-guessing parameter, and $u_j$ is the upper bound parameter. To avoid estimation difficulties the $g_j$'s and $u_j$'s must be pre-specified by the user. The polychotomous functions has a similar form that can be found in Muraki and Carlson (1995).
Estimation begins by computing a matrix of
quasi-tetrachoric correlations, potentially with
Carroll's (1945) adjustment for chance responds. A MINRES
factor analysis with nfact is then extracted and
item parameters are estimated by $a_{ij} =
f_{ij}/u_j$, where $f_{ij}$ is the factor loading for
the jth item on the ith factor, and
$u_j$ is the square root of the factor uniqueness,
$\sqrt{1 - h_j^2}$. The initial intercept parameters
are determined by calculating the inverse normal of the
item facility (i.e., item easiness), $q_j$, to obtain
$d_j = q_j / u_j$. A similar implementation is also
used for obtaining initial values for polychotomous
items. Following these initial estimates the model is
iterated using the EM estimation strategy with fixed
quadrature points. Implicit equation accelerations
described by Ramsey (1975) are also added to facilitate
parameter convergence speed, and these are adjusted every
third cycle.
Factor scores are estimated assuming a normal prior
distribution and can be appended to the input data matrix
(full.data = TRUE) or displayed in a summary table
for all the unique response patterns. summary and
coef allow for all the rotations available from
the GPArotation package as well as a
'promax' rotation.
Using plot will plot the either the test surface
function or the test information function for 1 and 2
dimensional solutions. To examine individual item plots
use itemplot. Residuals are computed using
the LD statistic (Chen & Thissen, 1997) in the lower
diagonal of the matrix returned by residuals, and
Cramer's V above the diagonal.
Bock, R. D., Gibbons, R., & Muraki, E. (1988). Full-Information Item Factor Analysis. Applied Psychological Measurement, 12(3), 261-280.
Carroll, J. B. (1945). The effect of difficulty and chance success on correlations between items and between tests. Psychometrika, 26, 347-372.
Chalmers, R., P. (2012). mirt: A Multidimensional Item Response Theory Package for the R Environment. Journal of Statistical Software, 48(6), 1-29.
Muraki, E. & Carlson, E. B. (1995). Full-information factor analysis for polytomous item responses. Applied Psychological Measurement, 19, 73-90.
Ramsay, J. O. (1975). Solving implicit equations in psychometric data analysis. Psychometrika, 40(3), 337-360.
Wood, R., Wilson, D. T., Gibbons, R. D., Schilling, S. G., Muraki, E., & Bock, R. D. (2003). TESTFACT 4 for Windows: Test Scoring, Item Statistics, and Full-information Item Factor Analysis [Computer software]. Lincolnwood, IL: Scientific Software International.
expand.table, key2binary,
polymirt, confmirt,
bfactor, itemplot#load LSAT section 7 data and compute 1 and 2 factor models
data(LSAT7)
data <- expand.table(LSAT7)
(mod1 <- mirt(data, 1))
summary(mod1)
residuals(mod1)
plot(mod1) #test information function
(mod2 <- mirt(data, 2))
summary(mod2)
coef(mod2)
residuals(mod2)
plot(mod2)
anova(mod1, mod2) #compare the two models
scores <- fscores(mod2) #save factor score table
###########
#data from the 'ltm' package in numeric format
pmod1 <- mirt(Science, 1)
plot(pmod1)
summary(pmod1)
pmod2 <- mirt(Science, 2)
coef(pmod2)
residuals(pmod2)
plot(pmod2)
itemplot(pmod2)
anova(pmod1, pmod2)
###########
data(SAT12)
data <- key2binary(SAT12,
key = c(1,4,5,2,3,1,2,1,3,1,2,4,2,1,5,3,4,4,1,4,3,3,4,1,3,5,1,3,1,5,4,5))
mod1 <- mirt(data, 1)
mod2 <- mirt(data, 2)
mod3 <- mirt(data, 3)
anova(mod1,mod2)
anova(mod2, mod3) #negative AIC, 2 factors probably best
#with guessing
mod1g <- mirt(data, 1, guess = .1)
coef(mod1g)
mod2g <- mirt(data, 2, guess = .1)
coef(mod2g)
anova(mod1g, mod2g)
summary(mod2g, rotate='promax')Run the code above in your browser using DataLab