plmodel
Parametric Logistic Model Fitting Function
plmodel
is a basic fitting function for parametric logistic IRT models
(2PL, 3PL, 3PLu, 4PL, Rasch/1PL), providing a wrapper around
mirt
and multipleGroup
relying on
marginal maximum likelihood (MML) estimation via the standard EM algorithm.
- Keywords
- regression
Usage
plmodel(y, weights = NULL, impact = NULL,
type = c("2PL", "3PL", "3PLu", "4PL", "1PL", "RM"),
grouppars = FALSE, vcov = TRUE,
start = NULL, method = "BFGS", maxit = 500, reltol = 1e-5, …)
Arguments
- y
item response object that can be coerced (via
as.matrix
) to a numeric matrix with scores 0, 1. Typically, either already a matrix, data frame, or dedicated object of classitemresp
.- weights
an optional vector of weights (interpreted as case weights).
- impact
an optional
factor
allowing for grouping the subjects (rows). If specified, a multiple-group model is fitted to account for impact (see details below). By default, no impact is modelled, i.e., a single-group model is used.- type
character string, specifying the type of parametric logistic IRT model to be estimated (see details below).
- grouppars
logical. Should the estimated distributional group parameters of a multiple group model be included in the model parameters?
- vcov
logical or character specifying the type of variance-covariance matrix (if any) computed for the final model. The default
vcov = TRUE
corresponds tovcov = "Oakes"
, seemirt
for further options. If set tovcov = FALSE
(orvcov = "none"
),vcov()
will return a matrix ofNA
s only.- start
an optional vector or list of starting values (see examples below).
- method, maxit, reltol
control parameters for the optimizer employed by
mirt
for the EM algorithm.- …
further arguments passed to
mirt
ormultipleGroup
, respectively.
Details
plmodel
provides a basic fitting function for parametric logistic IRT
models (2PL, 3PL, 3PLu, 4PL, Rasch/1PL) providing a wrapper around
mirt
and multipleGroup
relying on
MML estimation via the standard EM algorithm (Bock & Aitkin, 1981). Models are
estimated under the slope/intercept parametrization, see e.g. Chalmers (2012).
The probability of person \(i\) ‘solving’ item \(j\) is modelled as:
$$P(X_{ij} = 1|\theta_{i},a_{j},d_{j},g_{j},u_{j}) =
g_{j} + \frac{(u_{j} - g_{j})}{1 + \exp{(-(a_{j}\theta_{i} + d_{j}))}}$$
A reparametrization of the intercepts to the classical IRT parametrization,
\(b_{j} = -\frac{d_{j}}{a_{j}}\), is provided via the corresponding
itempar
method.
If an optional impact
variable is supplied, a multiple-group model of
the following form is being fitted: Item parameters are fixed to be equal
across the whole sample. For the first group of the impact
variable the
person parameters are fixed to follow the standard normal distribution. In the
remaining impact
groups, the distributional parameters (mean and
variance of a normal distribution) of the person parameters are
estimated freely. See e.g. Baker & Kim (2004, Chapter 11) or Debelak & Strobl
(2018) for further details. To improve convergence of the model fitting
algorithm, the first level of the impact
variable should always correspond
to the largest group. If this is not the case, levels are re-ordered internally.
If grouppars
is set to TRUE
the freely estimated distributional
group parameters (if any) are returned as part of the model parameters.
By default, type
is set to "2PL"
. Therefore, all so-called
guessing parameters are fixed at 0 and all upper asymptotes are fixed at 1.
"3PL"
results in all upper asymptotes being fixed at 1 and "3PLu"
results in all all guessing parameters being fixed at 0. "4PL"
results
in a full estimated model as specified above. Finally, if type
is set to
"1PL"
(or equivalently "RM"
), an MML-estimated Rasch model is
being fitted. This means that all slopes are restricted to be equal across all
items, all guessing parameters are fixed at 0 and all upper asymptotes are
fixed at 1.
Note that internally, the so-called guessing parameters and upper asymptotes
are estimated on the logit scale (see also mirt
).
Therefore, most of the basic methods below include a logit
argument,
which can be set to TRUE
or FALSE
allowing for a retransformation
of the estimates and their variance-covariance matrix (if requested) using the
logistic function and the delta method if logit = FALSE
.
plmodel
returns an object of class "plmodel"
for which
several basic methods are available, including print
, plot
,
summary
, coef
, vcov
, logLik
, estfun
,
discrpar
, itempar
, threshpar
,
guesspar
, upperpar
, and personpar
.
Value
plmodel
returns an S3 object of class "plmodel"
,
i.e., a list of the following components:
estimated model parameters in slope/intercept parametrization,
covariance matrix of the model parameters,
modified data, used for model-fitting, i.e., without observations with zero weight,
logical vector of length ncol(y)
, indicating
which items were used during estimation,
number of observations (with non-zero weights),
original number of observations in y
,
the weights used (if any),
logical indicating whether the data contain NA
s,
either NULL
or the supplied impact
variable
with the levels reordered in decreasing order (if this has not been the case
prior to fitting the model),
log-likelihood of the fitted model,
number of estimated (more precisely, returned) model parameters,
convergence code from mirt
,
number of iterations used by mirt
,
convergence threshold passed to mirt
,
the logical grouppars
value,
the type
of model restriction specified,
the mirt
object fitted internally.
References
Baker FB, Kim SH (2004). Item Response Theory: Parameter Estimation Techniques. Chapman & Hall/CRC, Boca Raton.
Bock RD, Aitkin M (1981). Marginal Maximum Likelihood Estimation of Item Parameters: Application of an EM Algorithm. Psychometrika, 46(4), 443--459.
Chalmers RP (2012). mirt: A Multidimensional Item Response Theory Package for the R Environment. Journal of Statistical Software, 48(6), 1--29. 10.18637/jss.v048.i06
Debelak R, Strobl C (2018). Investigating Measurement Invariance by Means of Parameter Instability Tests for 2PL and 3PL Models. Educational and Psychological Measurement, forthcoming. 10.1177/0013164418777784
See Also
Examples
# NOT RUN {
if(requireNamespace("mirt")) {
o <- options(digits = 4)
## mathematics 101 exam results
data("MathExam14W", package = "psychotools")
## 2PL
twopl <- plmodel(y = MathExam14W$solved)
summary(twopl)
## how to specify starting values as a vector of model parameters
st <- coef(twopl)
twopl <- plmodel(y = MathExam14W$solved, start = st)
## or a list containing a vector of slopes and a vector of intercepts
set.seed(0)
st <- list(a = rlnorm(13, 0, 0.0625), d = rnorm(13, 0, 1))
twopl <- plmodel(y = MathExam14W$solved, start = st)
## visualizations
plot(twopl, type = "profile")
plot(twopl, type = "regions")
plot(twopl, type = "piplot")
plot(twopl, type = "curves", xlim = c(-6, 6))
plot(twopl, type = "information", xlim = c(-6, 6))
## visualizing the IRT parametrization
plot(twopl, type = "curves", xlim = c(-6, 6), items = 1)
abline(v = itempar(twopl)[1])
abline(h = 0.5, lty = 2)
## 2PL accounting for gender impact
table(MathExam14W$gender)
mtwopl <- plmodel(y = MathExam14W$solved, impact = MathExam14W$gender,
grouppars = TRUE)
summary(mtwopl)
plot(mtwopl, type = "piplot")
## specifying starting values as a vector of model parameters, note that in
## this example impact is being modelled and therefore grouppars must be TRUE
## to get all model parameters
st <- coef(mtwopl)
mtwopl <- plmodel(y = MathExam14W$solved, impact = MathExam14W$gender,
start = st)
## or a list containing a vector of slopes, a vector of intercepts and a vector
## of means and a vector of variances as the distributional group parameters
set.seed(1)
st <- list(a = rlnorm(13, 0, 0.0625), d = rnorm(13, 0, 1), m = 0, v = 1)
mtwopl <- plmodel(y = MathExam14W$solved, impact = MathExam14W$gender,
start = st)
## MML estimated Rasch model (1PL)
rm <- plmodel(y = MathExam14W$solved, type = "1PL")
summary(rm)
options(digits = o$digits)
}
# }