psychotools (version 0.5-1)

plmodel: Parametric Logistic Model Fitting Function

Description

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.

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 class itemresp.

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 to vcov = "Oakes", see mirt for further options. If set to vcov = FALSE (or vcov = "none"), vcov() will return a matrix of NAs 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 or multipleGroup, respectively.

Value

plmodel returns an S3 object of class "plmodel", i.e., a list of the following components:

coefficients

estimated model parameters in slope/intercept parametrization,

vcov

covariance matrix of the model parameters,

data

modified data, used for model-fitting, i.e., without observations with zero weight,

items

logical vector of length ncol(y), indicating which items were used during estimation,

n

number of observations (with non-zero weights),

n_org

original number of observations in y,

weights

the weights used (if any),

na

logical indicating whether the data contain NAs,

impact

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),

loglik

log-likelihood of the fitted model,

df

number of estimated (more precisely, returned) model parameters,

code

convergence code from mirt,

iterations

number of iterations used by mirt,

reltol

convergence threshold passed to mirt,

grouppars

the logical grouppars value,

type

the type of model restriction specified,

mirt

the mirt object fitted internally.

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.

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

raschmodel, gpcmodel, rsmodel, pcmodel, btmodel

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)
}
# }