R: Parametric Logistic Model (n-PL) Fitting Function

nplmodel {psychotools}

R Documentation

Parametric Logistic Model (n-PL) Fitting Function

Description

nplmodel is a basic fitting function for n-PL type 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

nplmodel(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 `NA`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` or `multipleGroup`, respectively.

Details

nplmodel (plmodel for backward compatibility with earlier psychotools versions) provides a basic fitting function for n-PL type 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), Debelak & Strobl (2019), or Schneider et al. (2022) 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.

nplmodel returns an object of class "nplmodel" for which several basic methods are available, including print, plot, summary, coef, vcov, logLik, estfun, discrpar, itempar, threshpar, guesspar, upperpar, and personpar.

Finally, if type is set to "1PL", a Rasch model is estimated. Here, a common slope parameter is estimated for all items, whereas the person parameters are assumed to follow a standard normal distribution. Please note that this variant of the Rasch model differs from the one used by mirt, which sets all slope parameters to 1, and estimates the variance of the person parameters instead. Both variants are mathematically equivalent and hence should lead to the same intercept parameter estimates. For numerical reasons, nplmodel and mirt can lead to slightly different item parameter estimates, though, under their respective default settings, in particular when some items are very easy or very difficult and the common slope parameter is large. A distinct advantage of the variant used by nplmodel is that it allows a direct comparison of the slope and intercept parameters with that estimated in more complex IRT models, such as the 2PL model.

Value

nplmodel returns an S3 object of class "nplmodel", 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 `NA`s,
`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,
`call`	original function call.

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. doi:10.18637/jss.v048.i06

Debelak R, Strobl C (2019). Investigating Measurement Invariance by Means of Parameter Instability Tests for 2PL and 3PL Models. Educational and Psychological Measurement, 79(2), 385–398. doi:10.1177/0013164418777784

Schneider L, Strobl C, Zeileis A, Debelak R (2022). An R Toolbox for Score-Based Measurement Invariance Tests in IRT Models. Behavior Research Methods, forthcoming. doi:10.3758/s13428-021-01689-0

Examples

if(requireNamespace("mirt")) {

o <- options(digits = 4)

## mathematics 101 exam results
data("MathExam14W", package = "psychotools")

## 2PL
twopl <- nplmodel(y = MathExam14W$solved)
summary(twopl)

## how to specify starting values as a vector of model parameters
st <- coef(twopl)
twopl <- nplmodel(y = MathExam14W$solved, start = st)
## or a list containing a vector of slopes and a vector of intercepts
set.seed(0)
st <- list(a = rlnorm(13, 0, 0.0625), d = rnorm(13, 0, 1))
twopl <- nplmodel(y = MathExam14W$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$gender)
mtwopl <- nplmodel(y = MathExam14W$solved, impact = MathExam14W$gender,
  grouppars = TRUE)
summary(mtwopl)
plot(mtwopl, type = "piplot")
## specifying starting values as a vector of model parameters, note that in
## this example impact is being modelled and therefore grouppars must be TRUE
## to get all model parameters
st <- coef(mtwopl)
mtwopl <- nplmodel(y = MathExam14W$solved, impact = MathExam14W$gender,
  start = st)
## or a list containing a vector of slopes, a vector of intercepts and a vector
## of means and a vector of variances as the distributional group parameters
set.seed(1)
st <- list(a = rlnorm(13, 0, 0.0625), d = rnorm(13, 0, 1), m = 0, v = 1)
mtwopl <- nplmodel(y = MathExam14W$solved, impact = MathExam14W$gender,
  start = st)

## MML estimated Rasch model (1PL)
rm <- nplmodel(y = MathExam14W$solved, type = "1PL")
summary(rm)

options(digits = o$digits)
}

[Package psychotools version 0.7-4 Index]