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 |

`weights` |
an optional vector of weights (interpreted as case weights). |

`impact` |
an optional |

`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 |

`start` |
an optional vector or list of starting values (see examples below). |

`method` , `maxit` , `reltol` |
control parameters for the optimizer employed
by |

`...` |
further arguments passed to |

### 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 |

`n` |
number of observations (with non-zero weights), |

`n_org` |
original number of observations in |

`weights` |
the weights used (if any), |

`na` |
logical indicating whether the data contain |

`impact` |
either |

`loglik` |
log-likelihood of the fitted model, |

`df` |
number of estimated (more precisely, returned) model parameters, |

`code` |
convergence code from |

`iterations` |
number of iterations used by |

`reltol` |
convergence threshold passed to |

`grouppars` |
the logical |

`type` |
the |

`mirt` |
the |

`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

### See Also

`raschmodel`

, `gpcmodel`

,
`rsmodel`

, `pcmodel`

, `btmodel`

### 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)
}
```

*psychotools*version 0.7-4 Index]