This model is a dichotomous response model originally proposed by Liang (2007) and is implemented using the parameterization by Falk & Cai (2016).
rpf.lmp(q = 0, multidimensional = FALSE)an item model
a non-negative integer that controls the order of the polynomial (2q+1) with a default of q=0 (1st order polynomial = 2PL).
whether to use a multidimensional model.
Defaults to FALSE. The multidimensional version is not yet
available.
The LMP model replaces the linear predictor part of the two-parameter logistic function with a monotonic polynomial, \(m(\theta,\omega,\xi,\mathbf{\alpha},\mathbf{\tau})\),
$$\mathrm P(\mathrm{pick}=1|\omega,\xi,\mathbf{\alpha},\mathbf{\tau},\theta) = \frac{1}{1+\exp(-(\xi + m(\theta;\omega,\mathbf{\alpha},\mathbf{\tau})))} $$
where \(\mathbf{\alpha}\) and \(\mathbf{\tau}\) are vectors of length q.
The order of the polynomial is always odd and is controlled by
the user specified non-negative integer, q. The model contains
2+2*q parameters and are used in conjunction with the rpf.prob
or rpf.dTheta function in the following order:
\(\omega\) - the natural log of the slope of the item model when q=0,
\(\xi\) - the intercept,
\(\alpha\) and \(\tau\) - two parameters that control bends in
the polynomial. These latter parameters are repeated in the same order for
models with q>0. For example, a q=2 polynomial with have an item
parameter vector of: \(\omega, \xi, \alpha_1, \tau_1, \alpha_2, \tau_2\).
In general, the polynomial looks like the following:
$$m(\theta;\omega,\alpha,\tau) = b_1\theta + b_2\theta^2 + \dots + b_{2q+1}\theta^{2q+1} $$
However, the coefficients, b, are not directly estimated, but are a function of the item parameters. In particular, the derivative \(m'(\theta;\omega,\alpha,\tau)\) is parameterized in the following way:
$$m'(\theta;\omega,\alpha,\tau) = \left\{\begin{array}{ll}\exp(\omega) \prod_{u=1}^q(1-2\alpha_{u}\theta + (\alpha_{u}^2 + \exp(\tau_{u}))\theta^2) & \mbox{if } q > 0 \\ \exp(\omega) & \mbox{if } q = 0\end{array} \right.$$
See Falk & Cai (2016) for more details as to how the polynomial is constructed. At the lowest order polynomial (q=0) the model reduces to the two-parameter logistic (2PL) model. However, parameterization of the slope parameter, \(\omega\), is currently different than the 2PL (i.e., slope = exp(\(\omega\))). This parameterization ensures that the response function is always monotonically increasing without requiring constrained optimization.
For an alternative parameterization that releases constraints
on \(\omega\), allowing for monotonically decreasing functions,
see rpf.grmp. And for polytomous items, see both
rpf.grmp and rpf.gpcmp.
Falk, C. F., & Cai, L. (2016). Maximum marginal likelihood estimation of a monotonic polynomial generalized partial credit model with applications to multiple group analysis. Psychometrika, 81, 434-460. tools:::Rd_expr_doi("10.1007/s11336-014-9428-7")
Liang (2007). A semi-parametric approach to estimating item response functions. Unpublished doctoral dissertation, Department of Psychology, The Ohio State University.
Other response model:
rpf.drm(),
rpf.gpcmp(),
rpf.grmp(),
rpf.grm(),
rpf.mcm(),
rpf.nrm()
spec <- rpf.lmp(1) # 3rd order polynomial
theta<-seq(-3,3,.1)
p<-rpf.prob(spec, c(-.11,.37,.24,-.21),theta)
spec <- rpf.lmp(2) # 5th order polynomial
p<-rpf.prob(spec, c(.69,.71,-.5,-8.48,.52,-3.32),theta)
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