rpf.lmp(k = 0, multidimensional = FALSE)FALSE. The multidimensional version is not yet
available.$$\mathrm P(\mathrm{pick}=1|\omega,\xi,\mathbf{\alpha},\mathbf{\tau},\theta) = \frac{1}{1+\exp(-(\xi + m(\theta,\omega,\xi,\mathbf{\alpha},\mathbf{\tau})))}$$
where $\mathbf{\alpha}$ and $\mathbf{\tau}$ are vectors of length k.
The order of the polynomial is always odd and is controlled by
the user specified non-negative integer, k. The model contains
2+2*k parameters and are used as input to the rpf.prob
function in the following order:
$\omega$ - the natural log of the slope of the item model when k=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 k>1. For example, a k=2 polynomial with have an item
parameter vector of: $\omega, \xi, \alpha_1, \tau_1, \alpha_2, \tau_2$.
See Falk & Cai (in press) for more details as to how the polynomial is constructed. In general, the polynomial looks like the following, but coefficients, b, are not directly estimated, but are a function of the item parameters.
$$\mathrm m(\theta) = \xi + b_1\theta + b_2\theta^2 + \dots + b_{2k+1}\theta^{2k+1}$$
At the lowest order polynomial (k=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.
Please note that the functions implementing this item model may eventually be replaced or subsumed by an alternative item model. That is, backwards compatability will not necessarily be guaranteed and this item model should be considered experimental until further notice.
For example, Falk & Cai present a polytomous item model derived from the generalized partial credit model that also uses a monotonic polynomial as the linear predictor, referred to as a GPC-MP item model. Since the GPC-MP reduces to the LMP when the number of categories is 2, this is a potential candidate for replacing the LMP item model. An alternative may include the retention of a dichotomous response model, but with a lower (and upper) asymptote that further reduces to the three-parameter logistic (or four-parameter logistic) item model when k=0. Finally, future versions may reparameterize $\omega$, or allow the option to release constraints on monotonicity. For instance, releasing constraints on $\omega$ may be desirable in cases where the user wishes to have the option of a monotonically decreasing response function. Further releasing constraints on $\tau$ would allow nonmontonicity and would be equivalent to replacing the linear predictor with a polynomial.
Liang (2007). A semi-parametric approach to estimating item response functions. Unpublished doctoral dissertation, Department of Psychology, The Ohio State University.
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)Run the code above in your browser using DataLab