# rpf.gpcmp: Create monotonic polynomial generalized partial credit... In rpf: Response Probability Functions

## Description

This model is a polytomous model proposed by Falk & Cai (2016) and is based on the generalized partial credit model (Muraki, 1992).

## Usage

 1 rpf.gpcmp(outcomes = 2, q = 0, multidimensional = FALSE) 

## Arguments

 outcomes The number of possible response categories. q a non-negative integer that controls the order of the polynomial (2q+1) with a default of q=0 (1st order polynomial = generalized partial credit model). multidimensional whether to use a multidimensional model. Defaults to FALSE. The multidimensional version is not yet available.

## Details

The GPC-MP replaces the linear predictor part of the generalized partial credit model with a monotonic polynomial, m(theta; omega, alpha, tau). The response function for category k is:

\frac{exp(sum_{v=0}^k (xi_k + m(theta;omega,xi,alpha,tau)))}{sum_{u=0}^{K-1}exp(sum_{v=0}^u (xi_u + m(theta;omega,xi,alpha,tau)))}

where alpha and tau are vectors of length q. The GPC-MP uses the same parameterization for the polynomial as described for the logistic function of a monotonic polynomial (LMP). See also (rpf.lmp).

The order of the polynomial is always odd and is controlled by the user specified non-negative integer, q. The model contains 1+(outcomtes-1)+2*q parameters and are used as input to the rpf.prob function in the following order: omega - natural log of the slope of the item model when q=0, xi - a (outcomes-1)-length vector of intercept parameters, 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 3 categories will have an item parameter vector of: omega, xi1, xi2, alpha1, tau1, alpha2, tau2.

Note that the GPC-MP reduces to the LMP when the number of categories is 2, and the GPC-MP reduces to the generalized partial credit model when the order of the polynomial is 1 (i.e., q=0).

an item model

## References

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. doi: 10.1007/s11336-014-9428-7

Muraki, E. (1992). A generalized partial credit model: Application of an EM algorithm. Applied Psychological Measurement, 16, 159–176.

Other response model: rpf.drm(), rpf.grmp(), rpf.grm(), rpf.lmp(), rpf.mcm(), rpf.nrm()
 1 2 3 spec <- rpf.gpcmp(5,2) # 5-category, 3rd order polynomial theta<-seq(-3,3,.1) p<-rpf.prob(spec, c(1.02,3.48,2.5,-.25,-1.64,.89,-8.7,-.74,-8.99),theta)