greek2b: Find the b Vector from a Greek-Letter Parameterization of...

In flexmet: Flexible Latent Trait Metrics using the Filtered Monotonic Polynomial Item Response Model

Description

Convert the Greek-letter parameterization of item parameters (used to ensure monotonicitiy) to the b-vector parameterization (polynomial coefficients).

Usage

greek2b(xi, omega, alpha = NULL, tau = NULL)

Arguments

xi	see details
omega	see details
alpha	see details, vector of length k, set to NULL if k = 0
tau	see details, vector of length k, set to NULL if k = 0

Details

For

m(θ) = b0 + b1θ + b2θ^2 + ... + b(2k+1)θ^{2k+1}

to be a monotonic function, a necessary and sufficient condition is that its first derivative,

p(θ) = a0 + a1θ + ... + a(2k)θ^{2k},

is nonnegative at all theta. Here, let

b0 = ξ

be the constant of integration and

b(s) = a(s-1)/s

for s = 1, 2, ..., 2k+1. Notice that p(θ) is a polynomial function of degree 2k. A nonnegative polynomial of an even degree can be re-expressed as the product of k quadratic functions.

If k >= 1:

p(θ) = exp{ω} Π_{s=1}^{k}[1 - 2α(s)θ + (α(s)^2+ exp(τ(s)))θ^2]

If k = 0:

p(θ) = 0.

Value

A vector of item parameters in the b parameterization.

References

Liang, L., & Browne, M. W. (2015). A quasi-parametric method for fitting flexible item response functions. Journal of Educational and Behavioral Statistics, 40, 5–34. doi: 10.3102/1076998614556816

See Also

b2greek

Examples

(bvec <- greek2b(xi = 0, omega = 1, alpha = .1, tau = -1))
## 0.0000000  2.7182818 -0.2718282  0.3423943

(b2greek(bvec))
##  0.0  1.0  0.1 -1.0

flexmet documentation built on July 14, 2021, 1:06 a.m.