greek2b {flexmet}

Find the b Vector from a Greek-Letter Parameterization of Item Parameters.

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(\theta) = b_{0} + b_{1}\theta + b_{2}\theta^2 + \cdots + b_{2k+1}\theta^{2k+1}

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

p(\theta) = a_{0} + a_{1}\theta + ... + a_{2k}\theta^{2k},

is nonnegative at all theta. Here, let

b_{0} = \xi

be the constant of integration and

b_{s} = a_{s-1}/s

for s = 1, 2, ..., 2k+1. Notice that p(\theta) 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 \geq 1:

p(\theta) = \exp{\omega} \Pi_{s=1}^{k}[1 - 2\alpha_{s}\theta + (\alpha_{s}^2+ \exp(\tau_{s}))\theta^2]

If k = 0:

p(\theta) = 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