irt.eq: IRT methods for Test Equating
In jagonzalb/SNSequate: Standard and Nonstandard Statistical Models and Methods for Test Equating

Description Usage Arguments Details Value Author(s) References See Also Examples

Implements methods to perform Test Equating over IRT models.

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irt.eq(n_items, param_x, param_y, theta_points=NULL, weights=NULL,
       n_points=10, w=1, A=NULL, B=NULL, link=NULL, 
       method_link=NULL, common=NULL, method="TS", D=1.7,...)

`n_items`	Number of items of the test
`param_x`	Estimated parameters for IRT model on test X. This list must have the following structure: list(a, b, c), where each parameter is a vector with the respective estimate for each subject. If you want to perform other models (i.e. Rasch), replace according with a vector of zeros.
`param_y`	Estimated parameters for IRT model on test Y. This list must have the following structure: list(a, b, c), where each parameter is a vector with the respective estimate for each subject. If you want to perform other models (i.e. Rasch), replace according with a vector of zeros.
`method`	A string, either "TS" or "OS". Each one stands for "True Score Equating" and "Observed score equating". Notice that OS requires the additional arguments "theta_points" and "weigths".
`theta_points`	For "OS" only. Points over a grid of possible values of θ to integrate out the ability term.
`weights`	For "OS" only. Weigths for integrate out the ability term. If is NULL, the method assumes the distribution of ability is characterized by a finite number of abilities (Kolen and Brennan 2013, pg 199).
`n_points`	In case theta_ponints is not provided, is the length of the grid for the gaussian quadrature.
`A, B`	Scaling parameters. In the case they are not provided, they will be calculated depending on the next described inputs.
`link`	An irt.link object.
`method_link`	Method used to estimate A and B. Default is "mean/sigma". Others are "mean/mean", "Haebara" and "Stocklord". For more information see irt.link
`common`	Common items to estimate A and B. Default asume all items are common.
`w`	Weight factor between real and synthetic population.
`D`	Scaling parameter. Default is 1.7
`...`	Further arguments.

This function implements two methods to perform Test Equating over Item Response Theory models (Kolen and Brennan 2013).

"True Score Equating" relate number-correct scores on Form X and Form Y. Assumes that the true score associated with each θ is equivalent to the true score on another form associated with that θ.

"Observed Score Equating" uses the IRT model to produce an estimated distribution of observed number-correct scores on each form. Using the compound binomial distribution (Lord and Wingersky 1984) to find the conditional distributions f(x|θ), and then integrate out the θ parameter. Afterwards, an Equipercentile Equating process is done over the estimated distributions.

An object of the clas irt.eq is returned. Depending on the method used, the outputs are:

True Score Equating: A list(n_items, theta_equivalent, tau_y) containing the number of items, the theta equivalent values on Form X to Form Y and the equivalent scores.
Observed Score Equating: A list(n_items, f_hat, g_hat, e_Y_x) containing the number of items, the estimated distributions and the equated values.

Daniel Leon A. dnacuna@mat.uc.cl

Kolen, M. J., & Brennan, R. L. (2013). Test Equating, Scaling, and Linking: Methods and Practices, Third Edition. Springer Science & Business Media.

irt.link

data(KB36_t)
dfo <- KB36_t

param_x <- list(a=dfo[,3],b=dfo[,4],c=dfo[,5])
param_y <- list(a=dfo[,7],b=dfo[,8],c=dfo[,9])

theta_points=c(-5.2086,-4.163,-3.1175,-2.072,-1.0269,0.0184,
               1.0635,2.109,3.1546,4.2001)
weights=c(0.000101,0.00276,0.03021,0.142,0.3149,0.3158,
         0.1542,0.03596,0.003925,0.000186)


irt.eq(36, param_x, param_y, method="TS", A=1, B=0)
irt.eq(36, param_x, param_y, theta_points, weights, method="OS",
       A=1, B=0)