SNSequate (version 1.0)

ker.eq: The Kernel method of test equating

Description

This function implements the kernel method of test equating as described in Holland and Thayer (1989), and Von Davier et al. (2004). Nonstandard kernels others than the gaussian are available. Associated standard error of equating are also provided.

Usage

ker.eq(scores, kert, hx = NULL, hy = NULL, degree, design, Kp = 1, scores2, 
degreeXA, degreeYA, J, K, L, wx, wy, w)

Arguments

Note that depending on the specified equating design, not all arguments are necessary as detailed below.

scores

If the "EG" design is specified, a two column matrix containing the raw sample frequencies coming from the two groups of scores to be equated. It is assumed that the data in the first and second columns come from tests $X$ and $Y$, respectively.

kert

A character string giving the type of kernel to be used for continuization. Current options include "gauss", "logis", and "uniform" for the gaussian, logistic and uniform kernels, respectively

hx

An integer indicating the value of the bandwidth parameter to be used for kernel continuization of $F(x)$. If not provided (Default), this value is automatically calculated (see details).

hy

An integer indicating the value of the bandwidth parameter to be used for kernel continuization of $G(y)$. If not provided (Default), this value is automatically calculated (see details).

degree

A vector indicating the number of power moments to be fitted to the marginal distributions ("EG" design), and/or the number or cross moments to be fitted to the joint distributions (see Details).

design

A character string indicating the equating design (one of "EG", "SG", "CB", "NEAT_CE", "NEAT_PSE")

Kp

A number which acts as a weight for the second term in the combined penalization function used to obtain h (see details).

scores2

Only used for the "CB", "NEAT_CE" and "NEAT_PSE" designs. See the description of scores.

degreeXA

A vector indicating the number of power moments to be fitted to the marginal distributions $X$ and $A$, and the number or cross moments to be fitted to the joint distribution $(X,A)$ (see details). Only used for the "NEAT_CE" and "NEAT_PSE" desi

degreeYA

Only used for the "NEAT_CE" and "NEAT_PSE" designs (see the description for degreeXA)

J

The number of possible $X$ scores. Only needed for "CB", "NEAT_CB" and "NEAT_PSE" designs

K

The number of possible $Y$ scores. Only needed for "CB", "NEAT_CB" and "NEAT_PSE" designs

L

The number of possible $A$ scores. Needed for "NEAT_CB" and "NEAT_PSE" designs

wx

A number that satisfies $0\leq w_X\leq 1$ indicating the weight put on the data that is not subject to order effects. Only used for the "CB" design.

wy

A number that satisfies $0\leq w_Y\leq 1$ indicating the weight put on the data that is not subject to order effects. Only used for the "CB" design.

w

A number that satisfies $0\leq w\leq 1$ indicating the weight given to population $P$. Only used for the "NEAT" design.

...

Further arguments currently not used.

Value

• An object of class ker.eq representing the kernel equating process. Generic functions such as print, and summary have methods to show the results of the equating. The results include summary statistics, equated values, standard errors of equating, and others. The function SEED can be used to obtain standard error of equating differences (SEED) of two objects of class ker.eq. The function PREp can be used on a ker.eq object to obtain the percentage relative error measure (see Von Davier et al, 2004).
• Scores
  The possible values of $x_j$ and $y_k$
• eqYx
  The equated values of test $X$ in test $Y$ scale
• eqXy
  The equated values of test $Y$ in test $X$ scale
• SEEYx
  The standard error of equating for equating $X$ to $Y$
• SEEXy
  The standard error of equating for equating $Y$ to $X$

Details

This is a generic function that implements the kernel method of test equating as described in Von Davier et al. (2004). Given test scores $X$ and $Y$, the functions calculates $$\hat{e}_Y(x)=G_{h_{Y}}^{-1}(F_{h_{X}}(x;\hat{r}),\hat{s})$$ where $\hat{r}$ and $\hat{s}$ are estimated score probabilities obtained via loglinear smoothing (see loglin.smooth). The value of $h_X$ and $h_Y$ can either be specified by the user or left unspecified (default) in which case they are automatically calculated. For instance, one can specifies large values of $h_X$ and $h_Y$, so that the $\hat{e}_Y(x)$ tends to the linear equating function (see Theorem 4.5 in Von Davier et al, 2004 for more details).

References

Holland, P. and Thayer, D. (1989). The kernel method of equating score distributions. (Technical Report No 89-84). Princeton, NJ: Educational Testing Service. Holland, P., King, B. and Thayer, D. (1989). The standard error of equating for the kernel method of equating score distributions (Tech. Rep. No. 89-83). Princeton, NJ: Educational Testing Service. Von Davier, A., Holland, P., and Thayer, D. (2004). The Kernel Method of Test Equating. New York, NY: Springer-Verlag.

See Also

loglin.smooth, SEED, PREp

Examples

#Kernel equating under the "EG" design
data(Math20EG)
mod<-ker.eq(scores=Math20EG,kert="gauss",hx=NULL,hy=NULL,degree=c(2,3),design="EG") 

summary(mod)

#Reproducing Table 7.6 in Von Davier et al, (2004)

scores<-0:20
SEEXy<-mod$SEEXy
SEEYx<-mod$SEEYx

Table7.6<-cbind(scores,SEEXy,SEEYx)
Table7.6

#Other nonstandard kernels. Table 10.3 in Von Davier (2011).

mod.logis<-ker.eq(scores=Math20EG,kert="logis",hx=NULL,hy=NULL,degree=c(2,3),design="EG") 
mod.unif<-ker.eq(scores=Math20EG,kert="unif",hx=NULL,hy=NULL,degree=c(2,3),design="EG") 
mod.gauss<-ker.eq(scores=Math20EG,kert="gauss",hx=NULL,hy=NULL,degree=c(2,3),design="EG") 

XtoY<-cbind(mod.logis$eqYx,mod.unif$eqYx,mod.gauss$eqYx)
YtoX<-cbind(mod.logis$eqXy,mod.unif$eqXy,mod.gauss$eqXy)

Table10.3<-cbind(XtoY,YtoX)
Table10.3