SNSequate (version 1.0)

loglin.smooth: Pre-smoothing using log-linear models.

Description

This function fits log-linear models to score data and provides estimates of the (vector of) score probabilities as well as the C matrix decomposition of their covariance matrix, according to the specified equating design (see Details).

Usage

loglin.smooth(scores, degree, design, 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 vector containing the raw sample frequencies coming from one group taking the test. If the "SG" design is specified, a matrix containing the (joint) bivariate sample frequencies for $X$ (raws) and $Y$

degree

Either a number or vector indicating the number of power moments to be fitted to the marginal distributions, or the number or cross moments to be fitted to the joint distributions, respectively. For the "EG" design it will be a number (see Detai

design

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

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" design

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

• sp.est
  The estimated score probabilities
• C
  The C matrix which is so that $\Sigma=CC^t$

Details

This function fits loglinear models as described in Holland and Thayer (1987), and Von Davier et al. (2004). The following general equation can be used to represent the models according to the different designs used, in which the vector $o$ (or matrix) of (marginal or bivariate) score probabilities satisfies the log-linear model: $$\log(o_{gh})=\alpha_m+Z_m(z_g)+W_m(w_h)+ZW_m(z_g,w_h)$$ where $Z_m(z_g)=\sum_{i=1}^{T_{Zm}}\beta_{zmi}(z_g)^i$, $W_m(w_h)=\sum_{i=1}^{T_{Wm}}\beta_{Wmi}(w_h)^i$, and, $ZW_m(z_g,w_h)=\sum_{i=1}^{I_{Zm}}\sum_{i'=1}^{I_{Wm}}\beta_{ZWmii'}(z_g)^i(w_h)^{i'}$. The symbols will vary according to the different equating designs specified. Possible values are: $o=p_{(12)}, p_{(21)}, p, q$; $Z=X, Y$; $W=Y, A$; $z=x, y$; $w=y, a$; $m=(12), (21), P, Q$; $g=j, k$; $h=l, k$. Particular cases of this general equation for each of the equating designs can be found in Von Davier et al (2004) (e.g., Equations (7.1) and (7.2) for the "EG" design, Equation (8.1) for the "SG" design, Equations (9,1) and (9.2) for the "CB" design).

References

Holland, P. and Thayer, D. (1987). Notes on the use of loglinear models for fitting discrete probability distributions. Research Report 87-31, 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

glm, ker.eq

Examples

#Table 7.4 from Von Davier et al. (2004)
data(Math20EG)
rj<-loglin.smooth(scores=Math20EG[,1],degree=2,design="EG")$sp.est
sk<-loglin.smooth(scores=Math20EG[,2],degree=3,design="EG")$sp.est
score<-0:20
Table7.4<-cbind(score,rj,sk)
Table7.4