gradedResponse: A link function based on Samejima's graded response
In ralmond/CPTtools: Tools for Creating Conditional Probability Tables

gradedResponse

R Documentation

A link function based on Samejima's graded response

Description

This function converts a matrix of effective theta values into a conditional probability table by applying Samejima's graded response model to each row of the table.

Usage

gradedResponse(et, linkScale = NULL, obsLevels = NULL)

Arguments

`et`	A matrix of effective theta values. There should be one row in this table for each configuration of the parent variables of the conditional probability table and one column for each state of the child variables except for the last.
`linkScale`	Unused. For compatibility with other link functions.
`obsLevels`	A character vector giving the names of the child variable states. If supplied, it should have length `ncol(et)+1`.

Details

This function takes care of the third step in the algorithm of calcDPCTable. Its input is a matrix of effective theta values (comparable to the last column of the output of eThetaFrame), one column for each of the child variable states (obsLevels) except for the last one. Each row represents a different configuration of the parent variables. The output is the conditional probability table.

Let X be the child variable of the distribution, and assume that it can take on M possible states labeled x_1 through x_M in increasing order. The graded response model defines a set of functions Z_m(\theta_k) for m=2,\ldots,M, where

Pr(X >= x_m | \theta_k) = logit^{-1} -D*Z_m(\theta_k)

The conditional probabilities for each child state given the effective thetas for the parent variables is then given by

Pr(X == x_m |\theta_k) \frac{\sum_{r=1}^m Z_r(\theta_k)}{\sum_{r=1}^M Z_r(\theta_k)}

The K \times M-1 matrix et is the values of Z_m(\theta_k). This function then performs the rest of the generalized partial credit model. This is a generalization of Muraki (1992), because the functions Z_m(\cdot) are not restricted to be the same functional form for all m.

If supplied obsLevels is used for the column names.

Value

A matrix with one more column than et giving the conditional probabilities for each configuration of the parent variables (which correspond to the rows).

Note

The linkScale parameter is unused. It is for compatibility with other link function choices.

Author(s)

Russell Almond

References

Almond, R.G., Mislevy, R.J., Steinberg, L.S., Yan, D. and Williamson, D.M. (2015). Bayesian Networks in Educational Assessment. Springer. Chapter 8.

Muraki, E. (1992). A Generalized Partial Credit Model: Application of an EM Algorithm. Applied Psychological Measurement, 16, 159-176. DOI: 10.1177/014662169201600206

Samejima, F. (1969) Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph No. 17, 34, (No. 4, Part 2).

I also have planned a manuscript that describes these functions in more detail.

Examples

## Set up variables
skill1l <- c("High","Medium","Low") 
correctL <- c("Correct","Incorrect") 
pcreditL <- c("Full","Partial","None")
gradeL <- c("A","B","C","D","E") 

## Get some effective theta values.
et <- effectiveThetas(3)

gradedResponse(matrix(et,ncol=1),NULL,correctL)

gradedResponse(outer(et,c(Full=1,Partial=-1)),NULL,pcreditL)

gradedResponse(outer(et,c(A=2,B=1,C=0,D=-1)),NULL,gradeL)

ralmond/CPTtools documentation built on Dec. 27, 2024, 7:15 a.m.