partialCredit: A link function based on the generalized partial credit model

In ralmond/CPTtools: Tools for Creating Conditional Probability Tables

A link function based on the generalized partial credit model

Description

This function converts a matrix of effective theta values into a conditional probability table by applying the generalized partial credit model to each row of the table.

Usage

partialCredit(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	An optional 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 generalized partial credit model defines a set of functions Z_m(\theta_k) for m=2,\ldots,M, where

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

The conditional probabilities for each child state is calculated by taking the differences between the curves.

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. The original Samejima (1969) development assumed that all of the functions Z_m(\cdot) had the same linear form a(\theta_k-b_m), with the b_m strictly increasing (note that in CPTtools, the states are ordered from highest to lowest, so that they should be strictly decreasing). This meant that the curves would never cross. The general notation of calcDPCTable does not ensure the curves do not cross, which could result in negative probabilities. This function handles this case by forcing negative probabilities to zero (and adjusting the probabilities for the other state to be properly normalized).

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 development here follows Muraki (1992) rather than Samejima (1969).

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

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

See Also

Other Link functions: gradedResponse,normalLink

Functions which directly use the link function: eThetaFrame, calcDPCTable, mapDPC

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)

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

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

partialCredit(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.