gradedResponse {CPTtools} | R Documentation |
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.
gradedResponse(et, linkScale = NULL, obsLevels = NULL)
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 |
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(θ_k) for m=2,…,M, where
Pr(X >= x_m | θ_k) = logit^{-1} -D*Z_m(θ_k)
The conditional probabilities for each child state given the effective thetas for the parent variables is then given by
Pr(X == x_m |θ_k) \frac{∑_{r=1}^m Z_r(θ_k)}{∑_{r=1}^M Z_r(θ_k)}
The K \times M-1 matrix et
is the values of
Z_m(θ_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.
A matrix with one more column than et
giving the conditional
probabilities for each configuration of the parent variables (which
correspond to the rows).
The linkScale
parameter is unused. It is for compatibility
with other link function choices.
Russell Almond
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.
Other Link functions:
gradedResponse
,normalLink
Functions which directly use the link function:
eThetaFrame
, calcDPCTable
,
mapDPC
Earlier version of the graded response link:
calcDSTable
## 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)