calcDSTable: Creates the probability table for DiBello-Samejima...

In ralmond/CPTtools: Tools for Creating Conditional Probability Tables

Creates the probability table for DiBello–Samejima distribution

Description

The calcDSTable function takes a description of input and output variables for a Bayesian network distribution and a collection of IRT-like parameter (discrimination, difficulty) and calculates a conditional probability table using the DiBello-Samejima distribution (see Details). The calcDSFrame function returns the value as a data frame with labels for the parent states.

Usage

calcDSTable(skillLevels, obsLevels, lnAlphas, beta, dinc = 0,
            rule = "Compensatory")
calcDSFrame(skillLevels, obsLevels, lnAlphas, beta, dinc = 0,
            rule = "Compensatory")

Arguments

skillLevels	A list of character vectors giving names of levels for each of the condition variables.
obsLevels	A character vector giving names of levels for the output variables from highest to lowest. As a special case, can also be a vector of integers.
lnAlphas	A vector of log slope parameters. Its length should be either 1 or the length of skillLevels, depending on the choice of rule.
beta	A vector of difficulty (-intercept) parameters. Its length should be either 1 or the length of skillLevels, depending on the choice of rule.
dinc	Vector of difficulty increment parameters (see Details).
rule	Function for computing effective theta (see Details).

Details

The DiBello–Samejima model is a mechanism for creating conditional probability tables for Bayesian network models using IRT-like parameters. The basic procedure unfolds in three steps.

1. Each level of each input variable is assigned an “effective theta” value — a normal value to be used in calculations.

2. For each possible skill profile (combination of states of the parent variables) the effective thetas are combined using a combination function. This produces an “effective theta” for that skill profile.

3. The effective theta is input into Samejima's graded-response model to produce a probability distribution over the states of the outcome variables.

The parent (conditioning) variables are described by the skillLevels argument which should provide for each parent variable in order the names of the states ranked from highest to lowest value. The original method (Almond et al., 2001) used equally spaced points on the interval [-1,1] for the effective thetas of the parent variables. The current implementation uses the function effectiveThetas to calculate equally spaced points on the normal curve.

The combination of the individual effective theta values into a joint value for effective theta is done by the function reference by rule. This should be a function of three arguments: theta — the vector of effective theta values for each parent, alphas — the vector of discrimination parameters, and beta — a scalar value giving the difficulty. The initial distribution supplies five functions appropriate for use with calcDSTable: Compensatory, Conjunctive, and Disjunctive, OffsetConjunctive, and OffsetDisjunctive. The last two have a slightly different parameterization: alpha is assumed to be a scalar and betas parameter is vector valued. Note that the discrimination and difficulty parameters are built into the structure function and not the IRT curve.

The Samejima graded response link function describes a series of curves:

P_m(\theta) = Pr(X >= x-m | \theta) = logit^{-1} (\theta - d_m)

for m>1, where D=1.7 (a scale factor to make the logistic curve match more closely with the probit curve). The probability for any given category is then the difference between two adjacent logistic curves. Note that because a difficulty parameter was included in the structure function, we have the further constraint that \sum d_m =0.

To remove the parameter restriction we work with the difference between the parameters: d_m-d_{m-1}. The value of d_2 is set at -sum(dinc)/2 to center the d values. Thus the dinc parameter (which is required only if length(obsLevels)>2) should be of length length(obsLevels)-2. The first value is the difference between the d values for the two highest states, and so forth.

Normally obslevel should be a character vector giving state names. However, in the special case of state names which are integer values, R will “helpfully” convert these to legal variable names by prepending a letter. This causes other functions which rely on the names() of the result being the state names to break. As a special case, if the value of obsLevel is of type numeric, then calcDSFrame() will make sure that the correct values are preserved.

Value

For calcDSTable, a matrix whose rows correspond configurations of the parent variable states (skillLevels) and whose columns correspond to obsLevels. Each row of the table is a probability distribution, so the whole matrix is a conditional probability table. The order of the parent rows is the same as is produced by applying expand.grid to skillLevels.

For calcDSFrame a CPF, a data frame with additional columns corresponding to the entries in skillLevels giving the parent value for each row.

Note

This distribution class is not suitable for modeling relationship among proficiency variable, primarily because the normal mapping used in the effective theta calculation and the Samejima graded response models are not inverses. For those model, the function calcDNTable, which uses a probit link function, is recommended instead.

This function has largely been superceeded by calls to calcDPCTable with gradedResponse as the link function.

Author(s)

Russell Almond

References

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

Almond, R.G., DiBello, L., Jenkins, F., Mislevy, R.J., Senturk, D., Steinberg, L.S. and Yan, D. (2001) Models for Conditional Probability Tables in Educational Assessment. Artificial Intelligence and Statistics 2001 Jaakkola and Richardson (eds)., Morgan Kaufmann, 137–143.

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

See Also

effectiveThetas,Compensatory, OffsetConjunctive,eThetaFrame, calcDNTable, calcDSllike, calcDPCTable, expand.grid

Examples

## Set up variables
skill1l <- c("High","Medium","Low") 
skill2l <- c("High","Medium","Low","LowerYet") 
correctL <- c("Correct","Incorrect") 
gradeL <- c("A","B","C","D","E") 

cptCorrect <- calcDSTable(list(S1=skill1l,S2=skill2l),correctL,
                          log(c(S1=1,S2=.75)),1.0,rule="Conjunctive")

cpfCorrect <- calcDSFrame(list(S1=skill1l,S2=skill2l),correctL,
                          log(c(S1=1,S2=.75)),1.0,rule="Conjunctive")


cptGraded <- calcDSTable(list(S1=skill1l),gradeL, 0.0, 0.0, dinc=c(.3,.4,.3))

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