eThetaFrame: Constructs a data frame showing the effective thetas for each...

In ralmond/CPTtools: Tools for Creating Conditional Probability Tables

Constructs a data frame showing the effective thetas for each parent combination.

Description

This evaluates the combination function but not the link function of of an effective theta distribution. It produces a table of effective thetas one for each configuration of the parent values according to the combination function given in the model argument.

Usage

eThetaFrame(skillLevels, lnAlphas, beta, rule = "Compensatory")

Arguments

skillLevels	A list of character vectors giving names of levels for each of the condition variables.
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.
rule	Function for computing effective theta (see Details).

Details

The DiBello framework for creating for creating conditional probability tables for Bayesian network models using IRT-like parameters 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. The function rule determines the rule for combination.

3. The effective theta is input into a link function (e.g., Samejima's graded-response function) to produce a probability distribution over the states of the outcome variables.

This function applies the first two of those steps and returns a data frame with the original skill levels and the effective thetas.

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.

Value

For a data frame with one column for each parent variable and an additional column for the effective theta values. The number of rows is the product of the number of states in each of the components of the skillLevels argument.

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.

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.

See Also

effectiveThetas,Compensatory, OffsetConjunctive,calcDNTable, calcDSTable, calcDPCTable, expand.grid

Examples

  skill <- c("High","Medium","Low")
  eThetaFrame(list(S1=skill,S2=skill), log(c(S1=1.25,S2=.75)), 0.33,
             "Compensatory") 
  eThetaFrame(list(S1=skill,S2=skill), log(c(S1=1.25,S2=.75)), 0.33,
              "Conjunctive")
  eThetaFrame(list(S1=skill,S2=skill), log(c(S1=1.25,S2=.75)), 0.33,
              "Disjunctive")
  eThetaFrame(list(S1=skill,S2=skill), log(1.0), c(S1=0.25,S2=-0.25),
              "OffsetConjunctive")
  eThetaFrame(list(S1=skill,S2=skill), log(1.0), c(S1=0.25,S2=-0.25),
              "OffsetDisjunctive")

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