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Bayesian Networks in Educational Assessment

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LCMCMC

A simple Latent class model.

This is a very simple latent class model used for MCMC learning exercises in the tutorial and book.

Purpose

This is a small example that was developed for Chapter 9 of Bayesian Networks in Educational Assessment. As the proficiency variable has two levels, it is a latent class? model. The example is fairly small as its purpose is to illustrate the Markov chain Monte Carlo (MCMC?) algorithm.

This example also illustrates the label swapping? problem. Although the intention is that the value 1 represents mastery? of the skill and 0 represents non-mastery, swapping the meanings of 0 and 1 produces a new model with an identical likelihood. In order to ensure that the algorithm converges to the desired state, either prior constraints must be placed on the parameter, or some additional steps are required after model fitting to untangle the dependencies.

Proficiency Models

The proficiency model consists of a single latent variable, theta, which has two possible states: 0 (non-mastery) and 1 (mastery?). The prior probability distribution for this variable is a mildly-informative Beta?(3,3) distribution, which has a mean of .5.

Proficiency model for two item latent class models.

Task Models

We posit two tasks: a simple dichotomously-scored task, and a partial credit task. Again, as this mostly a Bayes net calculation exercise, the task models are not fully developed.

Evidence Models

There are two evidence models, one for the dichotomously-scored tasks (with a two-level observable?) and one for the partial credit task (with a three-level observable).

Evidence .Evidence model for the dichotomously-scored tasks"
Evidence .Evidence model for 3-level partial credit tasks"

In both cases, the conditional probability tables are given a hyper-Dirichlet? distribution.

Assembly Model

A complete form of the assessment consists of two dichotomously-scored tasks and one partial credit task. Note that there is a second task data description file for creating a second parallel form.

Data Sets

The data can be found at http://pluto.coe.fsu.edu/BNinEA/lcmcmc/ (tarball)

The subfolder initialOut contains the output of a MCMC run.

References

Chapter 9 of Bayesian Networks in Educational Assessment.

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