Using Bayesian Networks to Predict Test Scores

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Using Bayesian Networks to Predict Test Scores

• by Zach Pardos
• Neil Heffernan, Advisor

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Introduction Overview

• ASSISTment tutoring system
• The Task
• Bayesian networks
• Platform selection

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ASSISTment Tutoring System

• Online tutoring system developed at WPI
• - Assess student knowledge/learning
• Assists and prepares students for the MCAS
• 2nd year of operation
• Participation includes over
• 2,000 students
• With 20 teachers/classes
• At 6 schools

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ASSISTment Tutoring System

• Students attempt to answer top level questions
  based on previous MCAS test questions
• If the student answers incorrectly or asks for a
  hint they are given supporting questions,
  called scaffolds, or hint text messages
• All answers and actions are logged on the server

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The Task

• To use Bayesian networks to assess students
  knowledge levels in the ASSISTment system and
  predict their performance on the MCAS test.
• Research topic Compare predictive performance of
  fine-grain vs. coarse-grain skill models.

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Bayesian Networks

• "The essence of the Bayesian approach is to
  provide a mathematical rule explaining how you
  should change your existing beliefs in the light
  of new evidence. In other words, it allows
  scientists to combine new data with their
  existing knowledge or expertise.
• - The Economist (9/30/00)

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Bayesian Networks

• New data
• 2,000 students answering questions online
• MCAS test results
• Existing knowledge or expertise
• Various grain skill models
• Prof. Neil Heffernan
• Bayes Rule
• Where R is a random variable with value r
  and evidence e

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Platform Selection

• Bayesian network software choices
• GeNIe
• MSBNx
• BayesiaLab
• Netica
• MATLAB with BNT (Bayes Net Toolkit)
• Java Bayes

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Platform selection

• Choice MATLAB with BNT
• Pros
• Provides wide selection of inference engines
• MATLABs robust programming environment
• Automation
• Runs on GNU/Linux
• Existing Perl interface for the many scripts that
  will perform data mining tasks.
• Cons
• Little Slow

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Project Overview

• The datasets
• Skill models
• Parameters
• Implementation
• Results

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The Datasets

• Student online response data
• 600 students from 2004-2005
• Student selection criteria
• Completed at least 100 items online
• Completed the 2005 MCAS test
• 2,568 question items
• Student state MCAS test scores for 05
• Used for calculating prediction accuracy
• No test data used for training/parameter learning

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Skill Models

• Skill models describe the skills which are
  related to the online and MCAS questions.
• Skill models used
• MCAS1
• MCAS5
• MCAS39
• WPI106

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Skill Models

• Skill models used for the MCAS test consisting of
  29 multiple choice questions
• MCAS1
• MCAS5

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Skill Models

• MCAS39
• WPI106
• The MCAS1 is a two layer network with skill nodes
  mapped to question nodes. The other 3 networks
  have a third, intermediary layer of AND nodes.
  This allows all question nodes to have the same
  number of parameters (slip/guess). The AND
  nodes also reflect the notion that a student must
  know all tagged skills to answer the item
  correct.

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Skill Models
Transfer table for skill models
WPI-106 WPI-39 WPI-5 WPI-1
Equation-concept setting-up-and-solving-equations Patterns-Relations-Algebra The skill of math
Plot Graph modeling-covariation Patterns-Relations-Algebra The skill of math
Slope understanding-line-slope-concept Patterns-Relations-Algebra The skill of math
Similar Triangles understanding-and-applying-congruence-and-similarity Geometry The skill of math
Perimeter Circumference Area using-measurement-formulas-and-techniques Perimeter The skill of math
Equation-Solving
Inequality-solving
X-Y-Graph
Congruence
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Parameters

• Parameters were set as a best guess starting
  point.
• Test model guess parameter is 0.25 because
  questions are multiple choice (out of four)

Original Parameters Online Model Test Model
Skills 0.50 Imported
Guess 0.10 0.25
Slip 0.05 0.05
Learned Parameters Online Model
Skills 0.44
Guess 0.30
Slip 0.38

• Preliminary learning of parameters using EM
• on the MCAS1 network indicates a guess of
• 0.30, slip of 0.38 and prior of 0.44 on the
  skills.
• These numbers were calculated recently and
• are not used in our prediction results thus far.

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Implementation

• The main routine bn_eval() takes in
• Name of skill model
• StudentID
• BNT object of the skill model bayes net
• bn_eval() outputs
• Status messages
• Predicted score/Actual score/Accuracy
• Logs prediction and skill assessment data

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Implementation

• The evaluation is a 2 stage process
• Stage 1
• Bayes skill model for the online data is loaded
• Students online results are compiled and
  sequenced for the network
• Student is given credit for all scaffold
  questions relating to a top level item answered
  correctly
• Results are entered into the network as evidence
• Marginals on the skill nodes are calculated using
  liklihood_weighting approximate inference .

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Implementation

• Stage 2 of evaluation
• Bayes skill model for the MCAS test is loaded
• Skill marginals calculated from stage 1 are
  entered into the test model as soft evidence
• Marginals on the question nodes are calculated
  using jtree (join-tree) exact inference.
• Test score points are summed by multiplying each
  marginal by 1 and then taking the ceiling of the
  total score.
• Predicted test score is compared to actual
  student test score.

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Implementation

• Example student run using MCAS1 model

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Implementation

• Assessed skill marginals using MCAS1

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Implementation

• Example student run using MCAS5 model

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Implementation

• Assessed skill marginals using MCAS5

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Implementation

• Example student run using MCAS39 model

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Implementation

• Assessed skill marginals using MCAS39

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Implementation

• Example student run using WPI106 model

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Implementation

• Assessed skill marginals using WPI106

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Results

• Model performance/accuracy results
• MAD is Mean Average Difference. The test is out
  of 29 points so a MAD score of 4.5 indicates that
  the model on average predicts a score that is 4.5
  points from the actual score.

MODEL MAD (RAW) ERROR
WPI-39 4.500 15.00
WPI-106 4.970 16.57
WPI-5 5.295 17.65
WPI-1 7.700 25.67
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Future Work

• Reduce runtime
• Optimize the number of samples used with
  liklihood_weighting inference for each model.
• Increase accuracy
• Learn full parameters in all models
• Use analysis to improve skill model tagging
• Experiment with alternative models
• Combine skill models into a hierarchy
• Introduce time as a variable (DBNs)

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References

• A copy of this presentation as well as our
  initial paper submitted to ITS2006 entitled
  Using Fine-Grained Skill Models to Fit Student
  Performance with Bayesian Networks can be found
  online at
• http//users.wpi.edu/zpardos/bayes.html
• Thanks to the WPI-CS department, Neil Heffernan,
  contributors at CMU and the ASSISTment
  developers.