Toward a Principled Sampling Theory for Quasi-Orders.

Area of Science:

• Mathematics
• Computer Science
• Educational Measurement

Background:

• Quasi-orders, reflexive and transitive binary relations, are vital in modeling mastery dependencies in educational testing.
• Current data mining methods for quasi-orders are sensitive to their structure, necessitating robust simulation studies.
• Existing methods for generating random quasi-orders struggle with larger item sets, limiting simulation scope.

Purpose of the Study:

• To develop novel techniques for generating representative samples of quasi-orders on finite item sets.
• To enable accurate comparison of data mining methods through unbiased simulation data.
• To address limitations of existing random quasi-order generation methods for larger datasets.

Main Methods:

• A discrete doubly inductive procedure for constructing the set of all quasi-orders.
• Randomization of the inductive procedure to generate representative samples.
• Outer and inner level inductive algorithms for extending and correcting quasi-orders, including bias correction techniques.

Main Results:

• The proposed algorithms generate quasi-order samples that are close to representative, even for up to 50 items.
• These methods operate within acceptable computing times.
• The new algorithms significantly improve upon existing techniques for generating uniform random quasi-orders on larger item sets.

Conclusions:

• The developed techniques provide a principled and efficient approach to generating random quasi-order samples.
• This advancement facilitates more reliable simulation studies in educational data mining and related fields.
• The algorithms overcome scalability issues, enabling analysis with reasonably large item sets.