Location-Scale Matching for Approximate Quasi-Order Sampling

Quasi-orders are reflexive and transitive binary relations and have many applications. Examples are the dependencies of mastery among the problems of a psychological test, or methods such as item tree or Boolean analysis that mine for quasi-orders in empirical data. Data mining techniques are typically tested based on simulation studies with unbiased samples of randomly generated quasi-orders. In this paper, we develop techniques for the approximately representative sampling of quasi-orders. Polynomial regression curves are fitted for the mean and standard deviation of quasi-order size as a function of item number. The resulting regression graphs are seen to be quadratic and linear functions, respectively. The extrapolated values for the mean and standard deviation are used to propose two quasi-order sampling techniques. The discrete method matches these location and scale measures with a transformed discrete distribution directly obtained from the sample. The continuous method uses the normal density function with matched expectation and variance. The quasi-orders are constructed according to the biased randomized doubly inductive construction, however they are resampled to become approximately representative following the matched discrete and continuous distributions. In simulations, we investigate the usefulness of these methods. The location-scale matching approach can cope with very large item sets. Close to representative samples of random quasi-orders are constructed for item numbers up to n = 400.