Abstract
Dependencies between educational test items can be represented as quasi-orders on the item set of a knowledge domain and used for an efficient adaptive assessment of knowledge. One approach to uncovering such dependencies is by exploratory algorithms of item tree analysis (ITA). There are several methods of ITA available. The basic tool to compare such algorithms concerning their quality are large-scale simulation studies that are crucially set up on a large collection of quasi-orders. A serious problem is that all known ITA algorithms are sensitive to the structure of the underlying quasi-order. Thus, it is crucial to base any simulation study that tries to compare the algorithms upon samples of quasi-orders that are representative, meaning each quasi-order is included in a sample with the same probability. Up to now, no method to create representative quasi-orders on larger item sets is known. Non-optimal algorithms for quasi-order generation were used in previous studies, which caused misinterpretations and erroneous conclusions. In this paper, we present a method for creating representative random samples of quasi-orders. The basic idea is to consider random extensions of quasi-orders from lower to higher dimension and to discard extensions that do not satisfy the transitivity property.
Highlights
Orders play an important role in various formal theories of behavioral, social, economic, or computer sciences
Other areas where order relations play an important role are in computer science, for example in database systems research (e.g., Rob and Coronel, 2009)
Large-scale simulation studies are required to examine whether such algorithms can handle different quasi-order structures underlying the data and various ranges of simulated response error probabilities
Summary
Orders play an important role in various formal theories of behavioral, social, economic, or computer sciences. The transitive closure of the binary relation corresponding to (rij) is the resulting random quasi-order This random process—existent in two variants, absolute and averaged; for details, see Sargin and Ünlü (2009)—is already an improvement of an older procedure (Schrepp, 1999) that drew x based on a uniform distribution on the interval 0–0.4 and that resulted in non-representative samples consisting of overly represented large quasi-orders. In a final step of the procedure, PQO(n + 1) is reduced to a required or reasonable size, for example by keeping only m randomly selected elements from PQO(n + 1) based on simple random sampling This step is necessary to have the number of elements of the PQO(n)’s reasonably limited, if this process is repeated several times; for instance, when creating a sample consisting of 1000 quasi-orders on 10 items starting with the set of all four quasi-orders on two items.
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