On Application of the Partition Distance Concept to a Comparative Analysis of Psychological or Sociological Tests

Abstract

We discuss two distance concepts between q-ary n-sequences, 2 ≤ q < n, called partition distances. This distances are metrics in the space of all partitions of a finite n-set. For the metrics, we study codes called q-partition codes and present a construction of these codes based on the first order Reed–Muller codes. A random coding bound is obtained. We also work out an application of q-partition codes to the statistical analysis of psychological or medical tests using questionnaires.

Keywords:

• Clusters analysis
• Medical tests
• Partition codes
• Partitions distance
• Partitions of sets
• Psychological tests

Mathematics Subject Classification:

• Primary 62P15
• Secondary 05A18, 94B60

Notes

¹Code X _m is a subcode of the first order Reed–Muller code C _m [4], where C _m is an q-ary linear (n,k)-code, k = m + 1, of length n = q ^m , size |C _m |=q ^m+1 and Hamming distance d = (q − 1) · q ^m−1.

²The conventional “parametric” (n,q)-questionnaire contains n questions and an individual (patient) gives one of q possible answers to each question. Each answer is estimated by the corresponding number 0, 1,…, q − 1 and the patient gets a tentative diagnosis identified by the sum (score) of n estimates. One has to take into account that this conventional approach cannot be applied to nonparametric questionnaires.