Abstract questionnaires and FS-decision digraphs

Jiaye Chen1, Suzan Kadri1, Mateja Šajna1, Ioana Schiopu-Kratina2
1Department of Mathematics and Statistics, University of Ottawa, 150 Louis-Pasteur Private, Ottawa, ON, K1N 6N5, Canada
2University of Ottawa, 150 Louis-Pasteur Private, Ottawa, ON, K1N 6N5, Canada

Abstract

A questionnaire is a sequence of multiple choice questions aiming to collect data on a population. We define an abstract questionnaire as an ordered pair \((N,\mathcal{M})\), where \(N\) is a positive integer and \(\mathcal{M}=(m_0,m_1,\ldots,m_{N-1})\) is an \(N\)-tuple of positive integers, with \(m_i\), for \(i \in \mathbb{Z}_N\), as the number of possible answers to question \(i\). An abstract questionnaire may be endowed with a skip-list (which tells us which questions to skip based on the sequence of answers to the earlier questions) and a flag-set (which tells us which sequences of answers are of special interest). An FS-decision tree is a decision tree of an abstract questionnaire that also incorporates the information contained in the skip-list and flag-set. The main objective of this paper is to represent the abstract questionnaire using a directed graph, which we call an FS-decision digraph, that contains the full information of an FS-decision tree, but is in general much more concise. We present an algorithm for constructing a fully reduced FS-decision digraph, and develop the theory that supports it. In addition, we show how to generate all possible orderings of the questions in an abstract questionnaire that respect a given precedence relation.

Keywords: survey, questionnaire, abstract questionnaire, decision tree, FS-decision tree, FS-decision digraph