Abstract questionnaires and FS-decision digraphs

TL;DR

We model a questionnaire as $(N,\mathcal{M})$ with per-question options and optionally a skip-list $\mathcal{S}$ and flag-set $F$. FS-decision trees and FS-decision digraphs encode the complete flow, with digraphs offering a compact representation that preserves all information; a fully reduced digraph is obtained by merging equivalent substructures. The paper provides formal definitions of skip-lists and flag-sets, algorithms to construct FS-decision trees and fully reduced FS-decision digraphs, and methods to generate compatible inputs from intuitive pre-skip-list and pre-flag-set data, along with procedures to enumerate question orderings under a precedence relation. A concrete example demonstrates the workflow and the resulting compact digraphs. Overall, the FS-decision framework enables efficient visualization, analysis, and design of complex questionnaire flows, with potential benefits for survey engineering and information retrieval.

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 $(N,{\cal M})$, where $N$ is a positive integer and ${\cal M}=(m_0,m_1,\ldots,m_{N-1})$ is an $N$-tuple of positive integers, with $m_i$, for $i \in \{0, 1, \ldots, N-1 \}$, 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.

Figures (5)

• Figure 1: The FS-decision tree $T$ (top) for a binary questionnaire ${\mathcal{Q}}$. If we imagine all edges directed downwards, this figure also represents the FS-decision digraph $D_T$ corresponding to $T$. Bottom left: an FS-decision digraph $D'$ for ${\mathcal{Q}}$ that is not fully reduced. Bottom right: the fully reduced FS-decision digraph $D"$ for ${\mathcal{Q}}$ . All edges are directed downwards, out-neighbours are ordered from left to right, and flagged and unflagged vertices are coloured white and black, respectively. Vertices in $D"$ are labelled in the order created by Algorithm \ref{['alg:S-dd']}.
• Figure 2: FS-decision trees for binary questionnaires with at most two questions, and the corresponding fully reduced FS-decision digraphs. All edges are directed downwards, out-neighbours are ordered from left to right, and flagged and unflagged vertices are coloured white and black, respectively.
• Figure 3: FS-decision trees for binary questionnaires with three questions, and the corresponding fully reduced FS-decision digraphs. All edges are directed downwards, out-neighbours are ordered from left to right, and flagged and unflagged vertices are coloured white and black, respectively. Continued in Figure \ref{['fig:level-3b']}.
• Figure 4: FS-decision trees for binary questionnaires with three questions, and the corresponding fully reduced FS-decision digraphs. All edges are directed downwards, out-neighbours are ordered from left to right, and flagged and unflagged vertices are coloured white and black, respectively. Continued from Figure \ref{['fig:level-3']}.
• Figure 5: The FS-decision tree (top) and the fully reduced FS-decision digraph (bottom) for the example in Section \ref{['sec:example']}. All edges are directed downwards, out-neighbours are ordered from left to right, and flagged and unflagged vertices are coloured white and black, respectively. Vertices in the FS-decision digraph are labelled in the order created by Algorithm \ref{['alg:S-dd']}.

Theorems & Definitions (21)

• Lemma 3.1
• Definition 4.1
• Definition 4.2
• Definition 4.3
• Definition 4.4
• Definition 4.5
• Lemma 4.6
• Lemma 4.7
• Definition 4.8
• Lemma 4.9