The Design and Optimality of Survey Counts: A Unified Framework Via the Fisher Information Maximizer

Abstract

Grouped and right-censored (GRC) counts have been used in a wide range of attitudinal and behavioural surveys yet they cannot be readily analyzed or assessed by conventional statistical models. This study develops a unified regression framework for the design and optimality of GRC counts in surveys. To process infinitely many grouping schemes for the optimum design, we propose a new two-stage algorithm, the Fisher Information Maximizer (FIM), which utilizes estimates from generalized linear models to find a global optimal grouping scheme among all possible $N$ -group schemes. After we define, decompose, and calculate different types of regressor-specific design errors, our analyses from both simulation and empirical examples suggest that: 1) the optimum design of GRC counts is able to reduce the grouping error to zero, 2) the performance of modified Poisson estimators using GRC counts can be comparable to that of Poisson regression, and 3) the optimum design is usually able to achieve the same estimation efficiency with a smaller sample size.

Keywords: Fisher Information Maximizer; grouped and right-censored count; modified Poisson estimator; optimum experimental design; survey methodology.

Figure 1.

An illustration of the synthesis stage of the FIM: synthesizing ${\mathbb{F}}_X^{\mathcal{G}}$ with $\mathcal{G}=\{0,2,4,\mathrm{\infty }\}$ .

Figure 2.

An illustration of the validation stage of the FIM: validating maximizers with $M=N=3$ .

Figure 3.

Boundaries of optimal grouping schemes (solid lines) and relative efficiencies of modified Poisson estimators (the axis of dashed lines starts from 60%) with combinations of $\lambda$ (evenly spaced from $0.1$ to $20$ ) and $N$ (3, 5, 7, and 9).