Statistical matching using fractional imputation 1. IntroductionStatistical matching using fractional imputation 1. Introduction

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Survey sampling is a scientific tool for making inference about the target population. However, we often do not collect all the necessary information in a single survey, due to time and cost constraints. In this case, we wish to exploit, as much as possible, information already available from different data sources from the same target population. Statistical matching, sometimes called data fusion (Baker, Harris and O’Brien 1989) or data combination (Ridder and Moffit 2007), aims to integrate two or more data sets when information available for matching records for individual participants across data sets is incomplete. D’Orazio, Zio and Scanu (2006) and Leulescu and Agafitei (2013) provide comprehensive overviews of the statistical matching techniques in survey sampling.

Statistical matching can be viewed as a missing data problem where a researcher wants to perform a joint analysis of variables that are never jointly observed. Moriarity and Scheuren (2001) provide a theoretical framework for statistical matching under a multivariate normality assumption. Rässler (2002) develops multiple imputation techniques for statistical matching with pre-specified parameter values for non-identifiable parameters. Lahiri and Larsen (2005) address regression analysis with linked data. Ridder and Moffit (2007) provide a rigorous treatment of the assumptions and approaches for statistical matching in the context of econometrics.

Statistical matching aims to construct fully augmented data files to perform statistically valid joint analyses. To simplify the setup, suppose that two surveys, Survey A and Survey B, contain partial information about the population. Suppose that we observe $x$ and $y_1$ from the Survey A sample and observe $x$ and $y_2$ from the Survey B sample. Table 1.1 illustrates a simple data structure for matching. If the Survey B sample (Sample B) is a subset of the Survey A sample (Sample A), then we can apply record linkage techniques (Herzog, Scheuren and Winkler 2007) to obtain values of $y_1$ for the survey B sample. However, in many cases, such perfect matching is not possible (for instance, because the samples may contain non-overlapping subsets), and we may rely on a probabilistic way of identifying the “statistical twins” from the other sample. That is, we want to create $y_1$ for each element in sample B by finding the nearest neighbor from Sample A. Nearest neighbor imputation has been discussed by many authors, including Chen and Shao (2001) and Beaumont and Bocci (2009), in the context of missing survey items.

Finding the nearest neighbor is often based on “how close” they are in terms of $x’\text{s}$ only. Thus, in many cases, statistical matching is based on the assumption that $y_1$ and $y_2$ are independent, conditional on $x.$ That is,

$$y_1\perp y_2|x\mathrm{.}(1.1)$$

Assumption (1.1) is often referred to as the conditional independence (CI) assumption and is heavily used in practice.

In this paper, we consider an alternative approach that does not rely on the CI assumption. After we discuss the assumptions in Section 2, we present the proposed methods in Section 3. Furthermore, we consider two extensions, one to split questionnaire designs (in Section 4) and the other to measurement error models (in Section 5). Results from two simulation studies are presented in Section 6. Section 7 concludes the paper.

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