# Analysis of Linked Files: A Missing Data Perspective

Gauri<EMAIL_ADDRESS>[ Roee
<EMAIL_ADDRESS>[ Gauri Kamat is PhD candidate in
Biostatistics, Brown Universitypresep= ]e1.  Roee Gutman is Associate
Professor of Biostatistics, Brown Universitypresep= ]e2.

# Supplementary Material

Gauri<EMAIL_ADDRESS>[ Roee
<EMAIL_ADDRESS>[ Gauri Kamat is PhD candidate in
Biostatistics, Brown Universitypresep= ]e1.  Roee Gutman is Associate
Professor of Biostatistics, Brown Universitypresep= ]e2.

###### Abstract

In many applications, researchers seek to identify overlapping entities across
multiple data files. Record linkage algorithms facilitate this task, in the
absence of unique identifiers. As these algorithms rely on semi-identifying
information, they may miss records that represent the same entity, or
incorrectly link records that do not represent the same entity. Analysis of
linked files commonly ignores such linkage errors, resulting in biased, or
overly precise estimates of the associations of interest. We view record
linkage as a missing data problem, and delineate the linkage mechanisms that
underpin analysis methods with linked files. Following the missing data
literature, we group these methods under three categories: likelihood and
Bayesian methods, imputation methods, and weighting methods. We summarize the
assumptions and limitations of the methods, and evaluate their performance in
a wide range of simulation scenarios.

record linkage,

missing data,

Bayesian,

imputation,

likelihood,

###### keywords:

and

##  1 Introduction

In many healthcare and social science applications, information about entities
is dispersed across multiple data files. To investigate the associations of
interest, researchers seek to link records representing the same entity across
these files. When unique identifiers, like social security numbers, are
available, linking records is a relatively straightforward task. However,
privacy regulations often limit access to such identifiers. In these
situations, researchers rely on semi-identifying linking variables, such as
names and addresses. Record linkage (data matching or entity resolution) is a
statistical technique that identifies records representing the same entity
across multiple files, in the absence of unique identifiers. Record linkage
techniques have found applications in epidemiology and healthcare (Stern,,
1986; Brenner et al.,, 1997; Golden and Mirel,, 2021), official statistics
(Winkler and Thibaudeau,, 1991; Dasylva et al.,, 2014), economics (Meyer and
Mittag,, 2019, 2021), and studies of human rights violations (Sadinle,, 2014).

Record linkage methods can be broadly categorized as being deterministic or
probabilistic. Deterministic linkage methods are based on non-random
similarity functions defined on the linking variables. These methods can link
records with high accuracy, when the linking variables are informative and
accurately reported (Gomatam et al.,, 2002). However, when these variables are
subject to variations in spelling, data entry errors, or incompleteness,
deterministic methods may perform inadequately (Campbell et al.,, 2008; Newman
et al.,, 2009; Zhu et al.,, 2015). Probabilistic linkage is based on the
likelihood that a given pair of records represents the same entity. This
likelihood can be computed in several ways, such as using mixture models
(Fellegi and Sunter,, 1969), microclustering models (Zanella et al.,, 2016),
or classification algorithms (Cochinwala et al.,, 2001).

A widely-used approach to probabilistic record linkage is the mixture model
proposed by Fellegi and Sunter, (1969). For computational convenience, this
model relies on several tacit independence assumptions. In addition,
identification of linked records is based on subjective thresholds (Winkler,,
2014; Binette and Steorts,, 2022). These features have often been criticized
(Tancredi and Liseo,, 2011), motivating the development of parametric Bayesian
approaches for probabilistic linkage (Tancredi and Liseo,, 2011; Gutman et
al.,, 2013; Steorts et al.,, 2016; Sadinle,, 2017).

Probabilistic record linkage has also been viewed as a prediction problem (Lee
et al.,, 2022). In this approach, classification algorithms, like decision
trees, are used to determine if a pair of records represents the same entity
(Cochinwala et al.,, 2001; Bilenko and Mooney,, 2003). Such algorithms rely on
the availability of training data, i.e., a known set of links, to estimate
parameters in the classification algorithm. These estimates are then used to
predict the unknown linkage status of record pairs. In the absence of training
data, unsupervised classification algorithms, such as $k$-means clustering,
can be employed to link records (Liu et al.,, 2003; Gu and Baxter,, 2006;
Christen,, 2007; Christen, 2008a, ; Christen, 2008b, ). The performance of
both supervised and unsupervised classification algorithms can depend on the
distance functions used to compare the records (Asher et al.,, 2020). When
reliable training data is available, supervised algorithms can outperform
unsupervised algorithms in terms of linkage accuracy (Elfeky et al.,, 2003;
Liu et al.,, 2003).

In this article, we exclusively focus on probabilistic record linkage when no
training data is available. We also do not focus on predictive classification
approaches to probabilistic linking, which have been comprehensively reviewed
in Binette and Steorts, (2022).

When the linking variables contain limited information, probabilistic linking
algorithms may miss links between records representing the same entity, or
falsely link records representing different entities. When analyzing linked
files, such errors can lead to substantial biases and a loss in efficiency
(Neter et al.,, 1965). In this article, we review statistical methods to
propagate the uncertainty stemming from linkage errors into the post-linkage
analysis. We do so by formulating record linkage within a missing data
framework (Rubin,, 1976), and formalizing linkage mechanisms that underlie
inferential methods with linked files. In addition, we characterize and
compare the performance of selected linkage and inference methods under a wide
variety of scenarios.

This article is organized as follows. The remainder of Section 1 provides
notation and background on probabilistic record linkage. Sections 2 and 3
identify and discuss inferential methods with linked files. Section 4
describes simulations to compare the performance of selected methods from
Sections 2 and 3. Section 5 concludes with a discussion.

###  1.1 Notation and background

Let $\mathbf{A}$ and $\mathbf{B}$ represent two files comprising $n_{A}$ and
$n_{B}$ records, respectively. Unless mentioned otherwise, assume that there
are no duplicates within each file. Let $n_{AB}$ be the number of records
common to both the files. Without loss of generality, assume that $n_{AB}\leq
n_{A}\leq n_{B}$. Further, let $\mathbf{L}_{Ai}=(L_{Ai1},\dots,L_{AiK})$
represent $K$ linking variables for record $i\in\mathbf{A}$, where
$i=1,\dots,n_{A}$. Similarly, let $\mathbf{L}_{Bj}=(L_{Bj1},\dots,L_{BjK})$
represent the same $K$ linking variables for record $j\in\mathbf{B}$, where
$j=1,\dots,n_{B}$. Let $\mathbf{Y}_{Ai}=(Y_{Ai1},\dots,Y_{AiP_{A}})$ denote
$P_{A}$ variables that are exclusive to record $i\in\mathbf{A}$, and let
$\mathbf{X}_{Bj}=(X_{Bj1},\dots,X_{BjP_{B}})$ denote $P_{B}$ variables that
are exclusive to record $j\in\mathbf{B}$.

Probabilistic record linkage seeks to identify the $n_{AB}$ overlapping
records, using $\mathbf{L}_{A}=\\{\mathbf{L}_{Ai}\\}$ and
$\mathbf{L}_{B}=\\{\mathbf{L}_{Bj}\\}$, and potentially
$\mathbf{Y}_{A}=\\{\mathbf{Y}_{Ai}\\}$ and
$\mathbf{X}_{B}=\\{\mathbf{X}_{Bj}\\}$. Alternatively, each pair in the set
$\mathbf{A}\times\mathbf{B}=\\{(i,j);i\in\mathbf{A}$, $j\in\mathbf{B}\\}$ is
classified as a pair that represents the same entity (link), or a pair that
does not represent the same entity (non-link). Formally, the parameter of
interest is the linkage structure between files $\mathbf{A}$ and $\mathbf{B}$.

####  1.1.1 The linkage structure

The linkage structure $\boldsymbol{\Delta}=[\Delta_{ij}]$ is a binary
configuration matrix (Green and Mardia,, 2006) of order $n_{A}\times n_{B}$,
where $\Delta_{ij}=1$, if records $i\in\mathbf{A}$ and $j\in\mathbf{B}$ are
linked, and $\Delta_{ij}=0$, otherwise (Fortini et al.,, 2001; Larsen,, 2002,
2005; Tancredi and Liseo,, 2011). The elements of $\boldsymbol{\Delta}$ can
also be stacked to form an $n_{A}n_{B}$-dimensional vector of binary indicator
variables (Fienberg and Manrique-Vallier,, 2009; Hof and Zwinderman,, 2014;
Hof et al.,, 2017).

The linkage is said to be bipartite, when (i) there are no duplicates within
each of the two files; and (ii) a record in file $\mathbf{A}$ can be linked to
at most one record in file $\mathbf{B}$, and vice versa. Formally,
$\sum_{i=1}^{n_{A}}\Delta_{ij}\leq 1\text{ }\forall j=1,\dots,n_{B}$, and
$\sum_{j=1}^{n_{B}}\Delta_{ij}\leq 1\text{ }\forall i=1,\dots,n_{A}$. Under
bipartite linkage, a computationally efficient and compact linkage structure
is the matching labeling (Sadinle,, 2017),
$\mathbf{Z}=(Z_{1},\dots,Z_{n_{A}})$, where

$\displaystyle Z_{i}=\begin{cases}j,&\text{if records }i\in\mathbf{A}\text{
and }j\in\mathbf{B}\text{ are linked;}\\\ n_{B}+i,&\text{ if no record in file
}\mathbf{B}\text{ can be linked to record }i\in\mathbf{A}.\end{cases}$ (1)

A one-to-one map between $\mathbf{Z}$ and $\boldsymbol{\Delta}$ exists, in
that $Z_{i}=j$ whenever $\Delta_{ij}=1$, and $Z_{i}=n_{B}+i$ whenever
$\Delta_{ij}=0$ $\forall j=1,\dots,n_{B}$. If we link all $n_{A}$ records from
file $\mathbf{A}$, and leave $n_{B}-n_{A}$ records from file $\mathbf{B}$
unlinked, then $\mathbf{Z}$ represents a matching permutation (Gutman et al.,,
2013). In this context, $\mathbf{Z}$ indicates the order in which a subset of
$n_{A}$ records from file $\mathbf{B}$ are linked to the records in file
$\mathbf{A}$.

An alternate form of the linkage structure is based on a group of latent
population entities. Each latent entity represents the “true” population unit,
whose characteristics are distorted to generate a record in one of the files.
Formally, let $\omega_{Ai}$ be an arbitrary, but fixed, integer representing
the latent entity for record $i\in\mathbf{A}$. Similarly, let $\omega_{Bj}$ be
an integer representing the latent entity for record $j\in\mathbf{B}$. If
$\omega_{Ai}=\omega_{Bj}$, the pair $(i\in\mathbf{A},j\in\mathbf{B})$ is said
to be a link. The set
$\boldsymbol{\Omega}=\\{\omega_{Ai},i\in\mathbf{A}\\}\cup\\{\omega_{Bj},j\in\mathbf{B}\\}$
represents the complete linkage structure between files $\mathbf{A}$ and
$\mathbf{B}$. This construct enables identification of duplicates within the
same file, as well as multi-file linkage. It has been adopted by Steorts,
(2015), Steorts et al., (2016), and Tancredi et al., (2020), among others. In
the case of two files with no duplicates within a file, a one-to-one map
between $\boldsymbol{\Delta}$ and $\boldsymbol{\Omega}$ exists, in that
$\Delta_{ij}=1$ whenever $\omega_{Ai}=\omega_{Bj}$, and $\Delta_{ij}=0$
whenever $\omega_{Ai}\neq\omega_{Bj}$.

For ease of exposition, we use the matrix $\boldsymbol{\Delta}$ as a
representation of the linkage structure. Given the direct correspondence
between the various representations, adapting to other forms can be
straightforward.

####  1.1.2 The Fellegi-Sunter model

A commonly used model to estimate the linkage structure is that of Fellegi and
Sunter, (1969) (FS), which builds on the work of Newcombe et al., (1959). This
model considers the set of record pairs
$\\{(i,j):i\in\mathbf{A},j\in\mathbf{B}\\}$ to be the union of the set of
links, $\mathbf{M}=\\{(i,j):i\in\mathbf{A},j\in\mathbf{B},\Delta_{ij}=1\\}$,
and the set of non-links,
$\mathbf{U}=\\{(i,j):i\in\mathbf{A},j\in\mathbf{B},\Delta_{ij}=0\\}$. The sets
$\mathbf{M}$ and $\mathbf{U}$ are identified using the linking variables
$\mathbf{L}_{A}$ and $\mathbf{L}_{B}$.

To assess the similarity between records $i\in\mathbf{A}$ and
$j\in\mathbf{B}$, comparison vectors
$\gamma_{ij}=(\gamma_{ij1},\dots,\gamma_{ijK})$ are constructed. The
similarity on the $k^{th}$ linking variable is defined using $l_{k}$ levels of
ordinal agreement, where the lowest level indicates the strongest agreement,
and $l_{k}$ indicates the strongest disagreement (Winkler,, 1990). Formally,
$\gamma_{ijk}$ is represented by a set of indicator variables
$\\{\gamma_{ijkl}:k=1,\dots,K,l=1,\dots,l_{k}\\}$, where

$\displaystyle\gamma_{ijkl}=\begin{cases}1,&\text{if $L_{Aik}$ and $L_{Bjk}$
have the $l^{th}$ level of agreement;}\\\ 0,&\text{otherwise. }\end{cases}$
(2)

The comparison vectors follow a two-class mixture distribution of the form
$\text{Pr}(\gamma_{ij})=\nu\text{}m(\gamma_{ij})+(1-\nu)\text{}u(\gamma_{ij})$,
where $m(\gamma_{ij})=\text{Pr}(\gamma_{ij}\text{ }|\text{
}(i,j)\in\mathbf{M},\boldsymbol{\theta_{M}})$, and
$u(\gamma_{ij})=\text{Pr}(\gamma_{ij}\text{ }|\text{
}(i,j)\in\mathbf{U},\boldsymbol{\theta_{U}})$, and $\nu$ is the marginal
probability that a pair of records is a link. The parameters
$\boldsymbol{\theta_{M}}$ and $\boldsymbol{\theta_{U}}$ govern the
distributions $m(\gamma_{ij})$ and $u(\gamma_{ij})$ among the links and non-
links, respectively.

Here, $m(\gamma_{ij})$ and $u(\gamma_{ij})$ represent general probability
distributions on the comparison space. Fellegi and Sunter, (1969) specify them
assuming conditional independence of the components of the comparison vectors,
given the linkage status of a record pair. Other specifications allow for
dependence between the components of $\gamma_{ij}$, using log-linear models
(Larsen and Rubin,, 2001) or multilevel normal models (Daggy et al.,, 2014).

The probabilities $m(\gamma_{ij})$ and $u(\gamma_{ij})$ are unknown, and are
estimated using the Expectation-Maximization (EM) algorithm (Dempster et al.,,
1977; Belin and Rubin,, 1995). Let $\hat{m}(\gamma_{ij})$ and
$\hat{u}(\gamma_{ij})$ be the estimates of $m(\gamma_{ij})$ and
$u(\gamma_{ij})$, respectively. The FS model assigns an estimated weight to
record pair $(i,j)$ as $\hat{w}_{ij}=\text{log
}\frac{\hat{m}(\gamma_{ij})}{\hat{u}(\gamma_{ij})}$.

Each record pair is classified as a link, non-link, or a possible link, by
comparing $\hat{w}_{ij}$ to two fixed thresholds (Tepping,, 1968; Binette and
Steorts,, 2022). Choosing optimal thresholds is a subjective process, based on
manual review of record pairs (Dusetzina et al.,, 2014), or on the evaluation
of linkage quality measures at different threshold values (Christen and
Goiser,, 2005, 2007).

The original FS model classifies the linkage status of each record pair
independently, often resulting in many-to-one links. Jaro, (1989) addressed
this, by introducing a linear sum optimization as a precursor to the FS model.

Multiple extensions to the FS model have been proposed. These include
incorporating information from clerical reviews (Larsen and Rubin,, 2001),
modeling the dependence between linking variables (Winkler,, 1993; Schürle,,
2005; Xu et al.,, 2019), handling continuous-valued linking variables (Vo et
al.,, 2023), handling missing data (Li et al.,, 2022; Sadinle,, 2017), testing
the conditional independence assumption (Xu et al.,, 2022), improved
scalability (Enamorado et al.,, 2019), and extensions to multi-file linkage
(Sadinle and Fienberg,, 2013; Aleshin-Guendel and Sadinle,, 2022; Taylor et
al.,, 2024).

####  1.1.3 Linkage errors

Given that probabilistic linkage relies on semi-identifying information, the
estimated $\mathbf{\Delta}$ is seldom error-free. In particular, a linked file
may contain two kinds of linkage errors: (i) false links between records
representing different entities; and (ii) false non-links between records
representing the same entity.

False links can lead to erroneous estimates of the associations of interest
(Neter et al.,, 1965). For instance, in linear regression models, false links
can induce attenuation bias when estimating the regression coefficients (Wang
et al.,, 2022; Patki and Shapiro,, 2023). False links can also have
implications when estimating causal effects (Wortman and Reiter,, 2018),
mortality rates (Krewski et al.,, 2005), and population sizes (Isaki and
Schultz,, 1987; Di Consiglio and Tuoto,, 2015, 2018). False non-links reduce
the number of records available for analysis, leading to a loss in statistical
power and increased variability in estimates (Haas et al.,, 1994; Harron et
al.,, 2017). False non-links can also induce selection bias in the post-
linkage inferences. Selection bias occurs when particular groups of records
are less likely to link, and are thus excluded from the analysis (Lariscy,,
2011; Bohensky,, 2015; Reiter,, 2021).

The extent of errors in a linked file is determined by the degree of
information contained in the linking variables. This information can be
summarized in terms of two metrics: the reliability and the discriminatory
power (Shlomo,, 2019).

The reliability is the probability that a linking variable is similar, given
that a record pair is a link. Reliability reflects the quality of the linking
variable, and is diminished when there is measurement error or missingness.
For records $i\in\mathbf{A}$ and $j\in\mathbf{B}$, the aggregate reliability
across all linking variables is quantified by $m(\gamma_{ij})$ (Shlomo,,
2019). Low reliability of the linking variables is typically associated with a
higher proportion of false non-links (Tromp et al.,, 2006, 2011). The
discriminatory power is the probability that a linking variable is similar,
given that a record pair is a non-link (Doidge and Harron,, 2018). The
theoretical number of possible values of the linking variable, and their
empirical distribution, determines this probability. The combined
discriminatory power of all linking variables is quantified by
$u(\gamma_{ij})$ (Shlomo,, 2019). If the combined discriminatory power is low,
the record linkage algorithm may identify a high proportion of false links
(Tromp et al.,, 2006, 2011).

Both these metrics have been used to assess the feasibility and performance of
the FS model (Roos et al.,, 1992; Cook et al.,, 2001). Modifications to the FS
weights based on the discriminatory power have also been proposed (Yancey,,
2000; Winkler,, 2000; Zhu et al.,, 2009; Xu et al.,, 2021; Vo et al.,, 2023).

####  1.1.4 Blocking

With large file sizes, comparing all record pairs across the two files becomes
computationally cumbersome. Large files also prompt a higher chance of
occurrence of linkage errors (Wang et al.,, 2022). To improve the scalability
and accuracy of the linkage process, deterministic blocking is a commonly
implemented pre-processing technique. In this technique, only those records
that agree on predefined, highly accurate blocking variables are compared.
Records that do not agree on these variables are considered non-links.

When utilizing deterministic blocking, it is important to ascertain an optimal
block size, which often involves a cost-benefit trade-off (Baxter and
Christen,, 2003). Large blocks generate a correspondingly large record
comparison space, and may not provide substantial gains in computational
efficiency or accuracy. When the blocks are too small, the record linking
algorithm may miss many true links (Baxter and Christen,, 2003). Methods that
utilize training data to determine optimum blocking schemes have been proposed
as possible solutions (Bilenko et al.,, 2006; Michelson and Knoblock,, 2006;
Campbell et al.,, 2021).

When blocking variables are error-prone, records may be placed in the wrong
block, leading to linkage errors. Possible solutions include using multiple
blocking passes (Baxter and Christen,, 2003), complex indexing schemes
(Murray,, 2015), or a combination of deterministic and data-driven blocking
(McVeigh et al.,, 2020). Other alternatives include probabilistic blocking,
which simultaneously infers the blocking scheme and the linkage parameters
(Steorts et al., 2014a, ). Probabilistic blocking can effectively propagate
blocking uncertainty into the downstream statistical analysis (Marchant et
al.,, 2021).

When files are partitioned into deterministic blocks, typical practice is to
implement the linkage algorithm independently within each block (Tancredi and
Liseo,, 2011; Murray,, 2015; Sadinle,, 2017). However, it is also possible to
allow the linkage algorithm to borrow information across blocks (Gutman et
al.,, 2013; Shan et al.,, 2022).

####  1.1.5 Post-linkage analysis

Linking two files is commonly not the end goal in a study; rather, researchers
may seek to estimate the population association between
$\mathbf{Y}_{A}=\\{\mathbf{Y}_{Ai}\\}$ and
$\mathbf{X}_{B}=\\{\mathbf{X}_{Bj}\\}$ among the $n_{AB}$ overlapping records.
One approach to summarizing this association is describing the conditional
mean
$\operatorname{\mathbb{E}}(\mathbf{Y}_{A}|\mathbf{X}_{B})=\mu(\mathbf{X}_{B};\boldsymbol{\beta})$,
where $\mu(.)$ denotes an inverse link function of known form, and
$\boldsymbol{\beta}=(\beta_{1},\dots,\beta_{P{{}_{B}}})$ is a set of
parameters that includes the estimand of interest. Alternatively, the
association can be summarized using correlation coefficients, principal
components analysis, or other multivariate analysis techniques (Rencher,,
2002).

###  1.2 A missing data formulation

Inference on $\boldsymbol{\beta}$ using the $n_{AB}$ overlapping records
involves analysis of “complete data”, given by
$\mathcal{D}=\big{[}\mathbf{L}_{Ai},\mathbf{L}_{Bj},\mathbf{Y}_{Ai},\mathbf{X}_{Bj},\Delta_{ij}\big{]}_{(i\in\mathbf{A},j\in\mathbf{B})}$.
Assuming that the variables
$(\mathbf{L}_{A},\mathbf{L}_{B},\mathbf{Y}_{A},\mathbf{X}_{B})$ are fully
observed, the observed data are
$\mathcal{D}^{obs}=\big{[}\mathbf{L}_{Ai},\mathbf{L}_{Bj},\mathbf{Y}_{Ai},\mathbf{X}_{Bj}\big{]}_{(i\in\mathbf{A},j\in\mathbf{B})}$,
and the missing data are
$\mathcal{D}^{mis}=\\{\Delta_{ij}\\}_{(i\in\mathbf{A},j\in\mathbf{B})}$. The
linkage status $\Delta_{ij}$ can be viewed as a discrete latent variable that
is never observed, but explains the observed data (Fienberg and Manrique-
Vallier,, 2009). This view motivates inference using likelihood-based and
Bayesian methods commonly employed in latent class modeling, such as the EM
(Dempster et al.,, 1977) and data augmentation (DA) (Tanner and Wong,, 1987)
algorithms.

Alternatively, one can impute the missing $\boldsymbol{\Delta}$, using a
stochastic model based on $\mathcal{D}^{obs}$. The resultant “complete data”
comprises a set of identified links, which can be analyzed using standard
methods. Uncertainty around the imputed $\boldsymbol{\Delta}$ can be accounted
for using multiple imputation (MI) (Rubin,, 1987; Little and Rubin,, 2019).
Inference can also be accomplished by weighting the post-linkage analysis
model, with weights estimated from models based on $\mathcal{D}^{obs}$.

####  1.2.1 A taxonomy of linkage mechanisms

A linkage mechanism characterizes the relationship between the linkage
structure and the variables in the individual files. There are broad parallels
between missing data mechanisms (Rubin,, 1976), and linkage mechanisms. For a
given prior on $\boldsymbol{\Delta}$, we define four mechanisms that describe
$f(\boldsymbol{\Delta}|\mathbf{L}_{A},\mathbf{L}_{B},\mathbf{Y}_{A},\mathbf{X}_{B},\boldsymbol{\theta})$,
where $\boldsymbol{\theta}$ is a vector parameter governing the distribution.
The definitions assume that
$\mathbf{L}_{A},\mathbf{L}_{B},\mathbf{Y}_{A},\text{ and }\mathbf{X}_{B}$ are
fully observed.

######  Definition 1.1.

Strongly non-informative linkage (SNL): The linkage mechanism is said to be
strongly non-informative, if the conditional distribution of
$\boldsymbol{\Delta}$ given
$\mathbf{L}_{A},\mathbf{L}_{B},\mathbf{Y}_{A},\mathbf{X}_{B}$ depends
exclusively on $\mathbf{L}_{A}\text{ and }\mathbf{L}_{B}$. Formally,
$f(\boldsymbol{\Delta}|\mathbf{L}_{A},\mathbf{L}_{B},\mathbf{Y}_{A},\mathbf{X}_{B},\boldsymbol{\theta})=f(\boldsymbol{\Delta}|\mathbf{L}_{A},\mathbf{L}_{B},\boldsymbol{\theta})$
for all $\mathbf{Y}_{A},\mathbf{X}_{B},$ and $\boldsymbol{\theta}$.

The SNL mechanism asserts that variables exclusive to each of the files do not
inform the linkage. This has analogies to when the data are missing at random
(MAR) (Rubin,, 1976). Specifically, SNL can be viewed as a special case of
MAR, where the missingness is determined by a set of auxiliary variables
$\mathbf{L}_{A}$ and $\mathbf{L}_{B}$. These variables may not inherently be
of interest in the analysis, but predict the missingness (Schafer,, 1997;
Collins et al.,, 2001).

A more restrictive version of SNL states that the linkage is completely
independent of $\mathbf{L}_{A},\mathbf{L}_{B},\mathbf{Y}_{A},\mathbf{X}_{B}$.
Thus,
$f(\boldsymbol{\Delta}|\mathbf{L}_{A},\mathbf{L}_{B},\mathbf{Y}_{A},\mathbf{X}_{B},\boldsymbol{\theta})=f(\boldsymbol{\Delta}|\boldsymbol{\theta})$
for all $\mathbf{L}_{A},\mathbf{L}_{B},\mathbf{Y}_{A}$, $\mathbf{X}_{B},$ and
$\boldsymbol{\theta}$. Han and Lahiri, (2019) refer to this as linkage
completely at random (LCAR), similar to when the data are missing completely
at random (MCAR) (Rubin,, 1976). A special case of the LCAR mechanism is when
$f(\boldsymbol{\Delta}|\boldsymbol{\theta})$ is a constant. This mechanism is
seldom plausible in practice, as it entails that every record pair has an
equal probability of being a link.

######  Definition 1.2.

Non-informative linkage (NL): The linkage mechanism is said to be non-
informative, if the conditional distribution of $\boldsymbol{\Delta}$ given
$\mathbf{L}_{A},\mathbf{L}_{B},\mathbf{Y}_{A},\mathbf{X}_{B}$ depends on
$\mathbf{L}_{A},\mathbf{L}_{B}$ and $\mathbf{X}_{B}$, but not the outcome
$\mathbf{Y}_{A}$. Formally,
$f(\boldsymbol{\Delta}|\mathbf{L}_{A},\mathbf{L}_{B},\mathbf{Y}_{A},\mathbf{X}_{B},\boldsymbol{\theta})=f(\boldsymbol{\Delta}|\mathbf{L}_{A},\mathbf{L}_{B},\mathbf{X}_{B},\boldsymbol{\theta})$
for all $\mathbf{Y}_{A}$ and $\boldsymbol{\theta}$.

Post-linkage analysis methods with limited access to the linkage process (for
e.g., Lahiri and Larsen,, 2005) typically rely on this assumption for
inference. Han and Lahiri, (2019) refer to the NL mechanism as linkage at
random (LAR).

To exemplify an NL scenario, suppose that file $\mathbf{B}$ contains the
covariate age, and file $\mathbf{A}$ is a subset of units in file $\mathbf{B}$
that are less than 60 years old. While age is not a linking variable common to
both files, it implicitly informs the linkage. The NL mechanism is also a
special case of the MAR mechanism, where auxiliary variables $\mathbf{L}_{A}$
and $\mathbf{L}_{B}$, and substantively important variables $\mathbf{X}_{B}$,
inform the linkage.

######  Definition 1.3.

Weakly non-informative linkage (WNL): The linkage mechanism is said to be
weakly non-informative, if
$f(\boldsymbol{\Delta}|\mathbf{L}_{A},\mathbf{L}_{B},\mathbf{Y}_{A},\mathbf{X}_{B},\boldsymbol{\theta})$
depends on $\mathbf{L}_{A},\mathbf{L}_{B},\mathbf{Y}_{A},\text{ and
}\mathbf{X}_{B}$, for all $\boldsymbol{\theta}$.

The WNL mechanism is also comparable to the MAR mechanism, as only fully
observed variables inform the linkage.

######  Definition 1.4.

Informative linkage (IL): The linkage mechanism is said to be informative, if
none of the above linkage mechanisms hold. For instance, the conditional
distribution of $\boldsymbol{\Delta}$ may depend on extraneous variables not
observed in either file.

IL may induce bias in the post-linkage inferences, particularly if such
extraneous variables are associated with the substantively important
variables. The IL mechanism is comparable to the missing not at random (MNAR)
mechanism (Rubin,, 1976), as the linkage status depends on unobserved
quantities.

In what follows, we discuss linked-data inferential methods that rely on one
or more of the linkage mechanisms as their underlying assumptions.

##  2 Primary analysis of linked files

Primary analysis refers to the situation when the record linkage and the post-
linkage analysis are performed by the same individual. Alternatively, the
analyst has access to files $\mathbf{A}$ and $\mathbf{B}$, or to the
likelihood of every record pair $(i\in\mathbf{A},j\in\mathbf{B})$ being a
link. Following the missing data literature (Molenberghs et al.,, 2014; Little
and Rubin,, 2019), we classify primary analysis methods into three broad
categories: likelihood and Bayesian methods, imputation methods, and weighting
methods. We note that this classification can be overlapping. For instance,
some methods can be classified as being both Bayesian and imputation-based, as
proper MI methods commonly rely on sampling from (approximate) Bayesian
posterior distributions (Rubin,, 1996).

###  2.1 Likelihood and Bayesian methods

Likelihood-based and Bayesian methods rely on the specification of the
complete-data likelihood, namely
$\mathcal{L}_{c}(\mathbf{L}_{A},\mathbf{L}_{B},\mathbf{Y}_{A},\mathbf{X}_{B},\boldsymbol{\Delta}|\boldsymbol{\beta},\boldsymbol{\theta})$,
where $\boldsymbol{\theta}$ denotes a set of parameters governing the linkage
process. It is often reasonable to assume that $\boldsymbol{\theta}$ and
$\boldsymbol{\beta}$ have distinct parameter spaces (Rubin,, 1976; Little and
Rubin,, 2019). This allows for $\mathcal{L}_{c}$ to be factored as

$\displaystyle\begin{split}\mathcal{L}_{c}(\mathbf{L}_{A},\mathbf{L}_{B},\mathbf{Y}_{A},\mathbf{X}_{B},\boldsymbol{\Delta}|\boldsymbol{\beta},\boldsymbol{\theta})&=\mathcal{L}_{l}(\mathbf{L}_{A},\mathbf{L}_{B}|\boldsymbol{\Delta},\boldsymbol{\theta})\text{
}\mathcal{L}_{a}(\mathbf{Y}_{A},\mathbf{X}_{B}|\mathbf{L}_{A},\mathbf{L}_{B},\boldsymbol{\Delta},\boldsymbol{\beta}),\end{split}$
(3)

where $\mathcal{L}_{l}$ denotes the record linkage likelihood, and
$\mathcal{L}_{a}$ denotes the likelihood for associations between
$\mathbf{Y}_{A}\text{ and }\mathbf{X}_{B}$. Equation (3) treats the missing
$\boldsymbol{\Delta}$ as a parameter in the likelihood. The joint posterior
distribution of $(\boldsymbol{\Delta},\boldsymbol{\beta},\boldsymbol{\theta})$
is

$\displaystyle\begin{split}\mathcal{P}(\boldsymbol{\Delta},\boldsymbol{\beta},\boldsymbol{\theta}|\mathbf{L}_{A},\mathbf{L}_{B},\mathbf{Y}_{A},\mathbf{X}_{B})&\propto\mathcal{L}_{l}(\mathbf{L}_{A},\mathbf{L}_{B}|\boldsymbol{\Delta},\boldsymbol{\theta})\text{
}\mathcal{L}_{a}(\mathbf{Y}_{A},\mathbf{X}_{B}|\mathbf{L}_{A},\mathbf{L}_{B},\boldsymbol{\Delta},\boldsymbol{\beta})\text{
}\pi_{\Delta}(\boldsymbol{\Delta})\text{
}\pi_{\theta,\beta}(\boldsymbol{\theta},\boldsymbol{\beta}),\end{split}$ (4)

where $\pi_{\Delta}$ denotes a prior on $\boldsymbol{\Delta}$, and
$\pi_{\theta,\beta}$ denotes a joint prior on
$(\boldsymbol{\theta},\boldsymbol{\beta})$.

For Bayesian inference, distinctness of $\boldsymbol{\theta}$ and
$\boldsymbol{\beta}$ requires that these parameters are a priori independent
(Rubin,, 1976; van Buuren,, 2018). Hence,
$\pi_{\theta,\beta}(\boldsymbol{\theta},\boldsymbol{\beta})=\pi_{\theta}(\boldsymbol{\theta})\text{
}\pi_{\beta}(\boldsymbol{\beta})$, where $\pi_{\theta}$ and $\pi_{\beta}$
represent prior distributions on $\boldsymbol{\theta}$ and
$\boldsymbol{\beta}$, respectively. Thus,

$\displaystyle\begin{split}\mathcal{P}(\boldsymbol{\Delta},\boldsymbol{\beta},\boldsymbol{\theta}|\mathbf{L}_{A},\mathbf{L}_{B},\mathbf{Y}_{A},\mathbf{X}_{B})&\propto\mathcal{L}_{l}(\mathbf{L}_{A},\mathbf{L}_{B}|\boldsymbol{\Delta},\boldsymbol{\theta})\text{
}\mathcal{L}_{a}(\mathbf{Y}_{A},\mathbf{X}_{B}|\mathbf{L}_{A},\mathbf{L}_{B},\boldsymbol{\Delta},\boldsymbol{\beta})\text{
}\pi_{\Delta}(\boldsymbol{\Delta})\text{
}\pi_{\theta}(\boldsymbol{\theta})\text{
}\pi_{\beta}(\boldsymbol{\beta}).\end{split}$ (5)

We note that the distinctness assumption is more plausible under the SNL and
NL mechanisms. It may not hold under WNL, when some components of
$\boldsymbol{\beta}$ may inform the linkage, and affect $\boldsymbol{\theta}$.
Moreover, $\boldsymbol{\theta}$ and $\boldsymbol{\beta}$ may be correlated a
priori; for example, when the linking and substantive variables represent
longitudinal versions of the same characteristic. Here and throughout, we
assume that the $\boldsymbol{\beta}$ represent the parameters of scientific
interest, and the $\boldsymbol{\theta}$ are nuisance parameters in the
likelihood. For notational simplicity, we henceforth suppress the dependence
of Equation (5) on $\boldsymbol{\theta}$.

####  2.1.1 Likelihood and Bayesian inference under SNL

Under the SNL assumption, Bayesian inference on $\boldsymbol{\beta}$ can be
accomplished using an approximation to Equation (5), called linkage averaging
(Sadinle,, 2018). Formally, under SNL, the posterior distribution of
$\boldsymbol{\beta}$ is

$\displaystyle\begin{split}\mathcal{P}(\boldsymbol{\beta}|\mathbf{L}_{A},\mathbf{L}_{B},\mathbf{Y}_{A},\mathbf{X}_{B})&=\sum_{\boldsymbol{\Delta}}\mathcal{P}(\boldsymbol{\beta}|\mathbf{L}_{A},\mathbf{L}_{B},\mathbf{Y}_{A},\mathbf{X}_{B},\boldsymbol{\Delta})\text{
}\mathcal{P}(\boldsymbol{\Delta}|\mathbf{L}_{A},\mathbf{L}_{B}).\end{split}$
(6)

Summing over all possible values of $\boldsymbol{\Delta}$ can be
computationally prohibitive. One solution is to approximate Equation (6) using
a method akin to Markov Chain Monte Carlo model composition ($MC^{3}$)
(Madigan et al.,, 1995, 1996). Specifically, let
$\boldsymbol{\Delta}^{(1)},\dots,\boldsymbol{\Delta}^{(M)}$ be $M$ samples of
the linkage structure drawn from
$\mathcal{P}(\boldsymbol{\Delta}|\mathbf{L}_{A},\mathbf{L}_{B})$. For a well-
behaved function $g(.)$ defined on the space $\\{0,1\\}^{n_{A}\times n_{B}}$,
the average

${G}=\frac{1}{M}\sum_{m=1}^{M}g(\boldsymbol{\Delta}^{(m)})$ (7)

converges almost surely to
$\operatorname{\mathbb{E}}(g(\boldsymbol{\Delta}))$, as $M\rightarrow\infty$
(Smith and Roberts,, 1993). Thus, Equation (6) can be approximated by setting
$g(\boldsymbol{\Delta}^{(m)})\equiv\mathcal{P}(\boldsymbol{\beta}|\mathbf{L}_{A},\mathbf{L}_{B},\mathbf{Y}_{A},\mathbf{X}_{B},\boldsymbol{\Delta}^{(m)})$
in Equation (7).

For large $M$, point and interval estimates for $\boldsymbol{\beta}$ can be
obtained using
$\hat{\boldsymbol{\beta}}^{(1)},\dots,\hat{\boldsymbol{\beta}}^{(M)}$, where
$\hat{\boldsymbol{\beta}}^{(m)}$ is a draw from
$\mathcal{P}(\boldsymbol{\beta}|\mathbf{L}_{A},\mathbf{L}_{B},\mathbf{Y}_{A},\mathbf{X}_{B},\boldsymbol{\Delta}^{(m)})$,
$m=1,\ldots,M$. Alternatively, each $\boldsymbol{\hat{\beta}}^{(m)}$ can
represent a set of samples from the posterior
$\mathcal{P}(\boldsymbol{\beta}|\mathbf{L}_{A},\mathbf{L}_{B},\mathbf{Y}_{A},\mathbf{X}_{B},\boldsymbol{\Delta}^{(m)})$;
i.e.,
$\hat{\boldsymbol{\beta}}^{(m)}=\\{\hat{\boldsymbol{\beta}}_{s}^{(m)}\\}$,
$s=1,\ldots,S$. Stacking these samples across the $M$ linkage structures
enables us to obtain point and interval estimates for $\boldsymbol{\beta}$
(Zhou and Reiter,, 2010). In some cases, it is possible to compute the
posterior mean and variance of $\boldsymbol{\hat{\beta}}^{(m)}$ analytically.
Estimation can then follow from the identities in Sadinle, (2018), which we
summarize in the supplementary materials.

The posterior distribution of the linkage structure,
$\mathcal{P}(\boldsymbol{\Delta}|\mathbf{L}_{A},\mathbf{L}_{B})\propto\mathcal{L}_{l}(\mathbf{L}_{A},\mathbf{L}_{B}|\boldsymbol{\Delta})\pi_{\Delta}(\boldsymbol{\Delta})$,
can be specified using one of two approaches: direct modeling of the
information in $\mathbf{L}_{A}\text{ and }\mathbf{L}_{B}$, or using comparison
vectors.

The direct modeling approach for
$\mathcal{P}(\boldsymbol{\Delta}|\mathbf{L}_{A},\mathbf{L}_{B})$ assumes that
the linking variables $\mathbf{L}_{A}$ and $\mathbf{L}_{B}$ are erroneous
versions of their true latent values. Under the assumption of bipartite
linkage, Tancredi and Liseo, (2011) utilize a hierarchical generative model to
specify $\mathcal{L}_{l}(\mathbf{L}_{A},\mathbf{L}_{B}|\boldsymbol{\Delta})$.
The model posits that $\mathbf{L}_{Ai}$ and $\mathbf{L}_{Bj}$ are noisy
realizations of the variables $\boldsymbol{\eta}_{A}$ and
$\boldsymbol{\eta}_{B}$, which are independent realizations of a latent
variable $\boldsymbol{\eta}$, with a discrete and finite support. Conditional
on $\boldsymbol{\eta}_{A}$ and $\boldsymbol{\eta}_{B}$, $\mathbf{L}_{Ai}$ and
$\mathbf{L}_{Bj}$ follow an identifiable variant of the hit-and-miss model
(Copas and Hilton,, 1990; Swartz et al.,, 2004). This is a mixture
distribution with two components: a point mass at the true values of
$\mathbf{L}_{Ai}$ or $\mathbf{L}_{Bj}$, and a uniform distribution over the
remaining points in the support of $\boldsymbol{\eta}$. At the top level of
the hierarchy, a distribution for $\boldsymbol{\eta}_{A}$ and
$\boldsymbol{\eta}_{B}$ given $\boldsymbol{\Delta}$ is postulated. The number
of links $n_{AB}=\sum_{i}\sum_{j}\Delta_{ij}$ is treated as a random quantity.
Under diffuse prior distributions on $n_{AB}$ and $\boldsymbol{\Delta}$, the
joint posterior distribution is explored via Metropolis-within-Gibbs
algorithms (Robert and Casella,, 1999).

Steorts et al., (2016) build on the measurement error model of Hall and
Fienberg, (2012), and posit a likelihood for a bipartite graph that maps every
record to a latent population entity. Akin to Tancredi and Liseo, (2011), the
likelihood postulates that the linking variables are noisy representations of
latent, multinomial random variables. The prior distribution on the linkage
structure is assumed to be uniform, and posterior sampling is accomplished via
Metropolis-within-Gibbs algorithms with split-merge proposal moves (Steorts et
al., 2014b, ). Linking records to latent entities, rather than to other
records, allows the model to handle linkage of multiple files, and duplicates
within a file.

The original methodologies of Tancredi and Liseo, (2011) and Steorts et al.,
(2016) are restricted to categorical linking variables. Extensions to
incorporate multivariate normal linking variables (Liseo and Tancredi,, 2011),
and string-valued linking variables (Steorts,, 2015) have also been proposed.

In some applications, it may be difficult to directly model complex linking
variables, like addresses or telephone numbers. A solution is to define the
likelihood
$\mathcal{L}_{l}(\mathbf{L}_{A},\mathbf{L}_{B}|\boldsymbol{\Delta})$ using
comparison vectors $\gamma_{ij}$ (Section 1.1.2). Akin to the FS model, the
comparison vectors are assumed to follow a mixture distribution over the set
of links and non-links. However, bipartite linkage is explicitly enforced via
a prior distribution on $\boldsymbol{\Delta}$. This approach was developed by
Fortini et al., (2001) and Larsen, (2002, 2005) for the case of binary
comparison vectors. Sadinle, (2017) extended it to include partial agreements
on complex linking variables, and to accommodate ignorable missingness
(Rubin,, 1976) in the linking variables. Formally, the model is

$\displaystyle\begin{split}\gamma_{ij}|\Delta_{ij}=1\sim m(\gamma_{ij})\\\
\gamma_{ij}|\Delta_{ij}=0\sim u(\gamma_{ij})\\\
\boldsymbol{\Delta}\sim\pi_{\Delta}(\boldsymbol{\Delta}),\end{split}$ (8)

where $m(\gamma_{ij})$ and $u(\gamma_{ij})$ are as defined in Section 1.1.2.
The prior $\pi_{\Delta}(\boldsymbol{\Delta})$ is defined over the space of all
bipartite linkage structures.

One possible joint prior distribution is
$\pi_{\Delta}(\boldsymbol{\Delta},n_{AB})=\pi_{1}(n_{AB})\text{
}\pi_{2}(\boldsymbol{\Delta}|n_{AB})$, where $\pi_{1}$ and $\pi_{2}$ are prior
distributions on the number of links, and the space of bipartite linkage
structures with $n_{AB}$ links, respectively (Larsen,, 2002; Sadinle,, 2017).
The joint posterior distribution of $(n_{AB},\boldsymbol{\Delta})$ can be
explored using a Gibbs sampler based on an adaptive multinomial distribution
(Sadinle,, 2017), or a Metropolis-Hastings algorithm with proposals akin to
Fisher-Yates shuffling (Fisher and Yates,, 1963; Larsen,, 2005).

####  2.1.2 Likelihood and Bayesian inference under WNL

The linkage averaging approach in Equation (6) can propagate linkage
uncertainty to the downstream analysis. However, information from the analysis
models is not utilized in the linkage process. An alternative is to allow the
analysis models to inform the linkage. In a Bayesian framework, this can be
accomplished by sampling $\boldsymbol{\Delta}$ and $\boldsymbol{\beta}$
simultaneously from their joint posterior distribution (Equation (5)).

It has been shown analytically, and via simulations, that incorporating
substantive relationships into the linkage process can facilitate correct
identification of links and non-links (Kamat et al.,, 2023). These
improvements are commonly called the “feedback effect” of the post-linkage
analysis model (Steorts et al.,, 2018). The magnitude of the improvements can
depend on (i) fit of the specified likelihood
$\mathcal{L}_{a}(\mathbf{Y}_{A},\mathbf{X}_{B}|\mathbf{L}_{A},\mathbf{L}_{B},\boldsymbol{\Delta},\boldsymbol{\beta})$;
and (ii) the strength of the association between $\mathbf{Y}_{A}$ and
$\mathbf{X}_{B}$.

Sampling from the joint posterior distribution in Equation (5) can be
accomplished via the DA algorithm. At the $m^{th}$ iteration, this algorithm
iterates between two steps: (i) I-Step: Impute $\boldsymbol{\Delta}^{(m)}$
from the posterior full conditional distribution
$\mathcal{P}(\boldsymbol{\Delta}|\mathbf{L}_{A},\mathbf{L}_{B},\mathbf{Y}_{A},\mathbf{X}_{B},\boldsymbol{\beta}^{(m-1)})$;
and (ii) P-step: Sample $\boldsymbol{\beta}^{(m)}$ from the posterior full
conditional distribution
$\mathcal{P}(\boldsymbol{\beta}|\mathbf{L}_{A},\mathbf{L}_{B},\mathbf{Y}_{A},\mathbf{X}_{B},\boldsymbol{\Delta}^{(m)})$.
The resultant samples of $\boldsymbol{\beta}$ incorporate the uncertainty
around $\boldsymbol{\Delta}$, and can be used for point and interval
estimation.

Gutman et al., (2013) model the associations between variables common to both
files, and those exclusive to either file. High quality linking variables are
utilized in an informative blocking step, which reduces the computational
complexity. Within each block, the linkage is assumed to be bipartite and
complete, so that every record in the smaller block can be linked to a unique
record in the larger block. The linkage likelihood
$\mathcal{L}_{l}(\mathbf{L}_{A},\mathbf{L}_{B}|\boldsymbol{\Delta})$ is
defined using a joint distribution that describes inconsistently recorded
linking variables. The distributional form of
$\mathcal{L}_{a}(\mathbf{Y}_{A},\mathbf{X}_{B}|\mathbf{L}_{A},\mathbf{L}_{B},\boldsymbol{\Delta},\boldsymbol{\beta})$
is specific to the application, and can include models of scientific interest,
and models for relationships that only inform the linkage. For unlinked
records in the larger block, variables in the smaller block are treated as
missing data, and imputed as part of an overarching DA scheme. Under a uniform
prior on $\boldsymbol{\Delta}$, posterior sampling for $\boldsymbol{\Delta}$
is accomplished via Metropolis algorithms. Dalzell and Reiter, (2018) extend
the model of Gutman et al., (2013) to incorporate erroneous blocking
variables, and to allow individual records to shift between blocks.

A limitation of the approach of Gutman et al., (2013) is the assumption of
complete linkage, which can be violated in practice. An alternative approach
augments the Bayesian FS likelihood (Equation (8)) with relationships between
variables exclusive to either file (Tang et al.,, 2020; Kamat et al.,, 2023).
Bayesian generative models have also been applied in a joint modeling
framework, in the context of linear regression (Tancredi and Liseo,, 2015;
Steorts et al.,, 2018), and small area estimation (Briscolini et al.,, 2018).

Hof and Zwinderman, (2014) propose to estimate $\boldsymbol{\beta}$ by
maximizing the complete-data likelihood $\mathcal{L}_{c}$, over all possible
values of $\boldsymbol{\Delta}$. Since the parameter space of
$\boldsymbol{\Delta}$ is fairly large, maximizing $\mathcal{L}_{c}$ can be
computationally prohibitive. As a solution, Hof and Zwinderman, (2014) propose
a two-step pseudo-likelihood approach to estimate $\boldsymbol{\beta}$. In the
first step, estimates of the linking probabilities for all record pairs are
computed under the FS model. These estimates are treated as fixed constants in
the pseudo-likelihood. In the second step, the pseudo-likelihood is maximized
using the EM algorithm, to obtain an estimate $\boldsymbol{\hat{\beta}}$. The
sampling variance of $\boldsymbol{\hat{\beta}}$, and confidence intervals for
$\boldsymbol{\beta}$, can be computed using the Bootstrap, or an asymptotic
normal approximation for the sampling distribution of
$\boldsymbol{\hat{\beta}}$. A similar inferential approach for time-to-event
data has been proposed in Hof et al., (2017).

###  2.2 Imputation

Fully Bayesian approaches to estimating $\boldsymbol{\beta}$ can propagate
linkage uncertainty in a principled manner, but are computationally intensive.
Moreover, in a joint modeling framework, the models used in the linkage,
namely
$\mathcal{L}_{a}(\mathbf{Y}_{A},\mathbf{X}_{B}|\mathbf{L}_{A},\mathbf{L}_{B},\boldsymbol{\Delta},\boldsymbol{\beta})$,
may differ from the scientific model of interest. In such situations, an
alternative is to approximate a fully Bayesian analysis using multiple
imputation (MI) (Rubin,, 1987).

In MI, $M>1$ imputations of the linkage structure are generated from its
posterior distribution. Each imputation is analyzed separately, and estimates
are pooled across the imputations using common combination rules (Rubin,,
1987; Little and Rubin,, 2019). This procedure can propagate linkage
uncertainty into the downstream analysis, while simplifying the computation.
Application of MI to linked-data inference has been exemplified in Gutman et
al., (2013), Abowd et al., (2021), and Shan et al., (2021), among others.

Formally, in each of the $M$ posterior samples of the linkage structure,
$\boldsymbol{\Delta}^{(1)},\dots,\boldsymbol{\Delta}^{(M)}$, we obtain a point
estimate, $\boldsymbol{\hat{\beta}}^{(m)}$, and its sampling variance
$U^{(m)}$, for $m=1,\dots,M$. A point estimate for $\boldsymbol{\beta}$ across
the $M$ linked files is
$\boldsymbol{\hat{\beta}}=M^{-1}\sum_{m=1}^{M}\boldsymbol{\hat{\beta}}^{(m)}$.
Its estimated variance is $T_{M}=\bar{U}_{M}+(1+M^{-1})B_{M}$, where
$\bar{U}_{M}=M^{-1}\sum_{m=1}^{M}U^{(m)}$ and
$B_{M}=(M-1)^{-1}\sum_{m=1}^{M}(\boldsymbol{\hat{\beta}}^{(m)}-\boldsymbol{\hat{\beta}})^{2}$.
A confidence interval for $\boldsymbol{\hat{\beta}}$ can be constructed using
a Student’s-$t$ approximation,
$(\boldsymbol{\hat{\beta}}-\boldsymbol{\beta})/\sqrt{T_{M}}\sim t_{\nu}$,
where $\nu=(M-1)(1+\bar{U}_{M}/((1+M^{-1})B_{M}))^{2}$ (Rubin,, 1987).

A limitation of MI is that the imputation model for $\boldsymbol{\Delta}$ may
be uncongenial (Meng,, 1994; Xie and Meng,, 2017) to the post-linkage analysis
model. The analysis model is congenial to the imputation model, if there
exists a unifying Bayesian model for the complete data that embeds both of
them (Murray,, 2018). An uncongenial imputation model may lead to biased point
estimates, and interval estimates with invalid coverages (van Buuren,, 2018).
Since congeniality is hard to ascertain in practice, a general solution is to
use an imputation model that is less restrictive than the analysis model.
Thus, it may be advisable to include as many relationships of substantive
interest as possible in the imputation model for $\boldsymbol{\Delta}$
(Rubin,, 1996; Xie and Meng,, 2017). The joint modeling approaches in Section
2.1.2 can provide a way to accomplish this.

Goldstein et al., (2012) propose to cast linking two files in a statistical
matching framework (D’Orazio et al.,, 2006). In this framework, all records in
file $\mathbf{A}$ have $\mathbf{X}_{B}$ missing. For record $i\in\mathbf{A}$,
all records in file $\mathbf{B}$ serve as a pool of candidate donors. Assuming
that a set of known links is available, Goldstein et al., (2012) construct a
posterior distribution, $p^{*}_{ij}\propto p_{ij}\text{
}p_{B|A}(\mathbf{X}_{B}|\mathbf{Y}_{A},\mathbf{L}_{A},\mathbf{L}_{B})$, where
$p_{ij}\propto\hat{w}_{ij}$ (Section 1.1.2) are prior linking probabilities,
and $p_{B|A}$ denotes the imputation model. For record $i\in\mathbf{A}$,
records $j\in\mathbf{B}$ with $p^{*}_{ij}$ below a predefined threshold are
considered non-links. If multiple records exceed the threshold, the value of
$\mathbf{X}_{Bj}$ such that $p^{*}_{ij}>p^{*}_{ij^{\prime}}\forall
j^{\prime}\neq j$ is used for imputing the missing $\mathbf{X}_{Bi}$. If no
$p^{*}_{ij}$ reaches the threshold, the missing $\mathbf{X}_{Bi}$ is multiply
imputed by sampling from
$p_{B|A}(\mathbf{X}_{B}|\mathbf{Y}_{A},\mathbf{L}_{A},\mathbf{L}_{B})$. Each
imputed file is analyzed separately, and estimates are pooled using MI
combining rules.

The imputation model $p_{B|A}$ assumes joint multivariate normality of
$(\mathbf{Y}_{A},\mathbf{X}_{B},\mathbf{L}_{A},\mathbf{L}_{B})$, when all
variables are continuous. When the data comprise a mix of continuous and
categorical variables, $p_{B|A}$ is derived using latent normal models
(Goldstein et al.,, 2009).

One limitation of the algorithm is that a set of known links is required to
estimate the parameters in
$p_{B|A}(\mathbf{X}_{B}|\mathbf{Y}_{A},\mathbf{L}_{A},\mathbf{L}_{B})$. Such a
set is not always available in practice. Moreover, the imputation procedure
produces statistically valid estimates, as long as the linkage errors are
independent of $\mathbf{X}_{B}$ given $\mathbf{Y}_{A},\mathbf{L}_{A},\text{
and }\mathbf{L}_{B}$. This is equivalent to $\mathbf{X}_{B}$ being missing at
random (Goldstein et al.,, 2012). Violation of this assumption can lead to
substantial bias in point estimates (Harron et al.,, 2014). A possible
mitigation strategy is to make the MAR assumption more plausible, by
specifying rich imputation models that accommodate variables from auxiliary
data sources (Goldstein and Harron,, 2015).

###  2.3 Weighting

Weighting methods aim to obtain unbiased parameter estimates of generalized
linear models fit with linked data. These methods are confined to scenarios
where a univariate response $\mathbf{Y}_{A}$ is regressed on a set of
covariates $\mathbf{X}_{B}$. Weighting methods rely on two assumptions. The
first assumption is that the linkage is complete, i.e., the smaller file
$\mathbf{A}$ is a subset of the larger file $\mathbf{B}$, so that
$n_{AB}=n_{A}$. The second assumption is that the linkage mechanism is NL.

Let $\mathbf{Y}^{R}_{A}=\\{\mathbf{Y}^{R}_{Aj}:j=1,\ldots,n_{B}\\}$ be the
correct (but unobserved) permutation of records in file $\mathbf{A}$ to those
in file $\mathbf{B}$. Then, a probabilistically linked file is of the form
$(\mathbf{Y}^{*}_{A},\mathbf{X}_{B})$, where
$\mathbf{Y}^{*}_{A}=\boldsymbol{\Delta}\mathbf{Y}_{A}^{R}$. Under the NL
assumption,

$\displaystyle\begin{split}\operatorname{\mathbb{E}}(\mathbf{Y}_{A}^{*}|\mathbf{L}_{A},\mathbf{L}_{B},\mathbf{X}_{B})&=\operatorname{\mathbb{E}}(\operatorname{\mathbb{E}}(\mathbf{Y}^{*}_{A}|\mathbf{Y}^{R}_{A},\mathbf{L}_{A},\mathbf{L}_{B},\mathbf{X}_{B})|\mathbf{L}_{A},\mathbf{L}_{B},\mathbf{X}_{B})\\\
&=\operatorname{\mathbb{E}}(\mathbf{Q}\mathbf{Y}^{R}_{A}|\mathbf{L}_{A},\mathbf{L}_{B},\mathbf{X}_{B})=\mathbf{Q}\mu(\mathbf{X}_{B};\boldsymbol{\beta}),\end{split}$
(9)

where
$\mathbf{Q}=[q_{ij}]=[\text{Pr}(\Delta_{ij}=1|\mathbf{L}_{Ai},\mathbf{L}_{Bj},\mathbf{X}_{Bj})]_{(i\in\mathbf{A},j\in\mathbf{B})}$
is either known, or can be estimated during the linkage process. Under the
one-to-one linkage constraint, the row sums of $\mathbf{Q}$,
$\sum_{j\in\mathbf{B}}q_{ij}=1$.

A common model to describe the association between $\mathbf{Y}^{R}_{A}$ and
$\mathbf{X}_{B}$ is the estimating equation,

$\kappa(\mathbf{X}_{B};\boldsymbol{\beta})\equiv\upsilon(\mathbf{X}_{B};\boldsymbol{\beta})\Big{[}\mathbf{Y}^{R}_{A}-\mu(\mathbf{X}_{B};\boldsymbol{\beta})\Big{]}=0,$
(10)

where $\upsilon(\mathbf{X}_{B};\boldsymbol{\beta})$ is the weighting matrix,
which is independent of $\mathbf{Y}^{R}_{A}$. Ignoring linkage errors and
solving the estimating equation

$\kappa^{*}(\mathbf{X}_{B};\boldsymbol{\beta})\equiv\upsilon(\mathbf{X}_{B};\boldsymbol{\beta})\Big{[}\mathbf{Y}^{*}_{A}-\mu(\mathbf{X}_{B};\boldsymbol{\beta})\Big{]}=0$
(11)

yields the naive coefficient estimate $\hat{\boldsymbol{\beta}}^{*}$.

######  Proposition 1.

Under the NL assumption, the following holds:

  1. (i)

If $\tilde{\boldsymbol{\beta}}$ is the true value of $\boldsymbol{\beta}$,
then
$\operatorname{\mathbb{E}}(\kappa^{*}(\mathbf{X}_{B};\tilde{\boldsymbol{\beta}})|\mathbf{X}_{B})\neq
0$, and

  2. (ii)

$\operatorname{\mathbb{E}}(\hat{\boldsymbol{\beta}}^{*}-\boldsymbol{\beta}|\mathbf{X}_{B})\approx-\Big{[}\frac{\partial\kappa^{*}(\mathbf{X}_{B};\boldsymbol{\beta})}{\partial\boldsymbol{\beta}}\Big{]}^{-1}\upsilon(\mathbf{X}_{B};\boldsymbol{\beta})\text{
}(\mathbf{Q}-\mathbf{I})\text{ }\mu(\mathbf{X}_{B};\boldsymbol{\beta})$.

We defer the proof to the supplementary materials. Proposition 1 indicates
that $\kappa^{*}(\mathbf{X}_{B};\boldsymbol{\beta})$ is a biased estimating
equation, and $\hat{\boldsymbol{\beta}}^{*}$ is a biased estimator for
$\boldsymbol{\beta}$.

Scheuren and Winkler, (1993) describe an approach to mitigate the bias in
$\hat{\boldsymbol{\beta}}^{*}$ for linear regression, when $n_{A}=n_{B}$,
$\mu(\mathbf{X}_{B};{\boldsymbol{\beta}})=\mathbf{X}_{B}{\boldsymbol{\beta}}$
and $\upsilon(\mathbf{X}_{B};{\boldsymbol{\beta}})=\mathbf{X}_{B}^{T}$. Their
bias-adjusted estimator is
$\hat{{\boldsymbol{\beta}}}_{SW}=\hat{{\boldsymbol{\beta}}}^{*}-(\mathbf{X}_{B}^{T}\mathbf{X}_{B})^{-1}\mathbf{X}_{B}^{T}(\mathbf{Q}-\boldsymbol{\Delta})\mathbf{Y}^{*}_{A}$.
As most record pairs from files $\mathbf{A}$ and $\mathbf{B}$ do not represent
links, Scheuren and Winkler, (1993) truncate the $i^{th}$ row of $\mathbf{Q}$,
so that all entries except the two highest $q_{ij}$ are set to null. Scheuren
and Winkler, (1997) utilize $\hat{{\boldsymbol{\beta}}}_{SW}$ in a procedure
that recursively iterates between identifying non-links, and improving the
estimates of ${\boldsymbol{\beta}}$.

Although $\boldsymbol{\hat{{\boldsymbol{\beta}}}}_{SW}$ has a lower bias
compared to the naive estimator, it is not unbiased for ${\boldsymbol{\beta}}$
(Lahiri and Larsen,, 2005). An alternative is to solve the adjusted estimating
equation (Han and Lahiri,, 2019)

$\tilde{\kappa}(\mathbf{X}_{B};{\boldsymbol{\beta}})\equiv\upsilon(\mathbf{X}_{B};{\boldsymbol{\beta}})\Big{[}\mathbf{Y}^{*}_{A}-\mathbf{Q}\mu(\mathbf{X}_{B};{\boldsymbol{\beta}})\Big{]}=0.$
(12)

For the identity link function,
$\mu(\mathbf{X}_{B};{\boldsymbol{\beta}})=\mathbf{X}_{B}{\boldsymbol{\beta}}$,
and equally sized files, setting
$\upsilon(\mathbf{X}_{B};{\boldsymbol{\beta}})=\mathbf{X}_{B}^{T}\mathbf{Q}^{T}$,
yields the estimator
$\hat{{\boldsymbol{\beta}}}^{LL}=(\mathbf{X}_{B}^{T}\mathbf{Q}^{T}\mathbf{Q}\mathbf{X}_{B})^{-1}\mathbf{X}_{B}^{T}\mathbf{Q}^{T}\mathbf{Y}_{A}^{*}$
(Lahiri and Larsen,, 2005). Alternatively, $\hat{\boldsymbol{\beta}}^{LL}$ can
be obtained using a least squares regression of $\mathbf{Y}_{A}^{*}$ on a
weighted combination of the covariates, $\mathbf{Q}\mathbf{X}_{B}$.

The point estimator $\hat{{\boldsymbol{\beta}}}^{LL}$ is unbiased when
$\mathbf{Q}$ is known. However, in most applications, only an estimate of
$\mathbf{Q}$ is available. When ${\mathbf{Q}}$ is estimated using the FS
model, its rows may not sum to one, as the one-to-one linkage constraint is
not enforced. A possible solution is to normalize each entry in $\mathbf{Q}$
as $q^{{}^{\prime}}_{ij}=q_{ij}/\sum_{j\in\mathbf{B}}q_{ij}$.

Chipperfield and Chambers, (2015) argue that using $q^{{}^{\prime}}_{ij}$ can
lead to biased inferences under one-to-one linkage. This limitation can be
overcome by estimating $\mathbf{Q}$ using a bootstrap for the entire linkage
process (Winglee et al.,, 2005; Chipperfield and Chambers,, 2015). Other
variations of the weighting approach have been proposed, including using
weighted least squares when covariates are spread over multiple files (Hof and
Zwinderman,, 2012), and relaxing the assumption of complete linkage (Hof and
Zwinderman,, 2012; Chipperfield and Chambers,, 2015).

Computation of variances with estimated weights is generally not
straightforward (Little and Rubin,, 2019). Valid variance estimators require
elaborate theoretical calculations (Tsiatis,, 2006), or computationally-
intensive resampling techniques (Little and Rubin,, 2019). For particular
choices of $\upsilon(\mathbf{X}_{B};\boldsymbol{\beta})$, Han, (2018) derives
theoretical plug-in estimators for the sampling variance of
$\hat{\boldsymbol{\beta}}$ under Equation (12). To simplify calculations, Han,
(2018) suggests truncating every row of $\mathbf{Q}$ to include the highest
(or two highest) values. These estimators, however, do not account for the
uncertainty arising from using an estimate of $\mathbf{Q}$. This can be
addressed by using resampling methods that re-estimate $\mathbf{Q}$ at every
iteration. Two such methods are based on the unified Jackknife theory (Jiang
et al.,, 2002; Han and Lahiri,, 2019), and the Bootstrap (Lahiri and Larsen,,
2005).

###  2.4 Interlude: Estimating complex models with linked files

A majority of the linked-data inferential methods in the literature center
around estimating parameters of generalized linear models. Some work has also
focused on inferring the size of a closed population, using capture-recapture
(Fienberg,, 1972) or multiple-systems estimation (Bird and King,, 2018). For
example, Bayesian models for concurrently inferring the linkage and the
population size have been developed, for the case of two files (Tancredi and
Liseo,, 2011), and multiple files (Tancredi et al.,, 2020).

The Bayesian and imputation-based approaches also offer a flexible framework
to estimate other complex models, such as hierarchical models (Gelman et al.,,
2013), or multivariate analyses like finite mixture models (Peel and
MachLachlan,, 2000). As some multivariate analysis methods are exploratory in
nature (for e.g, factor analysis), posterior sampling within a fully Bayesian
framework can be computationally daunting (Conti et al.,, 2014). In such
cases, an MI approach can offer more feasibility. In Section 4.2, we provide
an example of using MI to perform model-based cluster analysis with a linked
file.

##  3 Secondary analysis of linked files

Secondary analysis refers to settings when the linkage and the data analysis
are performed by different individuals. Specifically, the data analyst is
provided a linked file, but does not have access to the individual files, the
linking variables, or linking probabilities for record pairs. This situation
arises when the original data comprise sensitive information, and releasing
individual files interferes with privacy regulations, or poses considerable
disclosure risks (Gilbert et al.,, 2017; Christen et al.,, 2020).

In secondary analysis, the analyst does not have access to records that were
left unlinked during the linkage process. Consequently, most analysis methods
can only propagate the uncertainty due to false links. Inferences from the
linked file may not be generalizable to the entire population of entities,
without assumptions on the relationship between the linked and unlinked
records, or availability of additional information about the unlinked records.

Formally, secondary analysis is based on the file
$\mathbf{F}\equiv\\{\mathbf{Y}_{Al},\mathbf{X}_{Bl}\\}_{l=1}^{n_{AB}}$,
obtained by linking files $\mathbf{A}$ and $\mathbf{B}$. The file $\mathbf{F}$
can be partitioned into true and false links, but this partitioning is
unknown. In this case, we reformulate the linkage structure using a vector of
indicators $\mathbf{C}=\\{C_{l}\\}$, where $C_{l}=1$ if record pair $l$ is a
true link, and $C_{l}=0$, otherwise, where $l=1,\dots n_{AB}$.

###  3.1 Likelihood, Bayesian, and imputation methods

Under the WNL assumption, Gutman et al., (2016) postulate that record pairs in
$\mathbf{F}$ are derived from a two-class mixture model. Formally, the
likelihood is

$\displaystyle\begin{split}\mathbf{Y}_{Al},\mathbf{X}_{Bl}|C_{l}=1&\sim\mathcal{L}_{M}(\mathbf{Y}_{Al},\mathbf{X}_{Bl}|{\boldsymbol{\beta}}_{M}),\\\
\mathbf{Y}_{Al},\mathbf{X}_{Bl}|C_{l}=0&\sim\mathcal{L}_{U}(\mathbf{Y}_{Al},\mathbf{X}_{Bl}|{\boldsymbol{\beta}}_{U}),\\\
C_{l}&\overset{\mathrm{iid}}{\sim}\text{ Bernoulli }(\lambda),\end{split}$
(13)

where $\boldsymbol{{\boldsymbol{\beta}}}_{M}$ and
$\boldsymbol{{\boldsymbol{\beta}}}_{U}$ are parameters governing the joint
distribution of $(\mathbf{Y}_{A},\mathbf{X}_{B})$ among the true and false
links, respectively, and $\lambda$ is the marginal probability of a true link.
The models $\mathcal{L}_{M}$ and $\mathcal{L}_{U}$ can be augmented to include
additional variables present in the linked file.

To avoid label switching, Gutman et al., (2016) rely on informative prior
distributions for
$({\boldsymbol{\beta}}_{M},{\boldsymbol{\beta}}_{U},\lambda)$. These priors
are constructed using a validation sample, and other available linkage
paradata. The missing $\mathbf{C}$ is imputed using a DA scheme, that iterates
between sampling $\mathbf{C}$, and the parameters
$(\boldsymbol{\beta}_{M},\boldsymbol{\beta}_{U},\lambda)$. When
$\mathcal{L}_{M}$ represents the scientific model of interest, inference based
on posterior samples of ${\boldsymbol{\beta}}_{M}$ can account for the
uncertainty around $\mathbf{C}$. Alternatively, each DA iteration provides a
set of links, that can be used to fit the analysis model of interest. Point
and interval estimates across posterior samples of $\mathbf{C}$ can be
obtained using the MI combining rules (Section 2.2).

Slawski et al., (2021) consider the case of linear regression, and assume
independence between $\mathbf{Y}_{Al}$ and $\mathbf{X}_{Bl}$ when postulating
$\mathcal{L}_{U}$ in Equation (13). An estimate $\hat{\boldsymbol{\beta}}_{M}$
is obtained by maximizing the likelihood in (13) using the EM algorithm. The
variance of $\hat{\boldsymbol{\beta}}_{M}$ is approximated using the Huber-
Sandwich estimator (Freedman,, 2006), and confidence intervals are constructed
under an asymptotic normal approximation for the sampling distribution of
$\hat{\boldsymbol{\beta}}_{M}$.

###  3.2 Weighting

In a secondary analysis setting, the matrix $\mathbf{Q}$ (Equation (9)) can
only be modeled using coarse summary statistics from the linkage process. One
such statistic is the overall proportion of correct links, $\lambda$. This can
be calculated, for example, by comparing a representative audit sample from
the linked file to a gold-standard linked file (Chambers and Diniz da Silva,,
2020).

Under the assumption that $n_{A}=n_{B}$, and that the linkage is complete and
one-to-one, Chambers, (2009) imposes an “exchangeable” structure on
$\mathbf{Q}$. Formally, the exchangeable linkage error (ELE) model is
(Chambers,, 2009)

$\displaystyle\begin{split}[\mathbf{Q}]_{(i\in\mathbf{A},j\in\mathbf{B})}&=[\text{Pr
}(\Delta_{ij}=1|\mathbf{L}_{Ai},\mathbf{L}_{Bj},\mathbf{X}_{Bj})]_{(i\in\mathbf{A},j\in\mathbf{B})}\\\
&=\begin{cases}\lambda,\text{ if }i=j\\\ \frac{1-\lambda}{n_{A}-1},\text{ if
}i\neq j.\end{cases}\end{split}$ (14)

This construct implies that (i) every record $j\in\mathbf{B}$ has probability
$\lambda$ of being the correct link for record $i\in\mathbf{A}$, and (ii) any
two records that are not linked to $i\in\mathbf{A}$ are equally likely to be
the correct links.

Exchangeability implies that the linkage is independent of the linking and
substantive variables, so that the linkage mechanism is LCAR. In many
applications, this assumption may be overly stringent, as the probability of
correct linkage can be heterogeneous across records (Winkler,, 2021). A
possible solution is to assume that the linkage probabilities are homogeneous
within disjoint subgroups (or blocks) of the linked file, rather than across
all the linked records (Chambers and Diniz da Silva,, 2020).

For inference on $\boldsymbol{\beta}$ in Equation (12), Chambers, (2009)
proposes different types of weighting matrices
$\upsilon(\mathbf{X}_{B};\boldsymbol{\beta})$. In the case of linear
regression, setting
$\upsilon(\mathbf{X}_{B};\boldsymbol{\beta})=\mathbf{X}_{B}^{\top}$ yields an
estimator with a ratio-type bias correction, similar in spirit to Scheuren and
Winkler, (1993). Setting
$\upsilon(\mathbf{X}_{B};\boldsymbol{\beta})=\mathbf{X}_{B}^{\top}\mathbf{Q}^{\top}$,
yields the Lahiri-Larsen estimator, with $\mathbf{Q}$ defined in Equation
(14). Setting
$\upsilon(\mathbf{X}_{B};\boldsymbol{\beta})=\mathbf{X}_{B}^{\top}\mathbf{Q}^{\top}(\sigma^{2}\mathbf{I}_{n_{A}}+\mathbf{\Sigma})^{-1}$,
where $\sigma^{2}=\operatorname{\mathbb{V}}(\mathbf{Y}_{A}|\mathbf{X}_{B})$
and
$\mathbf{\Sigma}=\operatorname{\mathbb{V}}(\boldsymbol{\Delta}\operatorname{\mathbb{E}}(\mathbf{Y}_{A}|\mathbf{X}_{B})|\mathbf{X}_{B})$,
yields the asymptotically efficient best linear unbiased estimator (BLUE) of
$\boldsymbol{\beta}$. Corresponding estimators for the case of logistic
regression have also been proposed in Chambers, (2009, pp.22-23). Chambers,
(2009) derives Sandwich-type asymptotic sampling variances of these
estimators, which account for the variability from using an estimate of
$\lambda$.

Kim and Chambers, 2012a  extend Equation (12) to the case of incomplete
linkage, when some records from both files may remain unlinked. For valid
inference, Kim and Chambers, 2012a  assume ignorable non-linkage, which
implies that unlinked records in each file are a random sample of all the
records in that file. Equivalently, the relationship between
$\mathbf{Y}^{*}_{A}$ and $\mathbf{X}_{B}$ is the same among linked and
unlinked records. This assumption is equivalent to unit non-response in
surveys occurring completely at random (Rubin,, 1976). When the non-linkage is
not ignorable, Kim and Chambers, 2012a  propose to adjust regression
coefficient estimates using inverse probability weighting (Seaman and White,,
2011).

Zhang and Tuoto, (2021) use the ELE model to develop pseudo-OLS estimators for
linear regression, that accommodate incomplete linkage and heterogeneous
linkage probabilities. Other extensions of the ELE model include its
application to multi-file linkage (Kim and Chambers, 2012b, ; Kim and
Chambers,, 2015), to estimation of survey-weighted population totals
(Chipperfield,, 2019), to survival analysis (Vo et al.,, 2024), to robust
regression in the presence of outliers (Chambers et al.,, 2022), and to
estimation using calibrated auxiliary information (Chambers and Diniz da
Silva,, 2020). Sensitivity of the linear regression estimators to the ELE
assumption has also been demonstrated, theoretically (Zhang and Tuoto,, 2021),
and via simulations (Chambers et al.,, 2019).

##  4 Simulation studies

We examine the performance of selected methods from Sections 2 and 3 using
extensive simulations. These methods encompass a wide range of linkage and
inference techniques, and depict the main performance trends. We consider two
simulation scenarios, corresponding to the primary and secondary analysis
settings.

###  4.1 Scenario 1: Primary analysis

####  4.1.1 Simulation design

We generate files $\mathbf{A}$ and $\mathbf{B}$ of sizes $n_{A}=600$ and
$n_{B}=900$, respectively. The simulations are designed as a factorial
experiment with six factors classified into five main categories:

  1. 1.

Overlap: The overlap represents the proportion of true links. It is defined as
the proportion of records from file $\mathbf{A}$ that exist in file
$\mathbf{B}$, as a fraction of the smaller file. We examine three levels of
the overlap: 100% (complete overlap), 50% (medium overlap), and 20% (low
overlap).

  2. 2.

Blocking: We consider two blocking scenarios. In the first scenario, both
files comprise 6 blocks. Consequently, there are 100 records per block in file
$\mathbf{A}$, and 150 records per block in file $\mathbf{B}$. In the second
scenario, both files comprise 60 blocks, so that each block in file
$\mathbf{A}$ contains 10 records, and each block in file $\mathbf{B}$ contains
15 records.

  3. 3.

Linking variables: We examine the effect of the discriminatory power (DP) of
the linking variables under two scenarios. In the high DP scenario, we
simulate four linking variables that represent an individual’s zip code
stratum (with 30 categories), last name (with $n_{A}+n_{B}-n_{AB}$ unique
values), race (with 5 categories), and year of birth. In the low DP scenario,
we simulate four variables to represent individual’s gender (with 2
categories), last name (with $n_{A}$ unique values), race (with 5 categories),
and year of birth. In both cases, we generate the year of birth and race from
the same distributions, but vary how the other linking variables are
generated.

  4. 4.

Variables for post-linkage analysis: For record pairs that are links, we
generate $\mathbf{Y}_{A}$ and $\mathbf{X}_{B}$ from a bivariate normal
distribution

$\displaystyle\begin{pmatrix}Y_{Al}\\\
X_{Bl}\end{pmatrix}\overset{iid}{\sim}N_{2}\left(\begin{pmatrix}0\\\
0\end{pmatrix},\begin{pmatrix}4&2\rho\\\ 2\rho&1\end{pmatrix}\right).$ (15)

Under this setup, a linear regression of $\mathbf{Y}_{A}$ on $\mathbf{X}_{B}$
is of the form ${Y}_{Al}=\beta X_{Bl}+\epsilon_{l}$, where $\beta=2\rho$, and
$\epsilon_{l}\overset{iid}{\sim}N(0,4(1-\rho^{2}))$. We vary the value of
$\rho$, which controls the signal-to-noise ratio (SNR) of the regression
model. In the high SNR scenario, we set $\rho=0.9$, and in the low SNR
scenario, we set $\rho=0.4$. Among non-links, we generate $\mathbf{Y}_{A}$ and
$\mathbf{X}_{B}$ independently as $Y_{Ai}\overset{iid}{\sim}N(0,1)$ and
$X_{Bj}\overset{iid}{\sim}N(0,1)$.

  5. 5.

Measurement error: We induce measurement errors in two out of the four linking
variables: the last name and the year of birth. We vary the level and the
mechanism of error generation.

    1. (a)

Error level: We consider two error levels: 10% (low) and 40% (high).

    2. (b)

Error mechanism: Within the high and low error levels, we generate measurement
errors according to the LCAR, SNL, NL, WNL, and IL mechanisms. Under LCAR, we
randomly select records from the two files and introduce errors in the last
name and the year of birth. Under SNL, we let the probability of error in a
record to depend on an individual’s race. Under NL and WNL, we let the
probability of error in a record depend on $\mathbf{X}_{B}$, and
$\mathbf{Y}_{A}$, respectively. Under IL, this probability depends on an
unobserved variable which is associated with $\mathbf{Y}_{A}$ and
$\mathbf{X}_{B}$.

In summary, we generate $2^{4}\times 3\times 5=240$ simulation configurations.
For each configuration, we generate 100 simulation replications. We describe
the data generation process in the supplementary materials.

####  4.1.2 Evaluation metrics

We evaluate the performance of seven methods when estimating the regression
slope, $\beta$. For each method, in each simulation replication, we record the
point estimate $\hat{\beta}$, its estimated variance, and whether a 95%
interval estimate covers the true value of $\beta$. We summarize the results
across all replications by calculating the mean bias of the point estimate,
the average estimated standard error (SE), and the coverage rate of the 95%
interval estimate.

####  4.1.3 Record linkage and inference methods

We link files $\mathbf{A}$ and $\mathbf{B}$ using the FS method, implemented
in the fastlink package (Enamorado et al.,, 2019) in R. Using the linked file,
we obtain an estimate of the slope, $\hat{\beta}$, its sampling variance, and
a 95% confidence interval for $\beta$.

We implement the weighted estimated equations approach of Han and Lahiri,
(2019) in the complete overlap scenario. We estimate the matrix of posterior
matching probabilities, $\mathbf{\hat{Q}}$, using the fastLink package in R.
We consider three versions of the matrix $\mathbf{\hat{Q}}$: (i) the full
matrix without any modifications (HLF); (ii) in every row, we set all entries
except the two highest to zero (HL2); and (iii) in every row, we set all
entries except the highest to zero (HL1). We obtain the estimate $\hat{\beta}$
using the blocked version of Equation (12) (Han and Lahiri,, 2019, p.5). To
estimate the sampling error of $\hat{\beta}$, we follow the jacknife
resampling approach in Han and Lahiri, (2019). In each iteration of the
jacknife, we leave out records from a block in both files, so that the number
of jacknife replicates is equal to the number of blocks. We aggregate results
across all jacknife replicates to obtain the sampling variance of
$\hat{\beta}$, and a 95% confidence interval for $\beta$.

We also examine the pseudo-likelihood method of Hof and Zwinderman, (2012)
(abbreviated as HZ). To construct the pseudo-likelihood for the distribution
of $\mathbf{Y}_{A}$ given $\mathbf{X}_{B}$, we use Equation (8) from Hof and
Zwinderman, (2012, p.79). We calculate the probabilities for the observed
agreement patterns using the fastLink package in R. We obtain the point
estimate $\hat{\beta}$ using the EM algorithm (Hof and Zwinderman,, 2012,
Appendix B). To compute the sampling variance of $\hat{\beta}$, we invert the
observed Fisher information matrix (Hof and Zwinderman,, 2012, p.80). We
compute a 95% confidence interval for $\beta$ under asymptotic normality of
the sampling distribution of $\hat{\beta}$.

We use four Bayesian methods to link files $\mathbf{A}$ and $\mathbf{B}$, and
estimate the slope $\beta$. The first method is that of Sadinle, (2017)
(abbreviated as SL), which we apply independently within each block. We assume
a uniform prior distribution over the parameters of the record linkage
likelihood in Equation (8). We use exact agreement to compare the zip stratum,
gender, race, and the year of birth. We model the similarity in the last names
using the Levenshtein edit $(LD)$ distance (Levenshtein,, 1966). We
standardize the $LD$ distance to obtain a final measure in the range of
$[0,1]$, where 0 represents total agreement and 1 represents total
disagreement. Following Sadinle, (2017), we categorize $LD$ into four levels
of disagreement: perfect agreement ($LD$ = 0), mild disagreement ($0<LD\leq
0.25$), low disagreement ($0.25<LD\leq 0.5$), and extreme disagreement
($LD\geq 0.5$). We generate $1000$ samples from the posterior distribution of
the linkage structure, using a Gibbs sampler implemented in the R package BRL
(Sadinle,, 2020).

The second Bayesian method is that of Steorts, (2015) (abbreviated as ST),
which we apply independently within each block. To implement the linking
algorithm, we concatenate records from each block in the two files into a
single file, and perform deduplication on the concatenated file. In the record
linkage model, we assume that the year of birth, zip stratum, race, and gender
are categorical variables, and use the $LD$ distance to model similarities on
the last name. We set prior distributions and hyperparameters following the
recommendations in Steorts, (2015, pp.9-17). We provide specific details of
the hyperparameter values in the supplementary materials. We generate 1000
samples from the posterior distribution of the linkage structure, using the
Gibbs sampler implemented in the R package blink (Steorts,, 2020).

The third Bayesian method is that of Kamat et al., (2023) (abbreviated as
KSG), which augments the likelihood in Equation (8) with associations between
$\mathbf{Y}_{A}$ and $\mathbf{X}_{B}$. For record pair $(i,j)$, we assume
$Y_{Ai}|X_{Bj}\sim N(\beta_{0M}+\beta_{1M}X_{Bj},\sigma_{M}^{2})$ if the pair
is a link, and $Y_{Ai}|X_{Bj}\sim
N(\beta_{0U}+\beta_{1U}X_{Bj},\sigma_{U}^{2})$ if the pair is a non-link. As
in the SL algorithm, we assume uniform prior distributions on the parameters
defining the similarity on the zip stratum, gender, race, year of birth and
last name. To complete the Bayesian model, we assume the improper prior
distributions $p(\boldsymbol{\beta}_{M},\sigma_{M})\propto\sigma_{M}^{-2}$ and
$p(\boldsymbol{\beta}_{U},\sigma_{U})\propto\sigma_{U}^{-2}$. We adjust for
blocking following the method described in Kamat et al., (2023). We generate
1000 samples from the posterior distribution of the linkage structure using
the DA algorithm.

The fourth Bayesian method is that of Gutman et al., (2013) (abbreviated as
GT). We apply this method in the complete overlap scenario only. We extend the
likelihood in Gutman et al., (2013) to account for associations between
variables common to both the files. We model the agreements on the year of
birth, zip stratum, race, and gender as Bernoulli random variables. We specify
Beta(1,1) prior distributions on the parameters of the Bernoulli distribution.
In addition, we model the number of record pairs with LD distances in the
ranges $(0,0.25)$, $(0.25,0.5)$, and $(0.5,1)$ using a multinomial
distribution with three parameters. We specify a Dirichlet (1,1,1) prior on
each of the parameters. We describe the complete model specification in the
supplementary materials. We generate 1000 samples from the posterior
distribution of the linkage structure, using the DA algorithm in Gutman et
al., (2013). In each DA iteration, we perform $50,000$ record-pair-switches
per block.

For all Bayesian methods, in each posterior sample of the linkage structure,
we compute the estimated regression coefficient $\hat{\beta}$ among the
identified links, and its sampling variance. We obtain point and interval
estimates across all posterior samples using the MI combining rules described
in Section 2.2.

####  4.1.4 Results

Within each level of the overlap, we identify three simulation factors that
have the largest effect on the bias, SE, and coverage, using ANOVA (Cangul et
al.,, 2009). The error mechanism, error level, and the discriminatory power
are generally the most influential factors across the different evaluation
metrics. We report results by averaging over levels of the remaining factors
(the block size and the SNR).

Table 1 summarizes the performance for the complete overlap configuration,
when the linking variables have high discriminatory power, and there is low
measurement error. Across all linkage mechanisms, the bias of $\hat{\beta}$ is
generally negative, as false links attenuate the estimates towards zero. The
FS estimator of $\beta$ underestimates the true value, and has coverage below
the nominal. When the mechanism is LCAR, SNL, or NL, the Bayesian, weighting,
and likelihood-based methods have generally small biases. For these linkage
mechanisms, the Bayesian methods have the smallest biases, with SL being the
most efficient. Moreover, their interval estimates have coverages at or above
the nominal. Weighting methods have relatively larger biases, but produce
interval estimates with near-nominal coverage. Across the LCAR, SNL, and NL
mechanisms, HZ exhibits small biases, but also has the largest SEs among all
methods, which results in coverage rates well above 95%.

Table 1: Bias, estimated SE, and coverage probabilities for estimates of
$\beta$ in the primary analysis scenario with complete overlap. The
measurement error level is low, the linking variables have high discriminatory
power.
Linkage Mechanism | Method | Bias | SE | Coverage
---|---|---|---|---
LCAR | FS | -0.099 | 0.039 | 0.64
SL | -0.002 | 0.059 | 0.94
ST | -0.007 | 0.065 | 0.94
GT | 0.010 | 0.069 | 0.94
KSG | 0.009 | 0.069 | 0.94
HLF | -0.058 | 0.066 | 0.93
| HL2 | -0.058 | 0.066 | 0.92
| HL1 | -0.044 | 0.066 | 0.94
| HZ | 0.011 | 0.137 | 1.00
SNL | FS | -0.094 | 0.039 | 0.64
SL | -0.003 | 0.059 | 0.94
ST | -0.007 | 0.065 | 0.94
GT | 0.011 | 0.068 | 0.94
KSG | 0.013 | 0.068 | 0.94
HLF | -0.064 | 0.066 | 0.93
| HL2 | -0.064 | 0.066 | 0.93
| HL1 | -0.049 | 0.066 | 0.93
| HZ | 0.014 | 0.127 | 1.00
NL | FS | -0.097 | 0.039 | 0.54
SL | -0.003 | 0.059 | 0.94
ST | -0.006 | 0.069 | 0.95
GT | 0.009 | 0.069 | 0.94
KSG | 0.011 | 0.068 | 0.94
HLF | -0.050 | 0.067 | 0.95
| HL2 | -0.050 | 0.067 | 0.95
| HL1 | -0.034 | 0.067 | 0.95
| HZ | -0.084 | 0.139 | 0.99
WNL | FS | -0.125 | 0.061 | 0.42
SL | -0.085 | 0.060 | 0.82
ST | -0.143 | 0.061 | 0.31
GT | -0.070 | 0.067 | 0.88
KSG | -0.072 | 0.069 | 0.90
HLF | -0.094 | 0.068 | 0.90
| HL2 | -0.095 | 0.068 | 0.89
| HL1 | -0.076 | 0.068 | 0.90
| HZ | -0.111 | 0.142 | 0.99
IL | FS | -0.172 | 0.059 | 0.16
SL | -0.091 | 0.060 | 0.70
ST | -0.204 | 0.058 | 0.04
GT | -0.103 | 0.066 | 0.74
KSG | -0.124 | 0.068 | 0.63
HLF | -0.111 | 0.068 | 0.88
| HL2 | -0.112 | 0.068 | 0.86
| HL1 | -0.088 | 0.068 | 0.88
| HZ | -0.162 | 0.157 | 1.00

The bias under all the methods increases when the linkage mechanism is either
WNL or IL. Under WNL, GT, KSG, and HL1 produce estimates with relatively
smaller bias, but have coverage rates below the nominal. Estimates under HZ
have high bias, but inflated SEs result coverage probabilities above the
nominal. Under IL, the bias under all the methods increases considerably. The
coverage is below-nominal across all the Bayesian methods. The weighting
methods have coverage near 88%, while HZ has above-nominal coverage.

Increasing the measurement error has a noticeable impact on the performance of
all the methods. When the linking variables are highly discriminatory,
increase in the error level leads to larger biases under HZ and the weighting
methods (Table S1 in the supplementary materials). In spite of the increased
error, the Bayesian methods have relatively small biases under LCAR, SNL, and
NL. Under WNL and IL, there is a substantial decrease in performance across
all the methods. Under these linkage mechanisms, HZ provides valid coverage at
the cost of higher SEs, and ST has the lowest coverage. The results are
qualitatively similar when the discriminatory power is low (Tables S2, S3 in
the supplementary materials).

Table 2: Bias, estimated SE, and coverage probabilities for estimates of
$\beta$ in the primary analysis scenario with medium and low overlap. The
measurement error level is low, the linking variables have high discriminatory
power.
|  | Medium Overlap | Low Overlap
---|---|---|---
Linkage Mechanism | Method |  Bias | SE | Coverage |  Bias | SE | Coverage
LCAR | FS | -0.158 | 0.098 | 0.51 | -0.314 | 0.132 | 0.38
SL | -0.109 | 0.115 | 0.90 | -0.280 | 0.207 | 0.79
| ST | 0.002 | 0.093 | 0.96 | 0.003 | 0.147 | 0.95
| KSG | -0.095 | 0.117 | 0.92 | -0.242 | 0.210 | 0.82
| HZ | 0.017 | 0.172 | 0.99 | 0.006 | 0.163 | 0.97
SNL | FS | -0.147 | 0.087 | 0.58 | -0.293 | 0.134 | 0.44
SL | -0.110 | 0.115 | 0.89 | -0.282 | 0.207 | 0.78
| ST | 0.003 | 0.092 | 0.95 | 0.010 | 0.148 | 0.94
| KSG | -0.103 | 0.118 | 0.91 | -0.236 | 0.209 | 0.80
| HZ | 0.012 | 0.178 | 1.00 | 0.079 | 0.185 | 0.95
NL | FS | -0.208 | 0.091 | 0.53 | -0.295 | 0.139 | 0.36
SL | -0.125 | 0.120 | 0.89 | -0.325 | 0.216 | 0.70
| ST | -0.005 | 0.098 | 0.91 | -0.001 | 0.158 | 0.96
| KSG | -0.108 | 0.128 | 0.90 | -0.279 | 0.224 | 0.72
| HZ | 0.010 | 0.175 | 0.98 | -0.001 | 0.194 | 0.96
WNL | FS | -0.239 | 0.085 | 0.29 | -0.472 | 0.134 | 0.19
SL | -0.137 | 0.118 | 0.85 | -0.335 | 0.213 | 0.66
| ST | -0.153 | 0.088 | 0.55 | -0.170 | 0.143 | 0.78
| KSG | -0.130 | 0.129 | 0.87 | -0.321 | 0.217 | 0.69
| HZ | -0.127 | 0.198 | 0.98 | -0.160 | 0.200 | 0.92
IL | FS | -0.279 | 0.083 | 0.15 | -0.532 | 0.123 | 0.17
SL | -0.160 | 0.120 | 0.79 | -0.377 | 0.218 | 0.61
| ST | -0.254 | 0.085 | 0.14 | -0.280 | 0.141 | 0.46
| KSG | -0.154 | 0.126 | 0.80 | -0.351 | 0.226 | 0.64
| HZ | -0.204 | 0.187 | 0.94 | -0.246 | 0.207 | 0.84

Table S5 depicts the results under the medium and low overlap scenarios, when
the linking variables have high discriminatory power, and there is low
measurement error. Under LCAR, SNL, and NL, estimates under ST have the lowest
bias, near-nominal coverage rates, and are the most efficient. Estimates under
HZ also have low bias and valid coverage, but with larger standard errors
compared to ST. The performance of SL and KSG declines when the overlap is not
complete, with KSG offering lower biases and better coverage rates than SL.
Under WNL and IL, estimates under ST have larger biases, and below-nominal
coverage. Under the medium overlap scenario, KSG and HZ have relatively low
biases, with KSG exhibiting smaller SEs, but below-nominal coverage. The
relative performance of the methods remains qualitatively unchanged when the
error level increases (Table S4), or when the linking variables have low
discriminatory power (Tables S5, S6).

We next examine the performance of the methods when the relative sizes of the
two files change. Specifically, we let $n_{A}=600$ and $n_{B}=1800$, and
consider the scenario when the linkage mechanism is IL, the error level is
high, and the linking variables are highly discriminatory. We present results
in Table S7 of the supplementary materials. Across all levels of the overlap,
the biases under all methods generally persist at the same level as when
$n_{B}=900$ (Tables S1 and S4). However, the estimated SEs are lower, leading
to reduced coverage rates.

We also evaluate the performance of GT and KSG, when the dependence between
$\mathbf{Y}_{A}$ and the comparison vectors is incorporated into the models.
For both methods, we let $Y_{Ai}|X_{Bj}\sim
N(\beta_{0M}+\beta_{1M}X_{Bj}+\beta_{2M}\gamma_{ijl},\sigma_{M}^{2})$ among
the links. Under KSG, we also model $Y_{Ai}|X_{Bj}\sim
N(\beta_{0U}+\beta_{1U}X_{Bj}+\beta_{2U}\gamma_{ijl},\sigma_{U}^{2})$ among
the non-links. Here, $\gamma_{ijl}$ denotes the comparison on the last name.
We do not include comparisons on the other linking variables, to avoid
singularity in the regression model. We consider the scenario when the overlap
is complete, the error level is high, and the linking variables are highly
discriminatory. Table S8 in the supplementary materials depicts the bias,
estimated SEs and coverage probabilities. Under the LCAR and SNL mechanisms,
the results are qualitatively similar to Table S1. Under NL, WNL, and IL, both
methods show improvement in the bias and the coverage.

###  4.2 Example: Model-based clustering with linked data

We illustrate the use of MI with linked data, when the downstream analysis
involves complex modeling. We consider files $\mathbf{A}$ and $\mathbf{B}$,
comprising $n_{A}=n_{B}=600$ records, with complete overlap. For record $i$ in
both files, we generate a random vector from a mixture of four-dimensional
Normal distributions. For mixture component $c\in\\{1,\dots,4\\}$, let
$(Y_{A1i},Y_{A2i},X_{B1i},X_{B2i})^{{}^{\prime}(c)}\sim
N_{4}(c\mathbf{u}_{c},c\mathbf{\Sigma}_{c})$, where $u_{c}$ is a unit vector
with 1 at the $c^{th}$ position, and 0 elsewhere, and
$\mathbf{\Sigma}_{c}=\bigg{(}\begin{smallmatrix}0.5&0.1&0.1&0.1\\\
0.1&0.5&0.1&0.1\\\ 0.1&0.1&0.5&0.1\\\
0.1&0.1&0.1&0.5\end{smallmatrix}\bigg{)}$. The proportions of records assigned
to the four clusters are $\\{0.4,0.3,0.2,0.1\\}$. This design ensures that the
clusters are relatively well-separated (Lee and Harel,, 2022). We assign the
variables $\mathbf{Y}_{A1}$ and $\mathbf{Y}_{A2}$ to file $\mathbf{A}$, and
$\mathbf{X}_{B1}$ and $\mathbf{X}_{B2}$ to file $\mathbf{B}$. The goal of the
analysis is to estimate (i) the population mean vector; and (ii) the
correlation between $\mathbf{Y}_{A1}$ and $\mathbf{X}_{B1}$ within each
cluster, after the files have been linked.

We let both files comprise three linking variables drawn from a discrete
uniform distribution over 5, 10, and 20 categories, respectively. We link the
two files using the SL algorithm, using binary agreement to construct
comparison vectors. Within each posterior sample of the linkage structure, we
fit a four-component Gaussian mixture model via the EM algorithm implemented
in the mclust package (Scrucca et al.,, 2016) in R. We estimate the cluster
means and correlations within each posterior sample of the linkage structure,
and combine these estimates using the MI combining rules from Section 2.2.
Table 3 depicts the results across 1000 posterior samples. The imputation
procedure recovers the true cluster means, correlations, and mixing
proportions with reasonable accuracy.

In this example, the mixture components can be recovered well, as all
components are well-separated. In more complex scenarios, label-invariance of
the mixture model may require restrictions on the ordering of the cluster
means. This issue is also relevant to other multivariate techniques like
factor analysis (Nassiri et al.,, 2018), where factor loadings may change
order across posterior samples of the linkage structure. With added
computation, fully Bayesian estimation can also be performed using a version
of the DA algorithm. As the weighting methods are restricted to generalized
linear models, they cannot be directly applied to this example.

Table 3: Cluster means, correlations, and mixing proportions across 1000
posterior samples of the linkage structure. The linkage structure is estimated
using the SL algorithm.
Cluster |  Estimated mean vector |  True mean vector |  Estimated correlation |  True correlation |  Estimated mixing proportion |  True mixing proportion
---|---|---|---|---|---|---
1 | (1.191, 0.046, -0.021, -0.059) | (1, 0, 0, 0) | 0.195 | 0.2 | 0.375 | 0.4
2 | (-0.019, 2.020, 0.031, 0.020) | (0, 2, 0, 0) | 0.195 | 0.2 | 0.295 | 0.3
3 | (0.021, -0.026, 2.930, -0.024) | (0, 0, 3, 0) | 0.195 | 0.2 | 0.213 | 0.2
4 | (0.058, -0.013, 0.068, 3.736) | (0, 0, 0, 4) | 0.195 | 0.2 | 0.112 | 0.1

###  4.3 Scenario 2: Secondary analysis

####  4.3.1 Simulation design

In this scenario, we generate a linked file of size $n_{AB}=600$. For record
$l$ in the linked file, we generate $(Y_{Al},X_{Bl})$ based on Equation (15).
We examine two levels of $\rho$: $\rho=0.9$ (high SNR) and $\rho=0.4$ (low
SNR).

The $n_{AB}$ linked records are partitioned into three blocks of sizes
$n_{1}=100$, $n_{2}=200$, and $n_{3}=300$. Following Chambers, (2009), we
constrain the linkage errors to be within blocks. In the low linkage error
scenario, we let the error rates within the three blocks to be
$\\{5\%,10\%,15\%\\}$. In the high linkage error scenario, we let the error
rates to be $\\{15\%,20\%,25\%\\}$. We consider four mechanisms for generating
the linkage errors: (i) the LCAR/ELE mechanism; (ii) the NL mechanism; (iii)
the WNL mechanism, and (iv) the IL mechanism, where the linkage errors depend
on an unobserved variable associated with $\mathbf{Y}_{A}\text{ and
}\mathbf{X}_{B}$. Under each mechanism, we generate errors by shuffling the
values of the outcome $\mathbf{Y}_{A}$. We provide a detailed description of
the data generation process in the supplementary materials.

In summary, we generate $2^{4}=16$ simulation configurations, which are
implemented as a full factorial design. In each configuration, we generate 100
simulation replications.

####  4.3.2 Inference methods

We compare five methods for inference on $\beta$, the slope of the regression
of $Y_{A}$ on $X_{B}$. We first consider the naive estimator of $\beta$, that
does not account for possible linkage errors. The second inferential method is
the weighted estimating equations approach of Chambers, (2009). We consider
three weighting estimators: the bias-adjusted ratio type estimator (ChR), the
Lahiri-Larsen estimator under the ELE assumption (ChL), and the BLUE (ChB).
These estimators require an estimate of the probability of correct linkage per
block, $\hat{\lambda}_{b},b=1,2,3$. To obtain $\hat{\lambda}_{b}$, we use a
random audit sample of size $n_{b}/2$ in block $b$, and record the proportion
of correct links $n_{bc}$. We set
$\hat{\lambda}_{b}=\min\\{n_{b}^{-1}(n_{b}-1),\max\\{n_{b}^{-1},n_{bc}\\}\\}$.
This method avoids overly optimistic estimates of $\lambda_{b}$ when it is
close to one (Chambers,, 2009). We compute standard errors for each estimator
using the Sandwich variance formulae in Chambers, (2009, pp.13-17).

The third inferential method is that of Gutman et al., (2016) (abbreviated as
GT). We fit a mixture of linear regressions independently within each block.
We assume diffuse proper prior distributions on the regression model
parameters and the mixture class probabilities. We generate $1000$ samples
from the posterior distribution of the mixture model parameters, using the DA
algorithm. In each posterior sample, we estimate $\beta$, and its sampling
variance among record pairs classified as links. We compute point and interval
estimates using MI (Section 2.2).

We also examine the likelihood-based method of Slawski et al., (2021)
(abbreviated as SLW). Under this method, we fit a two-class mixture model
using the EM algorithm, and obtain an estimate $\hat{\beta}$ as described in
Slawski et al., (2021, pp. 996-997). We compute the covariance matrix as
described in Appendix C of Slawski et al., (2021). We obtain confidence
intervals for $\beta$ under an asymptotic normal approximation for the
sampling distribution of $\hat{\beta}$.

We summarize results across all simulation replications in terms of the mean
bias, the average estimated standard error (SE), and the coverage rate of the
95% interval estimate.

####  4.3.3 Results

Table S11 depicts the results when the linkage error levels are low, and the
SNR is high. The naive estimator underestimates $\beta$ across all the linkage
mechanisms, and has below-nominal coverage. All the other estimation methods
reduce the linkage error bias. Under the ELE mechanism, all methods (except
the naive) yield estimates with a small bias and above-nominal coverage, with
GT and SLW having the smallest SEs. Under the NL mechanism, all methods yield
coverage rates close to 95%, with SLW being the most efficient.

The bias under all methods increases when the linkage mechanism is WNL. The
increased bias leads to interval estimates with coverages close to 70% for
ChR, ChL, and 50% for ChB. GT and SLW have smaller biases and SEs, with
coverage rates of 77% and 73%, respectively. The decrease in performance is
more apparent under IL, where ChR, ChL, and ChB have biases that are larger
than the naive estimator. The biases and SEs under GT and SLW are lower than
the weighting methods, but coverage is below the nominal.

Table S9 in the supplementary materials displays the results when the linkage
error level and the SNR are low. Across all the linkage mechanisms, the naive
estimator continues to show large negative biases, with coverage rates below
the nominal. Under the ELE mechanism, the weighting methods have low biases
with coverage rates near 95%. Biases under the weighting methods increase
under the NL, WNL, and IL mechanisms, with coverage rates near 95% under NL,
and near 85% under WNL and IL. Across all linkage mechanisms, the performance
of the weighting methods is better when the SNR is low compared to when it is
high. Evidently, the linkage mechanism has a less noticeable impact on the
inferences when the association between $Y_{A}\text{ and }X_{B}$ is low. Under
all linkage mechanisms, the bias under GT and SLW is higher compared to the
high SNR scenario. This is because both models rely on the association between
$Y_{A}\text{ and }X_{B}$, which is attenuated in this scenario. Nonetheless,
the coverage under GT and SLW is close to nominal.

Table 4: Bias, estimated SE, and coverage probabilities for estimates of
$\beta$ in the secondary analysis scenario, when linkage error levels are low,
and the SNR is high.
Linkage Mechanism | Method | Bias | SE | Coverage
---|---|---|---|---
| Naive | -0.208 | 0.050 | 0.00
ELE (LCAR) | ChR | 0.004 | 0.077 | 0.97
ChL | 0.003 | 0.076 | 0.97
ChB | -0.001 | 0.070 | 0.99
GT | -0.011 | 0.047 | 0.97
SLW | 0.000 | 0.039 | 0.95
| Naive | -0.174 | 0.048 | 0.06
NL | ChR | 0.035 | 0.076 | 0.96
ChL | 0.033 | 0.074 | 0.95
ChB | 0.051 | 0.067 | 0.91
GT | -0.020 | 0.048 | 0.92
SLW | -0.007 | 0.040 | 0.91
| Naive | -0.115 | 0.044 | 0.22
WNL | ChR | 0.107 | 0.075 | 0.73
ChL | 0.104 | 0.074 | 0.74
ChB | 0.130 | 0.065 | 0.50
GT | -0.076 | 0.051 | 0.77
SLW | -0.047 | 0.041 | 0.73
| Naive | -0.098 | 0.038 | 0.47
IL | ChR | 0.195 | 0.076 | 0.11
ChL | 0.191 | 0.074 | 0.13
ChB | 0.183 | 0.063 | 0.13
GT | -0.078 | 0.061 | 0.80
SLW | -0.049 | 0.041 | 0.71

Table S10 in the supplementary materials depicts the results when the error
level is high and the SNR is high. Due to the higher error level, biases under
all the methods increase, especially under the WNL and IL mechanisms. The
weighting methods show the highest increase in bias, with coverage near 57%
under WNL, and 3% under IL. SLW has the smallest bias across all the linkage
mechanisms. GT and SLW have coverages near the nominal under the ELE and NL
mechanisms, but close to 50% when the linkage mechanism is WNL or IL. For the
same error level, the relative performance of the methods is qualitatively
similar when the SNR is low (Table S11).

##  5 Concluding remarks

In this article, we review inferential methods with linked data files.
Following the missing data literature, we group these methods into three
categories: likelihood and Bayesian methods, imputation methods, and weighting
methods. We outline the assumptions underlying the methods, and discuss their
possible benefits and shortcomings.

Using simulations, we evaluate the performance of selected methods for primary
and secondary analysis of linked files. In the primary analysis setting, our
simulations identify two factors that significantly affect performance: the
amount of overlap between the files, and the linkage mechanism. With high
overlap, and under LCAR, SNL, or NL, all methods yield generally valid
inferences. Under these linkage mechanisms, the model of Steorts, (2015) holds
up well with decreasing levels of overlap. Under WNL, the joint modeling
approaches from Section 2.1.2 offer some advantages over the other methods.
When the linkage mechanism is IL, there is a noticeable decline in performance
across most of the methods. The linkage mechanism also has a considerable
impact on inferences in the secondary analysis setting.

We conclude by pointing out some important and open avenues for research.
First, the area of post-linkage analysis has remained generally underdeveloped
beyond generalized linear models. In Section 4.2, we provide an illustrative
example of applying MI to cluster analysis with linked files. However, other
multivariate analysis techniques call for more sophisticated methodological
developments. Second, as indicated in the simulations, linkage mechanisms can
heavily influence post-linkage inferences. These mechanisms underpin every
record linkage algorithm, and are untestable in practice. Thus, it is crucial
to assess the sensitivity of inferences to alternative linkage mechanisms than
those assumed in the linkage process. As there are parallels between the
linkage and missing data mechanisms, sensitivity analysis methods for record
linkage can borrow from the rich missing data literature. Finally, as
demonstrated in the article, MI offers an appealing alternative to fully
Bayesian inference with linked data. However, the validity of MI inferences
under more elaborate scenarios, for example, with multiple analysis models and
uncongeniality, are unstudied.

This work was supported by the National Institute on Aging (R21AG059120) and
the Patient-Centered Outcomes Research Institute (ME-2017C3-10241). All
statements in this report, including its findings and conclusions, are solely
those of the authors and do not necessarily represent the views of the PCORI,
its Board of Governors, or the Methodology Committee.

We thank the Editor, Associate Editor, and two referees for their thoughtful
comments that improved the quality of the article.

## Supplementary Material

The supplementary material describes the data generation process for the
simulation study, presents additional simulation results, and provides
derivations referenced in the main text.

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