* Tancredi and Liseo, (2011) Tancredi, A. and Liseo, B. (2011).  A hierarchical bayesian approach to record linkage and population size problems.  The Annals of Applied Statistics, 5(2B):1553–1585.
  * Tancredi and Liseo, (2015) Tancredi, A. and Liseo, B. (2015).  Regression analysis with linked data: Problems and possible solutions.  Statistica, 75(1):19–35.
  * Tancredi et al., (2020) Tancredi, A., Steorts, R. C., and Liseo, B. (2020).  A unified framework for de-duplication and population size estimation (with Discussion).  Bayesian Analysis, 15:633 – 682.
  * Tang et al., (2020) Tang, J., Reiter, J., and Steorts, R. (2020).  Bayesian modeling for simultaneous regression and record linkage.  In Domingo-Ferrer, J. and Muralidhar, K., editors, Proceedings of the International Conference on Privacy in Statistical Databases, pages 209–223.
  * Tanner and Wong, (1987) Tanner, M. and Wong, W. (1987).  The calculation of posterior distributions by data augmentation.  Journal of the American Statistical Association, 82:528–540.
  * Taylor et al., (2024) Taylor, I., Kaplan, A., and Betancourt, B. (2024).  Fast Bayesian record linkage for streaming data contexts.  Journal of Computational and Graphical Statistics, 0(0):1–12.
  * Tepping, (1968) Tepping, B. J. (1968).  A model for optimum linkage of records.  Journal of the American Statistical Association, 63:1321–1332.
  * Tromp et al., (2011) Tromp, M., Ravelli, A. C., Bonsel, G., Hasman, A., and Reitsma, J. B. (2011).  Results from simulated data sets: Probabilistic record linkage outperforms deterministic record linkage.  Journal of Clinical Epidemiology, 64:564–572.
  * Tromp et al., (2006) Tromp, M., Reitsma, J. B., Ravelli, A. C., Méray, N., and Bonsel, G. J. (2006).  Record linkage: Making the most out of errors in linking variables.  In Proceedings of the AMIA Annual Symposium, page 779–783.
  * Tsiatis, (2006) Tsiatis, A. A. (2006).  Semiparametric Theory and Missing Data.  Springer.
  * van Buuren, (2018) van Buuren, S. (2018).  Flexible Imputation of Missing Data.  CRC press.
  * Vo et al., (2023) Vo, T. H., Chauvet, G., Happe, A., Oger, E., Paquelet, S., and Garès, V. (2023).  Extending the fellegi-sunter record linkage model for mixed-type data with application to the french national health data system.  Computational Statistics & Data Analysis, 179:107656.
  * Vo et al., (2024) Vo, T. H., Garès, V., Zhang, L.-C., Happe, A., Oger, E., Paquelet, S., and Chauvet, G. (2024).  Cox regression with linked data.  Statistics in Medicine, 43(2):296–314.
  * Wang et al., (2022) Wang, Z., Ben-David, E., Diao, G., and Slawski, M. (2022).  Regression with linked datasets subject to linkage error.  WIREs Computational Statistics, 14:e1570.
  * Winglee et al., (2005) Winglee, M., Valliant, R., and Scheuren, F. (2005).  A case study in record linkage.  Survey Methodology, 31:3–11.
  * Winkler, (1990) Winkler, W. (1990).  String comparator metrics and enhanced decision rules in the fellegi-sunter model of record linkage.  In Proceedings of the Section on Survey Research Methods, American Statistical Association, page 354–359.
  * Winkler, (1993) Winkler, W. (1993).  Improved decision rules in the fellegi–sunter model of record linkage.  In Proceedings of the Section on Survey Research Methods, American Statistical Association, pages 274–279.
  * Winkler, (2000) Winkler, W. (2000).  Frequency-based matching in fellegi-sunter model of record linkage.  In Proceedings of the Section on Survey Research Methods, American Statistical Association, pages 778–783.
  * Winkler, (2014) Winkler, W. (2014).  Matching and record linkage.  WIREs Computational Statistics, 6:313–325.
  * Winkler, (2021) Winkler, W. E. (2021).  Cleaning and using administrative lists: Enhanced practices and computational algorithms for record linkage and modeling/editing/imputation.  In Chun, A. Y., Larsen, M., Durrant, G., and Reiter, J. P., editors, Administrative Records for Survey Methodology, pages 271–296. Wiley.
  * Winkler and Thibaudeau, (1991) Winkler, W. E. and Thibaudeau, Y. (1991).  An application of the Fellegi-Sunter model of record linkage to the 1990 US decennial census.  Technical report, US Bureau of the Census.
  * Wortman and Reiter, (2018) Wortman, J. H. and Reiter, J. P. (2018).  Simultaneous record linkage and causal inference with propensity score subclassification.  Statistics in Medicine, 37(24):3533–3546.
  * Xie and Meng, (2017) Xie, X. and Meng, X.-L. (2017).  Dissecting multiple imputation from a multi-phase inference perspective: What happens when god’s, imputer’s and analyst’s models are uncongenial?  Statistica Sinica, 27:1485–1594.
  * Xu et al., (2021) Xu, H., Li, X., and Grannis, S. (2021).  A simple two-step procedure using the fellegi–sunter model for frequency-based record linkage.  Journal of Applied Statistics, 49(11):2789–2804.
  * Xu et al., (2019) Xu, H., Li, X., Shen, C., Hui, S. L., and Grannis, S. (2019).  Incorporating conditional dependence in latent class models for probabilistic record linkage: Does it matter?  The Annals of Applied Statistics, 13(3):1753 – 1790.
  * Xu et al., (2022) Xu, H., Li, X., Zhang, Z., and Grannis, S. (2022).  Score test for assessing the conditional dependence in latent class models and its application to record linkage.  Journal of the Royal Statistical Society, Series C (Applied Statistics), 71(5):1663–1687.
  * Yancey, (2000) Yancey, W. E. (2000).  Frequency-dependent probability measures for record linkage.  In Proceedings of the Section on Survey Research Methods, American Statistical Association, page 752–757.
  * Zanella et al., (2016) Zanella, G., Betancourt, B., Wallach, H., Miller, J., Zaidi, A., and Steorts, R. C. (2016).  Flexible models for microclustering with application to entity resolution.  In Proceedings of the 30th International Conference on Neural Information Processing Systems, page 1425–1433, Red Hook, NY, USA.
  * Zhang and Tuoto, (2021) Zhang, L.-C. and Tuoto, T. (2021).  Linkage-data linear regression.  Journal of the Royal Statistical Society Series A, 184(2):522–547.
  * Zhou and Reiter, (2010) Zhou, X. and Reiter, J. P. (2010).  A note on Bayesian inference after multiple imputation.  The American Statistician, 64(2):159–163.
  * Zhu et al., (2009) Zhu, V. J., Overhage, M. J., Egg, J., Downs, S. M., and Grannis, S. J. (2009).  An empiric modification to the probabilistic record linkage algorithm using frequency-based weight scaling.  Journal of the American Medical Informatics Association, 16(5):738–745.
  * Zhu et al., (2015) Zhu, Y., Matsuyama, Y., Ohashi, Y., and Setoguchi, S. (2015).  When to conduct probabilistic linkage vs. deterministic linkage? A simulation study.  Journal of Biomedical Informatics, 56:80–86.

##  1 Likelihood/Bayesian inference under SNL

Let $\hat{\boldsymbol{\beta}}^{(m)}$ be the posterior mean of
$\boldsymbol{\beta}$ obtained using the $m^{th}$ sample of the linkage
structure, $\boldsymbol{\Delta}^{(m)}$, for $m=1,\ldots,M$. A point estimate
$\hat{\boldsymbol{\beta}}$ across all $M$ samples can be computed as

$\displaystyle\hat{\boldsymbol{\beta}}$
$\displaystyle=\operatorname{\mathbb{E}}(\boldsymbol{\beta}|\mathbf{Y}_{A},\mathbf{X}_{B},\mathbf{L}_{A},\mathbf{L}_{B})=\operatorname{\mathbb{E}}(\operatorname{\mathbb{E}}(\boldsymbol{\beta}|\mathbf{Y}_{A},\mathbf{X}_{B},\mathbf{L}_{A},\mathbf{L}_{B},\boldsymbol{\Delta})|\mathbf{Y}_{A},\mathbf{X}_{B},\mathbf{L}_{A},\mathbf{L}_{B})\approx\frac{1}{M}\sum_{m=1}^{M}\hat{\boldsymbol{\beta}}^{(m)}.$
(16)

The variance of $\hat{\boldsymbol{\beta}}$ can be obtained using

$\displaystyle\begin{split}\operatorname{\mathbb{V}}(\hat{\boldsymbol{\beta}}|\mathbf{Y}_{A},\mathbf{X}_{B},\mathbf{L}_{A},\mathbf{L}_{B})=&\operatorname{\mathbb{E}}(\operatorname{\mathbb{V}}(\boldsymbol{\beta}|\mathbf{Y}_{A},\mathbf{X}_{B},\mathbf{L}_{A},\mathbf{L}_{B},\boldsymbol{\Delta})|\mathbf{Y}_{A},\mathbf{X}_{B},\mathbf{L}_{A},\mathbf{L}_{B})\\\
&+\operatorname{\mathbb{V}}(\operatorname{\mathbb{E}}(\boldsymbol{\beta}|\mathbf{Y}_{A},\mathbf{X}_{B},\mathbf{L}_{A},\mathbf{L}_{B},\boldsymbol{\Delta})|\mathbf{Y}_{A},\mathbf{X}_{B},\mathbf{L}_{A},\mathbf{L}_{B}).\end{split}$
(17)

The first term in Equation (17) is the posterior variance of
$\boldsymbol{\beta}$ averaged across all $M$ samples of $\boldsymbol{\Delta}$.
The second term in Equation (17) is given by
$M^{-1}\sum_{m=1}^{M}(\hat{\boldsymbol{\beta}}^{(m)}-\hat{\boldsymbol{\beta}})^{2}$.
This term represents the contribution of the linkage uncertainty, i.e., the
added variability due to the missing $\boldsymbol{\Delta}$. It is possible to
obtain a confidence interval for $\boldsymbol{\beta}$ analytically, using an
asymtptotic normal or a Student’s-t approximation from Section 2.2 of the main
text.

##  2 Proof of Proposition 1

The proof of part (i) is trivial, since

$\displaystyle\operatorname{\mathbb{E}}(\kappa^{*}(\mathbf{X}_{B};\tilde{\boldsymbol{\beta}})|\mathbf{X}_{B})=\upsilon(\mathbf{X}_{B};\tilde{\boldsymbol{\beta}})[\operatorname{\mathbb{E}}(\mathbf{Y}_{A}^{*}|\mathbf{X}_{B})-\mu(\mathbf{X}_{B},\tilde{\boldsymbol{\beta}})]=\upsilon(\mathbf{X}_{B};\tilde{\boldsymbol{\beta}})[(\mathbf{Q}-\mathbf{I})\mu(\mathbf{X}_{B},\tilde{\boldsymbol{\beta}})]\neq
0.$ (18)

To prove part (ii), we consider the first-order Taylor series expansion of
$\kappa^{*}(\mathbf{X}_{B};\hat{\boldsymbol{\beta}}^{*})$ about
${\boldsymbol{\beta}}$,

$\displaystyle\kappa^{*}(\mathbf{X}_{B};\hat{\boldsymbol{\beta}}^{*})\approx\kappa^{*}(\mathbf{X}_{B};{\boldsymbol{\beta}})+\frac{\partial\kappa^{*}(\mathbf{X}_{B};{\boldsymbol{\beta}})}{\partial{\boldsymbol{\beta}}}(\hat{\boldsymbol{\beta}}^{*}-{\boldsymbol{\beta}}).$
(19)

Equation (19) implies that

$\displaystyle\begin{split}\hat{\boldsymbol{\beta}}^{*}-{\boldsymbol{\beta}}&\approx\big{[}\frac{\partial\kappa^{*}(\mathbf{X}_{B};{\boldsymbol{\beta}})}{\partial{\boldsymbol{\beta}}}\big{]}^{-1}[\kappa^{*}(\mathbf{X}_{B};\hat{\boldsymbol{\beta}}^{*})-\kappa^{*}(\mathbf{X}_{B};{\boldsymbol{\beta}})]\\\
&=\big{[}\frac{\partial\kappa^{*}(\mathbf{X}_{B};{\boldsymbol{\beta}})}{\partial{\boldsymbol{\beta}}}\big{]}^{-1}[-\kappa^{*}(\mathbf{X}_{B};{\boldsymbol{\beta}})],\end{split}$
(20)

as $\kappa^{*}(\mathbf{X}_{B};\hat{\boldsymbol{\beta}}^{*})=0$. The bias will
be

$\displaystyle\begin{split}\operatorname{\mathbb{E}}(\hat{\boldsymbol{\beta}}^{*}-{\boldsymbol{\beta}}|\mathbf{X}_{B})&\approx-\big{[}\frac{\partial\kappa^{*}(\mathbf{X}_{B};{\boldsymbol{\beta}})}{\partial{\boldsymbol{\beta}}}\big{]}^{-1}\operatorname{\mathbb{E}}(\kappa^{*}(\mathbf{X}_{B};{\boldsymbol{\beta}})|\mathbf{X}_{B})\\\
&=-\big{[}\frac{\partial\kappa^{*}(\mathbf{X}_{B};{\boldsymbol{\beta}})}{\partial{\boldsymbol{\beta}}}\big{]}^{-1}\upsilon(\mathbf{X}_{B};{\boldsymbol{\beta}})[(\mathbf{Q}-\mathbf{I})\mu(\mathbf{X}_{B},{\boldsymbol{\beta}})].\end{split}$
(21)

##  3 Details of simulation studies

In this section, we describe the setup and implementation of the simulation
studies in the main text.

###  3.1 Scenario 1: Primary analysis

We generate files $\mathbf{A}$ and $\mathbf{B}$ of sizes $n_{A}=600$ and
$n_{B}=900$, respectively. The number of true links $n_{AB}$ is determined by
the overlap between the two files. For true links, we generate the linking
variables in file $\mathbf{A}$, and replicate their values in file
$\mathbf{B}$. For the non-linking record pairs, linking variables in both
files are generated independently.

####  3.1.1 Data generation

##### Linking variables in the high discriminatory power setting:

In this setting, we generate four linking variables to represent an
individual’s zip stratum, year of birth, race, and last name. We generate the
zip stratum for each record from a discrete uniform distribution over
$\\{1,\dots,30\\}$. We generate the year of birth by simulating age from a
normal distribution, $N(30,10^{2})$, and converting it to the year of birth.
Specifically, for record $r$, we generate the age
$a_{r}\overset{iid}{\sim}N(30,10^{2})$,and compute the year of birth as
$YOB_{r}=2025-a_{r}$. For every record, we generate the race from a discrete
uniform distribution over $\\{1,\dots,5\\}$. To generate last names, we use
the randomNames package in R, and generate $n_{A}+n_{B}-n_{AB}$ unique last
names (so that every record has a unique last name).

##### Linking variables in the low discriminatory power setting:

In this setting, we generate four linking variables to represent an
individual’s gender, year of birth, race, and last name. We generate gender
for each record from a discrete uniform distribution over $\\{1,2\\}$. We
generate the year of birth and race in the same manner as in the high
discriminatory power setting. We generate a set of last names using the
randomNames package, with the number of unique values set to $n_{A}$. We
assign a last name to every record by sampling with replacement from this set.

##### Variables exclusive to files $\mathbf{A}$ and $\mathbf{B}$:

We assume that the larger file ($\mathbf{B}$) includes the covariate
$\mathbf{X}_{B}$ and the smaller file ($\mathbf{A}$) includes the outcome
$\mathbf{Y}_{A}$. For the true links, we generate

$\displaystyle\begin{pmatrix}Y_{Al}\\\
X_{Bl}\end{pmatrix}\overset{iid}{\sim}N_{2}\left(\begin{pmatrix}0\\\
0\end{pmatrix},\begin{pmatrix}4&2\rho\\\ 2\rho&1\end{pmatrix}\right).$ (22)

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
$\epsilon_{l}\overset{iid}{\sim}N(0,\sigma^{2}=4(1-\rho^{2}))$. The parameter
$\rho$ controls the signal-to-noise ratio, which is defined as
$\text{SNR}=||\beta||_{2}^{2}/\sigma^{2}$. We consider $\rho\in\\{0.9,0.4\\}$.
When $\rho=0.9$, SNR $\approx 4$, indicating a strong linear relationship
between $\mathbf{Y}_{A}$ and $\mathbf{X}_{B}$, and a low noise variance. When
$\rho=0.4$, SNR $\approx 0.2$, representing increased noise variance and a
weak linear relationship between $\mathbf{Y}_{A}$ and $\mathbf{X}_{B}$. Among
non-links, we generate $\mathbf{Y}_{A}$ and $\mathbf{X}_{B}$ independently as
$Y_{Ai}\overset{iid}{\sim}N(0,1)$ and $X_{Bj}\overset{iid}{\sim}N(0,1)$.

##### Generating measurement error:

We induce measurement error in two linking variables: the last name and the
year of birth. We consider two error levels ($e$): 40% (high), and 10% (low).
To generate errors as per the LCAR mechanism, we randomly choose $en_{A}$
records from file $\mathbf{A}$, and $en_{B}$ records from file $\mathbf{B}$,
and randomly perturb the aforementioned linking variables in the chosen
records. For the SNL, NL, WNL, and IL mechanisms, we model the propensity of
each record having measurement error using logistic regression. Let
$\zeta_{r}$ represent the probability of record $r$ having measurement error.
We have

$\displaystyle\zeta_{r}$
$\displaystyle=\frac{1}{1+exp{(-(c_{1}+c_{2}V_{r}))}},$ (23)

where we set $c_{2}=1$, and choose $c_{1}$ using bisection to get the desired
error level $e$. The variable $\mathbf{V}=\\{V_{r}\\}$ is chosen based on the
linkage mechanism. Under SNL, we let $\mathbf{V}$ be a set of indicator
variables, where $V_{r}=1$, if the race for record $r$ is either category 4 or
5, and $V_{r}=0$, otherwise. Under NL, we let
$\mathbf{V}\equiv\mathbf{X}_{B}$. Under WNL, we let
$\mathbf{V}\equiv\mathbf{Y}_{A}$. Under IL, the probability of measurement
error depends on an extraneous unobserved variable associated with the
analysis variables, $\mathbf{W}=-0.3+\exp(\mathbf{Y}_{A})+0.5\mathbf{X}_{B}$.
In this case, we set $\mathbf{V}\equiv\mathbf{W}$ in Equation (23). After
computing $\zeta_{r}$, we generate an indicator variable $I_{r}\sim
Bernoulli(\zeta_{r})$, and induce errors in records with $I_{r}=1$.

We generate errors in the year of birth by shuffling its values among the
records containing error. We generate two types of errors in the last name:
(i) deletion, where a randomly chosen character is deleted from the string;
and (ii) transposition, where two characters from the string are randomly
chosen, and their position is interchanged to form a new string. We generate
each type of error with probability $e/2$.

####  3.1.2 Implementation of record linkage algorithms

##### Parameters in the Steorts, (2015) model:

The entity resolution model of Steorts, (2015) entails specification of four
parameters: (i) the maximum number of latent population entities; (ii) a
“steepness” parameter, that governs how likely it is for a string field to be
distorted to values that are far from the latent entity’s value; and (iii) two
hyperparameters $(b_{1},b_{2})$ of the Beta prior on the probability that a
record in a file is distorted on the $k^{th}$ linking variable. We set the
maximum number of latent entities to be the total sample size of the
concatenated, block-level file. Following Steorts, (2015), we set the
steepness parameter equal to 1. We set the hyperparameters $b_{1}=1$ and
$b_{2}=99$, so that the prior in (iii) has a small mean, and is reasonably
diffuse (Steorts,, 2015).

##### Modifying the Gutman et al., (2013) model:

We modify the likelihood of Gutman et al., (2013) to include comparisons
between variables common to both files. Suppose that files $\mathbf{A}$ and
$\mathbf{B}$ are divided into $H$ blocks, where each block in file
$\mathbf{B}$ is larger than the corresponding block in file $\mathbf{A}$. Let
$\boldsymbol{\theta}$ denote parameters governing the distribution of the
linking variable comparisons, and let $\boldsymbol{\tau}=(\beta,\sigma^{2})$
denote the linear regression parameters. Further, let $\mathbf{Z}_{h}$ denote
the matching permutation for block $h\in\\{1,\dots,H\\}$. We modify the
likelihood in Equations (2) and (5) of Gutman et al., (2013) as

$\mathcal{L}(\boldsymbol{\theta},\boldsymbol{\tau},\mathbf{Z}_{h})=\prod_{h=1}^{H}\big{(}\prod_{(i,j)\in\mathbf{M}_{h}}f_{1}(\mathbf{Y}_{Ai},\mathbf{X}_{Bj}|\boldsymbol{\tau},\mathbf{Z}_{h})\text{
}f_{2}(\mathbf{L}_{Ai},\mathbf{L}_{Bj}|\boldsymbol{\theta},\mathbf{Z}_{h})\big{)}\times\big{(}\prod_{j\in\mathbf{U}_{h}}f_{3}(\mathbf{X}_{Bj})f_{4}(\mathbf{L}_{Bj})\big{)},$
(24)

where $\mathbf{M}_{h}$ denotes the set of linked record pairs in block $h$,
and $\mathbf{U}_{h}$ denotes the set of unlinked records from File
$\mathbf{B}$ in block $h$. For records in $\mathbf{M}_{h}$, we model the
agreements on categorical linking variables as Bernoulli random variables.
Formally, if $((i,j):i\in\mathbf{A},j\in\mathbf{B})$ is a link, for the
$k^{th}$ categorical linking variable, we let
$\mathbbm{1}(L_{Aik}=L_{Bjk})\sim Bernoulli(\theta_{k})$. We model the number
of record pairs with Levenshtein distances in the ranges $(0,0.25)$,
$(0.25,0.5)$, and $(0.5,1)$ using a multinomial distribution with parameters
$\alpha_{L}=(\alpha_{L1},\alpha_{L2},\alpha_{L3})$. It is straightforward to
impose conjugate priors on
$\boldsymbol{\theta}=(\\{\theta_{k}\\},\alpha_{L})$, and sample from the
posterior distribution as part of the data augmentation algorithm in Gutman et
al., (2013). It is not essential to model the marginal distributions $f_{3}$
and $f_{4}$ due to the monotone pattern of missing records in every block.

###  3.2 Scenario 2: Secondary analysis

####  3.2.1 Data Generation

We generate a linked file of size $n_{AB}=600$, comprising three blocks of
sizes $n_{1}=100$, $n_{2}=200$, and $n_{3}=300$. For all records, we generate
$\mathbf{Y}_{A}$ and $\mathbf{X}_{B}$ using Equation (22), with
$\rho\in\\{0.9,0.4\\}$.

##### Generating linkage errors:

We assume that linkage errors are confined within the blocks. In the low
linkage error scenario, the block-specific linkage error rates are
$\\{e_{1},e_{2},e_{3}\\}=\\{5\%,10\%,15\%\\}$, and in the high linkage error
scenario, they are $\\{e_{1},e_{2},e_{3}\\}=\\{15\%,20\%,25\%\\}$. For each
error level, we generate linkage errors by shuffling values of the outcome. We
choose erroneous record-pairs according to three error mechanisms, as
described below.

  1. 1.

Exchangeable linkage error (ELE) or LCAR: In block $h$, we generate errors so
that for every record in one file, the probability of linking with the correct
record from the other file is $1-e_{h}$, and the probability of linking with
any incorrect record is $\frac{e_{h}}{n_{h}-1},h=1,\dots,3$.

  2. 2.

NL, WNL, and IL mechanisms: In block $h$, let $\zeta_{rh}$ represent the
probability of a linked record pair $r$ being a false link. We let

$\displaystyle\zeta_{rh}$
$\displaystyle=\frac{1}{1+exp({-(c_{1}+c_{2}V_{r})})},$ (25)

where we set $c_{2}=1$, and choose $c_{1}$ using bisection to get the desired
error level $e_{h}$. The variable $\mathbf{V}\equiv\\{V_{r}\\}$ is chosen
based on the linkage mechanism. Under NL, we let
$\mathbf{V}\equiv\mathbf{X}_{B}$. Under WNL, we let
$\mathbf{V}\equiv\mathbf{Y}_{A}$. Under IL, we set
$\mathbf{V}\equiv\mathbf{W}$, where
$\mathbf{W}=-0.3+\exp(\mathbf{Y}_{A})+0.5\mathbf{X}_{B}$. Next, we simulate
indicator variables $I_{rh}\sim Bernoulli(\zeta_{rh})$. For records with
$I_{rh}=1$, we shuffle the values of the outcome.

##  4 Additional simulation results

###  4.1 Primary analysis

Table S1: Bias, SE, and coverage probability for estimates of $\boldsymbol{\beta}$ in the primary analysis scenario with complete overlap. The measurement error level is high, the linking variables have high discriminatory power. Linkage Mechanism | Method | Bias | SE | Coverage
---|---|---|---|---
LCAR | FS | -0.097 | 0.079 | 0.67
SL | -0.008 | 0.060 | 0.94
ST | -0.011 | 0.098 | 0.93
GT | -0.020 | 0.072 | 0.91
KSG | -0.029 | 0.070 | 0.91
HLF | -0.352 | 0.076 | 0.62
| HL2 | -0.339 | 0.076 | 0.62
| HL1 | -0.327 | 0.076 | 0.62
| HZ | 0.120 | 0.141 | 0.84
SNL | FS | -0.081 | 0.080 | 0.71
SL | -0.008 | 0.060 | 0.94
ST | -0.004 | 0.096 | 0.96
GT | -0.032 | 0.075 | 0.91
KSG | -0.034 | 0.076 | 0.91
HLF | -0.338 | 0.075 | 0.67
| HL2 | -0.323 | 0.075 | 0.67
| HL1 | -0.308 | 0.076 | 0.69
| HZ | 0.111 | 0.142 | 0.90
NL | FS | -0.029 | 0.077 | 0.93
SL | -0.013 | 0.061 | 0.94
ST | 0.006 | 0.107 | 0.96
GT | -0.039 | 0.079 | 0.93
KSG | -0.032 | 0.080 | 0.93
HLF | -0.312 | 0.074 | 0.63
| HL2 | -0.305 | 0.074 | 0.63
| HL1 | -0.289 | 0.074 | 0.64
| HZ | 0.105 | 0.143 | 0.88
WNL | FS | -0.097 | 0.075 | 0.83
SL | -0.121 | 0.061 | 0.63
ST | -0.276 | 0.078 | 0.04
GT | -0.112 | 0.076 | 0.82
KSG | -0.106 | 0.076 | 0.82
HLF | -0.279 | 0.073 | 0.74
| HL2 | -0.274 | 0.074 | 0.74
| HL1 | -0.254 | 0.074 | 0.76
| HZ | -0.117 | 0.148 | 0.96
IL | FS | -0.121 | 0.073 | 0.58
SL | -0.280 | 0.062 | 0.53
ST | -0.339 | 0.078 | 0.02
GT | -0.262 | 0.081 | 0.76
KSG | -0.266 | 0.084 | 0.76
HLF | -0.278 | 0.073 | 0.75
| HL2 | -0.274 | 0.073 | 0.75
| HL1 | -0.244 | 0.073 | 0.78
| HZ | -0.157 | 0.153 | 0.94
Table S2: Bias, SE, and coverage probability for estimates of $\boldsymbol{\beta}$ in the primary analysis scenario with complete overlap. The measurement error level is low, the linking variables have low discriminatory power. Linkage Mechanism | Method | Bias | SE | Coverage
---|---|---|---|---
LCAR | FS | -0.028 | 0.065 | 0.88
SL | -0.011 | 0.062 | 0.96
ST | 0.000 | 0.066 | 0.95
GT | -0.018 | 0.069 | 0.93
KSG | -0.019 | 0.070 | 0.93
HLF | -0.064 | 0.074 | 0.84
| HL2 | -0.063 | 0.074 | 0.84
| HL1 | -0.042 | 0.075 | 0.89
| HZ | 0.177 | 0.136 | 0.72
SNL | FS | -0.028 | 0.065 | 0.88
SL | -0.010 | 0.062 | 0.96
ST | -0.001 | 0.065 | 0.97
GT | -0.018 | 0.071 | 0.93
KSG | -0.018 | 0.070 | 0.93
HLF | -0.058 | 0.074 | 0.87
| HL2 | -0.057 | 0.074 | 0.87
| HL1 | -0.038 | 0.074 | 0.91
| HZ | 0.178 | 0.136 | 0.70
NL | FS | -0.030 | 0.069 | 0.89
SL | -0.019 | 0.064 | 0.96
ST | -0.078 | 0.058 | 0.72
GT | -0.020 | 0.068 | 0.94
KSG | -0.022 | 0.070 | 0.94
HLF | -0.037 | 0.079 | 0.93
| HL2 | -0.036 | 0.079 | 0.94
| HL1 | -0.013 | 0.080 | 0.94
| HZ | 0.165 | 0.137 | 0.77
WNL | FS | -0.143 | 0.063 | 0.32
SL | -0.121 | 0.066 | 0.80
ST | -0.390 | 0.049 | 0.15
GT | -0.093 | 0.069 | 0.83
KSG | -0.090 | 0.069 | 0.83
HLF | -0.116 | 0.088 | 0.79
| HL2 | -0.115 | 0.088 | 0.79
| HL1 | -0.089 | 0.090 | 0.85
| HZ | 0.046 | 0.144 | 0.91
IL | FS | -0.179 | 0.061 | 0.16
SL | -0.127 | 0.069 | 0.60
ST | -0.410 | 0.047 | 0.03
GT | -0.109 | 0.074 | 0.74
KSG | -0.104 | 0.077 | 0.74
HLF | -0.129 | 0.090 | 0.70
| HL2 | -0.128 | 0.090 | 0.70
| HL1 | -0.097 | 0.092 | 0.79
| HZ | 0.006 | 0.146 | 0.91
Table S3: Bias, SE, and coverage probability for estimates of $\boldsymbol{\beta}$ in the primary analysis scenario with complete overlap. The measurement error level is high, the linking variables have low discriminatory power. Linkage Mechanism | Method | Bias | SE | Coverage
---|---|---|---|---
LCAR | FS | -0.219 | 0.094 | 0.45
SL | -0.038 | 0.066 | 0.90
ST | 0.002 | 0.098 | 0.94
GT | 0.017 | 0.072 | 0.93
KSG | 0.009 | 0.070 | 0.93
HLF | -0.340 | 0.096 | 0.44
| HL2 | -0.334 | 0.096 | 0.45
| HL1 | -0.325 | 0.097 | 0.44
| HZ | 0.326 | 0.127 | 0.53
SNL | FS | -0.227 | 0.092 | 0.45
SL | -0.039 | 0.065 | 0.88
ST | -0.258 | 0.111 | 0.71
GT | 0.036 | 0.069 | 0.91
KSG | 0.029 | 0.071 | 0.92
HLF | -0.329 | 0.102 | 0.43
| HL2 | -0.324 | 0.102 | 0.43
| HL1 | -0.314 | 0.103 | 0.44
| HZ | 0.202 | 0.133 | 0.59
NL | FS | -0.227 | 0.092 | 0.45
SL | -0.060 | 0.071 | 0.84
ST | 0.964 | 0.095 | 0.48
GT | -0.060 | 0.081 | 0.89
KSG | -0.063 | 0.080 | 0.88
HLF | -0.238 | 0.091 | 0.53
| HL2 | -0.236 | 0.089 | 0.52
| HL1 | -0.219 | 0.089 | 0.54
| HZ | 0.298 | 0.129 | 0.57
WNL | FS | -0.186 | 0.093 | 0.49
SL | -0.080 | 0.073 | 0.82
ST | -0.487 | 0.063 | 0.02
GT | -0.088 | 0.081 | 0.84
KSG | -0.081 | 0.084 | 0.86
HLF | -0.172 | 0.103 | 0.58
| HL2 | -0.170 | 0.103 | 0.58
| HL1 | -0.147 | 0.105 | 0.59
| HZ | 0.064 | 0.136 | 0.79
IL | FS | -0.225 | 0.083 | 0.31
SL | -0.092 | 0.075 | 0.77
ST | -0.503 | 0.064 | 0.01
GT | -0.104 | 0.088 | 0.77
KSG | -0.120 | 0.092 | 0.77
HLF | -0.172 | 0.104 | 0.58
| HL2 | -0.170 | 0.104 | 0.58
| HL1 | -0.145 | 0.105 | 0.59
| HZ | 0.017 | 0.140 | 0.82
Table S4: Bias, SE, and coverage probability for estimates of $\boldsymbol{\beta}$ in the primary analysis scenario with medium and low overlap. The measurement error level is high, and the linking variables have high discriminatory power. |  | Medium Overlap | Low Overlap
---|---|---|---
Linkage Mechanism | Method |  Mean Bias | SE | Coverage |  Mean Bias | SE | Coverage
LCAR | FS | -0.236 | 0.111 | 0.46 | -0.595 | 0.146 | 0.21
SL | -0.189 | 0.137 | 0.78 | -0.405 | 0.221 | 0.53
| ST | 0.020 | 0.140 | 0.94 | -0.009 | 0.227 | 0.93
| KSG | -0.169 | 0.141 | 0.82 | -0.319 | 0.227 | 0.60
| HZ | 0.089 | 0.159 | 0.88 | 0.020 | 0.188 | 0.88
SNL | FS | -0.246 | 0.109 | 0.48 | -0.565 | 0.146 | 0.26
SL | -0.184 | 0.136 | 0.78 | -0.404 | 0.221 | 0.55
| ST | -0.018 | 0.136 | 0.92 | 0.011 | 0.223 | 0.94
| KSG | -0.166 | 0.142 | 0.83 | -0.319 | 0.227 | 0.60
| HZ | 0.061 | 0.162 | 0.91 | 0.057 | 0.179 | 0.87
NL | FS | -0.131 | 0.106 | 0.62 | -0.432 | 0.143 | 0.30
SL | -0.221 | 0.143 | 0.68 | -0.462 | 0.224 | 0.38
| ST | 0.005 | 0.151 | 0.95 | -0.013 | 0.248 | 0.95
| KSG | -0.189 | 0.145 | 0.72 | -0.361 | 0.229 | 0.50
| HZ | 0.071 | 0.162 | 0.90 | 0.022 | 0.183 | 0.89
WNL | FS | -0.261 | 0.103 | 0.40 | -0.561 | 0.145 | 0.18
SL | -0.194 | 0.135 | 0.76 | -0.416 | 0.224 | 0.53
| ST | -0.284 | 0.112 | 0.28 | -0.278 | 0.186 | 0.65
| KSG | -0.182 | 0.138 | 0.77 | -0.314 | 0.229 | 0.56
| HZ | -0.140 | 0.166 | 0.93 | -0.164 | 0.182 | 0.88
IL | FS | -0.283 | 0.102 | 0.34 | -0.564 | 0.138 | 0.20
SL | -0.207 | 0.138 | 0.72 | -0.441 | 0.225 | 0.46
| ST | -0.337 | 0.115 | 0.15 | -0.334 | 0.189 | 0.52
| KSG | -0.185 | 0.141 | 0.75 | -0.378 | 0.231 | 0.51
| HZ | -0.166 | 0.172 | 0.91 | -0.187 | 0.190 | 0.82
Table S5: Bias, SE, and coverage probability for estimates of $\boldsymbol{\beta}$ in the primary analysis scenario with medium and low overlap. The measurement error level is low, the linking variables have low discriminatory power.  |  | Medium Overlap | Low Overlap
---|---|---|---
Linkage Mechanism | Method |  Mean Bias | SE | Coverage |  Mean Bias | SE | Coverage
LCAR | FS | -0.207 | 0.086 | 0.46 | -0.510 | 0.116 | 0.20
SL | -0.246 | 0.146 | 0.61 | -0.541 | 0.216 | 0.24
| ST | -0.013 | 0.092 | 0.91 | -0.018 | 0.152 | 0.95
| KSG | -0.212 | 0.151 | 0.66 | -0.456 | 0.226 | 0.43
| HZ | 0.122 | 0.155 | 0.81 | 0.091 | 0.166 | 0.71
SNL | FS | -0.205 | 0.086 | 0.45 | -0.498 | 0.116 | 0.23
SL | -0.246 | 0.146 | 0.61 | -0.545 | 0.216 | 0.26
| ST | -0.016 | 0.092 | 0.93 | -0.022 | 0.153 | 0.96
| KSG | -0.213 | 0.152 | 0.66 | -0.453 | 0.225 | 0.45
| HZ | 0.113 | 0.156 | 0.81 | 0.094 | 0.166 | 0.74
NL | FS | -0.220 | 0.091 | 0.45 | -0.555 | 0.120 | 0.17
SL | -0.288 | 0.158 | 0.53 | -0.628 | 0.231 | 0.23
| ST | -0.089 | 0.082 | 0.67 | -0.091 | 0.139 | 0.74
| KSG | -0.231 | 0.159 | 0.54 | -0.511 | 0.240 | 0.38
| HZ | 0.114 | 0.156 | 0.81 | 0.078 | 0.168 | 0.77
WNL | FS | -0.344 | 0.085 | 0.15 | -0.667 | 0.114 | 0.07
SL | -0.352 | 0.156 | 0.30 | -0.700 | 0.225 | 0.08
| ST | -0.405 | 0.069 | 0.26 | -0.412 | 0.117 | 0.40
| KSG | -0.333 | 0.158 | 0.32 | -0.623 | 0.229 | 0.12
| HZ | -0.022 | 0.166 | 0.89 | -0.069 | 0.182 | 0.77
IL | FS | -0.411 | 0.084 | 0.09 | -0.730 | 0.113 | 0.03
SL | -0.420 | 0.160 | 0.17 | -0.786 | 0.229 | 0.03
| ST | -0.443 | 0.068 | 0.08 | -0.452 | 0.116 | 0.28
| KSG | -0.412 | 0.162 | 0.20 | -0.708 | 0.232 | 0.10
| HZ | -0.079 | 0.170 | 0.83 | -0.151 | 0.190 | 0.68
Table S6: Bias, SE, and coverage probability for estimates of $\boldsymbol{\beta}$ in the primary analysis scenario with medium and low overlap. The measurement error level is high, and the linking variables have low discriminatory power. |  | Medium Overlap | Low Overlap
---|---|---|---
Linkage Mechanism | Method |  Mean Bias | SE | Coverage |  Mean Bias | SE | Coverage
LCAR | FS | -0.511 | 0.111 | 0.19 | -0.895 | 0.114 | 0.03
SL | -0.423 | 0.170 | 0.23 | -0.717 | 0.221 | 0.12
| ST | -0.029 | 0.143 | 0.95 | -0.105 | 0.238 | 0.91
| KSG | -0.389 | 0.173 | 0.38 | -0.607 | 0.229 | 0.20
| HZ | 0.231 | 0.145 | 0.58 | 0.163 | 0.159 | 0.56
SNL | FS | -0.504 | 0.110 | 0.17 | -0.895 | 0.112 | 0.03
SL | -0.419 | 0.171 | 0.26 | -0.712 | 0.222 | 0.12
| ST | -0.285 | 0.161 | 0.70 | -0.374 | 0.270 | 0.69
| KSG | -0.347 | 0.174 | 0.33 | -0.604 | 0.230 | 0.23
| HZ | 0.102 | 0.154 | 0.40 | 0.081 | 0.166 | 0.41
NL | FS | -0.458 | 0.111 | 0.21 | -0.871 | 0.115 | 0.02
SL | -0.484 | 0.176 | 0.18 | -0.804 | 0.226 | 0.08
| ST | -0.110 | 0.129 | 0.70 | -0.178 | 0.233 | 0.74
| KSG | -0.400 | 0.179 | 0.22 | -0.712 | 0.228 | 0.15
| HZ | 0.212 | 0.146 | 0.63 | 0.119 | 0.165 | 0.63
WNL | FS | -0.476 | 0.104 | 0.14 | -0.811 | 0.116 | 0.03
SL | -0.456 | 0.180 | 0.18 | -0.819 | 0.242 | 0.04
| ST | -0.503 | 0.092 | 0.12 | -0.516 | 0.162 | 0.37
| KSG | -0.321 | 0.185 | 0.26 | -0.709 | 0.250 | 0.19
| HZ | -0.035 | 0.157 | 0.71 | -0.108 | 0.175 | 0.62
IL | FS | -0.492 | 0.102 | 0.12 | -0.854 | 0.112 | 0.01
SL | -0.484 | 0.180 | 0.14 | -0.831 | 0.240 | 0.04
| ST | -0.517 | 0.096 | 0.07 | -0.548 | 0.170 | 0.32
| KSG | -0.412 | 0.187 | 0.23 | -0.800 | 0.244 | 0.11
| HZ | -0.047 | 0.161 | 0.64 | -0.120 | 0.179 | 0.58
Table S7: Bias, SE, and coverage probability for estimates of $\boldsymbol{\beta}$ in the primary analysis scenario, when $n_{A}=600$ and $n_{B}=1800$. The error level is high, the linking variables are highly discriminatory, and the error mechanism is IL. Overlap | Method | Bias | SE | Coverage
---|---|---|---|---
Complete | SL | -0.327 | 0.051 | 0.50
ST | -0.342 | 0.047 | 0.00
GT | -0.264 | 0.079 | 0.75
KSG | -0.269 | 0.079 | 0.75
HLF | -0.278 | 0.067 | 0.70
HL2 | -0.275 | 0.067 | 0.70
HL1 | -0.241 | 0.069 | 0.72
HZ | -0.165 | 0.144 | 0.88
Medium | SL | -0.212 | 0.110 | 0.62
ST | -0.329 | 0.102 | 0.09
KSG | -0.282 | 0.119 | 0.68
HZ | -0.167 | 0.159 | 0.84
Low | SL | -0.436 | 0.196 | 0.39
ST | -0.330 | 0.176 | 0.49
KSG | -0.370 | 0.200 | 0.49
HZ | -0.187 | 0.181 | 0.75
Table S8: Results under GT and KSG, when the record linkage models allow for dependence of $Y_{A}$ on the comparison vectors. The overlap is complete, the measurement error level is high, and the linking variables have high discriminatory power. Overlap | Method | Bias | SE | Coverage
---|---|---|---|---
LCAR | GT | -0.032 | 0.074 | 0.91
KSG | -0.033 | 0.074 | 0.91
SNL | GT | -0.033 | 0.075 | 0.91
KSG | -0.036 | 0.075 | 0.91
NL | GT | -0.028 | 0.076 | 0.93
KSG | -0.029 | 0.077 | 0.93
WNL | GT | -0.088 | 0.077 | 0.89
KSG | -0.090 | 0.079 | 0.88
IL | GT | -0.189 | 0.101 | 0.82
KSG | -0.195 | 0.099 | 0.77

###  4.2 Secondary analysis

Table S9: Bias, estimated SE, and coverage probability for estimates of $\boldsymbol{\beta}$ in the secondary analysis scenario, when linkage error levels are low, and the SNR is low. Linkage Mechanism | Method | Bias | SE | Coverage
---|---|---|---|---
| Naive | -0.102 | 0.077 | 0.76
ELE (LCAR) | ChR | 0.002 | 0.090 | 0.95
ChL | 0.001 | 0.090 | 0.94
ChB | 0.001 | 0.089 | 0.93
GT | -0.089 | 0.103 | 0.96
SLW | -0.078 | 0.103 | 0.96
| Naive | -0.088 | 0.565 | 0.91
NL | ChR | 0.024 | 0.090 | 0.97
ChL | 0.023 | 0.089 | 0.98
ChB | 0.025 | 0.089 | 0.97
GT | -0.064 | 0.103 | 0.91
SLW | -0.056 | 0.548 | 0.94
| Naive | -0.065 | 0.076 | 0.78
WNL | ChR | 0.048 | 0.090 | 0.89
ChL | 0.047 | 0.089 | 0.87
ChB | 0.047 | 0.089 | 0.88
GT | -0.045 | 0.097 | 0.94
SLW | -0.034 | 0.587 | 0.97
| Naive | -0.027 | 0.075 | 0.88
IL | ChR | 0.093 | 0.088 | 0.82
ChL | 0.091 | 0.088 | 0.84
ChB | 0.090 | 0.088 | 0.85
GT | 0.017 | 0.092 | 0.97
SLW | -0.005 | 0.487 | 1.00
Table S10: Bias, estimated SE, and coverage probability for estimates of $\boldsymbol{\beta}$ in the secondary analysis scenario, when linkage error levels are high and the SNR is high. Linkage Mechanism | Method | Bias | SE | Coverage
---|---|---|---|---
| Naive | -0.387 | 0.058 | 0.00
ELE (LCAR) | ChR | 0.006 | 0.107 | 0.99
ChL | 0.004 | 0.106 | 0.99
ChB | -0.003 | 0.099 | 1.00
GT | -0.016 | 0.055 | 0.99
SLW | 0.002 | 0.050 | 0.99
| Naive | -0.312 | 0.055 | 0.00
NL | ChR | 0.088 | 0.106 | 0.97
ChL | 0.086 | 0.105 | 0.97
ChB | 0.121 | 0.098 | 0.84
GT | -0.038 | 0.061 | 0.90
SLW | -0.015 | 0.047 | 0.93
| Naive | -0.213 | 0.050 | 0.00
WNL | ChR | 0.213 | 0.113 | 0.57
ChL | 0.207 | 0.111 | 0.57
ChB | 0.260 | 0.104 | 0.15
GT | -0.132 | 0.059 | 0.61
SLW | -0.081 | 0.044 | 0.52
| Naive | -0.104 | 0.044 | 0.26
IL | ChR | 0.351 | 0.121 | 0.03
ChL | 0.346 | 0.119 | 0.03
ChB | 0.399 | 0.113 | 0.01
GT | -0.137 | 0.073 | 0.49
SLW | -0.080 | 0.045 | 0.54
Table S11: Bias, estimated SE, and coverage probability for estimates of $\boldsymbol{\beta}$ in the secondary analysis scenario, when linkage error levels are high and the SNR is low. Linkage Mechanism | Method | Bias | SE | Coverage
---|---|---|---|---
| Naive | -0.170 | 0.078 | 0.44
ELE (LCAR) | ChR | 0.004 | 0.106 | 0.96
ChL | 0.002 | 0.105 | 0.96
ChB | 0.002 | 0.105 | 0.97
GT | -0.165 | 0.103 | 0.96
SLW | -0.152 | 0.847 | 0.82
| Naive | -0.135 | 0.078 | 0.58
NL | ChR | 0.046 | 0.105 | 0.97
ChL | 0.044 | 0.104 | 0.96
ChB | 0.048 | 0.104 | 0.96
GT | -0.064 | 0.103 | 0.91
SLW | -0.121 | 0.704 | 0.89
| Naive | -0.374 | 0.042 | 0.59
WNL | ChR | -0.181 | 0.070 | 0.76
ChL | -0.183 | 0.070 | 0.76
ChB | -0.183 | 0.069 | 0.73
GT | -0.045 | 0.097 | 0.94
SLW | -0.063 | 0.980 | 0.93
| Naive | -0.029 | 0.076 | 0.95
IL | ChR | 0.176 | 0.104 | 0.61
ChL | 0.173 | 0.104 | 0.63
ChB | 0.174 | 0.103 | 0.63
GT | 0.017 | 0.092 | 0.97
SLW | -0.012 | 0.493 | 1.00