# Unveiling Challenges in Mendelian Randomization for Gene-Environment
Interaction

Malka Gorfine1, Conghui Qu2, Ulrike Peters2, Li Hsu2
1 Department of Statistics and Operations Research, Tel Aviv University,
Tel Aviv, Israel
2 Division of Public Health Sciences, Fred Hutchinson Cancer Center,
Seattle, USA

###### Abstract

Many diseases and traits involve a complex interplay between genes and
environment, generating significant interest in studying gene-environment
interaction through observational data. However, for lifestyle and
environmental risk factors, they are often susceptible to unmeasured
confounding factors and as a result, may bias the assessment of the joint
effect of gene and environment. Recently, Mendelian randomization (MR) has
evolved into a versatile method for assessing causal relationships based on
observational data to account for unmeasured confounders. This approach
utilizes genetic variants as instrumental variables (IVs) and aims to offer a
reliable statistical test and estimation of causal effects. MR has gained
substantial popularity in recent years largely due to the success of large-
scale genome-wide association studies in identifying genetic variants
associated with lifestyle and environmental factors. Many methods have been
developed for MR; however, little work has been done for evaluating gene-
environment interaction. In this paper, we focus on two primary IV approaches:
the 2-stage predictor substitution (2SPS) and the 2-stage residual inclusion
(2SRI), and extend them to accommodate gene-environment interaction under both
the linear and logistic regression models for the continuous and binary
outcomes, respectively. Extensive simulation and analytical derivations show
that finding solutions in the linear regression model setting is relatively
straightforward; however, the logistic regression model is significantly more
complex and demands additional effort.

Key words: linear regression; logistic regression; measurement error;
interaction effect; instrumental variable; colorectal cancer; polygenic risk
score

##  1 Introduction

There is a great interest to study the interaction between genes and
environmental risk factors in complex diseases (Virolainen et al., 2023).
However, environmental risk factors are often susceptible to unmeasured
confounding, which, if not properly accounted, can yield under- or over-
estimation of the effect. To study gene-environment interaction, it is
inevitable to study the joint effect of gene and environment. If these
factors, especially environmental factors, are subject to confounding, then
proper inference of gene-environment interaction may be compromised.

In epidemiology, the approach of Mendelian randomization (MR) has evolved into
a versatile method for evaluating causal associations based on observational
data, when unmeasured confounding variables are abundant. The approach uses
genetic variants as instrumental variables (IVs), and the goal is to provide a
reliable statistical test and estimation of causal effects when unmeasured
confounding factors are present. MR capitalizes on a natural experiment in
which the genotypes are randomly assigned during meiosis, given the parents’
genes, and it is assumed that genotypes are indirectly impact the disease
status, independently from any potential confounders. A partial list of
econometrics, causal and epidemiological works presenting the basic ideas and
important discussions includes Bowden and Turkington (1984); Angrist et al.
(1996); Davey Smith and Ebrahim (2003); Katan (2004); Pearl (2000); Greenland
(2000); Martens et al. (2006); Hernán and Robins (2006) and Didelez and
Sheehan (2007).

Consider the variables $X$ and $Y$, where $X$ is the non-randomized exposure
or treatment and $Y$ is the response. Intervening on $X$ refers to the act of
setting $X$ to a specific value of our choice, which does not alter the
distributions of the other variables in the system except through the effects
induced by the changes made to $X$. Also, consider a variable denoted as
$G_{IV}$, which we aim to utilize as the IV. In our context $G_{IV}$ is the
genotype or a polygenic risk score (PRS). An unobservable variable $U$
represents the potential confounding that may exist between $X$ and $Y$. The
following are the required assumptions related to $G_{IV}$ which define an IV:

A.1

IV is associated with the exposure of interest, i.e.,
$G_{IV}\not\\!\perp\\!\\!\\!\perp X$.

A.2

IV is independent of unmeasured confounders of the exposure and outcome, i.e.,
$G_{IV}\perp\\!\\!\\!\perp U$.

A.3

IV is independent of the outcome outside of the mediating effects of the
exposure, i.e., $Y\perp\\!\\!\\!\perp G_{IV}|(X,U)$.

A wealth of literature is available reviewing various aspects of MR and IVs,
such as the underlying MR assumptions (Hernán and Robins, 2006; Didelez and
Sheehan, 2007; Glymour et al., 2012; Didelez et al., 2010), the available
methods (Angrist et al., 1996; Baiocchi et al., 2014), multiple IVs (Palmer et
al., 2012; Clarke and Windmeijer, 2012), and non-linear regression models such
as logistic, Poisson, Cox proportional hazards and additive hazards (Palmer et
al., 2011; Burgess et al., 2017; Wan et al., 2018). In short, two main IV
approaches have been extensively developed within the context of linear
models. The first approach, known as 2-stage predictor substitution (2SPS) or
2-stage least squares, starts with a linear regression model of the exposure
on the IV. The fitted value obtained from this first-stage regression is
replacing the exposure in the second-stage linear regression of the outcome
$Y$. The second approach, equally popular, is called 2-stage residual
inclusion (2SRI). In this approach, the residuals from the first-stage
regression are included as an additional covariate, along with the exposure
$X$, in the second-stage linear regression of the outcome $Y$. Both
estimators, 2SPS and 2SRI, are consistent when applied to linear models (Terza
et al., 2008; Cai et al., 2011). Extending 2SPS and 2SRI to address non-linear
models simply involves replacing the second-stage linear regression model with
alternative models such as Poisson, logistic, or Cox proportional hazard
models.

As nicely summarized by Wan et al. (2018), the literature has conflicting
views on the consistency of two-stage IV methods for nonlinear models.
Focusing on the conditional treatment effect based on observational data,
Terza et al. (2008) showed the consistency of 2SRI in a wide range of
nonlinear models and under restrictive assumptions, while this consistency has
not been established for 2SPS. As a consequence, 2SRI has gained widespread
acceptance as the preferred method in studies that deal with discrete and
survival outcomes (Wan et al., 2018). Conversely, within the context of MR,
both the 2SRI and 2SPS approaches have demonstrated bias when estimating the
log odds ratio, where a dichotomous genetic marker was utilized as IV and the
exposure $X$ was a continuous phenotype. Studies have revealed that this bias
amplifies with a greater magnitude of unmeasured confounding (Palmer et al.,
2008; Burgess and Collaboration, 2013; Wan et al., 2018). Methods leveraging
gene-environment interactions within MR analyses (Spiller et al., 2019;
Tchetgen Tchetgen et al., 2021; Spiller et al., 2022) estimate causal
associations while correcting for instrumental invalidity, in particular when
assumptions A.2 or A.3 are violated.

Because of conflicting recommendations in the existing literature and because
of the increasing popularity of applying 2SRI to control for unmeasured
confounding in clinical studies, Wan et al. (2018) revisited the conditions
under which one can establish the consistency of 2-stage IV nonlinear models.
Their main conclusions include:

  1. 1.

Previous findings on 2SRI (Terza et al., 2008) rely on an (unrealistic)
underlying assumption that only one unmeasured confounder exists.

  2. 2.

When estimating non-null conditional treatment effect, 2SRI suffers from the
same problem as 2SPS does and is not consistent in non-collapsible models even
given a perfect IV.

  3. 3.

2SRI and 2SPS estimators are consistent when estimating collapsible effect
measures (such as risk rates or risk differences) using Poisson or additive
hazards models.

  4. 4.

In practice, one may consider using collapsible models instead of non-
collapsible ones. For example, in case of rare binary events, one may consider
log-linear models instead of logistic models, if possible.

Terza et al. (2008) and Burgess et al. (2017) also considered an adjusted 2SPS
method where the fitted value of exposure $X$ and the residual from the first-
stage regression are included additively in the second-stage regression of
$Y$. Some investigators have therefore recommended this adjusted two-stage
method when the second-stage regression is logistic on the premise that it is
less biased than the unadjusted two-stage method (Burgess et al., 2017). The
discussion so far assumed no interaction between the exposure $X$ and any
other observed variables, here, genes, in the regression model of $Y$.

The present study focuses on testing and estimating the effect of interaction
between $G$ and $X$ on $Y$, as well as the main effects of $G$ and $X$, which
are necessary to facilitate the inference of the joint effect size of $G$ and
$X$. Here, $Y$ may either be a continuous or a dichotomous variable, and
linear and logistic regression models are adopted, respectively. For each
model we explore the naive estimators that ignore the existence of unmeasured
confounders, and also straightforward modified versions of 2SPS and 2SRI to
accommodate effect modification. By extensive simulation the biases of causal
effects and type-I error of these estimators are demonstrated.

Our work offers three key contributions: (i) We demonstrate the validity of
straightforward extensions of MR methods designed for linear outcomes $Y$ when
dealing with gene-environment interaction. These extensions are valid in terms
of bias and type-I error rates. (ii) In the context of binary $Y$ and logistic
regression, we elucidate the significant distinctions between the 2SPS and
2SRI methods. Furthermore, we reveal that none of these methods exhibit
consistency; however, they generally maintained type I error for gene-
environment interaction or with limited inflation, except for some scenarios.
The quest for a valid approach in the realm of non-linear regression models
remains an unresolved challenge. (iii) We apply the MR methods to a large-
scale gene-environment interaction analysis of BMI in association with
colorectal cancer risk, encompassing 44,500 cases and 52,235 controls, and
compare the performance of these methods in a real data setting. While there
is no evidence of type-I error inflation for gene-BMI interaction, it remains
unclear what the true effect sizes of gene-BMI interaction should be from the
MR results. Interestingly, we demonstrate that the true causal effect size of
BMI is likely greater than the effect size based on measured BMI.

##  2 Linear Regression

We start with a continuous outcome $Y$ such that the true model is given by

$Y=\beta_{0}+\beta_{1}X+\beta_{2}G+\beta_{3}XG+\beta_{Z}Z+\beta_{U}U+\epsilon_{Y},$
(1)

where $U$ represents unmeasured confounding variable(s) that is also affecting
$X$, $Z$ is an observed covariate affecting $Y$ and $X$,
$\beta_{j},j=1,\ldots,3,$ and $\beta_{Z}$ are regression coefficients, and
$\epsilon_{Y}$ is the residual error with mean 0 and variance
$\sigma_{Y}^{2}$, and is independent of $(X,G,Z,U)$. For simplicity of
presentation, a univariate $Z$ is considered. Without loss of generality, it
is assumed that $U$ is a mean 0 and variance $\sigma_{U}^{2}$ random variable
and is independent of $Z$. By the MR assumption, $U$ is independent of $G$ as
well as the genetic instruments defined below for the model of the exposure
$X$. The coefficient $\beta_{3}$ represents the effect-modification parameter.
Our goal is estimating $\beta=(\beta_{1},\beta_{2},\beta_{3})^{T}$ and testing
the null hypothesis $H_{0}:\beta_{3}=0$ against a two-sided alternative.

In this work we focus on the setting in which there are an observed confounder
$Z$ and an unobserved confounder $U$. In addition, it is assumed genetic
instrument variable $G_{IV}$ is available and has a relationship with $X$ as
described in the following

$X=\gamma_{0}+\gamma_{IV}G_{IV}+\gamma_{Z}Z+\gamma_{U}U+\epsilon_{X},$ (2)

where $\gamma_{0},\gamma_{IV}$, $\gamma_{Z}$, and $\gamma_{U}$ are the
regression coefficients, $\epsilon_{X}$ is the residual error with mean 0 and
variance 1 and is independent of $(G_{IV},Z,U)$ and $Y$.

As $U$ is unobserved, modeling $X$ relies on the observable data by fitting a
linear regression model, namely,

$E(X|G_{IV},Z)=\alpha_{0}+\alpha_{1}G_{IV}+\alpha_{Z}Z\,$

and getting $(\widehat{\alpha}_{0},\widehat{\alpha}_{1},\widehat{\alpha}_{Z})$
by using e.g., ordinary least squares. Given that $U$ and $\epsilon_{X}$ are
independent, the estimator
$(\widehat{\alpha}_{0},\widehat{\alpha}_{1},\widehat{\alpha}_{Z})$ is
consistent to the true value of $(\gamma_{0},\gamma_{IV},\gamma_{Z})$ in model
(2) and has the usual asymptotic normality property following the Central
Limit Theorem. Then, $X$ can be predicted by

$\widehat{X}=\widehat{\alpha}_{0}+\widehat{\alpha}_{1}G_{IV}+\widehat{\alpha}_{Z}Z$

and the residuals are given by

$\widehat{\delta}=X-\widehat{X}\,.$

Alternatively, we can consider $X$ to be predicted only by $G_{IV}$, i.e.,

$\widehat{X}_{a}=\widehat{\alpha}_{1}G_{IV}\,,$

so the residual is given by

$\widehat{\delta}_{a}=X-\widehat{X}_{a}\,.$

In the subsequent discussion, we consider six methods to estimate $\beta$,
denoted as
$\widehat{\theta}=(\widehat{\theta}_{1},\widehat{\theta}_{2},\widehat{\theta}_{3})^{T}$.
It is important to note that each approach computes $\widehat{\theta}$
differently, although we use a unified notation for ease of presentation:

1\. Naive:

The naive estimator ignores the fact that there could be unobserved
confounders and uses the following model
$E(Y|X,G,Z)=\theta_{0}+\theta_{1}X+\theta_{2}G+\theta_{3}XG+\theta_{Z}Z$ to
get $\widehat{\theta}$.

2\. 2SPS:

The two-stage prediction substitution (2SPS) under Eq. (1) starts with the
first-stage regression, where the exposure $X$ is regressed on the IVs,
$G_{IV}$, and known confounders $Z$ to obtain fitted values of the exposure,
$\widehat{X}$. In the second-stage regression, the outcome $Y$ is regressed on
the fitted values for the exposure $\widehat{X}$ from the first-stage
regression, i.e.,
$E(Y|\widehat{X},G,Z)=\theta_{0}+\theta_{1}\widehat{X}+\theta_{2}G+\theta_{3}\widehat{X}G+\theta_{Z}Z$.

3\. 2SPSadj:

Based on the regression model of $X$, the residual,
$\widehat{\delta}=X-\widehat{X}$, is obtained, which comprises of both the
unobserved confounders $U$ and the error $\epsilon_{X}$. In the second-stage
regression, the outcome $Y$ is regressed not only on the fitted values for the
exposure $\widehat{X}$ and known confounders $Z$, but also $\widehat{\delta}$
in order to capture some of the unobserved confounders $U$. That is, the
second-stage model is given by
$E(Y|\widehat{X},G,Z,\widehat{\delta})=\theta_{0}+\theta_{1}\widehat{X}+\theta_{2}G+\theta_{3}\widehat{X}G+\theta_{Z}Z+\theta_{4}\widehat{\delta}$,
with further adjustment of $\widehat{\delta}$.

4\. 2SPSa:

Sometimes only the regression coefficient associated with $G_{IV}$ is
available. An alternative to the 2SPS method is to obtain the fitted value for
$X$ based only on $G_{IV}$, i.e.,
$\widehat{X}_{a}=\widehat{\alpha}_{1}G_{IV}$. In the second-stage regression
for $Y$, the incorporation of the interaction terms between $Z$ and $G$ is
required in order to absorb the effect of $Z$ on $X$. Hence, the outcome
regression model is given by
$E(Y|\widehat{X}_{a},G,Z)=\theta_{0}+\theta_{1}\widehat{X}_{a}+\theta_{2}G+\theta_{3}\widehat{X}_{a}G+\theta_{Z}Z+\theta_{GZ}ZG$.

5\. 2SPSadj-a:

The second stage regression will additionally include the residuals
$\delta_{a}=X-\widehat{X}_{a}$ similarly to the adjusted 2SPS estimation
method, namely,
$E(Y|\widehat{X}_{a},G,Z,\widehat{\delta}_{a})=\theta_{0}+\theta_{1}\widehat{X}_{a}+\theta_{2}G+\theta_{3}\widehat{X}_{a}G+\theta_{Z}Z+\theta_{GZ}ZG+\theta_{4}\widehat{\delta}_{a}$.

6\. 2SRI:

The two-stage residual inclusion (2SRI) under Eq. (1) is an adjusted two-stage
method, where the residual $\widehat{\delta}$ from the first-stage regression
(2) is included in the second-stage regression (i.e., Eq. (1)) (Terza et al.,
2008, and references therein). The second-stage model is given by
$E(Y|X,G,Z,\widehat{\delta})=\theta_{0}+\theta_{1}X+\theta_{2}G+\theta_{3}XG+\theta_{Z}Z+\theta_{4}\widehat{\delta}$.

###  2.1 Justification of 2SPS and 2SRI approaches

We begin by taking a closer look at the 2SPS approach. Plugging in Eq. (2)
into Eq. (1), we obtain

$\displaystyle Y$ $\displaystyle=$
$\displaystyle\beta_{0}+\beta_{1}(\gamma_{0}+\gamma_{IV}G_{IV}+\gamma_{Z}Z+\gamma_{U}U+\epsilon_{X})+\beta_{2}G$
$\displaystyle+\beta_{3}(\gamma_{0}+\gamma_{IV}G_{IV}+\gamma_{Z}Z+\gamma_{U}U+\epsilon_{X})G+\beta_{Z}Z+\beta_{U}U+\epsilon_{Y},$
$\displaystyle=$
$\displaystyle{\beta}_{0}+\beta_{1}\widetilde{X}+(\beta_{2}+\beta_{3}\gamma_{U}U+\beta_{3}\epsilon_{X})G+\beta_{3}\widetilde{X}G$
$\displaystyle+\beta_{Z}Z+(\beta_{1}\gamma_{U}U+\beta_{U}U)+\beta_{1}\epsilon_{X}+\epsilon_{Y}$

where

$\widetilde{X}=\gamma_{0}+\gamma_{IV}G_{IV}+\gamma_{Z}Z\,.$

Then, taking the conditional expectation of $Y$ given $G$, $G_{IV}$, and $Z$,
we have

$\displaystyle E(Y|G,G_{IV},Z)$ $\displaystyle=$
$\displaystyle{\beta}_{0}+\beta_{1}\widetilde{X}+\beta_{2}G+\beta_{3}\widetilde{X}G+\beta_{Z}Z\,.$

This supports the use of the 2SPS approach, wherein fitting a model of $Y$
with $(\widehat{X},G,\widehat{X}G,Z)$ results in a consistent estimator for
$\beta=(\beta_{1},\beta_{2},\beta_{3})$. This justification stems from the
collapsibility property of the linear regression model, allowing for the
substitution of $\widetilde{X}$ with $\widehat{X}$. Furthermore, including
$\widehat{\delta}$ in the model for the adjusted 2SPS approach does not
compromise the consistency of the estimators for $\beta$; however, it has the
potential to decrease the standard errors. This is due to the assumption that
$\widehat{\delta}$ is independent of $\widehat{X}$ and encompasses the effects
of $U$. If the influence of $U$ is large, substantial gain in efficiency can
be achieved without jeopardizing the consistency of the estimators.

Next, when we contemplate an alternative predictor
$\widetilde{X}_{a}=\gamma_{IV}G_{IV}$, it yields

$\displaystyle Y$ $\displaystyle=$
$\displaystyle\beta_{0}+\beta_{1}(\gamma_{0}+\widetilde{X}_{a}+\gamma_{Z}Z+\gamma_{U}U+\epsilon_{X})+\beta_{2}G$
$\displaystyle+\beta_{3}(\gamma_{0}+\widetilde{X}_{a}+\gamma_{Z}Z+\gamma_{U}U+\epsilon_{X})G+\beta_{Z}Z+\beta_{U}U+\epsilon_{Y},$
$\displaystyle=$
$\displaystyle\widetilde{\beta}_{0}+\beta_{1}\widetilde{X}_{a}+(\beta_{2}+\beta_{3}\gamma_{0}+\beta_{3}\gamma_{Z}Z+\beta_{3}\gamma_{U}U+\beta_{3}\epsilon_{X})G+\beta_{3}\widetilde{X}_{a}G$
$\displaystyle+\widetilde{\beta}_{Z}Z+(\beta_{1}\gamma_{U}U+\beta_{U}U)+\beta_{1}\epsilon_{X}+\epsilon_{Y},$

where $\widetilde{\beta}_{0}$ and $\widetilde{\beta}_{Z}$ are generic symbols
for the intercept and the regression coefficient associated with $Z$. The
expectation of $Y$ conditional on $G$, $G_{IV}$ and $Z$, gives

$\displaystyle E(Y|G,G_{IV},Z)$ $\displaystyle=$
$\displaystyle\widetilde{\beta}_{0}+\beta_{1}\widetilde{X}_{a}+(\beta_{2}+\beta_{3}\gamma_{0})G+\beta_{3}\widetilde{X}_{a}G+\widetilde{\beta}_{Z}Z+\beta_{GZ}GZ\,,$

where $\beta_{GZ}$ is the regression coefficent for the interaction term
between $G$ and $Z$. Drawing from the expectation outlined above, modeling $Y$
with $(G,\widehat{X}_{a},G\widehat{X}_{a},Z,GZ)$ produces estimates that are
consistent for $\beta_{1}$ and the interaction term $\beta_{3}$. However, it
does not yield consistent estimates for the main effect of $G$, except for
when $\gamma_{0}=0$. Notably, in contrast to the adjusted 2SPS approach, this
method necessitates the inclusion of an additional interaction term $GZ$.

Moving on, let us delve into the 2SRI approach. We make an assumption that the
joint distribution of $(X,U)$, given $G_{IV}$ and $Z$, conforms to a bivariate
normal distribution with

$\displaystyle\left(\begin{array}[]{c}U\\\ X\end{array}\right)|\,G_{IV},Z$
$\displaystyle\sim$ $\displaystyle N\left(\left(\begin{array}[]{c}0\\\
\mu_{1}\end{array}\right),\left(\begin{array}[]{cc}\sigma_{U}^{2}&\gamma_{U}\sigma_{U}^{2}\\\
\gamma_{U}\sigma_{U}^{2}&\sigma_{X}^{2}\end{array}\right)\right)$

where $\mu_{1}=\gamma_{0}+\gamma_{IV}G_{IV}+\gamma_{Z}Z$ and
$\sigma_{X}^{2}=\gamma_{U}^{2}\sigma_{U}^{2}+\sigma_{\epsilon_{X}}^{2}$.
Hence, the conditional distribution of $U$ given $X$ is

$\displaystyle U|X\sim
N\left(\gamma_{U}\sigma_{U}^{2}/\sigma_{X}^{2}(X-\mu_{1}),\sigma_{U}^{2}(1-\gamma_{U}^{2}\sigma_{U}^{2}/\sigma_{X}^{2})\right)\,.$

Lastly, in the context of Eq. (1), we can calculate the expected value of $Y$
given $(X,G,Z)$, resulting in

$\displaystyle E(Y|X,G,Z)$ $\displaystyle=$
$\displaystyle\beta_{0}+\beta_{1}X+\beta_{2}G+\beta_{3}XG+\beta_{U}E(U|X,G)+\beta_{Z}Z$
$\displaystyle=$
$\displaystyle\beta_{0}+\beta_{1}X+\beta_{2}G+\beta_{3}XG+\beta_{U}\frac{\gamma_{U}\sigma_{U}^{2}}{\sigma_{X}^{2}}(X-\mu_{1})+\beta_{Z}Z$

and $\delta=X-\mu_{1}$. This computation provides a rationale for utilizing
the 2SRI estimator in the linear model. This also suggests that the naive
method provides a consistent estimator for the main effect of $G$,
$\beta_{2}$, and the interaction effect $\beta_{3}$, while it does not yield a
consistent estimator for the main effect $X$, $\beta_{1}$, the latter of which
has also been proposed by VanderWeele et al. (2012). It is worth noting that
even if we substitute $\mu_{1}$ with $\gamma_{IV}G_{IV}$, the 2SRI estimator
remains consistent for $\beta$, as the influence of $Z$ is absorbed into the
main effect of $Z$.

###  2.2 Simulation Study - Linear Regression

A total of $10,000$ observations were considered for Model (2) for the
exposure $X$ with $\gamma_{0}=0$, $\gamma_{Z}=\gamma_{IV}=0.5$,
$G_{IV}\sim\mbox{N}(0,1)$, $U\sim\mbox{N}(0,1)$, $Z\sim\mbox{N}(0,1)$ and
$\epsilon_{X}\sim N(0,1)$. The instrumental variable $G_{IV}$ represents a
polygenic risk score. Multiple values of $\gamma_{U}$ were explored,
$\gamma_{U}\in\\{0,0.5,1,2,4\\}$. The values of $X$ underwent a transformation
to standardize their variance to 1 before being used in the primary $Y$
outcome model to facilitate ease of comparison. For the linear regression
model of Eq. (1) for the outcome $Y$, $10,000$ observations were generated
with $G\sim\mbox{Bin}(2,0.3)$ representing the number of copies (0, 1 or 2) of
a single bi-allelic SNP. Also,
$(\beta_{0},\beta_{1},\beta_{2},\beta_{Z})=(0,1,0.5,0.5)$, multiple values of
$\beta_{U}$, $\beta_{U}\in\\{0,1.5,3\\}$ and $\epsilon_{Y}\sim N(0,1)$. Each
configuration was examined twice: once with $\beta_{3}=0.5$ and once under the
null hypothesis, $H_{0}:\beta_{3}=0$, to explore whether the presence of the
unmeasured variable $U$ has an impact on the type-I error. Unless stated
otherwise, the simulation results are derived from 500 repetitions of each
configuration. We also included a scenario in which $X$ and $G$ are dependent
through a dependence between $G_{IV}$ and $G$. In particular, the following
conditional distribution is adopted

$G|G_{IV}\sim\mbox{Bin}(2\,,\,0.3+0.3I\\{G_{IV}>0\\})\,.$

Table 1 provides a summary of the configurations studied.

Figures 1-2, Tables 2-3 and Tables S1-S4 of the Supporting Information (SI)
summarise the results. Figures 1-2 display results specifically for
$\gamma_{U}=1$ as the overall findings remain consistent across various values
of $\gamma_{U}$. The complete results are in Tables S1-S4 of SI. Biased
results are marked in bold.

As expected, when $\beta_{U}=0$, all the methods yield unbiased estimators for
$\beta=(\beta_{1},\beta_{2},\beta_{3})$ across all scenarios. Notably, in many
cases, the naive approach demonstrates significantly higher efficiency
compared to the other five estimators. In cases where $\beta_{U}>0$ and there
is independence between $G$ and $G_{IV}$, the naive approach exhibits bias in
estimating $\beta_{1}$ while maintaining its unbiasedness in estimating
$(\beta_{2},\beta_{3})$. All MR-based estimators are unbiased for $\beta$.
When $G$ and $G_{IV}$ are dependent, the naive estimator exhibits bias in
estimating $(\beta_{1},\beta_{2})$, while the MR estimators provide unbiased
estimations for $\beta$. Among the five MR-based estimators, 2SRI demonstrates
the highest level of efficiency, comparable to the naive estimator. For the
2SPS estimators, adjusting for $\widehat{\delta}$ improves the efficiency and
2SPS estimators that include both genetic instrument and non-genetic
predictors have better efficiency. The type-I error rates of the naive and the
MR-based approaches for testing $H_{0}:\beta_{3}=0$ against a two-sided
alternative are reasonably close to the nominal level, 0.05.

In conclusion, all methods, including the naive approach, are suitable for
conducting an interaction test. However, the naive estimator yields biased
estimators for main effects, especially the main effect of the exposure, which
makes it difficult to gauge the joint effect of exposure and $G$ accurately.
Each of the five MR-based techniques demonstrates consistency in estimating
both the main effects and interaction effect. However, among them, 2SRI
emerges as the favored option owing to its superior efficiency.

##  3 Logistic Regression

We are interested in assessing the association of exposure $X$, genotype $G$
and their interaction with binary outcome $Y$. Define
$\text{logit}(p)=\log\\{p/(1-p)\\}$. Suppose the true model is

$\displaystyle\text{logit}\\{\Pr(Y=1|X,G,Z,U)\\}$ $\displaystyle=$
$\displaystyle\beta_{0}+\beta_{1}X+\beta_{2}G+\beta_{3}XG+\beta_{Z}Z+\beta_{U}U,$
(4)

where $(X,G,Z,U)$ are defined in Section 2, and we have a genetic instrument
variable $G_{IV}$ for exposure $X$ and the model of Eq. (2) holds. Similar to
the linear setting, we adapted the naive and the five MR-based approaches to
fit the logistic regression model. Specifically, when performing the second-
stage regression of $Y$ for each of the five estimators discussed in Section
2, the linear model should be substituted with a logistic regression model
incorporating the relevant components.

While modifying the MR-based estimators to fit the logistic regression model
is simple, justifying these MR-based estimators is challenging due to the to
non-linear structure of the logit link. Here we examine the analytical form
for the 2SRI estimator . As $U$ is not directly observable, we focus on the
expected value of $Y$ given $(X,G,Z)$ using an approximation method inspired
by Carroll et al. (2006). This approximation leverages the relationship
between probit and logit models, particularly in the context of measurement
error. Namely,

$\displaystyle\text{logit}\\{E(Y|X,G,Z)\\}$ $\displaystyle\approx$
$\displaystyle\left\\{{\beta}_{0}+\beta_{1}X+\beta_{2}G+\beta_{3}GX+\beta_{U}E(U|X,G,Z)+\beta_{Z}Z\right\\}/\phi$
$\displaystyle=$
$\displaystyle\left\\{{\beta}_{0}+\beta_{1}X+\beta_{2}G+\beta_{3}GX+\beta_{U}\frac{\gamma_{U}\sigma_{U}^{2}}{\sigma_{X}^{2}}(X-\mu_{1})+\beta_{Z}Z\right\\}/\phi\,,$

where

$\phi=\left(1+\beta_{U}^{2}\operatorname{var}(U|X)/1.7^{2}\right)^{1/2}\,,$

and
$\operatorname{var}(U|X)=\sigma_{U}^{2}(1-{\gamma_{U}^{2}\sigma_{U}^{2}}/{\sigma_{X}^{2}})$.
Testing, for instance, the hypothesis that $\beta_{3}=0$ is approximately
equivalent to testing $\beta_{3}/\phi_{1}=0$. This may also explain that for
the naive approach where $\delta=X-\mu_{1}$ is not included, the bias only
impacts the main effect of $X$, but not the main effect of $G$ and the
interaction coefficient $\beta_{3}$. Adding $\widehat{\delta}$, as in the 2SRI
approach, corrects the bias for the main effect of $X$ to some extent. Even in
this situation, all parameters are attenuated by
$\left(1+\beta_{U}^{2}\operatorname{var}(U|X)/1.7^{2}\right)^{1/2}$ and when
$\beta_{U}=0$, the 2SRI estimator is consistent. We attempted to derive a
similar analytical approximation for the 2SPS estimators; however, the
approximation is rather complex and does not have a close form.

###  3.1 Simulation Study - Logistic Regression

A case-control study design with logistic regression model was employed,
involving 5,000 cases and 5,000 controls, unless otherwise specified. The
exposure $X$ was generated based on the regression model of Eq. (2) as
described in Section 2.2. For the outcome of logistic regression model of Eq.
(4), $(\beta_{0},\beta_{1},\beta_{2},\beta_{Z})=(\log(0.01/0.99),1,0.5,0.5)$
and $\beta_{U}\in\\{0,1.5,3\\}$. The remaining elements adhere to the
description provided in Section 2.2.

The results are summarized in Figures 3-8. The naive estimator for
$(\beta_{1},\beta_{2},\beta_{3})$ exhibits bias if $\beta_{U}\neq 0$, even
when $\gamma_{U}=0$. Notably, unlike the linear model, this bias pertains to
each parameter, including $\beta_{3}$ for the effect of GxE, and is not
limited to just $\beta_{1}$ for the effect of exposure $X$. This bias arises
from the non-collapsibility property inherent to logistic regression. While
the naive estimator for $\beta_{2}$ and $\beta_{3}$ is generally biased
downward, when $\gamma_{U}>0$ and $\beta_{U}=1.5$ or $3$, the naive estimator
overestimates $\beta_{1}$ (Figures 3-5). When $\beta_{U}=-3.0,-1.5$, or $-0.5$
in the opposite direction of $\gamma_{U}>0$, the naive estimator
underestimates $\beta_{1}$ (Figure 7). A similar observation is for when
$\beta_{U}$ is positive and $\gamma_{U}$ is negative (Figure 8).

A concerning aspect of the 2SPS estimators is their notable bias, even when
the naive approach exhibits only a minor empirical bias for the interaction
parameter $\beta_{3}$ (Figure 3 and 5). The adjusted 2SPS estimators, 2SPSadj
and 2SPSadj-a, improves the bias to some extent, especially when $\gamma_{U}$
is large. It appears that the only potentially valuable MR-based estimator is
the 2SRI, where it occasionally improves bias for $\beta_{3}$. For the main
effects, $\beta_{1}$ and $\beta_{2}$, the 2SPS estimators again exhibit
notable bias. In contrast, the 2SRI estimator that includes the additional
residual term $\widehat{\delta}$ frequently demonstrates lower bias for
$\beta_{1}$ compared to the naive estimator, which can be substantial, and
improves bias for $\beta_{2}$ occasionally. The naive and 2SRI estimators
display a high degree of correlation. As in the linear model of $Y$, among all
the MR-based estimators, 2SRI has the smallest standard error.

Tables 4-5 show the results of type-I error of testing the null hypothesis of
$\beta_{3}=0$. The type-I error of the naive and MR-based approaches exhibit
occasional inflation when $\beta_{U}>0$, particularly noticeable in Setting
IV. Consequently, we increased the overall sample size from 10,000 to 30,000
and expanded the number of replications from 500 to 1000 and we also explored
additional scenarios, summarized in Table 6. Our primary observation is that,
under small values of $\beta_{U}$ (e.g., $\beta_{U}=0.5$), the naive and 2SRI
estimators tend to closely approximate the nominal type-I error. However,
under large values of $\beta_{U}$, the statistical tests of the naive and 2SRI
approaches may become invalid. The 2SPS estimator sometimes have noticeable
type-I error inflation when the effect for the known confounder $Z$ is nonzero
(e.g., $\beta_{U}=0.5,\beta_{Z}=\gamma_{Z}=0.5,\beta_{2}=0.5$), but the
adjusted 2SPS estimator reduces the type-I error inflation somewhat, except
for when $\gamma_{U}=0$. The 2SPS alternative estimator based on only the
genetic instrument generally approximate the type-I error well with
occasionally limited inflation. When the effect of $U$ on $X$, i.e.,
$\gamma_{U}$, is large, all estimators appear to have a reasonable type-I
error. Interestingly, when the main effect of $G$ is null, all estimators,
including the naive estimator, maintains the correct type-I error.

##  4 An Application to Colorectal Cancer

We applied the naive (or conventional) and the five MR-based methods to assess
the genetic interaction with the body mass index (BMI) in association with
colorectal cancer (CRC) risk using the pooled data of studies from the Colon
Cancer Family Registry, the Colorectal Transdisciplinary Study, the Genetics
and Epidemiology of Colorectal Cancer Consortium, and the United Kingdom
Biobank. The details of these studies have been previously published (Huyghe
et al., 2019; Schmit et al., 2019; Schumacher et al., 2015). Briefly, the
studies were either population-based case-control studies or nested case-
control studies assembled from cohort studies via risk-set sampling. Cases and
controls were matched on age, sex, race, and enrollment date or trial group,
when applicable. Colorectal adenocarcinoma cases were confirmed by medical
records, pathological reports, or death certificate information. All
participants gave written informed consent and studies were approved by their
respective Institutional Review Boards.

Analyses were limited to individuals of European ancestry based on self-
reported race and clustering of principal components with 1000 Genomes
European population. Individuals were further excluded based on cryptic
relatedness or duplicates (prioritizing cases and/or individuals genotyped on
the better platform), genotyping or imputation errors, non-CRC outcomes, and
age outliers. The final pooled sample size was 52,235 controls and 44,500
cases. Demographics and environmental exposures, here, BMI, were self-reported
either at in-person interviews or via structured questionnaires and were
harmonized across studies through a multi-step procedure (Hutter et al.,
2012). All individuals were genotyped by microarrays and the quality control
steps had been previously published (Huyghe et al., 2019). Briefly, genotyped
SNPs were excluded based on call-rate ($<95-98\%$), lack of Hardy-Weinberg
equilibrium (p-value $<10^{-4}$), discrepancies between reported and genotypic
sex, discordant calls between duplicates. All autosomal SNPs were imputed to
the Haplotype Reference Consortium r1.1 (2016) reference panel via the
Michigan Imputation Server (Das et al., 2016). Imputed common SNPs were
restricted based on a pooled MAF $\geq 5\%$ and imputation accuracy
(R${}^{2}>0.9$). After imputation and quality control analyses, a total of
about 5.5 million common SNPs were included.

To predict BMI, we fit a linear regression model including three PRS
previously developed by (Yengo et al., 2018; Kichaev et al., 2019) (1,368
SNPs), Bull et al. (2020) (312 SNPs), and Pulit et al. (2019) (1,092 SNPs)
using the 52,235 controls from the pooled dataset. The model also includes
study, age, sex, and three principal components (PCs) for genetic ancestry.
The R2 explained by all predictors is 9.3%. All three PRSs were significantly
associated with BMI (p-values are $1.6\text{e-}4$, $1.4\text{e-}4$, and
$<2\text{e-}16$, respectively) and the three PRSs alone explained 6.2% of
variation of BMI, which is considerably greater than typical biomarkers and
lifestyle factors, see e.g., Burgess and Thompson (2013) and references
therein. Based on this model, we constructed two scores for predicting BMI.
The first overall score was the weighted sum of the three PRSs as well as
other non-genetic predictors including age, sex, PCs, and study, with weights
being the regression coefficient estimates in the linear model. The second
score was based on only the three PRS. For the 2SPS and adjusted 2SPS
estimators, we used the first overall score, and for the 2SPSa and adjusted
2SPSa we used the second PRS-only score. For the 2SRI, we calculated the
residual by subtracting the first overall score from BMI. All models (naive
and MR) were adjusted for study, age, sex, and three PCs.

Figure 9 shows the boxplots of regression coefficient estimates and standard
errors for the main effects of BMI (i.e., $X$) and $G$ and the interaction
effect between BMI and $G$ for the naive and the five MR-based estimators.
Here we focus on the results for Chromosome 1 (in each run $G$ is one of the
414,107 SNPs of Chromosome 1) to compare the relative performance of all
estimators (the results for other chromosomes are similar). As it can be seen
from Figure 9, the estimates of interaction effect of GxBMI centered around 0
for all estimators, suggesting that there was no overall GxBMI interaction.
The QQplot that compared $-\log_{10}\text{p-value}$ with the expected under
the uniform distribution also showed no evidence of inflation for GxBMI
interaction for all estimators (Figure 10). As majority of SNPs are not
associated with CRC risk, this result is consistent with our simulation
results under the main effect for $G$ being null where the type-I error for
all estimators is generally maintained (Table 6). The QQplot also shows that
there was no evidence for GxBMI interaction under the overall null. The naive
and 2SRI estimators had the smallest standard errors with mean $\sim 0.015$
for both estimators. The four variations of 2SPS estimators all have greater
standard errors with 2SPSa and adjusted 2SPSa estimators having largest
standard errors $\sim 0.063$ vs $\sim 0.050$, which may be due to the fact
that the BMI predictor for the 2SPSa estimators based on only PRS was less
predictive than that for the 2SPS estimators that included both the PRS and
other non-genetic predictors. For the 2SPS and 2SPSa, adjusting for the
residuals did not improve the efficiency. The pairwise plots also showed that
the 2SPS and adjusted 2SPS were highly correlated in both the estimates and
p-values, so were the 2SPSa and adjusted 2SPSa estimators (Figures 11 and 12).
2SPS and 2SPSa were also correlated, but not as strongly. The naive and 2SRI
estimators were highly correlated in both the estimates and p-values. However,
the 2SPS and 2SPSa estimators had little correlation with the naive and 2SRI
estimators with the 2SPS and 2SPSa estimators having wider ranges, possibly
because of their greater variances. The patterns for the main effects of $G$
and BMI were similar (Figure 9, Figures S1–S4 of SI). The notable difference
is that when the main effect is non-zero, here, BMI, the p-values for various
estimators were more correlated than when the main effect is null, here, $G$.
We conducted a simulation study mimicking the real data and found the patterns
observed for the real data were consistent to the simulated data (Figure S5 of
SI).

It is worth noting that the main effect estimates of BMI based on all the
methods had a non-zero median. While it is generally not meaningful to examine
the main effects in the presence of an interaction effect, since there was no
evidence for GxBMI interaction, we could evaluate the effect of BMI on CRC
risk. All estimators suggested that higher BMI is associated with increased
CRC risk. Interestingly, the log-odds ratio estimates were 0.228, 0.234,
0.228, 0.235, and 0.234 per 5 unit increase in BMI for the 2SPS, adjusted
2SPS, 2SPSa, adjusted 2SPSa, and 2SRI estimators, respectively, all of which
were greater than the naive estimator, 0.139. This difference, especially the
MR estimate greater than the naive estimator based on conventional
observational studies, has been noted before (Suzuki et al., 2021). Possible
explanations include inherent limitations of conventional observational
studies such as reverse causation (Mandic et al., 2023) and measurement error
(i.e. residual confounding) (Davey Smith and Ebrahim, 2003). For example, if a
particular behavior is associated with increased risk but this behavior is
also associated with under reporting BMI, failing to adjust for this behavior
will result in attenuation of the effect of BMI. This is consistent with our
simulation result when the effect of the unobserved confounder $U$ has
opposite direction for the exposure and outcome (Figure 7). Other possible
explanations may be that genetically determined BMI is likely to reflect
lifelong exposure and there is empirical evidence showing that the magnitude
of the association for lifelong exposure for the risk of disease is larger
than that for the short-term exposure beginning later in life (Ference et al.,
2012). Another possibility is that the MR estimates might have been inflated,
see for example, Figure 7; however, such possibility seems to be less likely,
in part because the inflation, if exists, limits to the 2SPS estimators not
the 2SRI estimator and in our real data analysis, 2SPS and 2SRI estimates were
consistent and they were all greater than the naive estimate.

##  5 Discussion

This research focuses on practical MR models designed for both testing and
estimating the effect of GxE on $Y$. Extensions of the 2SPS and 2SRI methods
tailored for linear and logistic regression models of the outcome variable $Y$
have been introduced and extensively evaluated through a simulation study. In
the case of the linear model, all the MR-based approaches, as well as the
naive approach, provide a valid test for GxE. However, the naive estimators
often exhibit bias. Among the MR-based methods, 2SRI is recommended for
mitigating the bias associated with the naive approach, as it is shown to be
the most efficient approach. Regarding the logistic regression model, the MR-
based methods generally maintains type-I error with limited inflation, when
the effect of unobserved confounder $U$ on $Y$ is weak or when there is no
measured confounder $Z$. All estimators were biased, though the 2SRI estimator
has the smallest bias, especially for the main effect of the exposure, which
can be substantial under the naive method. This was demonstrated by our
application to colorectal study, where, in fact, the effect of observed BMI
was substantially attenuated compared to the MR estimators.

Our research can be expanded in several directions. First, we have considered
perhaps the simplest model for the unmeasured confounder. In practice, it is
likely that the unmeasured confounder interacts with the exposure or with $G$.
Then, some of the current observations may no longer hold. Second, in the real
data analysis, three PRSs were used, each involving hundreds to over a
thousand genetic variants, as genetic instruments for BMI in order to increase
the power of the MR analysis (Burgess and Thompson, 2013). However, it is
likely that some of the genetic variants in the PRSs may not be valid IVs, in
the sense that they may have direct effects on the outcome, beside mediating
through the BMI. Including invalid IVs may bias the results. Many methods have
been developed to relax this assumption by incorporating a horizontal
pleiotropy effect (Burgess and Thompson, 2017), allowing for some direct
effects under the InSide assumption (Bowden and Turkington, 1984), or
identifying invalid IVs (Verbanck et al., 2018), but not in the context of
gene-environment interaction. Third, the popular logistic regression model for
binary outcome clearly has limitation when it comes to the MR analysis due to
its non-collapsibility. Due to lack of a close form or approximation of a
close form for the commonly used MR estimators, especially the 2SPS-type
estimators, it is not easy to understand the magnitude and direction of the
biases based on entirely empirical results, the complexity of which have been
shown in our extensive simulation study. Future research towards better
understanding of the MR methods for binary outcome is needed.

As put by VanderWeele (2009), it is important to distinguish between
interaction and effect modification. In particular, “interaction is defined in
terms of the effects of two interventions whereas effect modification is
defined in terms of the effect of one intervention varying across strata of a
second variable”. Under the above effect-modification definition, there is
asymmetry between $G$ and $X$. The role of $G$ concerns whether the effect of
primary interest varies across strata defined by $G$. Effect modification can
have important implications for public health as the impact of the exposure
$X$ may differ across sub-populations defined by $G$. This may indicate the
necessity for different interventions among these sub-populations. Based on
causal DAGs and Rule 2 of Pearl’s do-calculus (Pearl, 1995), VanderWeele
(2009) provided the required condition for interaction and effect modification
to coincide. Hence, some of the results presented in this work are also
applicable in effect modification formulation, although, methods directly
targeting estimands of effect modification could be more efficient, and are
beyond the scope of this work.

##  6 Acknowledgements

The work is supported in part by grants from the National Institutes of Health
(R01CA189532, R01HL145806, U01 CA164930, R01 CA20140), the Israel Science
Foundation (ISF) grant number 767/21 and by a grant from the Tel-Aviv
University Center for AI and Data Science (TAD). The authors thank the
Genetics and Epidemiology of Colorectal Cancer Consortium (GECCO) and the
participating studies, investigators, staff, and study participants for their
dedication and contributions. A full list of these studies and funding is
provided in the Supporting Information.

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Table 1: Summary of simulation settings Setting |  Model of $Y$ |  Value of $\beta_{3}$ |  $G$ and $G_{IV}$
---|---|---|---
I.A | linear | 0.5 | independent
I.B | linear | 0.0 | independent
II.A | linear | 0.5 | dependent
II.B | linear | 0.0 | dependent
III.A | logistic | 0.5 | independent
III.B | logistic | 0.0 | independent
IV.A | logistic | 0.5 | dependent
IV.B | logistic | 0.0 | dependent
Table 2: Simulation results, linear regression, Setting I.B - Type I error of $H_{0}:\beta_{3}=0$ vs a two-sided alternative, based on 500 simulated datasets.  | $\gamma_{U}=0$ | $\gamma_{U}=0.5$ | $\gamma_{U}=1$ | $\gamma_{U}=2$ | $\gamma_{U}=4$
---|---|---|---|---|---
| Mean | SD | Mean | SD | Mean | SD | Mean | SD | Mean | SD
$\beta_{U}=0$
Naive | 0.04 | 0.20 | 0.04 | 0.20 | 0.05 | 0.21 | 0.06 | 0.23 | 0.05 | 0.21
2SPS | 0.05 | 0.21 | 0.06 | 0.23 | 0.06 | 0.24 | 0.07 | 0.26 | 0.07 | 0.26
2SPSadj | 0.04 | 0.19 | 0.04 | 0.19 | 0.04 | 0.19 | 0.04 | 0.19 | 0.03 | 0.18
2SPSa | 0.05 | 0.21 | 0.05 | 0.21 | 0.05 | 0.21 | 0.04 | 0.20 | 0.04 | 0.20
2SPSadj-a | 0.05 | 0.22 | 0.05 | 0.21 | 0.05 | 0.21 | 0.05 | 0.21 | 0.05 | 0.21
2SRI | 0.04 | 0.20 | 0.04 | 0.20 | 0.05 | 0.21 | 0.06 | 0.23 | 0.05 | 0.21
$\beta_{U}=1.5$
Naive | 0.05 | 0.23 | 0.06 | 0.24 | 0.04 | 0.21 | 0.03 | 0.18 | 0.05 | 0.21
2SPS | 0.07 | 0.25 | 0.07 | 0.25 | 0.07 | 0.26 | 0.08 | 0.27 | 0.07 | 0.26
2SPSadj | 0.07 | 0.26 | 0.06 | 0.24 | 0.05 | 0.21 | 0.04 | 0.20 | 0.04 | 0.19
2SPSa | 0.04 | 0.20 | 0.04 | 0.21 | 0.04 | 0.20 | 0.04 | 0.20 | 0.05 | 0.21
2SPSadj-a | 0.05 | 0.21 | 0.06 | 0.23 | 0.05 | 0.23 | 0.05 | 0.21 | 0.04 | 0.20
2SRI | 0.05 | 0.23 | 0.06 | 0.23 | 0.04 | 0.21 | 0.04 | 0.19 | 0.05 | 0.21
$\beta_{U}=3$
Naive | 0.04 | 0.20 | 0.05 | 0.22 | 0.06 | 0.23 | 0.05 | 0.22 | 0.05 | 0.21
2SPS | 0.07 | 0.26 | 0.07 | 0.26 | 0.06 | 0.24 | 0.06 | 0.24 | 0.07 | 0.25
2SPSadj | 0.07 | 0.25 | 0.06 | 0.23 | 0.06 | 0.24 | 0.05 | 0.22 | 0.04 | 0.19
2SPSa | 0.05 | 0.22 | 0.05 | 0.21 | 0.05 | 0.21 | 0.05 | 0.21 | 0.05 | 0.21
2SPSadj-a | 0.05 | 0.22 | 0.06 | 0.24 | 0.06 | 0.24 | 0.05 | 0.21 | 0.04 | 0.21
2SRI | 0.04 | 0.20 | 0.05 | 0.23 | 0.06 | 0.23 | 0.05 | 0.21 | 0.05 | 0.21
Table 3: Simulation results, linear regression, Setting II.B - Type I error of $H_{0}:\beta_{3}=0$ vs a two-sided alternative, based on 500 simulated datasets.  | $\gamma_{U}=0$ | $\gamma_{U}=0.5$ | $\gamma_{U}=1$ | $\gamma_{U}=2$ | $\gamma_{U}=4$
---|---|---|---|---|---
| Mean | SD | Mean | SD | Mean | SD | Mean | SD | Mean | SD
$\beta_{U}=0$
Naive | 0.05 | 0.21 | 0.05 | 0.21 | 0.06 | 0.23 | 0.06 | 0.23 | 0.06 | 0.23
2SPS | 0.05 | 0.22 | 0.06 | 0.24 | 0.06 | 0.23 | 0.05 | 0.21 | 0.05 | 0.21
2SPSadj | 0.07 | 0.26 | 0.07 | 0.26 | 0.07 | 0.26 | 0.07 | 0.26 | 0.07 | 0.26
2SPSa | 0.05 | 0.21 | 0.04 | 0.21 | 0.04 | 0.20 | 0.05 | 0.22 | 0.06 | 0.23
2SPSadj-a | 0.05 | 0.22 | 0.05 | 0.23 | 0.05 | 0.23 | 0.05 | 0.22 | 0.05 | 0.22
2SRI | 0.05 | 0.21 | 0.05 | 0.21 | 0.06 | 0.23 | 0.06 | 0.23 | 0.06 | 0.23
$\beta_{U}=1.5$
Naive | 0.03 | 0.18 | 0.04 | 0.19 | 0.05 | 0.22 | 0.05 | 0.23 | 0.06 | 0.23
2SPS | 0.06 | 0.23 | 0.06 | 0.24 | 0.06 | 0.24 | 0.05 | 0.23 | 0.06 | 0.23
2SPSadj | 0.04 | 0.20 | 0.05 | 0.21 | 0.05 | 0.23 | 0.06 | 0.23 | 0.06 | 0.24
2SPSa | 0.05 | 0.22 | 0.05 | 0.22 | 0.05 | 0.22 | 0.04 | 0.21 | 0.04 | 0.21
2SPSadj-a | 0.04 | 0.20 | 0.04 | 0.19 | 0.04 | 0.20 | 0.06 | 0.23 | 0.06 | 0.23
2SRI | 0.03 | 0.18 | 0.04 | 0.19 | 0.05 | 0.21 | 0.06 | 0.23 | 0.06 | 0.23
$\beta_{U}=3$
Naive | 0.04 | 0.19 | 0.04 | 0.19 | 0.05 | 0.22 | 0.04 | 0.21 | 0.06 | 0.23
2SPS | 0.05 | 0.22 | 0.05 | 0.22 | 0.05 | 0.23 | 0.05 | 0.22 | 0.05 | 0.23
2SPSadj | 0.04 | 0.20 | 0.04 | 0.19 | 0.05 | 0.21 | 0.05 | 0.22 | 0.06 | 0.23
2SPS-a | 0.05 | 0.22 | 0.05 | 0.21 | 0.04 | 0.20 | 0.04 | 0.21 | 0.04 | 0.20
2SPSadj-a | 0.04 | 0.21 | 0.04 | 0.19 | 0.03 | 0.18 | 0.05 | 0.21 | 0.05 | 0.23
2SRI | 0.04 | 0.19 | 0.04 | 0.19 | 0.05 | 0.21 | 0.05 | 0.21 | 0.05 | 0.21
Table 4: Simulation results, logistic regression, Setting III.B - Type I error of $H_{0}:\beta_{3}=0$ vs a two-sided alternative, based on 500 simulated datasets.  | $\gamma_{U}=0$ | $\gamma_{U}=0.5$ | $\gamma_{U}=1$ | $\gamma_{U}=2$ | $\gamma_{U}=4$
---|---|---|---|---|---
| Mean | SD | Mean | SD | Mean | SD | Mean | SD | Mean | SD
$\beta_{U}=0$
Naive | 0.06 | 0.23 | 0.06 | 0.24 | 0.04 | 0.21 | 0.06 | 0.23 | 0.05 | 0.22
2SPS | 0.06 | 0.24 | 0.06 | 0.24 | 0.07 | 0.26 | 0.08 | 0.27 | 0.07 | 0.26
2SPSadj | 0.06 | 0.23 | 0.06 | 0.23 | 0.05 | 0.21 | 0.06 | 0.24 | 0.07 | 0.25
2SPSa | 0.07 | 0.26 | 0.07 | 0.26 | 0.06 | 0.23 | 0.07 | 0.26 | 0.08 | 0.27
2SPSadj-a | 0.06 | 0.24 | 0.06 | 0.24 | 0.06 | 0.23 | 0.05 | 0.22 | 0.05 | 0.23
2SRI | 0.05 | 0.23 | 0.06 | 0.24 | 0.04 | 0.21 | 0.06 | 0.23 | 0.05 | 0.22
$\beta_{U}=1.5$
Naive | 0.09 | 0.28 | 0.07 | 0.26 | 0.07 | 0.26 | 0.07 | 0.25 | 0.04 | 0.20
2SPS | 0.09 | 0.29 | 0.10 | 0.30 | 0.07 | 0.26 | 0.09 | 0.28 | 0.07 | 0.26
2SPSadj | 0.06 | 0.24 | 0.04 | 0.20 | 0.03 | 0.18 | 0.05 | 0.22 | 0.04 | 0.19
2SPS-a | 0.05 | 0.23 | 0.05 | 0.23 | 0.06 | 0.23 | 0.06 | 0.23 | 0.04 | 0.21
2SPSadj-a | 0.06 | 0.23 | 0.04 | 0.19 | 0.05 | 0.21 | 0.04 | 0.21 | 0.05 | 0.21
2SRI | 0.08 | 0.28 | 0.07 | 0.26 | 0.06 | 0.24 | 0.07 | 0.25 | 0.04 | 0.19
$\beta_{U}=3$
Naive | 0.08 | 0.27 | 0.08 | 0.27 | 0.05 | 0.22 | 0.04 | 0.20 | 0.07 | 0.25
2SPS | 0.04 | 0.20 | 0.06 | 0.23 | 0.07 | 0.26 | 0.05 | 0.22 | 0.05 | 0.23
2SPSadj | 0.04 | 0.19 | 0.04 | 0.21 | 0.06 | 0.23 | 0.05 | 0.21 | 0.06 | 0.24
2SPS-a | 0.04 | 0.20 | 0.06 | 0.23 | 0.05 | 0.22 | 0.03 | 0.18 | 0.05 | 0.21
2SPSadj-a | 0.04 | 0.19 | 0.07 | 0.26 | 0.04 | 0.20 | 0.03 | 0.18 | 0.04 | 0.21
2SRI | 0.08 | 0.27 | 0.07 | 0.26 | 0.05 | 0.22 | 0.05 | 0.21 | 0.05 | 0.21
Table 5: Simulation results, logistic regression, Setting IV.B - Type I error of $H_{0}:\beta_{3}=0$ vs a two-sided alternative, based on 500 simulated datasets.  | $\gamma_{U}=0$ | $\gamma_{U}=0.5$ | $\gamma_{U}=1$ | $\gamma_{U}=2$ | $\gamma_{U}=4$
---|---|---|---|---|---
| Mean | SD | Mean | SD | Mean | SD | Mean | SD | Mean | SD
$\beta_{U}=0$
Naive | 0.06 | 0.24 | 0.07 | 0.25 | 0.06 | 0.24 | 0.05 | 0.21 | 0.04 | 0.20
2SPS | 0.07 | 0.25 | 0.07 | 0.25 | 0.05 | 0.21 | 0.07 | 0.25 | 0.05 | 0.23
2SPSadj | 0.05 | 0.23 | 0.04 | 0.21 | 0.04 | 0.21 | 0.04 | 0.19 | 0.04 | 0.19
2SPSa | 0.06 | 0.24 | 0.05 | 0.22 | 0.05 | 0.22 | 0.04 | 0.20 | 0.05 | 0.23
2SPSadj-a | 0.05 | 0.21 | 0.05 | 0.22 | 0.04 | 0.21 | 0.04 | 0.19 | 0.05 | 0.23
2SRI | 0.06 | 0.23 | 0.07 | 0.26 | 0.06 | 0.24 | 0.05 | 0.21 | 0.04 | 0.20
$\beta_{U}=1.5$
Naive | 0.15 | 0.36 | 0.12 | 0.32 | 0.06 | 0.24 | 0.06 | 0.23 | 0.04 | 0.20
2SPS | 0.13 | 0.34 | 0.14 | 0.35 | 0.13 | 0.33 | 0.11 | 0.31 | 0.09 | 0.28
2SPSadj | 0.09 | 0.29 | 0.08 | 0.27 | 0.05 | 0.21 | 0.05 | 0.22 | 0.05 | 0.22
2SPSa | 0.08 | 0.27 | 0.06 | 0.24 | 0.06 | 0.24 | 0.05 | 0.21 | 0.05 | 0.22
2SPSadj-a | 0.06 | 0.23 | 0.05 | 0.21 | 0.06 | 0.24 | 0.05 | 0.23 | 0.04 | 0.19
2SRI | 0.16 | 0.36 | 0.14 | 0.35 | 0.07 | 0.25 | 0.05 | 0.21 | 0.04 | 0.21
$\beta_{U}=3$
Naive | 0.11 | 0.31 | 0.10 | 0.30 | 0.06 | 0.24 | 0.05 | 0.23 | 0.05 | 0.23
2SPS | 0.09 | 0.28 | 0.09 | 0.28 | 0.06 | 0.23 | 0.06 | 0.23 | 0.05 | 0.22
2SPSadj | 0.07 | 0.26 | 0.06 | 0.24 | 0.05 | 0.23 | 0.04 | 0.19 | 0.04 | 0.20
2SPSa | 0.04 | 0.21 | 0.07 | 0.25 | 0.05 | 0.21 | 0.05 | 0.23 | 0.05 | 0.23
2SPSadj-a | 0.04 | 0.21 | 0.05 | 0.23 | 0.04 | 0.20 | 0.05 | 0.22 | 0.05 | 0.22
2SRI | 0.11 | 0.31 | 0.11 | 0.32 | 0.08 | 0.27 | 0.06 | 0.24 | 0.05 | 0.21
Table 6: Simulation results, logistic regression, $n=30,000$ \- Type I error of $H_{0}:\beta_{3}=0$ vs a two-sided alternative, based on 1000 simulated datasets.  | $\gamma_{U}=0$ | $\gamma_{U}=0.5$ | $\gamma_{U}=1$ | $\gamma_{U}=2$ | $\gamma_{U}=4$
---|---|---|---|---|---
| Mean | SD | Mean | SD | Mean | SD | Mean | SD | Mean | SD
$\beta_{U}=0.5$, $\beta_{Z}=\gamma_{Z}=0$, $\beta_{2}=0.5$
Naive | 0.06 | 0.23 | 0.06 | 0.24 | 0.05 | 0.22 | 0.06 | 0.23 | 0.05 | 0.22
2SPS | 0.07 | 0.26 | 0.08 | 0.28 | 0.07 | 0.25 | 0.05 | 0.22 | 0.05 | 0.21
2SPSadj | 0.06 | 0.23 | 0.06 | 0.23 | 0.05 | 0.21 | 0.05 | 0.22 | 0.05 | 0.21
2SPSa | 0.07 | 0.26 | 0.09 | 0.28 | 0.07 | 0.25 | 0.05 | 0.21 | 0.05 | 0.22
2SPSadj-a | 0.06 | 0.24 | 0.06 | 0.23 | 0.05 | 0.21 | 0.05 | 0.22 | 0.05 | 0.21
2SRI | 0.06 | 0.23 | 0.06 | 0.23 | 0.05 | 0.21 | 0.06 | 0.23 | 0.05 | 0.22
$\beta_{U}=3$, $\beta_{Z}=\gamma_{Z}=0$, $\beta_{2}=0.5$
Naive | 0.13 | 0.33 | 0.14 | 0.35 | 0.11 | 0.32 | 0.07 | 0.25 | 0.05 | 0.21
2SPS | 0.05 | 0.22 | 0.06 | 0.23 | 0.07 | 0.26 | 0.04 | 0.20 | 0.04 | 0.20
2SPSadj | 0.05 | 0.21 | 0.05 | 0.21 | 0.05 | 0.21 | 0.05 | 0.21 | 0.04 | 0.19
2SPSa | 0.05 | 0.23 | 0.06 | 0.24 | 0.07 | 0.26 | 0.04 | 0.20 | 0.04 | 0.20
2SPSadj-a | 0.05 | 0.21 | 0.05 | 0.22 | 0.05 | 0.22 | 0.05 | 0.21 | 0.04 | 0.20
2SRI | 0.13 | 0.34 | 0.13 | 0.33 | 0.10 | 0.30 | 0.06 | 0.24 | 0.05 | 0.22
$\beta_{U}=0.5$, $\beta_{Z}=\gamma_{Z}=0.5$, $\beta_{2}=0.5$
Naive | 0.06 | 0.24 | 0.06 | 0.24 | 0.06 | 0.23 | 0.05 | 0.22 | 0.06 | 0.24
2SPS | 0.09 | 0.28 | 0.11 | 0.31 | 0.12 | 0.32 | 0.11 | 0.31 | 0.09 | 0.28
2SPSadj | 0.07 | 0.25 | 0.05 | 0.22 | 0.04 | 0.20 | 0.05 | 0.21 | 0.05 | 0.21
2SPSa | 0.06 | 0.24 | 0.07 | 0.25 | 0.07 | 0.25 | 0.06 | 0.24 | 0.05 | 0.21
2SPSadj-a | 0.05 | 0.22 | 0.06 | 0.23 | 0.05 | 0.21 | 0.06 | 0.23 | 0.04 | 0.21
2SRI | 0.06 | 0.24 | 0.06 | 0.24 | 0.06 | 0.23 | 0.05 | 0.23 | 0.06 | 0.24
$\beta_{U}=3$, $\beta_{Z}=\gamma_{Z}=0.5$, $\beta_{2}=0.5$
Naive | 0.15 | 0.35 | 0.14 | 0.35 | 0.10 | 0.30 | 0.06 | 0.24 | 0.04 | 0.20
2SPS | 0.10 | 0.31 | 0.09 | 0.28 | 0.08 | 0.27 | 0.07 | 0.26 | 0.05 | 0.22
2SPSadj | 0.09 | 0.29 | 0.05 | 0.22 | 0.05 | 0.22 | 0.04 | 0.19 | 0.07 | 0.25
2SPSa | 0.07 | 0.25 | 0.07 | 0.26 | 0.06 | 0.24 | 0.04 | 0.21 | 0.04 | 0.21
2SPSadj-a | 0.07 | 0.25 | 0.05 | 0.22 | 0.07 | 0.25 | 0.05 | 0.22 | 0.05 | 0.23
2SRI | 0.15 | 0.35 | 0.13 | 0.34 | 0.10 | 0.30 | 0.06 | 0.23 | 0.04 | 0.20
$\beta_{U}=3$, $\beta_{Z}=\gamma_{Z}=0.5$, $\beta_{2}=0$
Naive | 0.04 | 0.20 | 0.05 | 0.21 | 0.05 | 0.21 | 0.05 | 0.23 | 0.05 | 0.22
2SPS | 0.04 | 0.20 | 0.05 | 0.22 | 0.04 | 0.20 | 0.05 | 0.22 | 0.06 | 0.24
2SPSadj | 0.05 | 0.21 | 0.06 | 0.23 | 0.05 | 0.23 | 0.05 | 0.21 | 0.05 | 0.22
2SPSa | 0.04 | 0.20 | 0.04 | 0.21 | 0.04 | 0.19 | 0.04 | 0.20 | 0.06 | 0.23
2SPSadj-a | 0.04 | 0.20 | 0.04 | 0.20 | 0.05 | 0.22 | 0.06 | 0.24 | 0.06 | 0.23
2SRI | 0.04 | 0.20 | 0.05 | 0.23 | 0.05 | 0.22 | 0.06 | 0.23 | 0.05 | 0.23

Figure 1: Simulation results, linear regression, Setting I.A with
$\gamma_{U}=1$ (top sub-figure) and Setting II.A with $\gamma_{U}=1$ (bottom
sub-figure): within each sub-figure, top line $\beta_{U}=0$, middle line
$\beta_{U}=1.5$ and third line $\beta_{U}=3$.

Figure 2: Simulation results, linear regression, Setting I.B with
$\gamma_{U}=1$, $\beta_{3}=0$ (top sub-figure) and Setting II.B with
$\gamma_{U}=1$, $\beta_{3}=0$ (bottom sub-figure): within each sub-figure, top
line $\beta_{U}=0$, middle line $\beta_{U}=1.5$ and third line $\beta_{U}=3$.

Figure 3: Simulation results, logistic regression, Setting III.A. Figure 4:
Simulation results, logistic regression, Setting III.B. Figure 5: Simulation
results, logistic regression, Setting IV.A. Figure 6: Simulation results,
logistic regression, Setting IV.B. Figure 7: Simulation results, logistic
regression, Setting III.A with negative values of $\beta_{U}$. Figure 8:
Simulation results, logistic regression, Setting III.A with negative values of
$\gamma_{U}$.

Figure 9: Boxplots of the estimates for the main effects of BMI and G, and the
interaction GxBMI (left column), and the standard errors of corresponding
estimates (right column) for the naive estimator and 5 MR estimators.

Figure 10: QQplots of $-\log_{10}\text{p-value}$ for GxBMI for all estimators
versus the expected $-\log_{10}\text{p-value}$ under the uniform distribution.

Figure 11: Pairwise plots of the estimates for GxBMI for all estimators.

Figure 12: Pairwise plots of $-\log_{10}\text{p-value}$ for GxBMI for all
estimators.

##  S7 Supporting Information

Table S1: Simulation results, linear regression, Setting I.A | $\gamma_{U}=0$ | $\gamma_{U}=0.5$ | $\gamma_{U}=1$ | $\gamma_{U}=2$ | $\gamma_{U}=4$
---|---|---|---|---|---
| Mean | SD | Mean | SD | Mean | SD | Mean | SD | Mean | SD
$R^{2}_{X|G_{IV},Z,U}$ | 0.33 | 0.01 | 0.43 | 0.01 | 0.60 | 0.01 | 0.82 | 0.00 | 0.94 | 0.00
$R^{2}_{X|G_{IV},Z}$ | 0.33 | 0.01 | 0.29 | 0.01 | 0.20 | 0.01 | 0.09 | 0.01 | 0.03 | 0.00
$\beta_{U}=0$
$R^{2}_{Y|X,G,Z,U}$ | 0.73 | 0.00 | 0.73 | 0.00 | 0.72 | 0.00 | 0.71 | 0.00 | 0.70 | 0.00
$R^{2}_{Y|X,G,Z}$ | 0.73 | 0.00 | 0.73 | 0.00 | 0.72 | 0.00 | 0.71 | 0.00 | 0.70 | 0.00
Naive $\beta_{1}$ | 1.00 | 0.01 | 1.00 | 0.01 | 1.00 | 0.01 | 1.00 | 0.01 | 1.00 | 0.01
$\beta_{2}$ | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02
$\beta_{3}$ | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02
2SPS $\beta_{1}$ | 1.00 | 0.05 | 1.00 | 0.05 | 1.00 | 0.06 | 1.00 | 0.10 | 1.00 | 0.18
$\beta_{2}$ | 0.50 | 0.02 | 0.50 | 0.03 | 0.50 | 0.03 | 0.50 | 0.03 | 0.50 | 0.03
$\beta_{3}$ | 0.50 | 0.04 | 0.50 | 0.05 | 0.50 | 0.06 | 0.50 | 0.10 | 0.51 | 0.18
2SPSadj $\beta_{1}$ | 1.00 | 0.03 | 1.00 | 0.03 | 1.00 | 0.04 | 1.00 | 0.06 | 1.00 | 0.11
$\beta_{2}$ | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02
$\beta_{3}$ | 0.50 | 0.03 | 0.50 | 0.03 | 0.50 | 0.04 | 0.50 | 0.06 | 0.50 | 0.10
2SPSa $\beta_{1}$ | 1.00 | 0.05 | 1.00 | 0.05 | 1.00 | 0.06 | 0.99 | 0.10 | 0.99 | 0.19
$\beta_{2}$ | 0.50 | 0.02 | 0.50 | 0.03 | 0.50 | 0.03 | 0.50 | 0.03 | 0.50 | 0.03
$\beta_{3}$ | 0.50 | 0.06 | 0.50 | 0.07 | 0.50 | 0.08 | 0.51 | 0.13 | 0.52 | 0.24
2SPSadj-a $\beta_{1}$ | 1.00 | 0.03 | 1.00 | 0.04 | 1.00 | 0.04 | 1.00 | 0.07 | 1.00 | 0.12
$\beta_{2}$ | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02
$\beta_{3}$ | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.05 | 0.50 | 0.08 | 0.51 | 0.15
2SRI $\beta_{1}$ | 1.00 | 0.03 | 1.00 | 0.03 | 1.00 | 0.03 | 1.00 | 0.05 | 1.00 | 0.09
$\beta_{2}$ | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02
$\beta_{3}$ | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02
$\beta_{U}=1.5$
$R^{2}_{Y|X,G,Z,U}$ | 0.83 | 0.00 | 0.86 | 0.00 | 0.88 | 0.00 | 0.89 | 0.00 | 0.89 | 0.00
$R^{2}_{Y|X,G,Z}$ | 0.45 | 0.01 | 0.61 | 0.01 | 0.73 | 0.00 | 0.83 | 0.00 | 0.87 | 0.00
Naive $\beta_{1}$ | 1.00 | 0.03 | 1.66 | 0.02 | 2.05 | 0.02 | 2.34 | 0.02 | 2.46 | 0.02
$\beta_{2}$ | 0.50 | 0.03 | 0.50 | 0.03 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02
$\beta_{3}$ | 0.50 | 0.03 | 0.50 | 0.03 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02
2SPS $\beta_{1}$ | 1.00 | 0.06 | 1.00 | 0.07 | 0.99 | 0.09 | 0.99 | 0.15 | 0.98 | 0.28
$\beta_{2}$ | 0.50 | 0.03 | 0.50 | 0.04 | 0.50 | 0.05 | 0.50 | 0.05 | 0.50 | 0.05
$\beta_{3}$ | 0.50 | 0.06 | 0.50 | 0.08 | 0.51 | 0.10 | 0.51 | 0.16 | 0.52 | 0.30
2SPSadj $\beta_{1}$ | 1.00 | 0.05 | 1.00 | 0.06 | 1.00 | 0.07 | 1.00 | 0.10 | 0.99 | 0.18
$\beta_{2}$ | 0.50 | 0.03 | 0.50 | 0.03 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02
$\beta_{3}$ | 0.50 | 0.05 | 0.51 | 0.05 | 0.50 | 0.05 | 0.50 | 0.07 | 0.50 | 0.11
2SPSa $\beta_{1}$ | 1.00 | 0.07 | 0.99 | 0.08 | 0.99 | 0.11 | 0.98 | 0.17 | 0.97 | 0.32
$\beta_{2}$ | 0.50 | 0.03 | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.05 | 0.50 | 0.05
$\beta_{3}$ | 0.50 | 0.08 | 0.51 | 0.10 | 0.51 | 0.14 | 0.52 | 0.22 | 0.54 | 0.40
2SPSadj-a $\beta_{1}$ | 0.99 | 0.06 | 0.99 | 0.06 | 0.99 | 0.07 | 0.99 | 0.11 | 0.99 | 0.20
$\beta_{2}$ | 0.50 | 0.03 | 0.50 | 0.03 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02
$\beta_{3}$ | 0.51 | 0.07 | 0.51 | 0.07 | 0.51 | 0.07 | 0.51 | 0.09 | 0.51 | 0.15
2SRI $\beta_{1}$ | 1.00 | 0.05 | 1.00 | 0.05 | 1.00 | 0.06 | 1.00 | 0.09 | 0.99 | 0.16
$\beta_{2}$ | 0.50 | 0.03 | 0.50 | 0.03 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02
$\beta_{3}$ | 0.50 | 0.03 | 0.50 | 0.03 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02
$\beta_{U}=3$
$R^{2}_{Y|X,G,Z,U}$ | 0.92 | 0.00 | 0.94 | 0.00 | 0.94 | 0.00 | 0.95 | 0.00 | 0.95 | 0.00
$R^{2}_{Y|X,G,Z}$ | 0.21 | 0.01 | 0.46 | 0.01 | 0.66 | 0.01 | 0.84 | 0.00 | 0.92 | 0.00
Naive $\beta_{1}$ | 1.00 | 0.05 | 2.32 | 0.04 | 3.11 | 0.04 | 3.68 | 0.03 | 3.91 | 0.03
$\beta_{2}$ | 0.50 | 0.05 | 0.50 | 0.05 | 0.50 | 0.04 | 0.50 | 0.03 | 0.50 | 0.02
$\beta_{3}$ | 0.50 | 0.05 | 0.50 | 0.05 | 0.50 | 0.04 | 0.50 | 0.03 | 0.50 | 0.02
2SPS $\beta_{1}$ | 0.99 | 0.09 | 0.99 | 0.11 | 0.99 | 0.14 | 0.98 | 0.22 | 0.97 | 0.41
$\beta_{2}$ | 0.50 | 0.05 | 0.50 | 0.06 | 0.50 | 0.07 | 0.50 | 0.07 | 0.50 | 0.07
$\beta_{3}$ | 0.51 | 0.10 | 0.51 | 0.12 | 0.51 | 0.15 | 0.52 | 0.24 | 0.54 | 0.43
2SPSadj $\beta_{1}$ | 0.99 | 0.09 | 0.99 | 0.10 | 0.99 | 0.11 | 0.99 | 0.16 | 0.99 | 0.30
$\beta_{2}$ | 0.50 | 0.05 | 0.50 | 0.05 | 0.50 | 0.04 | 0.50 | 0.03 | 0.50 | 0.02
$\beta_{3}$ | 0.51 | 0.09 | 0.51 | 0.09 | 0.51 | 0.08 | 0.51 | 0.09 | 0.51 | 0.13
2SPS-a $\beta_{1}$ | 0.99 | 0.11 | 0.99 | 0.13 | 0.98 | 0.16 | 0.97 | 0.25 | 0.96 | 0.47
$\beta_{2}$ | 0.50 | 0.05 | 0.50 | 0.06 | 0.50 | 0.07 | 0.50 | 0.07 | 0.50 | 0.07
$\beta_{3}$ | 0.51 | 0.13 | 0.51 | 0.16 | 0.52 | 0.20 | 0.53 | 0.32 | 0.56 | 0.58
2SPSadj-a $\beta_{1}$ | 0.99 | 0.10 | 0.99 | 0.11 | 0.99 | 0.12 | 0.99 | 0.17 | 0.98 | 0.31
$\beta_{2}$ | 0.50 | 0.05 | 0.50 | 0.05 | 0.50 | 0.04 | 0.50 | 0.03 | 0.50 | 0.02
$\beta_{3}$ | 0.51 | 0.12 | 0.52 | 0.12 | 0.52 | 0.12 | 0.52 | 0.13 | 0.52 | 0.17
2SRI $\beta_{1}$ | 1.00 | 0.08 | 1.00 | 0.09 | 1.00 | 0.11 | 1.00 | 0.16 | 0.99 | 0.28
$\beta_{2}$ | 0.50 | 0.05 | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.03 | 0.50 | 0.02
$\beta_{3}$ | 0.50 | 0.05 | 0.50 | 0.05 | 0.50 | 0.04 | 0.50 | 0.03 | 0.50 | 0.02
Table S2: Simulation results, linear regression, Setting II.A | $\gamma_{U}=0$ | $\gamma_{U}=0.5$ | $\gamma_{U}=1$ | $\gamma_{U}=2$ | $\gamma_{U}=4$
---|---|---|---|---|---
| Mean | SD | Mean | SD | Mean | SD | Mean | SD | Mean | SD
$R^{2}_{X|G_{IV},Z,U}$ | 0.33 | 0.01 | 0.43 | 0.01 | 0.60 | 0.01 | 0.82 | 0.00 | 0.94 | 0.00
$R^{2}_{X|G_{IV},Z}$ | 0.33 | 0.01 | 0.29 | 0.01 | 0.20 | 0.01 | 0.09 | 0.01 | 0.03 | 0.00
$\beta_{U}=0$
$R^{2}_{Y|X,G,Z,U}$ | 0.77 | 0.00 | 0.77 | 0.00 | 0.76 | 0.00 | 0.75 | 0.00 | 0.74 | 0.00
$R^{2}_{Y|X,G,Z}$ | 0.77 | 0.00 | 0.77 | 0.00 | 0.76 | 0.00 | 0.75 | 0.00 | 0.74 | 0.00
Naive $\beta_{1}$ | 1.00 | 0.02 | 1.00 | 0.02 | 1.00 | 0.02 | 1.00 | 0.02 | 1.00 | 0.02
$\beta_{2}$ | 0.50 | 0.01 | 0.50 | 0.01 | 0.50 | 0.01 | 0.50 | 0.01 | 0.50 | 0.01
$\beta_{3}$ | 0.50 | 0.01 | 0.50 | 0.01 | 0.50 | 0.01 | 0.50 | 0.01 | 0.50 | 0.01
2SPS $\beta_{1}$ | 1.00 | 0.05 | 1.00 | 0.06 | 1.00 | 0.07 | 1.01 | 0.11 | 1.02 | 0.20
$\beta_{2}$ | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.03 | 0.50 | 0.03 | 0.50 | 0.03
$\beta_{3}$ | 0.50 | 0.04 | 0.49 | 0.04 | 0.49 | 0.06 | 0.49 | 0.09 | 0.48 | 0.16
2SPSadj $\beta_{1}$ | 1.00 | 0.04 | 1.00 | 0.04 | 1.00 | 0.05 | 1.00 | 0.07 | 1.00 | 0.13
$\beta_{2}$ | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02
$\beta_{3}$ | 0.50 | 0.03 | 0.50 | 0.03 | 0.50 | 0.03 | 0.50 | 0.05 | 0.50 | 0.10
2SPSa $\beta_{1}$ | 1.00 | 0.06 | 1.00 | 0.06 | 1.01 | 0.08 | 1.01 | 0.12 | 1.02 | 0.22
$\beta_{2}$ | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.03 | 0.50 | 0.03 | 0.50 | 0.03
$\beta_{3}$ | 0.50 | 0.06 | 0.49 | 0.06 | 0.49 | 0.08 | 0.49 | 0.13 | 0.48 | 0.23
2SPSadj-a $\beta_{1}$ | 1.00 | 0.04 | 1.00 | 0.05 | 1.00 | 0.06 | 1.00 | 0.09 | 1.01 | 0.16
$\beta_{2}$ | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02
$\beta_{3}$ | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.05 | 0.50 | 0.08 | 0.49 | 0.14
2SRI $\beta_{1}$ | 1.00 | 0.03 | 1.00 | 0.03 | 1.00 | 0.04 | 1.00 | 0.05 | 1.00 | 0.09
$\beta_{2}$ | 0.50 | 0.01 | 0.50 | 0.01 | 0.50 | 0.01 | 0.50 | 0.01 | 0.50 | 0.01
$\beta_{3}$ | 0.50 | 0.01 | 0.50 | 0.01 | 0.50 | 0.01 | 0.50 | 0.01 | 0.50 | 0.01
$\beta_{U}=1.5$
$R^{2}_{Y|X,G,Z,U}$ | 0.85 | 0.00 | 0.88 | 0.00 | 0.89 | 0.00 | 0.90 | 0.00 | 0.90 | 0.00
$R^{2}_{Y|X,G,Z}$ | 0.51 | 0.01 | 0.65 | 0.01 | 0.76 | 0.00 | 0.85 | 0.00 | 0.89 | 0.00
Naive $\beta_{1}$ | 1.00 | 0.03 | 1.67 | 0.03 | 2.06 | 0.02 | 2.35 | 0.02 | 2.46 | 0.02
$\beta_{2}$ | 0.50 | 0.03 | 0.39 | 0.02 | 0.35 | 0.02 | 0.37 | 0.02 | 0.42 | 0.01
$\beta_{3}$ | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02
2SPS $\beta_{1}$ | 1.00 | 0.07 | 1.01 | 0.09 | 1.01 | 0.11 | 1.01 | 0.18 | 1.02 | 0.33
$\beta_{2}$ | 0.50 | 0.03 | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.05 | 0.50 | 0.05
$\beta_{3}$ | 0.49 | 0.05 | 0.49 | 0.07 | 0.49 | 0.09 | 0.49 | 0.14 | 0.48 | 0.26
2SPSadj $\beta_{1}$ | 1.00 | 0.06 | 1.00 | 0.07 | 1.00 | 0.08 | 1.00 | 0.11 | 1.00 | 0.20
$\beta_{2}$ | 0.50 | 0.03 | 0.50 | 0.03 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02
$\beta_{3}$ | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.05 | 0.50 | 0.06 | 0.50 | 0.10
2SPSa $\beta_{1}$ | 1.01 | 0.08 | 1.01 | 0.10 | 1.01 | 0.13 | 1.01 | 0.21 | 1.02 | 0.40
$\beta_{2}$ | 0.50 | 0.03 | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.05 | 0.50 | 0.05
$\beta_{3}$ | 0.49 | 0.08 | 0.49 | 0.10 | 0.49 | 0.13 | 0.49 | 0.21 | 0.48 | 0.39
2SPSadj-a $\beta_{1}$ | 1.00 | 0.08 | 1.00 | 0.08 | 1.00 | 0.09 | 1.00 | 0.13 | 1.00 | 0.23
$\beta_{2}$ | 0.50 | 0.03 | 0.50 | 0.03 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02
$\beta_{3}$ | 0.50 | 0.07 | 0.50 | 0.07 | 0.50 | 0.07 | 0.50 | 0.09 | 0.50 | 0.15
2SRI $\beta_{1}$ | 1.00 | 0.05 | 1.00 | 0.06 | 1.00 | 0.06 | 1.00 | 0.09 | 1.00 | 0.16
$\beta_{2}$ | 0.50 | 0.03 | 0.50 | 0.03 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02
$\beta_{3}$ | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.01
$\beta_{U}=3$
$R^{2}_{Y|X,G,Z,U}$ | 0.93 | 0.00 | 0.94 | 0.00 | 0.95 | 0.00 | 0.95 | 0.00 | 0.95 | 0.00
$R^{2}_{Y|X,G,Z}$ | 0.25 | 0.01 | 0.49 | 0.01 | 0.68 | 0.00 | 0.85 | 0.00 | 0.92 | 0.00
Naive $\beta_{1}$ | 1.00 | 0.05 | 2.34 | 0.05 | 3.13 | 0.04 | 3.69 | 0.03 | 3.91 | 0.03
$\beta_{2}$ | 0.50 | 0.04 | 0.27 | 0.04 | 0.20 | 0.03 | 0.24 | 0.02 | 0.34 | 0.02
$\beta_{3}$ | 0.50 | 0.04 | 0.51 | 0.04 | 0.51 | 0.03 | 0.50 | 0.02 | 0.50 | 0.02
2SPS $\beta_{1}$ | 1.01 | 0.11 | 1.01 | 0.13 | 1.01 | 0.17 | 1.01 | 0.26 | 1.02 | 0.48
$\beta_{2}$ | 0.50 | 0.05 | 0.50 | 0.06 | 0.50 | 0.06 | 0.50 | 0.07 | 0.50 | 0.07
$\beta_{3}$ | 0.49 | 0.08 | 0.49 | 0.10 | 0.49 | 0.13 | 0.48 | 0.21 | 0.47 | 0.38
2SPSadj $\beta_{1}$ | 1.00 | 0.11 | 1.00 | 0.11 | 1.00 | 0.13 | 1.00 | 0.18 | 0.99 | 0.31
$\beta_{2}$ | 0.50 | 0.05 | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.03 | 0.50 | 0.02
$\beta_{3}$ | 0.49 | 0.08 | 0.50 | 0.07 | 0.50 | 0.07 | 0.50 | 0.08 | 0.50 | 0.11
2SPSa $\beta_{1}$ | 1.01 | 0.14 | 1.01 | 0.16 | 1.01 | 0.20 | 1.01 | 0.32 | 1.02 | 0.60
$\beta_{2}$ | 0.50 | 0.05 | 0.50 | 0.06 | 0.50 | 0.06 | 0.50 | 0.07 | 0.50 | 0.07
$\beta_{3}$ | 0.49 | 0.12 | 0.49 | 0.15 | 0.49 | 0.20 | 0.48 | 0.31 | 0.47 | 0.57
2SPSadj-a $\beta_{1}$ | 1.00 | 0.13 | 1.00 | 0.14 | 1.00 | 0.15 | 1.00 | 0.20 | 0.99 | 0.34
$\beta_{2}$ | 0.50 | 0.05 | 0.50 | 0.04 | 0.50 | 0.03 | 0.50 | 0.02 | 0.50 | 0.02
$\beta_{3}$ | 0.49 | 0.11 | 0.50 | 0.11 | 0.50 | 0.11 | 0.50 | 0.12 | 0.50 | 0.17
2SRI $\beta_{1}$ | 1.00 | 0.09 | 1.00 | 0.10 | 1.00 | 0.11 | 1.00 | 0.16 | 0.99 | 0.28
$\beta_{2}$ | 0.50 | 0.05 | 0.50 | 0.04 | 0.50 | 0.03 | 0.50 | 0.02 | 0.50 | 0.02
$\beta_{3}$ | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.03 | 0.50 | 0.02 | 0.50 | 0.02
Table S3: Simulation results, linear regression, Setting I.B | $\gamma_{U}=0$ | $\gamma_{U}=0.5$ | $\gamma_{U}=1$ | $\gamma_{U}=2$ | $\gamma_{U}=4$
---|---|---|---|---|---
| Mean | SD | Mean | SD | Mean | SD | Mean | SD | Mean | SD
$R^{2}_{X|G_{IV},Z,U}$ | 0.33 | 0.01 | 0.43 | 0.01 | 0.60 | 0.01 | 0.82 | 0.00 | 0.94 | 0.00
$R^{2}_{X|G_{IV},Z}$ | 0.33 | 0.01 | 0.29 | 0.01 | 0.20 | 0.01 | 0.09 | 0.01 | 0.03 | 0.00
$\beta_{U}=0$
$R^{2}_{Y|X,G,Z,U}$ | 0.64 | 0.01 | 0.63 | 0.01 | 0.63 | 0.01 | 0.61 | 0.01 | 0.60 | 0.01
$R^{2}_{Y|X,G,Z}$ | 0.64 | 0.01 | 0.63 | 0.01 | 0.63 | 0.01 | 0.61 | 0.01 | 0.60 | 0.01
Naive $\beta_{1}$ | 1.00 | 0.01 | 1.00 | 0.01 | 1.00 | 0.01 | 1.00 | 0.01 | 1.00 | 0.01
$\beta_{2}$ | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02
$\beta_{2}$ | -0.00 | 0.02 | -0.00 | 0.02 | -0.00 | 0.02 | -0.00 | 0.02 | -0.00 | 0.02
2SPS $\beta_{1}$ | 1.00 | 0.04 | 1.00 | 0.04 | 1.00 | 0.06 | 1.00 | 0.09 | 1.00 | 0.16
$\beta_{2}$ | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02
$\beta_{3}$ | -0.00 | 0.03 | -0.00 | 0.04 | 0.00 | 0.05 | 0.00 | 0.07 | 0.01 | 0.13
2SPSadj $\beta_{1}$ | 1.00 | 0.03 | 1.00 | 0.03 | 1.00 | 0.04 | 1.00 | 0.06 | 1.00 | 0.10
$\beta_{2}$ | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02
$\beta_{3}$ | 0.00 | 0.03 | 0.00 | 0.03 | 0.00 | 0.03 | 0.00 | 0.05 | 0.00 | 0.09
2SPSa $\beta_{1}$ | 1.00 | 0.04 | 1.00 | 0.05 | 1.00 | 0.06 | 0.99 | 0.10 | 0.99 | 0.18
$\beta_{2}$ | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02
$\beta_{3}$ | -0.00 | 0.05 | 0.00 | 0.05 | 0.00 | 0.07 | 0.01 | 0.10 | 0.02 | 0.18
2SPSadj-a $\beta_{1}$ | 1.00 | 0.03 | 1.00 | 0.04 | 1.00 | 0.04 | 1.00 | 0.06 | 1.00 | 0.11
$\beta_{2}$ | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02
$\beta_{3}$ | 0.00 | 0.04 | 0.00 | 0.04 | 0.00 | 0.05 | 0.00 | 0.07 | 0.01 | 0.13
2SRI $\beta_{1}$ | 1.00 | 0.03 | 1.00 | 0.03 | 1.00 | 0.03 | 1.00 | 0.05 | 1.00 | 0.09
$\beta_{2}$ | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02
$\beta_{3}$ | -0.00 | 0.02 | -0.00 | 0.02 | -0.00 | 0.02 | -0.00 | 0.02 | -0.00 | 0.02
$\beta_{U}=1.5$
$R^{2}_{Y|X,G,Z,U}$ | 0.80 | 0.00 | 0.84 | 0.00 | 0.85 | 0.00 | 0.86 | 0.00 | 0.87 | 0.00
$R^{2}_{Y|X,G,Z}$U | 0.35 | 0.01 | 0.53 | 0.01 | 0.67 | 0.00 | 0.79 | 0.00 | 0.85 | 0.00
Naive $\beta_{1}$ | 1.00 | 0.03 | 1.66 | 0.02 | 2.05 | 0.02 | 2.34 | 0.02 | 2.46 | 0.02
$\beta_{2}$ | 0.50 | 0.03 | 0.50 | 0.03 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02
$\beta_{2}$ | -0.00 | 0.03 | -0.00 | 0.03 | 0.00 | 0.02 | 0.00 | 0.02 | 0.00 | 0.02
2SPS $\beta_{1}$ | 1.00 | 0.06 | 0.99 | 0.07 | 0.99 | 0.09 | 0.99 | 0.14 | 0.98 | 0.27
$\beta_{2}$ | 0.50 | 0.03 | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.04
$\beta_{3}$ | 0.00 | 0.06 | 0.00 | 0.07 | 0.01 | 0.09 | 0.01 | 0.14 | 0.02 | 0.25
2SPSadj $\beta_{1}$ | 1.00 | 0.05 | 0.99 | 0.06 | 1.00 | 0.07 | 1.00 | 0.09 | 0.99 | 0.17
$\beta_{2}$ | 0.50 | 0.03 | 0.50 | 0.03 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02
$\beta_{3}$ | 0.00 | 0.05 | 0.01 | 0.05 | 0.01 | 0.05 | 0.01 | 0.06 | 0.01 | 0.10
2SPSa $\beta_{1}$ | 1.00 | 0.07 | 0.99 | 0.08 | 0.99 | 0.10 | 0.98 | 0.17 | 0.97 | 0.31
$\beta_{2}$ | 0.50 | 0.03 | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.04
$\beta_{3}$ | 0.01 | 0.07 | 0.01 | 0.09 | 0.01 | 0.12 | 0.02 | 0.19 | 0.04 | 0.34
2SPSadj-a $\beta_{1}$ | 0.99 | 0.06 | 0.99 | 0.06 | 0.99 | 0.07 | 0.99 | 0.10 | 0.99 | 0.18
$\beta_{2}$ | 0.50 | 0.03 | 0.50 | 0.03 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02
$\beta_{3}$ | 0.01 | 0.07 | 0.01 | 0.07 | 0.01 | 0.07 | 0.01 | 0.09 | 0.01 | 0.14
2SRI $\beta_{1}$ | 1.00 | 0.05 | 1.00 | 0.05 | 1.00 | 0.06 | 1.00 | 0.09 | 0.99 | 0.16
$\beta_{2}$ | 0.50 | 0.03 | 0.50 | 0.03 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02
$\beta_{3}$ | -0.00 | 0.03 | -0.00 | 0.03 | 0.00 | 0.02 | 0.00 | 0.02 | 0.00 | 0.02
$\beta_{U}=3$
$R^{2}_{Y|X,G,Z,U}$ | 0.92 | 0.00 | 0.93 | 0.00 | 0.94 | 0.00 | 0.94 | 0.00 | 0.94 | 0.00
$R^{2}_{Y|X,G,Z}$ | 0.15 | 0.01 | 0.39 | 0.01 | 0.61 | 0.01 | 0.81 | 0.00 | 0.90 | 0.00
Naive $\beta_{1}$ | 1.00 | 0.05 | 2.32 | 0.04 | 3.11 | 0.04 | 3.68 | 0.03 | 3.91 | 0.03
$\beta_{2}$ | 0.50 | 0.05 | 0.50 | 0.05 | 0.50 | 0.04 | 0.50 | 0.03 | 0.50 | 0.02
$\beta_{2}$ | -0.00 | 0.05 | 0.00 | 0.05 | 0.00 | 0.04 | 0.00 | 0.03 | 0.00 | 0.02
2SPS $\beta_{1}$ | 0.99 | 0.09 | 0.99 | 0.11 | 0.99 | 0.14 | 0.98 | 0.21 | 0.96 | 0.39
$\beta_{2}$ | 0.50 | 0.05 | 0.50 | 0.06 | 0.50 | 0.06 | 0.50 | 0.06 | 0.50 | 0.06
$\beta_{3}$ | 0.01 | 0.09 | 0.01 | 0.11 | 0.01 | 0.14 | 0.02 | 0.21 | 0.04 | 0.39
2SPSadj $\beta_{1}$ | 0.99 | 0.09 | 0.99 | 0.10 | 0.99 | 0.11 | 0.99 | 0.16 | 0.98 | 0.29
$\beta_{2}$ | 0.50 | 0.05 | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.03 | 0.50 | 0.02
$\beta_{3}$ | 0.01 | 0.09 | 0.01 | 0.09 | 0.01 | 0.08 | 0.01 | 0.09 | 0.01 | 0.11
2SPSa $\beta_{1}$ | 0.99 | 0.11 | 0.99 | 0.12 | 0.98 | 0.16 | 0.97 | 0.25 | 0.95 | 0.46
$\beta_{2}$ | 0.50 | 0.05 | 0.50 | 0.06 | 0.50 | 0.06 | 0.50 | 0.06 | 0.50 | 0.06
$\beta_{3}$ | 0.01 | 0.12 | 0.01 | 0.15 | 0.02 | 0.19 | 0.03 | 0.29 | 0.05 | 0.53
2SPSadj-a $\beta_{1}$ | 0.99 | 0.10 | 0.99 | 0.11 | 0.99 | 0.12 | 0.99 | 0.16 | 0.98 | 0.29
$\beta_{2}$ | 0.50 | 0.05 | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.03 | 0.50 | 0.02
$\beta_{3}$ | 0.01 | 0.12 | 0.02 | 0.12 | 0.02 | 0.11 | 0.02 | 0.12 | 0.02 | 0.16
2SRI $\beta_{1}$ | 1.00 | 0.08 | 1.00 | 0.09 | 1.00 | 0.11 | 1.00 | 0.16 | 0.99 | 0.28
$\beta_{2}$ | 0.50 | 0.05 | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.03 | 0.50 | 0.02
$\beta_{3}$ | -0.00 | 0.05 | 0.00 | 0.05 | 0.00 | 0.04 | 0.00 | 0.03 | 0.00 | 0.02
Table S4: Simulation results, linear regression, Setting II.B | $\gamma_{U}=0$ | $\gamma_{U}=0.5$ | $\gamma_{U}=1$ | $\gamma_{U}=2$ | $\gamma_{U}=4$
---|---|---|---|---|---
| Mean | SD | Mean | SD | Mean | SD | Mean | SD | Mean | SD
$R^{2}_{X|G_{IV},Z,U}$ | 0.33 | 0.01 | 0.43 | 0.01 | 0.60 | 0.01 | 0.82 | 0.00 | 0.94 | 0.00
$R^{2}_{X|G_{IV},Z}$ | 0.33 | 0.01 | 0.29 | 0.01 | 0.20 | 0.01 | 0.09 | 0.01 | 0.03 | 0.00
$\beta_{U}=0$
$R^{2}_{Y|X,G,Z,U}$ | 0.65 | 0.00 | 0.65 | 0.00 | 0.64 | 0.00 | 0.62 | 0.01 | 0.61 | 0.01
$R^{2}_{Y|X,G,Z}$ | 0.65 | 0.00 | 0.65 | 0.00 | 0.64 | 0.00 | 0.62 | 0.01 | 0.60 | 0.01
Naive $\beta_{1}$ | 1.00 | 0.02 | 1.00 | 0.02 | 1.00 | 0.02 | 1.00 | 0.02 | 1.00 | 0.02
$\beta_{2}$ | 0.50 | 0.01 | 0.50 | 0.01 | 0.50 | 0.01 | 0.50 | 0.01 | 0.50 | 0.01
$\beta_{2}$ | -0.00 | 0.01 | -0.00 | 0.01 | 0.00 | 0.01 | 0.00 | 0.01 | 0.00 | 0.01
2SPS $\beta_{1}$ | 1.00 | 0.05 | 1.00 | 0.05 | 1.00 | 0.06 | 1.00 | 0.09 | 1.01 | 0.17
$\beta_{2}$ | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02
$\beta_{3}$ | -0.00 | 0.03 | -0.00 | 0.03 | -0.00 | 0.04 | -0.00 | 0.06 | -0.01 | 0.12
2SPSadj $\beta_{1}$ | 1.00 | 0.03 | 1.00 | 0.04 | 1.00 | 0.04 | 1.00 | 0.07 | 1.00 | 0.12
$\beta_{2}$ | 0.50 | 0.01 | 0.50 | 0.01 | 0.50 | 0.01 | 0.50 | 0.01 | 0.50 | 0.01
$\beta_{3}$ | 0.00 | 0.02 | 0.00 | 0.03 | 0.00 | 0.03 | 0.00 | 0.05 | 0.00 | 0.08
2SPSa $\beta_{1}$ | 1.00 | 0.05 | 1.00 | 0.06 | 1.00 | 0.07 | 1.01 | 0.11 | 1.01 | 0.21
$\beta_{2}$ | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02
$\beta_{3}$ | -0.00 | 0.04 | -0.00 | 0.05 | -0.00 | 0.06 | -0.01 | 0.09 | -0.01 | 0.17
2SPSadj-a $\beta_{1}$ | 1.00 | 0.04 | 1.00 | 0.04 | 1.00 | 0.05 | 1.00 | 0.08 | 1.01 | 0.14
$\beta_{2}$ | 0.50 | 0.01 | 0.50 | 0.01 | 0.50 | 0.01 | 0.50 | 0.01 | 0.50 | 0.01
$\beta_{3}$ | -0.00 | 0.04 | -0.00 | 0.04 | -0.00 | 0.05 | -0.00 | 0.07 | -0.00 | 0.12
2SRI $\beta_{1}$ | 1.00 | 0.03 | 1.00 | 0.03 | 1.00 | 0.04 | 1.00 | 0.05 | 1.00 | 0.09
$\beta_{2}$ | 0.50 | 0.01 | 0.50 | 0.01 | 0.50 | 0.01 | 0.50 | 0.01 | 0.50 | 0.01
$\beta_{3}$ | -0.00 | 0.01 | -0.00 | 0.01 | 0.00 | 0.01 | 0.00 | 0.01 | 0.00 | 0.01
$\beta_{U}=1.5$
$R^{2}_{Y|X,G,Z,U}$ | 0.81 | 0.00 | 0.84 | 0.00 | 0.86 | 0.00 | 0.87 | 0.00 | 0.87 | 0.00
$R^{2}_{Y|X,G,Z}$ | 0.37 | 0.01 | 0.54 | 0.01 | 0.68 | 0.00 | 0.80 | 0.00 | 0.85 | 0.00
Naive $\beta_{1}$ | 1.00 | 0.03 | 1.67 | 0.03 | 2.06 | 0.02 | 2.35 | 0.02 | 2.46 | 0.02
$\beta_{2}$ | 0.50 | 0.03 | 0.39 | 0.02 | 0.35 | 0.02 | 0.37 | 0.02 | 0.42 | 0.01
$\beta_{2}$ | -0.00 | 0.02 | 0.00 | 0.02 | 0.00 | 0.02 | 0.00 | 0.02 | 0.00 | 0.02
2SPS $\beta_{1}$ | 1.00 | 0.07 | 1.00 | 0.08 | 1.00 | 0.10 | 1.01 | 0.16 | 1.01 | 0.30
$\beta_{2}$ | 0.50 | 0.03 | 0.50 | 0.03 | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.04
$\beta_{3}$ | -0.00 | 0.05 | -0.01 | 0.06 | -0.01 | 0.08 | -0.01 | 0.12 | -0.02 | 0.22
2SPSadj $\beta_{1}$ | 1.00 | 0.06 | 1.00 | 0.07 | 1.00 | 0.07 | 1.00 | 0.11 | 0.99 | 0.18
$\beta_{2}$ | 0.50 | 0.03 | 0.50 | 0.03 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02
$\beta_{3}$ | -0.00 | 0.04 | -0.00 | 0.04 | 0.00 | 0.05 | 0.00 | 0.06 | 0.00 | 0.09
2SPSa $\beta_{1}$ | 1.00 | 0.08 | 1.00 | 0.10 | 1.01 | 0.13 | 1.01 | 0.20 | 1.01 | 0.38
$\beta_{2}$ | 0.50 | 0.03 | 0.50 | 0.03 | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.04
$\beta_{3}$ | -0.01 | 0.07 | -0.01 | 0.09 | -0.01 | 0.11 | -0.01 | 0.18 | -0.02 | 0.33
2SPSadj-a $\beta_{1}$ | 1.00 | 0.07 | 1.00 | 0.08 | 1.00 | 0.09 | 1.00 | 0.12 | 1.00 | 0.20
$\beta_{2}$ | 0.50 | 0.03 | 0.50 | 0.03 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02
$\beta_{3}$ | -0.00 | 0.06 | -0.00 | 0.06 | 0.00 | 0.07 | 0.00 | 0.08 | 0.00 | 0.13
2SRI $\beta_{1}$ | 1.00 | 0.05 | 1.00 | 0.06 | 1.00 | 0.06 | 1.00 | 0.09 | 1.00 | 0.16
$\beta_{2}$ | 0.50 | 0.03 | 0.50 | 0.03 | 0.50 | 0.02 | 0.50 | 0.02 | 0.50 | 0.02
$\beta_{3}$ | -0.00 | 0.02 | 0.00 | 0.02 | 0.00 | 0.02 | 0.00 | 0.02 | 0.00 | 0.01
$\beta_{U}=3$
$R^{2}_{Y|X,G,Z,U}$ | 0.92 | 0.00 | 0.93 | 0.00 | 0.94 | 0.00 | 0.94 | 0.00 | 0.94 | 0.00
$R^{2}_{Y|X,G,Z}$U | 0.16 | 0.01 | 0.40 | 0.01 | 0.62 | 0.01 | 0.81 | 0.00 | 0.91 | 0.00
Naive $\beta_{1}$ | 1.00 | 0.05 | 2.34 | 0.05 | 3.13 | 0.04 | 3.69 | 0.03 | 3.91 | 0.03
$\beta_{2}$ | 0.50 | 0.04 | 0.27 | 0.04 | 0.20 | 0.03 | 0.24 | 0.02 | 0.34 | 0.02
$\beta_{2}$ | -0.00 | 0.04 | 0.01 | 0.04 | 0.01 | 0.03 | 0.00 | 0.02 | 0.00 | 0.02
2SPS $\beta_{1}$ | 1.01 | 0.11 | 1.00 | 0.13 | 1.01 | 0.16 | 1.01 | 0.25 | 1.01 | 0.46
$\beta_{2}$ | 0.50 | 0.05 | 0.50 | 0.05 | 0.50 | 0.06 | 0.50 | 0.06 | 0.50 | 0.06
$\beta_{3}$ | -0.01 | 0.08 | -0.01 | 0.09 | -0.01 | 0.12 | -0.01 | 0.18 | -0.02 | 0.33
2SPSadj $\beta_{1}$ | 1.00 | 0.11 | 1.00 | 0.11 | 1.00 | 0.13 | 1.00 | 0.17 | 0.98 | 0.30
$\beta_{2}$ | 0.50 | 0.05 | 0.50 | 0.04 | 0.50 | 0.03 | 0.50 | 0.02 | 0.50 | 0.02
$\beta_{3}$ | -0.01 | 0.08 | -0.00 | 0.07 | 0.00 | 0.07 | 0.00 | 0.08 | 0.01 | 0.10
2SPSa $\beta_{1}$ | 1.01 | 0.13 | 1.01 | 0.16 | 1.01 | 0.20 | 1.01 | 0.31 | 1.02 | 0.58
$\beta_{2}$ | 0.50 | 0.05 | 0.50 | 0.05 | 0.50 | 0.06 | 0.50 | 0.06 | 0.50 | 0.06
$\beta_{3}$ | -0.01 | 0.12 | -0.01 | 0.14 | -0.01 | 0.18 | -0.01 | 0.28 | -0.03 | 0.51
2SPSadj-a $\beta_{1}$ | 1.00 | 0.13 | 1.00 | 0.13 | 1.00 | 0.14 | 1.00 | 0.19 | 0.99 | 0.31
$\beta_{2}$ | 0.50 | 0.05 | 0.50 | 0.04 | 0.50 | 0.03 | 0.50 | 0.02 | 0.50 | 0.02
$\beta_{3}$ | -0.01 | 0.11 | -0.00 | 0.11 | 0.00 | 0.10 | 0.00 | 0.11 | 0.00 | 0.15
2SRI $\beta_{1}$ | 1.00 | 0.09 | 1.00 | 0.10 | 1.00 | 0.11 | 1.00 | 0.16 | 0.99 | 0.28
$\beta_{2}$ | 0.50 | 0.05 | 0.50 | 0.04 | 0.50 | 0.03 | 0.50 | 0.02 | 0.50 | 0.02
$\beta_{3}$ | -0.00 | 0.04 | 0.00 | 0.04 | 0.00 | 0.03 | 0.00 | 0.02 | 0.00 | 0.02
Table S5: Simulation results, logistic regression, Setting III.A | $\gamma_{U}=0$ | $\gamma_{U}=0.5$ | $\gamma_{U}=1$ | $\gamma_{U}=2$ | $\gamma_{U}=4$
---|---|---|---|---|---
| Mean | SD | Mean | SD | Mean | SD | Mean | SD | Mean | SD
$R^{2}_{X|G_{IV},Z,U}$ | 0.33 | 0.01 | 0.43 | 0.01 | 0.60 | 0.01 | 0.82 | 0.00 | 0.94 | 0.00
$R^{2}_{X|G_{IV},Z}$ | 0.33 | 0.01 | 0.29 | 0.01 | 0.20 | 0.01 | 0.09 | 0.01 | 0.03 | 0.00
$\beta_{U}=0$
$\Pr(Y=1)$ | 0.04 | 0.00 | 0.04 | 0.00 | 0.04 | 0.00 | 0.04 | 0.00 | 0.04 | 0.00
Naive $\beta_{1}$ | 1.00 | 0.05 | 1.00 | 0.05 | 1.00 | 0.04 | 1.00 | 0.04 | 1.00 | 0.04
$\beta_{2}$ | 0.50 | 0.05 | 0.50 | 0.05 | 0.50 | 0.05 | 0.50 | 0.04 | 0.50 | 0.05
$\beta_{2}$ | 0.50 | 0.05 | 0.50 | 0.05 | 0.50 | 0.05 | 0.50 | 0.05 | 0.50 | 0.05
2SPS $\beta_{1}$ | 1.03 | 0.08 | 1.03 | 0.09 | 1.04 | 0.10 | 1.09 | 0.15 | 1.15 | 0.25
$\beta_{2}$ | 0.81 | 0.04 | 0.83 | 0.04 | 0.85 | 0.04 | 0.88 | 0.03 | 0.90 | 0.03
$\beta_{3}$ | 0.29 | 0.07 | 0.27 | 0.07 | 0.24 | 0.08 | 0.17 | 0.12 | 0.07 | 0.20
2SPSadj $\beta_{1}$ | 1.10 | 0.09 | 1.10 | 0.10 | 1.13 | 0.11 | 1.18 | 0.16 | 1.24 | 0.29
$\beta_{2}$ | 0.73 | 0.04 | 0.74 | 0.04 | 0.75 | 0.04 | 0.77 | 0.04 | 0.78 | 0.04
$\beta_{3}$ | 0.33 | 0.08 | 0.31 | 0.08 | 0.29 | 0.09 | 0.22 | 0.14 | 0.13 | 0.22
2SPSa $\beta_{1}$ | 0.99 | 0.09 | 0.98 | 0.10 | 0.99 | 0.11 | 0.99 | 0.17 | 0.98 | 0.28
$\beta_{2}$ | 0.82 | 0.04 | 0.83 | 0.04 | 0.85 | 0.04 | 0.89 | 0.03 | 0.90 | 0.03
$\beta_{3}$ | 0.34 | 0.10 | 0.33 | 0.10 | 0.31 | 0.11 | 0.30 | 0.16 | 0.30 | 0.28
2SPSadj-a $\beta_{1}$ | 1.06 | 0.10 | 1.06 | 0.11 | 1.07 | 0.13 | 1.09 | 0.19 | 1.06 | 0.35
$\beta_{2}$ | 0.73 | 0.04 | 0.74 | 0.04 | 0.76 | 0.04 | 0.78 | 0.04 | 0.79 | 0.04
$\beta_{3}$ | 0.38 | 0.10 | 0.37 | 0.11 | 0.36 | 0.12 | 0.34 | 0.17 | 0.36 | 0.31
2SRI $\beta_{1}$ | 1.00 | 0.07 | 0.99 | 0.08 | 1.00 | 0.09 | 1.00 | 0.13 | 1.00 | 0.22
$\beta_{2}$ | 0.50 | 0.05 | 0.50 | 0.05 | 0.50 | 0.05 | 0.50 | 0.04 | 0.50 | 0.05
$\beta_{3}$ | 0.50 | 0.05 | 0.50 | 0.05 | 0.50 | 0.05 | 0.50 | 0.05 | 0.50 | 0.05
$\beta_{U}=1.5$
$\Pr(Y=1)$ | 0.07 | 0.00 | 0.08 | 0.00 | 0.09 | 0.00 | 0.10 | 0.00 | 0.10 | 0.00
Naive $\beta_{1}$ | 0.87 | 0.04 | 1.44 | 0.05 | 1.85 | 0.06 | 2.23 | 0.07 | 2.42 | 0.07
$\beta_{2}$ | 0.42 | 0.04 | 0.44 | 0.05 | 0.46 | 0.06 | 0.48 | 0.07 | 0.49 | 0.08
$\beta_{2}$ | 0.37 | 0.04 | 0.38 | 0.06 | 0.40 | 0.07 | 0.45 | 0.08 | 0.48 | 0.09
2SPS $\beta_{1}$ | 0.87 | 0.08 | 0.80 | 0.08 | 0.77 | 0.09 | 0.75 | 0.13 | 0.81 | 0.23
$\beta_{2}$ | 0.60 | 0.04 | 0.60 | 0.03 | 0.59 | 0.03 | 0.59 | 0.03 | 0.58 | 0.03
$\beta_{3}$ | 0.22 | 0.06 | 0.15 | 0.07 | 0.10 | 0.08 | 0.04 | 0.11 | -0.07 | 0.20
2SPSadj $\beta_{1}$ | 0.93 | 0.08 | 0.96 | 0.09 | 1.03 | 0.12 | 1.13 | 0.19 | 1.20 | 0.35
$\beta_{2}$ | 0.55 | 0.04 | 0.62 | 0.04 | 0.70 | 0.04 | 0.79 | 0.04 | 0.84 | 0.05
$\beta_{3}$ | 0.25 | 0.07 | 0.24 | 0.07 | 0.23 | 0.10 | 0.19 | 0.15 | 0.11 | 0.28
2SPSa $\beta_{1}$ | 0.84 | 0.08 | 0.76 | 0.09 | 0.72 | 0.10 | 0.67 | 0.14 | 0.65 | 0.27
$\beta_{2}$ | 0.60 | 0.04 | 0.60 | 0.03 | 0.60 | 0.03 | 0.59 | 0.03 | 0.58 | 0.03
$\beta_{3}$ | 0.27 | 0.08 | 0.21 | 0.09 | 0.18 | 0.11 | 0.16 | 0.15 | 0.16 | 0.28
2SPSadj-a $\beta_{1}$ | 0.90 | 0.09 | 0.92 | 0.11 | 0.98 | 0.14 | 1.04 | 0.23 | 1.02 | 0.43
$\beta_{2}$ | 0.55 | 0.04 | 0.62 | 0.04 | 0.70 | 0.04 | 0.79 | 0.04 | 0.84 | 0.05
$\beta_{3}$ | 0.30 | 0.09 | 0.29 | 0.10 | 0.30 | 0.13 | 0.32 | 0.21 | 0.37 | 0.40
2SRI $\beta_{1}$ | 0.86 | 0.07 | 0.88 | 0.08 | 0.92 | 0.11 | 0.97 | 0.16 | 0.97 | 0.29
$\beta_{2}$ | 0.42 | 0.04 | 0.44 | 0.05 | 0.46 | 0.06 | 0.48 | 0.07 | 0.50 | 0.08
$\beta_{3}$ | 0.37 | 0.04 | 0.38 | 0.06 | 0.41 | 0.07 | 0.46 | 0.08 | 0.49 | 0.09
$\beta_{U}=3$
$\Pr(Y=1)$ | 0.13 | 0.00 | 0.15 | 0.00 | 0.17 | 0.00 | 0.17 | 0.00 | 0.18 | 0.00
Naive $\beta_{1}$ | 0.55 | 0.03 | 1.34 | 0.04 | 2.02 | 0.06 | 2.93 | 0.08 | 3.61 | 0.11
$\beta_{2}$ | 0.26 | 0.03 | 0.29 | 0.04 | 0.33 | 0.05 | 0.41 | 0.07 | 0.46 | 0.10
$\beta_{2}$ | 0.24 | 0.04 | 0.25 | 0.05 | 0.29 | 0.07 | 0.37 | 0.09 | 0.45 | 0.13
2SPS $\beta_{1}$ | 0.55 | 0.06 | 0.51 | 0.07 | 0.47 | 0.08 | 0.47 | 0.12 | 0.48 | 0.22
$\beta_{2}$ | 0.32 | 0.03 | 0.34 | 0.03 | 0.34 | 0.03 | 0.34 | 0.03 | 0.34 | 0.03
$\beta_{3}$ | 0.19 | 0.06 | 0.13 | 0.06 | 0.11 | 0.07 | 0.08 | 0.11 | 0.01 | 0.18
2SPSadj $\beta_{1}$ | 0.58 | 0.07 | 0.63 | 0.08 | 0.73 | 0.12 | 0.91 | 0.21 | 1.08 | 0.45
$\beta_{2}$ | 0.30 | 0.03 | 0.38 | 0.04 | 0.47 | 0.04 | 0.63 | 0.05 | 0.75 | 0.05
$\beta_{3}$ | 0.20 | 0.06 | 0.18 | 0.07 | 0.21 | 0.09 | 0.23 | 0.15 | 0.16 | 0.32
2SPSa $\beta_{1}$ | 0.54 | 0.08 | 0.49 | 0.08 | 0.45 | 0.09 | 0.43 | 0.14 | 0.42 | 0.25
$\beta_{2}$ | 0.32 | 0.03 | 0.34 | 0.03 | 0.34 | 0.03 | 0.34 | 0.03 | 0.34 | 0.03
$\beta_{3}$ | 0.21 | 0.08 | 0.16 | 0.09 | 0.15 | 0.10 | 0.13 | 0.14 | 0.11 | 0.26
2SPSadj-a $\beta_{1}$ | 0.56 | 0.08 | 0.62 | 0.09 | 0.70 | 0.13 | 0.85 | 0.24 | 0.97 | 0.50
$\beta_{2}$ | 0.30 | 0.03 | 0.38 | 0.04 | 0.47 | 0.04 | 0.63 | 0.05 | 0.75 | 0.05
$\beta_{3}$ | 0.22 | 0.09 | 0.21 | 0.10 | 0.25 | 0.14 | 0.32 | 0.22 | 0.32 | 0.44
2SRI $\beta_{1}$ | 0.55 | 0.06 | 0.60 | 0.07 | 0.68 | 0.10 | 0.82 | 0.19 | 0.91 | 0.40
$\beta_{2}$ | 0.26 | 0.03 | 0.29 | 0.04 | 0.34 | 0.05 | 0.42 | 0.07 | 0.47 | 0.10
$\beta_{3}$ | 0.24 | 0.04 | 0.25 | 0.05 | 0.30 | 0.07 | 0.39 | 0.10 | 0.46 | 0.13
Table S6: Simulation results, logistic regression, Setting IV.A | $\gamma_{U}=0$ | $\gamma_{U}=0.5$ | $\gamma_{U}=1$ | $\gamma_{U}=2$ | $\gamma_{U}=4$
---|---|---|---|---|---
| Mean | SD | Mean | SD | Mean | SD | Mean | SD | Mean | SD
$R^{2}_{X|G_{IV},Z,U}$ | 0.33 | 0.01 | 0.43 | 0.01 | 0.60 | 0.01 | 0.82 | 0.00 | 0.94 | 0.00
$R^{2}_{X|G_{IV},Z}$ | 0.33 | 0.01 | 0.29 | 0.01 | 0.20 | 0.01 | 0.09 | 0.01 | 0.03 | 0.00
$\beta_{U}=0$
$\Pr(Y=1)$ | 0.06 | 0.00 | 0.06 | 0.00 | 0.06 | 0.00 | 0.05 | 0.00 | 0.05 | 0.00
Naive $\beta_{1}$ | 1.00 | 0.06 | 1.00 | 0.06 | 1.00 | 0.06 | 1.00 | 0.06 | 1.00 | 0.05
$\beta_{2}$ | 0.50 | 0.05 | 0.50 | 0.05 | 0.50 | 0.05 | 0.50 | 0.05 | 0.50 | 0.05
$\beta_{2}$ | 0.50 | 0.05 | 0.50 | 0.05 | 0.50 | 0.05 | 0.50 | 0.05 | 0.50 | 0.05
2SPS $\beta_{1}$ | 1.10 | 0.10 | 1.12 | 0.10 | 1.15 | 0.13 | 1.23 | 0.17 | 1.40 | 0.30
$\beta_{2}$ | 0.80 | 0.04 | 0.82 | 0.04 | 0.84 | 0.04 | 0.86 | 0.03 | 0.88 | 0.03
$\beta_{3}$ | 0.22 | 0.07 | 0.20 | 0.07 | 0.15 | 0.08 |  0.06 | 0.11 | -0.10 | 0.19
2SPSadj $\beta_{1}$ | 1.19 | 0.11 | 1.20 | 0.12 | 1.24 | 0.15 | 1.32 | 0.21 | 1.47 | 0.35
$\beta_{2}$ | 0.74 | 0.05 | 0.76 | 0.04 | 0.78 | 0.04 | 0.80 | 0.04 | 0.81 | 0.04
$\beta_{3}$ | 0.30 | 0.08 | 0.29 | 0.08 | 0.25 | 0.10 | 0.17 | 0.13 | 0.03 | 0.22
2SPSa $\beta_{1}$ | 1.05 | 0.11 | 1.06 | 0.12 | 1.07 | 0.15 | 1.10 | 0.21 | 1.18 | 0.36
$\beta_{2}$ | 0.81 | 0.04 | 0.82 | 0.04 | 0.84 | 0.04 | 0.87 | 0.03 | 0.88 | 0.03
$\beta_{3}$ | 0.27 | 0.09 | 0.25 | 0.09 | 0.22 | 0.11 | 0.18 | 0.16 | 0.11 | 0.27
2SPSadj-a $\beta_{1}$ | 1.14 | 0.14 | 1.15 | 0.14 | 1.17 | 0.18 | 1.20 | 0.26 | 1.24 | 0.44
$\beta_{2}$ | 0.75 | 0.05 | 0.76 | 0.04 | 0.78 | 0.04 | 0.80 | 0.04 | 0.81 | 0.04
$\beta_{3}$ | 0.35 | 0.10 | 0.33 | 0.11 | 0.31 | 0.13 | 0.29 | 0.18 | 0.24 | 0.32
2SRI $\beta_{1}$ | 1.00 | 0.09 | 1.00 | 0.09 | 1.00 | 0.11 | 1.00 | 0.14 | 1.00 | 0.25
$\beta_{2}$ | 0.50 | 0.05 | 0.50 | 0.05 | 0.50 | 0.05 | 0.50 | 0.05 | 0.50 | 0.05
$\beta_{3}$ | 0.50 | 0.05 | 0.50 | 0.05 | 0.50 | 0.05 | 0.50 | 0.05 | 0.50 | 0.05
2SRI alt $\beta_{1}$ | 1.00 | 0.09 | 1.00 | 0.09 | 1.00 | 0.11 | 1.00 | 0.14 | 1.00 | 0.25
2SRI alt $\beta_{2}$ | 0.50 | 0.05 | 0.50 | 0.05 | 0.50 | 0.05 | 0.50 | 0.05 | 0.50 | 0.05
2SRI alt $\beta_{3}$ | 0.50 | 0.05 | 0.50 | 0.05 | 0.50 | 0.05 | 0.50 | 0.05 | 0.50 | 0.05
$\beta_{U}=1.5$
$\Pr(Y=1)$ | 0.08 | 0.00 | 0.10 | 0.00 | 0.11 | 0.00 | 0.12 | 0.00 | 0.12 | 0.00
Naive $\beta_{1}$ | 0.88 | 0.05 | 1.46 | 0.06 | 1.86 | 0.07 | 2.24 | 0.08 | 2.42 | 0.09
$\beta_{2}$ | 0.43 | 0.04 | 0.35 | 0.05 | 0.33 | 0.06 | 0.36 | 0.06 | 0.42 | 0.07
$\beta_{2}$ | 0.35 | 0.05 | 0.36 | 0.05 | 0.40 | 0.07 | 0.46 | 0.07 | 0.48 | 0.08
2SPS $\beta_{1}$ | 0.93 | 0.09 | 0.88 | 0.09 | 0.84 | 0.11 | 0.86 | 0.16 | 0.95 | 0.27
$\beta_{2}$ | 0.59 | 0.03 | 0.58 | 0.03 | 0.58 | 0.03 | 0.56 | 0.03 | 0.56 | 0.03
$\beta_{3}$ | 0.17 | 0.06 | 0.09 | 0.06 | 0.05 | 0.08 | -0.03 | 0.10 | -0.15 | 0.18
2SPSadj $\beta_{1}$ | 1.00 | 0.09 | 1.05 | 0.11 | 1.12 | 0.15 | 1.27 | 0.23 | 1.45 | 0.43
$\beta_{2}$ | 0.56 | 0.04 | 0.63 | 0.04 | 0.71 | 0.05 | 0.80 | 0.05 | 0.86 | 0.05
$\beta_{3}$ | 0.21 | 0.06 | 0.19 | 0.08 | 0.19 | 0.10 | 0.13 | 0.16 | -0.02 | 0.28
2SPSa $\beta_{1}$ | 0.88 | 0.10 | 0.83 | 0.11 | 0.78 | 0.12 | 0.78 | 0.18 | 0.82 | 0.33
$\beta_{2}$ | 0.60 | 0.03 | 0.59 | 0.03 | 0.58 | 0.03 | 0.56 | 0.03 | 0.56 | 0.03
$\beta_{3}$ | 0.22 | 0.09 | 0.13 | 0.09 | 0.11 | 0.10 | 0.05 | 0.14 | -0.01 | 0.26
2SPSadj-a $\beta_{1}$ | 0.95 | 0.12 | 1.01 | 0.14 | 1.06 | 0.19 | 1.17 | 0.29 | 1.29 | 0.54
$\beta_{2}$ | 0.56 | 0.04 | 0.63 | 0.04 | 0.71 | 0.05 | 0.80 | 0.05 | 0.86 | 0.05
$\beta_{3}$ | 0.27 | 0.09 | 0.24 | 0.11 | 0.25 | 0.15 | 0.23 | 0.22 | 0.14 | 0.40
2SRI $\beta_{1}$ | 0.88 | 0.08 | 0.90 | 0.09 | 0.93 | 0.13 | 0.97 | 0.17 | 0.97 | 0.31
$\beta_{2}$ | 0.43 | 0.04 | 0.45 | 0.05 | 0.47 | 0.06 | 0.49 | 0.07 | 0.50 | 0.07
$\beta_{3}$ | 0.35 | 0.05 | 0.36 | 0.05 | 0.40 | 0.07 | 0.46 | 0.07 | 0.49 | 0.08
2SRI alt $\beta_{1}$ | 0.88 | 0.08 | 0.90 | 0.09 | 0.93 | 0.13 | 0.97 | 0.17 | 0.97 | 0.31
2SRI alt $\beta_{2}$ | 0.43 | 0.04 | 0.45 | 0.05 | 0.47 | 0.06 | 0.49 | 0.07 | 0.50 | 0.07
2SRI alt $\beta_{3}$ | 0.35 | 0.05 | 0.36 | 0.05 | 0.40 | 0.07 | 0.46 | 0.07 | 0.49 | 0.08
$\beta_{U}=3$
$\Pr(Y=1)$ | 0.15 | 0.00 | 0.17 | 0.00 | 0.18 | 0.00 | 0.19 | 0.00 | 0.19 | 0.00
Naive $\beta_{1}$ | 0.56 | 0.04 | 1.36 | 0.05 | 2.05 | 0.07 | 2.95 | 0.10 | 3.62 | 0.12
$\beta_{2}$ | 0.26 | 0.03 | 0.16 | 0.04 | 0.14 | 0.05 | 0.20 | 0.06 | 0.33 | 0.09
$\beta_{2}$ | 0.23 | 0.04 | 0.24 | 0.04 | 0.29 | 0.06 | 0.37 | 0.09 | 0.44 | 0.11
2SPS $\beta_{1}$ | 0.57 | 0.08 | 0.53 | 0.08 | 0.51 | 0.09 | 0.51 | 0.14 | 0.55 | 0.26
$\beta_{2}$ | 0.32 | 0.03 | 0.33 | 0.03 | 0.33 | 0.03 | 0.33 | 0.03 | 0.33 | 0.03
$\beta_{3}$ | 0.16 | 0.06 | 0.11 | 0.06 | 0.08 | 0.07 | 0.04 | 0.10 | -0.03 | 0.17
2SPSadj $\beta_{1}$ | 0.60 | 0.08 | 0.66 | 0.10 | 0.79 | 0.14 | 1.01 | 0.25 | 1.25 | 0.50
$\beta_{2}$ | 0.30 | 0.03 | 0.37 | 0.04 | 0.48 | 0.04 | 0.63 | 0.05 | 0.76 | 0.05
$\beta_{3}$ | 0.18 | 0.06 | 0.17 | 0.07 | 0.18 | 0.09 | 0.18 | 0.15 | 0.08 | 0.30
2SPSa $\beta_{1}$ | 0.55 | 0.09 | 0.50 | 0.09 | 0.49 | 0.11 | 0.47 | 0.17 | 0.50 | 0.31
$\beta_{2}$ | 0.32 | 0.03 | 0.33 | 0.03 | 0.33 | 0.03 | 0.33 | 0.03 | 0.33 | 0.03
$\beta_{3}$ | 0.19 | 0.08 | 0.14 | 0.08 | 0.11 | 0.09 | 0.08 | 0.14 | 0.03 | 0.26
2SPSadj-a $\beta_{1}$ | 0.58 | 0.10 | 0.64 | 0.12 | 0.76 | 0.16 | 0.94 | 0.30 | 1.15 | 0.60
$\beta_{2}$ | 0.30 | 0.03 | 0.37 | 0.04 | 0.48 | 0.04 | 0.64 | 0.05 | 0.76 | 0.05
$\beta_{3}$ | 0.21 | 0.08 | 0.20 | 0.09 | 0.21 | 0.13 | 0.25 | 0.22 | 0.18 | 0.45
2SRI $\beta_{1}$ | 0.56 | 0.07 | 0.61 | 0.09 | 0.70 | 0.11 | 0.83 | 0.21 | 0.92 | 0.41
$\beta_{2}$ | 0.26 | 0.03 | 0.29 | 0.04 | 0.34 | 0.05 | 0.42 | 0.07 | 0.48 | 0.09
$\beta_{3}$ | 0.23 | 0.04 | 0.24 | 0.05 | 0.29 | 0.06 | 0.39 | 0.09 | 0.45 | 0.11
Table S7: Simulation results, logistic regression, Setting III.B | $\gamma_{U}=0$ | $\gamma_{U}=0.5$ | $\gamma_{U}=1$ | $\gamma_{U}=2$ | $\gamma_{U}=4$
---|---|---|---|---|---
| Mean | SD | Mean | SD | Mean | SD | Mean | SD | Mean | SD
$R^{2}_{X|G_{IV},Z,U}$ | 0.33 | 0.01 | 0.43 | 0.01 | 0.60 | 0.01 | 0.82 | 0.00 | 0.94 | 0.00
$R^{2}_{X|G_{IV},Z}$ | 0.33 | 0.01 | 0.29 | 0.01 | 0.20 | 0.01 | 0.09 | 0.01 | 0.03 | 0.00
$\beta_{U}=0$
$\Pr(Y=1)$ | 0.03 | 0.00 | 0.03 | 0.00 | 0.03 | 0.00 | 0.03 | 0.00 | 0.03 | 0.00
Naive $\beta_{1}$ | 1.00 | 0.04 | 1.00 | 0.04 | 1.00 | 0.04 | 1.00 | 0.04 | 1.00 | 0.04
$\beta_{2}$ | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.04
$\beta_{2}$ | 0.00 | 0.04 | -0.00 | 0.04 | 0.00 | 0.04 | -0.00 | 0.04 | -0.00 | 0.04
2SPS $\beta_{1}$ | 0.98 | 0.07 | 0.98 | 0.08 | 0.98 | 0.10 | 0.98 | 0.13 | 0.98 | 0.24
$\beta_{2}$ | 0.49 | 0.04 | 0.49 | 0.04 | 0.49 | 0.03 | 0.48 | 0.03 | 0.48 | 0.03
$\beta_{3}$ | -0.01 | 0.07 | -0.02 | 0.07 | -0.02 | 0.08 | -0.03 | 0.12 | -0.02 | 0.21
2SPSadj $\beta_{1}$ | 1.00 | 0.08 | 1.00 | 0.08 | 1.00 | 0.10 | 1.00 | 0.13 | 0.99 | 0.26
$\beta_{2}$ | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.04
$\beta_{3}$ | 0.00 | 0.07 | -0.01 | 0.07 | -0.01 | 0.09 | -0.01 | 0.13 | 0.02 | 0.23
2SPSa $\beta_{1}$ | 0.98 | 0.09 | 0.97 | 0.09 | 0.97 | 0.11 | 0.96 | 0.15 | 0.95 | 0.26
$\beta_{2}$ | 0.49 | 0.04 | 0.49 | 0.04 | 0.49 | 0.03 | 0.49 | 0.03 | 0.49 | 0.03
$\beta_{3}$ | -0.01 | 0.09 | -0.01 | 0.09 | -0.01 | 0.11 | -0.01 | 0.16 | 0.03 | 0.28
2SPSadj-a $\beta_{1}$ | 1.00 | 0.09 | 1.00 | 0.10 | 1.00 | 0.11 | 0.99 | 0.15 | 0.98 | 0.28
$\beta_{2}$ | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.04
$\beta_{3}$ | 0.00 | 0.09 | -0.00 | 0.10 | -0.00 | 0.12 | 0.00 | 0.18 | 0.03 | 0.30
2SRI $\beta_{1}$ | 1.00 | 0.07 | 1.00 | 0.07 | 1.00 | 0.08 | 0.99 | 0.11 | 1.00 | 0.21
$\beta_{2}$ | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.04
$\beta_{3}$ | 0.00 | 0.04 | -0.00 | 0.04 | 0.00 | 0.04 | -0.00 | 0.04 | -0.00 | 0.04
2SRI alt $\beta_{1}$ | 1.00 | 0.07 | 1.00 | 0.07 | 1.00 | 0.08 | 0.99 | 0.11 | 1.00 | 0.21
2SRI alt $\beta_{2}$ | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.04
2SRI alt $\beta_{3}$ | 0.00 | 0.04 | -0.00 | 0.04 | 0.00 | 0.04 | -0.00 | 0.04 | -0.00 | 0.04
$\beta_{U}=1.5$
$\Pr(Y=1)$ | 0.06 | 0.00 | 0.07 | 0.00 | 0.08 | 0.00 | 0.08 | 0.00 | 0.08 | 0.00
Naive $\beta_{1}$ | 0.87 | 0.04 | 1.45 | 0.04 | 1.85 | 0.05 | 2.23 | 0.06 | 2.42 | 0.06
$\beta_{2}$ | 0.44 | 0.04 | 0.44 | 0.04 | 0.46 | 0.05 | 0.48 | 0.06 | 0.50 | 0.07
$\beta_{2}$ | -0.02 | 0.04 | -0.02 | 0.05 | -0.02 | 0.05 | -0.01 | 0.07 | -0.00 | 0.07
2SPS $\beta_{1}$ | 0.84 | 0.07 | 0.77 | 0.07 | 0.72 | 0.08 | 0.71 | 0.13 | 0.69 | 0.22
$\beta_{2}$ | 0.41 | 0.03 | 0.38 | 0.03 | 0.35 | 0.03 | 0.33 | 0.03 | 0.32 | 0.03
$\beta_{3}$ | -0.03 | 0.06 | -0.04 | 0.06 | -0.04 | 0.07 | -0.06 | 0.11 | -0.07 | 0.19
2SPSadj $\beta_{1}$ | 0.87 | 0.07 | 0.87 | 0.09 | 0.90 | 0.11 | 0.97 | 0.18 | 0.98 | 0.34
$\beta_{2}$ | 0.43 | 0.04 | 0.44 | 0.04 | 0.45 | 0.04 | 0.48 | 0.04 | 0.49 | 0.05
$\beta_{3}$ | -0.02 | 0.07 | -0.01 | 0.07 | -0.01 | 0.09 | -0.01 | 0.15 | 0.00 | 0.26
2SPSa $\beta_{1}$ | 0.83 | 0.08 | 0.76 | 0.08 | 0.71 | 0.10 | 0.68 | 0.15 | 0.66 | 0.24
$\beta_{2}$ | 0.42 | 0.03 | 0.38 | 0.03 | 0.35 | 0.03 | 0.33 | 0.03 | 0.32 | 0.03
$\beta_{3}$ | -0.02 | 0.08 | -0.02 | 0.08 | -0.02 | 0.10 | -0.02 | 0.15 | -0.01 | 0.26
2SPSadj-a $\beta_{1}$ | 0.86 | 0.08 | 0.87 | 0.09 | 0.90 | 0.13 | 0.97 | 0.21 | 0.98 | 0.37
$\beta_{2}$ | 0.43 | 0.04 | 0.44 | 0.04 | 0.45 | 0.04 | 0.48 | 0.04 | 0.49 | 0.05
$\beta_{3}$ | -0.01 | 0.09 | -0.01 | 0.10 | -0.01 | 0.13 | -0.00 | 0.20 | -0.00 | 0.37
2SRI $\beta_{1}$ | 0.87 | 0.06 | 0.88 | 0.07 | 0.91 | 0.09 | 0.97 | 0.16 | 0.98 | 0.30
$\beta_{2}$ | 0.44 | 0.04 | 0.45 | 0.04 | 0.47 | 0.05 | 0.49 | 0.06 | 0.50 | 0.07
$\beta_{3}$ | -0.02 | 0.04 | -0.02 | 0.05 | -0.02 | 0.05 | -0.01 | 0.07 | -0.00 | 0.07
$\beta_{U}=3$
$\Pr(Y=1)$ | 0.12 | 0.00 | 0.14 | 0.00 | 0.15 | 0.00 | 0.16 | 0.00 | 0.16 | 0.00
Naive $\beta_{1}$ | 0.55 | 0.03 | 1.34 | 0.04 | 2.03 | 0.05 | 2.94 | 0.07 | 3.62 | 0.10
$\beta_{2}$ | 0.27 | 0.03 | 0.29 | 0.04 | 0.33 | 0.05 | 0.41 | 0.07 | 0.47 | 0.09
$\beta_{2}$ | -0.02 | 0.04 | -0.02 | 0.05 | -0.02 | 0.06 | -0.02 | 0.08 | -0.01 | 0.12
2SPS $\beta_{1}$ | 0.54 | 0.06 | 0.48 | 0.07 | 0.46 | 0.08 | 0.44 | 0.12 | 0.44 | 0.21
$\beta_{2}$ | 0.26 | 0.03 | 0.24 | 0.03 | 0.22 | 0.03 | 0.21 | 0.03 | 0.21 | 0.03
$\beta_{3}$ | -0.02 | 0.05 | -0.02 | 0.06 | -0.02 | 0.07 | -0.02 | 0.10 | -0.03 | 0.18
2SPSadj $\beta_{1}$ | 0.55 | 0.06 | 0.58 | 0.08 | 0.67 | 0.11 | 0.82 | 0.20 | 0.94 | 0.42
$\beta_{2}$ | 0.27 | 0.03 | 0.29 | 0.04 | 0.33 | 0.04 | 0.41 | 0.05 | 0.47 | 0.05
$\beta_{3}$ | -0.02 | 0.05 | -0.00 | 0.07 | 0.00 | 0.09 | -0.00 | 0.16 | -0.01 | 0.32
2SPSa $\beta_{1}$ | 0.53 | 0.07 | 0.47 | 0.08 | 0.46 | 0.09 | 0.43 | 0.13 | 0.43 | 0.24
$\beta_{2}$ | 0.26 | 0.03 | 0.24 | 0.03 | 0.22 | 0.03 | 0.21 | 0.03 | 0.21 | 0.03
$\beta_{3}$ | -0.01 | 0.07 | -0.00 | 0.09 | -0.01 | 0.10 | 0.00 | 0.14 | -0.02 | 0.26
2SPSadj-a $\beta_{1}$ | 0.55 | 0.07 | 0.57 | 0.09 | 0.67 | 0.12 | 0.81 | 0.22 | 0.95 | 0.46
$\beta_{2}$ | 0.27 | 0.03 | 0.29 | 0.04 | 0.33 | 0.04 | 0.42 | 0.05 | 0.47 | 0.05
$\beta_{3}$ | -0.01 | 0.08 | 0.00 | 0.10 | 0.00 | 0.13 | 0.01 | 0.22 | -0.02 | 0.42
2SRI $\beta_{1}$ | 0.55 | 0.06 | 0.59 | 0.07 | 0.69 | 0.10 | 0.83 | 0.19 | 0.94 | 0.38
$\beta_{2}$ | 0.27 | 0.03 | 0.30 | 0.04 | 0.34 | 0.05 | 0.43 | 0.07 | 0.47 | 0.09
$\beta_{3}$ | -0.02 | 0.04 | -0.02 | 0.05 | -0.02 | 0.06 | -0.02 | 0.08 | -0.01 | 0.12
Table S8: Simulation results, logistic regression, Setting IV.B | $\gamma_{U}=0$ | $\gamma_{U}=0.5$ | $\gamma_{U}=1$ | $\gamma_{U}=2$ | $\gamma_{U}=4$
---|---|---|---|---|---
| Mean | SD | Mean | SD | Mean | SD | Mean | SD | Mean | SD
$R^{2}_{X|G_{IV},Z,U}$ | 0.33 | 0.01 | 0.43 | 0.01 | 0.60 | 0.01 | 0.82 | 0.00 | 0.94 | 0.00
$R^{2}_{X|G_{IV},Z}$ | 0.33 | 0.01 | 0.29 | 0.01 | 0.20 | 0.01 | 0.09 | 0.01 | 0.03 | 0.00
$\beta_{U}=0$
$\Pr(Y=1)$ | 0.03 | 0.00 | 0.03 | 0.00 | 0.03 | 0.00 | 0.03 | 0.00 | 0.03 | 0.00
Naive $\beta_{1}$ | 1.00 | 0.05 | 1.00 | 0.05 | 1.00 | 0.05 | 1.00 | 0.05 | 1.01 | 0.05
$\beta_{2}$ | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.04
$\beta_{2}$ | 0.00 | 0.04 | -0.00 | 0.04 | 0.00 | 0.04 | -0.00 | 0.04 | -0.00 | 0.04
2SPS $\beta_{1}$ | 0.99 | 0.09 | 0.99 | 0.09 | 0.98 | 0.10 | 0.99 | 0.15 | 1.02 | 0.27
$\beta_{2}$ | 0.49 | 0.04 | 0.49 | 0.04 | 0.48 | 0.03 | 0.48 | 0.03 | 0.48 | 0.03
$\beta_{3}$ | -0.02 | 0.06 | -0.02 | 0.07 | -0.02 | 0.08 | -0.03 | 0.11 | -0.06 | 0.18
2SPSadj $\beta_{1}$ | 1.00 | 0.09 | 1.00 | 0.10 | 1.00 | 0.11 | 1.00 | 0.16 | 1.01 | 0.28
$\beta_{2}$ | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.04
$\beta_{3}$ | -0.00 | 0.07 | -0.00 | 0.07 | 0.01 | 0.08 | 0.01 | 0.12 | -0.02 | 0.20
2SPSa $\beta_{1}$ | 0.98 | 0.10 | 0.98 | 0.11 | 0.97 | 0.12 | 0.97 | 0.18 | 0.99 | 0.33
$\beta_{2}$ | 0.49 | 0.04 | 0.49 | 0.04 | 0.48 | 0.03 | 0.48 | 0.03 | 0.48 | 0.03
$\beta_{3}$ | -0.01 | 0.08 | -0.01 | 0.09 | -0.00 | 0.10 | -0.01 | 0.14 | -0.03 | 0.26
2SPSadj-a $\beta_{1}$ | 1.00 | 0.11 | 1.00 | 0.12 | 0.99 | 0.13 | 0.99 | 0.19 | 1.01 | 0.35
$\beta_{2}$ | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.04
$\beta_{3}$ | 0.00 | 0.09 | 0.00 | 0.09 | 0.01 | 0.11 | 0.01 | 0.16 | -0.02 | 0.30
2SRI $\beta_{1}$ | 1.00 | 0.07 | 1.00 | 0.08 | 1.00 | 0.09 | 1.00 | 0.12 | 1.00 | 0.21
$\beta_{2}$ | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.04 | 0.50 | 0.04
$\beta_{3}$ | 0.00 | 0.04 | -0.00 | 0.04 | 0.00 | 0.04 | -0.00 | 0.04 | -0.00 | 0.04
$\beta_{U}=1.5$
$\Pr(Y=1)$ | 0.06 | 0.00 | 0.08 | 0.00 | 0.08 | 0.00 | 0.09 | 0.00 | 0.09 | 0.00
Naive $\beta_{1}$ | 0.88 | 0.05 | 1.46 | 0.06 | 1.86 | 0.07 | 2.24 | 0.07 | 2.43 | 0.08
$\beta_{2}$ | 0.43 | 0.04 | 0.35 | 0.04 | 0.32 | 0.05 | 0.36 | 0.06 | 0.42 | 0.06
$\beta_{2}$ | -0.03 | 0.04 | -0.03 | 0.05 | -0.02 | 0.05 | -0.00 | 0.06 | -0.00 | 0.06
2SPS $\beta_{1}$ | 0.86 | 0.08 | 0.79 | 0.08 | 0.74 | 0.10 | 0.72 | 0.15 | 0.72 | 0.25
$\beta_{2}$ | 0.41 | 0.03 | 0.37 | 0.03 | 0.35 | 0.03 | 0.33 | 0.03 | 0.32 | 0.03
$\beta_{3}$ | -0.05 | 0.06 | -0.05 | 0.06 | -0.06 | 0.07 | -0.07 | 0.10 | -0.08 | 0.18
2SPSadj $\beta_{1}$ | 0.88 | 0.09 | 0.88 | 0.10 | 0.91 | 0.13 | 0.96 | 0.21 | 0.99 | 0.38
$\beta_{2}$ | 0.43 | 0.03 | 0.43 | 0.04 | 0.45 | 0.04 | 0.48 | 0.04 | 0.50 | 0.04
$\beta_{3}$ | -0.04 | 0.06 | -0.02 | 0.07 | -0.01 | 0.09 | -0.01 | 0.14 | 0.00 | 0.25
2SPSa $\beta_{1}$ | 0.84 | 0.10 | 0.77 | 0.10 | 0.72 | 0.11 | 0.68 | 0.18 | 0.67 | 0.29
$\beta_{2}$ | 0.42 | 0.03 | 0.38 | 0.03 | 0.35 | 0.03 | 0.33 | 0.03 | 0.32 | 0.03
$\beta_{3}$ | -0.04 | 0.08 | -0.03 | 0.08 | -0.03 | 0.09 | -0.03 | 0.14 | -0.03 | 0.25
2SPSadj-a $\beta_{1}$ | 0.87 | 0.10 | 0.87 | 0.12 | 0.91 | 0.16 | 0.96 | 0.26 | 0.99 | 0.45
$\beta_{2}$ | 0.43 | 0.03 | 0.43 | 0.04 | 0.45 | 0.04 | 0.48 | 0.04 | 0.50 | 0.04
$\beta_{3}$ | -0.03 | 0.08 | -0.02 | 0.09 | -0.01 | 0.12 | 0.00 | 0.20 | -0.01 | 0.34
2SRI $\beta_{1}$ | 0.88 | 0.07 | 0.89 | 0.09 | 0.92 | 0.11 | 0.97 | 0.17 | 0.99 | 0.29
$\beta_{2}$ | 0.43 | 0.04 | 0.45 | 0.04 | 0.47 | 0.05 | 0.48 | 0.06 | 0.50 | 0.06
$\beta_{3}$ | -0.03 | 0.04 | -0.04 | 0.05 | -0.02 | 0.05 | -0.01 | 0.06 | -0.00 | 0.06
$\beta_{U}=3$
$\Pr(Y=1)$ | 0.13 | 0.00 | 0.15 | 0.00 | 0.16 | 0.00 | 0.17 | 0.00 | 0.17 | 0.00
Naive $\beta_{1}$ | 0.56 | 0.04 | 1.36 | 0.05 | 2.05 | 0.07 | 2.95 | 0.10 | 3.63 | 0.13
$\beta_{2}$ | 0.27 | 0.03 | 0.16 | 0.03 | 0.14 | 0.04 | 0.20 | 0.06 | 0.32 | 0.08
$\beta_{2}$ | -0.02 | 0.03 | -0.03 | 0.04 | -0.02 | 0.05 | -0.01 | 0.08 | -0.01 | 0.10
2SPS $\beta_{1}$ | 0.56 | 0.07 | 0.50 | 0.08 | 0.47 | 0.10 | 0.45 | 0.13 | 0.46 | 0.25
$\beta_{2}$ | 0.26 | 0.03 | 0.24 | 0.03 | 0.22 | 0.03 | 0.21 | 0.03 | 0.21 | 0.03
$\beta_{3}$ | -0.04 | 0.05 | -0.03 | 0.06 | -0.03 | 0.06 | -0.03 | 0.09 | -0.04 | 0.17
2SPSadj $\beta_{1}$ | 0.57 | 0.07 | 0.59 | 0.10 | 0.68 | 0.13 | 0.81 | 0.23 | 0.95 | 0.48
$\beta_{2}$ | 0.27 | 0.03 | 0.29 | 0.03 | 0.33 | 0.04 | 0.41 | 0.04 | 0.47 | 0.05
$\beta_{3}$ | -0.03 | 0.05 | -0.02 | 0.07 | -0.01 | 0.09 | 0.00 | 0.14 | -0.02 | 0.27
2SPSa $\beta_{1}$ | 0.54 | 0.08 | 0.49 | 0.10 | 0.46 | 0.11 | 0.43 | 0.16 | 0.44 | 0.30
$\beta_{2}$ | 0.26 | 0.03 | 0.24 | 0.03 | 0.22 | 0.03 | 0.21 | 0.03 | 0.21 | 0.03
$\beta_{3}$ | -0.02 | 0.07 | -0.02 | 0.08 | -0.01 | 0.09 | -0.01 | 0.14 | -0.02 | 0.25
2SPSadj-a $\beta_{1}$ | 0.56 | 0.08 | 0.59 | 0.12 | 0.68 | 0.15 | 0.81 | 0.26 | 0.94 | 0.56
$\beta_{2}$ | 0.27 | 0.03 | 0.29 | 0.03 | 0.33 | 0.04 | 0.41 | 0.04 | 0.47 | 0.05
$\beta_{3}$ | -0.02 | 0.07 | -0.01 | 0.10 | -0.01 | 0.12 | 0.01 | 0.21 | -0.01 | 0.41
2SRI $\beta_{1}$ | 0.56 | 0.06 | 0.61 | 0.08 | 0.70 | 0.11 | 0.83 | 0.20 | 0.94 | 0.42
$\beta_{2}$ | 0.27 | 0.03 | 0.30 | 0.04 | 0.35 | 0.04 | 0.42 | 0.06 | 0.47 | 0.08
$\beta_{3}$ | -0.02 | 0.03 | -0.03 | 0.04 | -0.03 | 0.05 | -0.01 | 0.08 | -0.01 | 0.10

Figure S1: Pairwise plots of the estimates for the main effect of BMI for the
naive estimators and the 5 MR estimators.

Figure S2: Pairwise plots of $-\log_{10}\text{p-value}$ for the main effect of
BMI for the naive estimators and the five MR-based estimators.

Figure S3: Pairwise plots of the estimates for the main effect of G for the
naive estimators and the five MR-based estimators.

Figure S4: Pairwise plots of $-\log_{10}\text{p-value}$ for the main effect of
G for the naive estimators and the five MR-based estimators.

Figure S5: Simulation results, logistic Regression, Setting III.A with
$\gamma_{U}=-1$ and $\beta_{U}=1.5$: estimates pairwise plots

FUNDING: Genetics and Epidemiology of Colorectal Cancer Consortium (GECCO):
National Cancer Institute, National Institutes of Health, U.S. Department of
Health and Human Services (U01 CA164930, R01 CA201407). Genotyping/Sequencing
services were provided by the Center for Inherited Disease Research (CIDR)
contract number HHSN268201200008I . This research was funded in part through
the NIH/NCI Cancer Center Support Grant P30 CA015704. Scientific Computing
Infrastructure at Fred Hutch funded by ORIP grant S10OD028685.

The ATBC Study is supported by the Intramural Research Program of the U.S.
National Cancer Institute, National Institutes of Health, Department of Health
and Human Services.

CLUE II funding was from the National Cancer Institute (U01 CA086308, Early
Detection Research Network; P30 CA006973), National Institute on Aging (U01
AG018033), and the American Institute for Cancer Research. The content of this
publication does not necessarily reflect the views or policies of the
Department of Health and Human Services, nor does mention of trade names,
commercial products, or organizations imply endorsement by the US government.
Maryland Cancer Registry (MCR) Cancer data was provided by the Maryland Cancer
Registry, Center for Cancer Prevention and Control, Maryland Department of
Health, with funding from the State of Maryland and the Maryland Cigarette
Restitution Fund. The collection and availability of cancer registry data is
also supported by the Cooperative Agreement NU58DP006333, funded by the
Centers for Disease Control and Prevention. Its contents are solely the
responsibility of the authors and do not necessarily represent the official
views of the Centers for Disease Control and Prevention or the Department of
Health and Human Services.

The Colon Cancer Family Registry (CCFR, www.coloncfr.org) is supported in part
by funding from the National Cancer Institute (NCI), National Institutes of
Health (NIH) (award U01 CA167551). Support for case ascertainment was provided
in part from the Surveillance, Epidemiology, and End Results (SEER) Program
and the following U.S. state cancer registries: AZ, CO, MN, NC, NH; and by the
Victoria Cancer Registry (Australia) and Ontario Cancer Registry (Canada). The
CCFR Set-1 (Illumina 1M/1M-Duo) and Set-2 (Illumina Omni1-Quad) scans were
supported by NIH awards U01 CA122839 and R01 CA143237 (to GC). The CCFR Set-3
(Affymetrix Axiom CORECT Set array) was supported by NIH award U19 CA148107
and R01 CA81488 (to SBG). The CCFR Set-4 (Illumina OncoArray 600K SNP array)
was supported by NIH award U19 CA148107 (to SBG) and by the Center for
Inherited Disease Research (CIDR), which is funded by the NIH to the Johns
Hopkins University, contract number HHSN268201200008I. Additional funding for
the OFCCR/ARCTIC was through award GL201-043 from the Ontario Research Fund
(to BWZ), award 112746 from the Canadian Institutes of Health Research (to
TJH), through a Cancer Risk Evaluation (CaRE) Program grant from the Canadian
Cancer Society (to SG), and through generous support from the Ontario Ministry
of Research and Innovation. The SFCCR Illumina HumanCytoSNP array was
supported in part through NCI/NIH awards U01/U24 CA074794 and R01 CA076366 (to
PAN). The content of this manuscript does not necessarily reflect the views or
policies of the NCI, NIH or any of the collaborating centers in the Colon
Cancer Family Registry (CCFR), nor does mention of trade names, commercial
products, or organizations imply endorsement by the US Government, any cancer
registry, or the CCFR.

COLO2&3: National Institutes of Health (R01 CA060987).

Colorectal Cancer Transdisciplinary (CORECT) Study: The CORECT Study was
supported by the National Cancer Institute, National Institutes of Health
(NCI/NIH), U.S. Department of Health and Human Services (grant numbers U19
CA148107, R01 CA081488, P30 CA014089, R01 CA197350; P01 CA196569; R01
CA201407; R01 CA242218), National Institutes of Environmental Health Sciences,
National Institutes of Health (grant number T32 ES013678) and a generous gift
from Daniel and Maryann Fong.

CORSA: The CORSA study was funded by Austrian Research Funding Agency (FFG)
BRIDGE (grant 829675, to Andrea Gsur), the “Herzfelder’sche Familienstiftung”
(grant to Andrea Gsur) and was supported by COST Action BM1206.

CPS-II: The American Cancer Society funds the creation, maintenance, and
updating of the Cancer Prevention Study-II (CPS-II) cohort. The study protocol
was approved by the institutional review boards of Emory University, and those
of participating registries as required.

CRCGEN: Colorectal Cancer Genetics & Genomics, Spanish study was supported by
Instituto de Salud Carlos III, co-funded by FEDER funds –a way to build
Europe– (grants PI14-613 and PI09-1286), Agency for Management of University
and Research Grants (AGAUR) of the Catalan Government (grant 2017SGR723),
Junta de Castilla y León (grant LE22A10-2), the Spanish Association Against
Cancer (AECC) Scientific Foundation grant GCTRA18022MORE and the Consortium
for Biomedical Research in Epidemiology and Public Health (CIBERESP), action
Genrisk. Sample collection of this work was supported by the Xarxa de Bancs de
Tumors de Catalunya sponsored by Pla Director d’Oncología de Catalunya (XBTC),
Plataforma Biobancos PT13/0010/0013 and ICOBIOBANC, sponsored by the Catalan
Institute of Oncology. We thank CERCA Programme, Generalitat de Catalunya for
institutional support.

Czech Republic CCS: This work was supported by the Czech Science Foundation
(21-04607X, 21-27902S), by the Grant Agency of the Ministry of Health of the
Czech Republic (grants AZV NU21-07-00247 and AZV NU21-03-00145), and Charles
University Research Fund (Cooperation 43-Surgical disciplines).

DACHS: This work was supported by the German Research Council (BR 1704/6-1, BR
1704/6-3, BR 1704/6-4, CH 117/1-1, HO 5117/2-1, HE 5998/2-1, KL 2354/3-1, RO
2270/8-1 and BR 1704/17-1), the Interdisciplinary Research Program of the
National Center for Tumor Diseases (NCT), Germany, and the German Federal
Ministry of Education and Research (01KH0404, 01ER0814, 01ER0815, 01ER1505A
and 01ER1505B).

DALS: National Institutes of Health (R01 CA048998 to M. L. Slattery).

EDRN: This work is funded and supported by the NCI, EDRN Grant (U01-CA152753).

EPIC: The coordination of EPIC is financially supported by International
Agency for Research on Cancer (IARC) and also by the Department of
Epidemiology and Biostatistics, School of Public Health, Imperial College
London which has additional infrastructure support provided by the NIHR
Imperial Biomedical Research Centre (BRC). The national cohorts are supported
by: Danish Cancer Society (Denmark); Ligue Contre le Cancer, Institut Gustave
Roussy, Mutuelle Générale de l’Education Nationale, Institut National de la
Santé et de la Recherche Médicale (INSERM) (France); German Cancer Aid, German
Cancer Research Center (DKFZ), German Institute of Human Nutrition Potsdam-
Rehbruecke (DIfE), Federal Ministry of Education and Research (BMBF)
(Germany); Associazione Italiana per la Ricerca sul Cancro-AIRC-Italy,
Compagnia di SanPaolo and National Research Council (Italy); Dutch Ministry of
Public Health, Welfare and Sports (VWS), Netherlands Cancer Registry (NKR), LK
Research Funds, Dutch Prevention Funds, Dutch ZON (Zorg Onderzoek Nederland),
World Cancer Research Fund (WCRF), Statistics Netherlands (The Netherlands);
Health Research Fund (FIS) - Instituto de Salud Carlos III (ISCIII), Regional
Governments of Andalucía, Asturias, Basque Country, Murcia and Navarra, and
the Catalan Institute of Oncology - ICO (Spain); Swedish Cancer Society,
Swedish Research Council and and Region Skåne and Region Västerbotten
(Sweden); Cancer Research UK (14136 to EPIC-Norfolk; C8221/A29017 to EPIC-
Oxford), Medical Research Council (1000143 to EPIC-Norfolk; MR/M012190/1 to
EPIC-Oxford). (United Kingdom).

ESTHER/VERDI. This work was supported by grants from the Baden-Württemberg
Ministry of Science, Research and Arts and the German Cancer Aid.

Harvard cohorts: HPFS is supported by the National Institutes of Health (P01
CA055075, UM1 CA167552, U01 CA167552, R01 CA137178, R01 CA151993, and R35
CA197735), NHS by the National Institutes of Health (P01 CA087969, UM1
CA186107, R01 CA137178, R01 CA151993, and R35 CA197735), and PHS by the
National Institutes of Health (R01 CA042182).

Hawaii Adenoma Study: NCI grants R01 CA072520.

Kentucky: This work was supported by the following grant support: Clinical
Investigator Award from Damon Runyon Cancer Research Foundation (CI-8); NCI
R01CA136726.

LCCS: The Leeds Colorectal Cancer Study was funded by the Food Standards
Agency and Cancer Research UK Programme Award (C588/A19167).

MCCS cohort recruitment was funded by VicHealth and Cancer Council Victoria.
The MCCS was further supported by Australian NHMRC grants 509348, 209057,
251553 and 504711 and by infrastructure provided by Cancer Council Victoria.
Cases and their vital status were ascertained through the Victorian Cancer
Registry (VCR) and the Australian Institute of Health and Welfare (AIHW),
including the National Death Index and the Australian Cancer Database. BMLynch
was supported by MCRF18005 from the Victorian Cancer Agency.

MEC: National Institutes of Health (R37 CA054281, P01 CA033619, and R01
CA063464).

MECC: This work was supported by the National Institutes of Health, U.S.
Department of Health and Human Services (R01 CA081488, R01 CA197350, U19
CA148107, R01 CA242218, and a generous gift from Daniel and Maryann Fong.

NCCCS I & II: We acknowledge funding support for this project from the
National Institutes of Health, R01 CA066635 and P30 DK034987.

NFCCR: This work was supported by an Interdisciplinary Health Research Team
award from the Canadian Institutes of Health Research (CRT 43821); the
National Institutes of Health, U.S. Department of Health and Human Serivces
(U01 CA074783); and National Cancer Institute of Canada grants (18223 and
18226). The authors wish to acknowledge the contribution of Alexandre Belisle
and the genotyping team of the McGill University and Génome Québec Innovation
Centre, Montréal, Canada, for genotyping the Sequenom panel in the NFCCR
samples. Funding was provided to Michael O. Woods by the Canadian Cancer
Society Research Institute.

PLCO: Intramural Research Program of the Division of Cancer Epidemiology and
Genetics and supported by contracts from the Division of Cancer Prevention,
National Cancer Institute, NIH, DHHS. Funding was provided by National
Institutes of Health (NIH), Genes, Environment and Health Initiative (GEI) Z01
CP 010200, NIH U01 HG004446, and NIH GEI U01 HG 004438.

SMS and REACH: This work was supported by the National Cancer Institute (grant
P01 CA074184 to J.D.P. and P.A.N., grants R01 CA097325, R03 CA153323, and K05
CA152715 to P.A.N., and the National Center for Advancing Translational
Sciences at the National Institutes of Health (grant KL2 TR000421 to
A.N.B.-H.)

The Swedish Low-risk Colorectal Cancer Study: The study was supported by
grants from the Swedish research council; K2015-55X-22674-01-4,
K2008-55X-20157-03-3, K2006-72X-20157-01-2 and the Stockholm County Council
(ALF project).

Swedish Mammography Cohort and Cohort of Swedish Men: This work is supported
by the Swedish Research Council /Infrastructure grant, the Swedish Cancer
Foundation, and the Karolinska Institute´s Distinguished Professor Award to
Alicja Wolk.

UK Biobank: This research has been conducted using the UK Biobank Resource
under Application Number 8614

VITAL: National Institutes of Health (K05 CA154337).

WHI: The WHI program is funded by the National Heart, Lung, and Blood
Institute, National Institutes of Health, U.S. Department of Health and Human
Services through contracts 75N92021D00001, 75N92021D00002, 75N92021D00003,
75N92021D00004, 75N92021D00005

Acknowledgements:

CCFR: The Colon CFR graciously thanks the generous contributions of their
study participants, dedication of study staff, and the financial support from
the U.S. National Cancer Institute, without which this important registry
would not exist. The authors would like to thank the study participants and
staff of the Seattle Colon Cancer Family Registry and the Hormones and Colon
Cancer study (CORE Studies).

CLUE II: We thank the participants of Clue II and appreciate the continued
efforts of the staff at the Johns Hopkins George W. Comstock Center for Public
Health Research and Prevention in the conduct of the Clue II Cohort Study.
Cancer data was provided by the Maryland Cancer Registry, Center for Cancer
Prevention and Control, Maryland Department of Health, with funding from the
State of Maryland and the Maryland Cigarette Restitution Fund. The collection
and availability of cancer registry data is also supported by the Cooperative
Agreement NU58DP006333, funded by the Centers for Disease Control and
Prevention. Its contents are solely the responsibility of the authors and do
not necessarily represent the official views of the Centers for Disease
Control and Prevention or the Department of Health and Human Services.

CORSA: We kindly thank all individuals who agreed to participate in the CORSA
study. Furthermore, we thank all cooperating physicians and students and the
Biobank Graz of the Medical University of Graz.

CPS-II: The authors express sincere appreciation to all Cancer Prevention
Study-II participants, and to each member of the study and biospecimen
management group. The authors would like to acknowledge the contribution to
this study from central cancer registries supported through the Centers for
Disease Control and Prevention’s National Program of Cancer Registries and
cancer registries supported by the National Cancer Institute’s Surveillance
Epidemiology and End Results Program. The authors assume full responsibility
for all analyses and interpretation of results. The views expressed here are
those of the authors and do not necessarily represent the American Cancer
Society or the American Cancer Society – Cancer Action Network.