Sensitivity Analyses for Robust Causal Inference from Mendelian Randomization Analyses with Multiple Genetic Variants

TL;DR: A range of sensitivity analyses are discussed that will either support or question the validity of causal inference from a Mendelian randomization analysis with multiple genetic variants, and those that can be undertaken using summarized data are focused on.

Abstract: Mendelian randomization investigations are becoming more powerful and simpler to perform, due to the increasing size and coverage of genome-wide association studies and the increasing availability of summarized data on genetic associations with risk factors and disease outcomes. However, when using multiple genetic variants from different gene regions in a Mendelian randomization analysis, it is highly implausible that all the genetic variants satisfy the instrumental variable assumptions. This means that a simple instrumental variable analysis alone should not be relied on to give a causal conclusion. In this article, we discuss a range of sensitivity analyses that will either support or question the validity of causal inference from a Mendelian randomization analysis with multiple genetic variants. We focus on sensitivity analyses of greatest practical relevance for ensuring robust causal inferences, and those that can be undertaken using summarized data. Aside from cases in which the justification of the instrumental variable assumptions is supported by strong biological understanding, a Mendelian randomization analysis in which no assessment of the robustness of the findings to violations of the instrumental variable assumptions has been made should be viewed as speculative and incomplete. In particular, Mendelian randomization investigations with large numbers of genetic variants without such sensitivity analyses should be treated with skepticism.

Summary

FIGURE 1.

• Diagram of instrumental variable assumptions for Mendelian randomization.
• The three assumptions (i, ii, iii) are illustrated by the presence of an arrow, indicating the effect of one variable on the other (assumption i), or by a dashed line with a cross, indicating that there is no direct effect of one variable on the other (assumptions ii and iii).

ASSESSING THE INSTRUMENTAL VARIABLE ASSUMPTIONS

• The first set of approaches the authors consider are those to assess whether the instrumental variable assumptions are likely to be satisfied or not for a set of genetic variants.
• The authors consider in turn the assessment of the association with measured confounders, the exploitation of a natural experiment in the form of a gene-environment interaction, examination of a scatter plot combined with a heterogeneity test, and of a funnel plot combined with a test for directional pleiotropy.

Use of Measured Covariates

• The assumption that an instrumental variable is not associated with confounders of the risk factor-outcome association is not fully testable, as not all confounders will be known or measured.
• Associations are no stronger than would be expected by chance alone.
• Inhibition of interleukin-1 by the drug anakinra has been observed to lead to decreased levels of c-reactive protein and interleukin-6 in clinical trials.
• In some cases, valid causal inference may still be possible even if a genetic variant has a pleiotropic association with a measured covariate; for instance, by adjusting for the covariate in the analysis model.
• An alternative approach with summarized data is a multivariable Mendelian randomization analysis, in which genetic associations with the outcome are regressed on the genetic associations with the risk factor and covariates in a multivariable weighted regression model.

Gene-Environment Interaction

• For some applications of Mendelian randomization, a further natural experiment may be available if the postulated causal effect is present in one stratum of the population, but absent in another.
• 32 For example, the association of alcoholrelated genetic variants with esophageal cancer risk is present in those who drink alcohol, but absent in abstainers.
• One potential complication of such an analysis is the possibility of collider bias; 34 by stratifying on the risk factor, associations between the genetic variants and the outcome may be distorted in the strata (in the examples above, in alcohol consumers/abstainers).
• 35, 36 Associations (estimates in standard deviation units and 95% confidence intervals) of four genetic variants in the CRP gene region with a range of covariates per C-reactive protein increasing allele.
• 16 Copyright © 2016 Wolters Kluwer Health, Inc. Unauthorized reproduction of this article is prohibited.

Scatter Plot and Test for Heterogeneity

• Even if the instrumental variable assumptions are in doubt for some or all of the variants, if several independent genetic variants in different gene regions are concordantly associated with the outcome, then a causal conclusion would seem reasonable.
• Any point that substantially deviates from this line should be investigated for potential pleiotropy.
• A statistical test for heterogeneity can be performed using cochran's Q test on the causal estimates from each 2 are the inverse-variance weights.
• This statistic can be calculated using only summarized data.

FIGURE 3.

• Diagram to illustrate the difference between pleiotropy (left, the association of the genetic variant with the covariate is independent of the risk factor) and mediation (right, the association of the genetic variant with the covariate is mediated entirely via the risk factor).
• Egger regression is a method for detecting small study bias (often interpreted as publication bias) in a meta-analysis of separate studies.
• 45 the method can also be used for detecting directional pleiotropy from separate genetic variants.
• 47 the genetic associations should be orientated so that the associations with the risk factor all have the same sign.
• If there is no intercept term in this regression, the slope parameter is the inverse-variance weighted causal estimate.

ROBUST ANALYSIS METHODS

• The second category of sensitivity analyses is that of robust analysis methods.
• Robust analysis methods allow different (and when the main purpose is to test the causal null hypothesis, weaker) assumptions than standard instrumental variable methods.
• In turn, the authors consider penalization methods, median-based methods, and egger regression.

Penalization Methods

• The authors first consider methods in which the contribution of some genetic variants (e.g., heterogeneous or outlying variants) to the analysis is downweighted (or penalized).
• The simplest way of performing a penalization method is to omit some of the variants from the analysis.
• With a small number of genetic variants, the causal estimates omitting one variant at a time could be considered.
• This sensitivity analysis has been undertaken for the effect of lDl-c on aortic stenosis.
• They require individual-level data and a one-sample setting (genetic variants, risk factor, and outcome measurements are available for the same individuals).

Median-based Methods

• An alternative family of methods that gives consistent estimates when up to half the genetic variants are not valid instrumental variables, but that can be performed using summarized data rather than individual-level data, are medianbased methods.
• The weighted median estimate is consistent under the assumption that genetic variants representing over 50% of the weight in the analysis are valid instruments.
• This is for Mendelian randomization analysis of C-reactive protein on coronary artery disease risk using genetic variants throughout the genome that have been demonstrated as associated with C-reactive protein at a genome-wide level of significance.
• Horizontal lines represent 95% confidence intervals for the instrumental variable estimates.
• Confidence intervals for the median and weighted estimates can be estimated using bootstrapping.

Egger Regression

• The egger regression method was introduced above as a test for directional pleiotropy; this test does not make any assumption about the genetic variants.
• Under an assumption that is weaker than standard instrumental variable assumptions, the slope coefficient from the egger regression method provides an estimate of the causal effect that is consistent asymptotically even if all the genetic variants have pleiotropic effects on the outcome.
• There is some evidence for the general plausibility of the inSiDe assumption, as associations of genetic variants with different phenotypic variables have been shown to be largely uncorrelated in an empirical study.
• The penalization and median-based methods allow more general departures from the instrumental variable assumptions for the invalid instruments.
• Using genetic variants chosen solely on the basis of their association with the risk factor, a broad range of methods affirmed that lDl-c was a causal risk factor for cAD risk.

Example: C-reactive Protein and Coronary Artery Disease Risk

• The inverse-variance weighted method was originally proposed as a fixed-effect meta-analysis of the causal estimates from each of the genetic variants.
• The authors consider fixed-effect and multiplicative random-effects models for both the inversevariance weighted and egger regression methods.
• 56 Also, the authors consider simple (i.e., unweighted) median and weighted median estimates.
• The corresponding randomeffects analyses imply that there is no convincing evidence for a causal effect.

DISCUSSION

• When multiple genetic variants from different gene regions are used in a Mendelian randomization analysis, it is highly implausible that all the genetic variants satisfy the instrumental variable assumptions.
• This does not preclude a causal conclusion; however, it means that a simple instrumental variable analysis alone should not be relied on to give a causal conclusion.
• Inappropriate and naive application of standard Mendelian randomization methods may lead to exactly the same problems of unmeasured confounding that the technique was designed to avoid.
• The authors have discussed a range of sensitivity analyses that can be used to question the plausibility of a Mendelian randomization analysis using multiple variants, focusing on those analyses that are judged to be most useful to an applied analyst and those that can be performed using summarized data.
• Not every sensitivity analysis may be appropriate for each case, but some effort should be made to investigate whether a causal finding is robust to violations of the instrumental variable assumptions.

Comparison with Previous Literature

• From its initial popularization, proponents of Mendelian randomization have been candid about the stringent and untestable assumptions required in Mendelian randomization.
• 3, 14 However, applied investigations have not always reflected this need for caution.
• In comparison with previous attempts to offer robust approaches for causal inference in Mendelian randomization, the authors have here repeated some of the guidance of Glymour et al., 32 specifically relating to the search for gene-environment interactions and to testing for heterogeneity between the estimates from different variants.
• Substantial attenuation of the association on adjustment for the risk factor is expected if the genetic variant is a valid instrumental variable; however, such attenuation may not occur in practice, for example, due to measurement error in the exposure 58 -conversely, some attenuation may occur for an invalid instrumental variable.
• Violations of the assumptions of homogeneity and/or linearity of the causal effect would also lead to difficulties in interpreting the causal estimate, although they are unlikely to lead to inappropriate causal inferences or inflated type 1 error rates under the null.

Summarized Data and Two-sample Mendelian Randomization

• All of the sensitivity analyses discussed in this article can equally be performed b.
• Odds ratio for coronary artery disease per 1-SD (1.05 unit) increase in log-transformed c-reactive protein concentration (equivalent to a 2.86-fold increase in c-reactive protein concentration).
• A further concern with summarized data is the use of two-sample analyses, in which data on the gene-risk factor and gene-outcome associations are taken from nonoverlapping datasets.
• This is not to discourage the use of summarized data or two-sample Mendelian randomization analyses, but to acknowledge that the bar for evidential quality is even higher in this case.

Genetic Variants with Different Functional Effects

• The authors have assumed that there is a single causal effect of the risk factor on the outcome, and interpreted deviation from this (i.e., heterogeneity of causal effect estimates) as evidence that the instrumental variable assumptions are violated for some of the genetic variants.
• In reality, if genetic variants have different functional effects on the risk factor, then different magnitudes of causal effect may be expected.
• Genetic variants associated with body mass index may have different biological mechanisms giving rise to the association, and may affect the outcome to different extents.
• Heterogeneity between causal estimates based on sets of genetic variants grouped according to their biological function may help reveal which mechanisms are causal.
• The causal estimates presented in this article still provide a valid test of the causal null hypothesis, but do not have an interpretation as estimates of a causal parameter.

Pleiotropy and Other Violations of the Instrumental Variable Assumptions

• The authors have discussed violations of the instrumental variable assumptions primarily using the language of pleiotropy.
• In particular, violations of the exclusion restriction assumption (i.e., no effect of the genetic variant on the outcome except for that via the risk factor) can be expressed as pleiotropic effects.
• 63 while this adjustment has proved successful in some cases, it is not guaranteed to eliminate population stratification.
• 32 classical (nondifferential, zero mean) measurement error in the risk factor does not lead to bias in instrumental variable estimates.
• If there are multiple versions of the risk factor, then this would lead to difficulties in interpreting the causal findings.

CONCLUSIONS

• The increasing size and coverage of genome-wide association studies and the increasing availability of summarized data on genetic associations are making the application of Mendelian randomization simpler.
• The methods for sensitivity analysis described in this article will help to judge whether a causal conclusion from a Mendelian randomization analysis is reasonable or not.
• Aside from cases in which the selection of the genetic variants and their justification as instrumental variables is motivated by strong biological understanding, a Mendelian randomization analysis in which no assessment of the robustness of the findings has been made should be viewed as speculative.

Figures

FIGURE 3. Diagram to illustrate the difference between pleiotropy (left, the association of the genetic variant with the covariate is independent of the risk factor) and mediation (right, the association of the genetic variant with the covariate is mediated entirely via the risk factor).

TABLE 2. Summary of Sensitivity Analyses Considered in this Article, and Limitations of Each of the Proposed Analyses

FIGURE 4. Scatter plots of genetic associations with the outcome against genetic associations with the risk factor (lines represent 95% confidence intervals) for Mendelian randomization analysis of CRP on coronary artery disease risk using genetic variants in the CRP gene region (left) and genetic variants throughout the genome (right) that have been demonstrated as associated with C-reactive protein at a genome-wide level of significance.

TABLE 1. Estimates of Causal Effect of C-reactive Protein on Coronary Artery Disease Risk Based on 17 Genome-wide Significant Variants

FIGURE 5. Funnel plot of instrument precision ˆ ( ˆ ) β β Xj

FIGURE 1. Diagram of instrumental variable assumptions for Mendelian randomization. The three assumptions (i, ii, iii) are illustrated by the presence of an arrow, indicating the effect of one variable on the other (assumption i), or by a dashed line with a cross, indicating that there is no direct effect of one variable on the other (assumptions ii and iii).

FIGURE 6. Estimates (ordered by magnitude) of causal effect of CRP on CAD risk from inverse-variance weighted method using 17 genome-wide significant genetic variants omitting variants systematically two at a time.

FIGURE 2. Associations (estimates in standard deviation units and 95% confidence intervals) of four genetic variants in the CRP gene region with a range of covariates per C-reactive protein increasing allele. Adapted from CRP CHD Genetics Collaboration.16

Full text

Burgess, S., Bowden, J., Fall, T., Ingelsson, E., & Thompson, S. G.
(2017). Sensitivity analyses for robust causal inference from
Mendelian randomization analyses with multiple genetic variants.
Epidemiology
,
28
(1), 30-42.
https://doi.org/10.1097/EDE.0000000000000559
Publisher's PDF, also known as Version of record
License (if available):
CC BY
Link to published version (if available):
10.1097/EDE.0000000000000559
Link to publication record in Explore Bristol Research
PDF-document
This is the final published version of the article (version of record). It first appeared online via Wolters Kluwer at
http://journals.lww.com/epidem/fulltext/2017/01000/Sensitivity_Analyses_for_Robust_Causal_Inference.6.aspx.
Please refer to any applicable terms of use of the publisher.
University of Bristol - Explore Bristol Research
General rights
This document is made available in accordance with publisher policies. Please cite only the
published version using the reference above. Full terms of use are available:
http://www.bristol.ac.uk/red/research-policy/pure/user-guides/ebr-terms/

Copyright © 2016 Wolters Kluwer Health, Inc. Unauthorized reproduction of this article is prohibited.
30 | www.epidem.com Epidemiology • Volume 28, Number 1, January 2017
REVIEW ARTICLE
Abstract: Mendelian randomization investigations are becoming
more powerful and simpler to perform, due to the increasing size
and coverage of genome-wide association studies and the increas-
ing availability of summarized data on genetic associations with risk
factors and disease outcomes. However, when using multiple genetic
variants from different gene regions in a Mendelian randomization
analysis, it is highly implausible that all the genetic variants satisfy
the instrumental variable assumptions. This means that a simple
instrumental variable analysis alone should not be relied on to give
a causal conclusion. In this article, we discuss a range of sensitivity
analyses that will either support or question the validity of causal
inference from a Mendelian randomization analysis with multiple
genetic variants. We focus on sensitivity analyses of greatest practi-
cal relevance for ensuring robust causal inferences, and those that can
be undertaken using summarized data. Aside from cases in which the
justification of the instrumental variable assumptions is supported by
strong biological understanding, a Mendelian randomization analysis
in which no assessment of the robustness of the findings to viola-
tions of the instrumental variable assumptions has been made should
be viewed as speculative and incomplete. In particular, Mendelian
randomization investigations with large numbers of genetic variants
without such sensitivity analyses should be treated with skepticism.
(Epidemiology 2017;28: 30–42)
A
n instrumental variable in an observational study behaves
similarly to random treatment assignment in an experi-
mental setting.
1
It provides a natural experiment, whereby
individuals with different levels of the instrumental variable
differ on average with respect to the putative risk factor, but
not with respect to any confounders of the risk factor–out-
come association.
2
Mendelian randomization is the use of a
genetic variant as a proxy for a modifiable risk factor.
3,4
If
a genetic variant satisfies the assumptions of an instrumental
variable for the risk factor, then whether there is an associa-
tion between the genetic variant and the outcome is a test of
whether the risk factor is a cause of the outcome.
5
The instrumental variable assumptions are satisfied for
a genetic variant if
(i) the genetic variant is associated with the risk factor;
(ii) the genetic variant is not associated with confound-
ers of the risk factor–outcome relationship; and
(iii) the genetic variant is not associated with the out-
come conditional on the risk factor and confound-
ers of the risk factor–outcome relationship.
6
These assumptions imply that the only causal pathway
from the genetic variant to the outcome is via the risk factor,
and there is no other causal pathway either directly to the out-
come or via a confounder.
7
A diagram corresponding to these
assumptions is presented in Figure 1.
We further assume that all valid instrumental vari-
ables identify the same causal parameter; we return to this
assumption in the discussion. For this interpretation to hold,
it is necessary for certain parametric assumptions to hold. In
this article, we assume that the effects of (i) the instrumen-
tal variables on the risk factor, (ii) the instrumental variables
on the outcome, (iii) the risk factor on the outcome are lin-
ear without effect modification; and (iv) the association of
the genetic variant with the risk factor is homogeneous in
the population.
5
These assumptions are not necessary for the
identification of a causal effect, but they ensure that the esti-
mate from each instrumental variable targets the same average
causal effect.
8
Weaker assumptions can identify a local aver-
age causal effect;
9
however, the local average causal effect is
likely to differ for each instrumental variable. Although these
Copyright © 2016 Wolters Kluwer Health, Inc. All rights reserved.This is
an open access article distributed under the Creative Commons Attribution
License 4.0 (CCBY), which permits unrestricted use, distribution, and repro-
duction in any medium, provided the original work is properly cited.
ISSN: 1044-3983/16/2801-0030
DOI: 10.1097/EDE.0000000000000559
Submitted 9 October 2015; accepted 13 September 2016.
From the
a
Cardiovascular Epidemiology Unit, Department of Public Health
and Primary Care, University of Cambridge, Cambridge, United King-
dom;
b
Medical Research Council Integrative Epidemiology Unit, School
of Social and Community Medicine, University of Bristol, Bristol, United
Kingdom; and
c
Department of Medical Sciences, Molecular Epidemiol-
ogy, Uppsala University, Uppsala, Sweden.
Stephen Burgess is funded by a fellowship from the Wellcome Trust (100114).
Jack Bowden is supported by a Methodology Research Fellowship from
the UK Medical Research Council (Grant Number MR/N501906/1).
Simon G. Thompson is supported by the British Heart Foundation (Grant
Number CH/12/2/29428).
The authors report no conflicts of interest.
Supplemental digital content is available through direct URL citations
in the HTML and PDF versions of this article (www.epidem.com).
Editor’s Note: A Commentary on this article appears on p. 43.
Correspondence: Stephen Burgess, Department of Public Health & Primary
Care, Strangeways Research Laboratory, 2 Worts Causeway, Cambridge,
CB1 8RN, United Kingdom. E-mail: sb452@medschl.cam.ac.uk.
Sensitivity Analyses for Robust Causal Inference from
Mendelian Randomization Analyses with Multiple
Genetic Variants
Stephen Burgess,
a
Jack Bowden,
b
Tove Fall,
c
Erik Ingelsson,
c
and Simon G. Thompson
a

Copyright © 2016 Wolters Kluwer Health, Inc. Unauthorized reproduction of this article is prohibited.
Epidemiology • Volume 28, Number 1, January 2017 Sensitivity Analyses for Mendelian Randomization
© 2016 Wolters Kluwer Health, Inc. All rights reserved. www.epidem.com | 31
assumptions are strict, the causal estimate from an instru-
mental variable analysis is a valid test statistic for the causal
null hypothesis without requiring the assumptions of linearity,
homogeneity, or monotonicity.
10
In any case, the causal effect
of intervention on a risk factor is likely to depend on several
aspects of the intervention (e.g., its magnitude, duration, and
pathway), and therefore will not precisely correspond to the
estimate from a Mendelian randomization analysis.
11
Hence,
we would urge practitioners to view the assessment of causal-
ity as the primary result of a Mendelian randomization, and
not to interpret any causal estimate too literally.
12
We also assume that the genetic variants are mutu-
ally independent in their distributions, although extensions
are available for most of the analysis methods in the case of
correlated variants, provided that the correlation structure is
known.
13
Genetic variants are particularly suitable candidate
instrumental variables, as they are fixed at conception, and
hence cannot be affected by environmental factors that could
otherwise lead to confounding or reverse causation.
14
How-
ever, there are many well-documented ways in which the
instrumental variable assumptions may be violated for any
particular genetic variant, such as pleiotropy, linkage disequi-
librium, and population stratification.
3,15
For risk factors that are soluble protein biomarkers,
there is often a gene region that encodes the protein (for exam-
ple, the CRP gene region for C-reactive protein
16
), or a regula-
tor or inhibitor of the protein (e.g., the IL6R gene region for
interleukin-6
17
). Using one or more variants from such a gene
region as instrumental variables would be ideal for a Mende-
lian randomization analysis, as these genetic variants would
be the most likely to satisfy the instrumental variable assump-
tions, and the most informative proxies for intervention on the
risk factor.
18
However, such genetic variants do not exist for
many risk factors.
The approach of using multiple genetic variants in dif-
ferent gene regions is particularly suitable for complex risk
factors that are multifactorial and polygenic, such as body
mass index,
19
height,
20
or blood pressure.
21
Summarized data
(in particular, beta-coefficients and standard errors) on genetic
associations with the risk factor can be combined with sum-
marized data on genetic associations with the outcome (that
are often publicly available for download) to provide causal
effect estimates, under the assumption that the genetic vari-
ants are all instrumental variables.
22,23
Using multiple genetic
variants increases the power of a Mendelian randomization
investigation compared with an analysis based on a single
variant.
24
However, even if only one of the genetic variants is
not a valid instrumental variable, the causal estimate based on
all the variants from a conventional Mendelian randomization
analysis will be biased and type 1 (false positive) error rates
will be inflated.
25,26
In this article, we describe a range of sensitivity analy-
ses that either support or question the validity of causal infer-
ence from a Mendelian randomization analysis with multiple
genetic variants. These sensitivity analyses will be useful for
judging whether a causal conclusion from such an analysis is
plausible or not. We focus on those sensitivity analyses that
can be implemented using summarized data only. We consider
approaches under two broad categories: methods for assess-
ing the instrumental variable assumptions, and robust analysis
methods that rely on a less stringent set of assumptions than a
conventional Mendelian randomization analysis.
We illustrate the approaches using the example of esti-
mating the causal effect of C-reactive protein (CRP) on coro-
nary artery disease (CAD) risk using four genetic variants in
the CRP gene region,
16
and using 17 genetic variants (eTable
A1; http://links.lww.com/EDE/B114) that have been shown to
be associated with CRP at a genome-wide level of significance
in a large meta-analysis—see eFigure in Ref. 27—beta-coef-
ficients represent per allele associations with log-transformed
CRP concentrations. Genetic associations with CAD risk
were taken from the CARDIoGRAM consortium;
28
beta-
coefficients represent per allele log odds ratios for CAD risk.
Ethical approval for the analyses using four genetic variants in
the CRP gene region was granted by the Cambridgeshire eth-
ics review committee; for the analyses using 17 genetic vari-
ants associated with CRP concentrations and with CAD risk,
ethical approval was granted to the constituent studies by local
institutional review boards.
For reference, the causal estimate based on the genetic
variants in the CRP gene region is null (odds ratio: 1.00, 95%
confidence interval: 0.90, 1.13 per 1-SD increase in CRP con-
centrations [equal to a 1.05-unit increase in log-transformed
CRP or a 2.86-fold increase]), whereas the “causal” estimate
using an inverse-variance weighted method based on the
genome-wide significant variants (a less reliable approach)
22
is negative (odds ratio: 0.87, 95% confidence interval: 0.79,
0.96 per 1-SD increase). Software code for performing the
proposed sensitivity analyses is provided in eAppendix A.1
and A.2 (http://links.lww.com/EDE/B114).
Genetic
variant
Risk factor
Confounders
Outcome
i.
iii.
ii.
FIGURE 1. Diagram of instrumental variable assumptions for
Mendelian randomization. The three assumptions (i, ii, iii) are
illustrated by the presence of an arrow, indicating the effect of
one variable on the other (assumption i), or by a dashed line
with a cross, indicating that there is no direct effect of one vari-
able on the other (assumptions ii and iii).

Copyright © 2016 Wolters Kluwer Health, Inc. Unauthorized reproduction of this article is prohibited.
Burgess et al. Epidemiology • Volume 28, Number 1, January 2017
32
| www.epidem.com © 2016 Wolters Kluwer Health, Inc. All rights reserved.
ASSESSING THE INSTRUMENTAL VARIABLE
ASSUMPTIONS
The first set of approaches we consider are those to
assess whether the instrumental variable assumptions are
likely to be satisfied or not for a set of genetic variants. We
consider in turn the assessment of the association with mea-
sured confounders, the exploitation of a natural experiment in
the form of a gene–environment interaction, examination of a
scatter plot combined with a heterogeneity test, and of a fun-
nel plot combined with a test for directional pleiotropy.
Use of Measured Covariates
The assumption that an instrumental variable is not
associated with confounders of the risk factor–outcome
association is not fully testable, as not all confounders will
be known or measured. However, the associations of genetic
variants with measured covariates can be assessed. Lack
of association of the instrumental variable with measured
covariates does not imply lack of association with all con-
founders; however, an association with a measured covariate
should be investigated carefully for a potential pleiotropic
effect of the genetic variant. Figure 2, adapted from Wens-
ley et al.,
16
shows the associations of the four variants in
the CRP gene region with a range of potential confound-
ers. Associations are no stronger than would be expected by
chance alone.
If there are covariates that by biological considerations
should be downstream consequences of the risk factor, then
the associations of genetic variants with these covariates can
be assessed as positive controls to give confidence that the
function of the genetic variants matches the known conse-
quences of the risk factor. For instance, inhibition of inter-
leukin-1 by the drug anakinra has been observed to lead to
decreased levels of C-reactive protein and interleukin-6 in
clinical trials. If genetic variants associated with interleukin-1
are also associated with both these covariates, this makes it
more plausible that the variants are good proxies of interven-
tion on interleukin-1 levels.
29
A benefit of the use of multiple genetic variants is the
possibility to differentiate between pleiotropy and mediation,
two mechanisms by which a genetic variant may be associated
with a measured covariate (Figure 3). If a genetic variant is
associated with a covariate independently of the risk factor
(pleiotropy, or “horizontal pleiotropy”), then the instrumental
variable assumptions are likely to be violated and the genetic
variant should be excluded from an instrumental variable
analysis, as the association with the covariate is likely to open
a causal pathway from the variant to the outcome not via the
risk factor. However, if the genetic variant is associated with a
covariate due to its association with the risk factor of interest
(mediation or “vertical pleiotropy”), and there is no alterna-
tive causal pathway from the variant to the outcome except
for that via the risk factor, then the genetic variant is a valid
instrumental variable.
23
For instance, if increasing body mass index leads to
increased blood pressure, then genetic variants that are instru-
mental variables for body mass index should also be associ-
ated with blood pressure. If multiple genetic variants that are
candidate instrumental variables for body mass index are all
concordantly associated with blood pressure, then it is plau-
sible that the associations are due to mediation, not pleiot-
ropy. In contrast, if only one or two variants are associated
with blood pressure, then this is likely to be a manifestation of
pleiotropy. Pleiotropy and mediation are not mutually exclu-
sive (both could occur for the same covariate); however, this
approach may give an insight into whether the association
relates to a single genetic variant or to variants associated with
the risk factor more widely.
In some cases, valid causal inference may still be pos-
sible even if a genetic variant has a pleiotropic association
with a measured covariate; for instance, by adjusting for the
covariate in the analysis model. However, if the Mendelian
randomization investigation is performed using summarized
data, then the investigator is unlikely to be able to adjust for
covariates. An alternative approach with summarized data is
a multivariable Mendelian randomization analysis, in which
genetic associations with the outcome are regressed on the
genetic associations with the risk factor and covariates in a
multivariable weighted regression model.
30
A practical difficulty of determining which variants to
include in a Mendelian randomization analysis using mea-
sured covariates, aside from that of distinguishing between
pleiotropy and mediation, is that of multiple testing. If there
are large numbers of genetic variants and several measured
covariates, then it is difficult to set a statistical significance
threshold for rejecting a genetic variant as pleiotropic to
balance between the desire to exclude invalid instrumental
variables and the need to acknowledge the multiple tests. A
sensible compromise is to consider multiple thresholds, for
example, a conservative threshold to maximize robustness (a
fixed threshold such as P < 0.01), and a liberal threshold to
maximize power (such as a Bonferroni-corrected threshold
taking into account the number of comparisons made).
23
A
similar approach was previously taken to assess the causal role
of lipid fractions on CAD risk.
31
If no causal effect is detected
even in a liberal analysis, then the plausibility of a null causal
finding increases.
Gene–Environment Interaction
For some applications of Mendelian randomization, a
further natural experiment may be available if the postulated
causal effect is present in one stratum of the population, but
absent in another.
32
For example, the association of alcohol-
related genetic variants with esophageal cancer risk is present
in those who drink alcohol, but absent in abstainers.
33
A gene–
environment interaction provides evidence that a genetic asso-
ciation with the outcome in the population is a result of the
risk factor; if it were a result of pleiotropy, then it would be

Copyright © 2016 Wolters Kluwer Health, Inc. Unauthorized reproduction of this article is prohibited.
Epidemiology • Volume 28, Number 1, January 2017 Sensitivity Analyses for Mendelian Randomization
© 2016 Wolters Kluwer Health, Inc. All rights reserved. www.epidem.com | 33
likely to be present in both strata of the population. Gene–
environment interactions may be difficult to find, but can pro-
vide convincing evidence of a causal effect.
One potential complication of such an analysis is the
possibility of collider bias;
34
by stratifying on the risk fac-
tor, associations between the genetic variants and the out-
come may be distorted in the strata (in the examples above,
in alcohol consumers/abstainers). To our knowledge, no sys-
tematic investigation has been conducted as to the degree
that collider bias may lead to inappropriate causal infer-
ences in a Mendelian randomization setting, although sen-
sitivity analyses to assess the potential bias in the context of
instrumental variable analysis with a single instrument are
available.
35,36
−0.10.0 0.10.2
0.13 ( 0.11 , 0.14 )
0.00 ( 0.00 , 0.01 )
0.01 ( 0.00 , 0.02 )
0.00 ( −0.01 , 0.01 )
0.00 ( 0.00 , 0.01 )
−0.01 ( −0.02 , 0.00 )
−0.01 ( −0.02 , 0.00 )
0.00 ( −0.01 , 0.01 )
0.00 ( −0.01 , 0.01 )
−0.01 ( −0.02 , 0.00 )
−0.01 ( −0.02 , 0.00 )
−0.01 ( −0.02 , 0.00 )
0.00 ( −0.02 , 0.02 )
0.00 ( −0.02 , 0.02 )
−0.01 ( −0.03 , 0.02 )
0.00 ( −0.01 , 0.01 )
−0.01 ( −0.03 , 0.02 )
0.01 ( 0.00 , 0.02 )
−0.02 ( −0.06 , 0.01 )
0.01 ( 0.00 , 0.02 )
0.00 ( −0.01 , 0.02 )
0.02 ( −0.01 , 0.04 )
Per allele effect
rs1130864
−0.2 0.00.1 0.20.3
0.21 ( 0.17 , 0.24 )
0.00 ( −0.02 , 0.02 )
0.01 ( −0.01 , 0.03 )
0.02 ( 0.00 , 0.05 )
0.01 ( −0.02 , 0.03 )
0.00 ( −0.03 , 0.02 )
0.00 ( −0.02 , 0.03 )
−0.01 ( −0.04 , 0.02 )
0.01 ( −0.01 , 0.03 )
0.00 ( −0.02 , 0.03 )
0.01 ( −0.03 , 0.05 )
0.01 ( −0.02 , 0.05 )
−0.10 ( −0.44 , 0.24 )
−0.02 ( −0.08 , 0.03 )
0.01 ( −0.04 , 0.06 )
0.01 ( −0.01 , 0.04 )
−0.08 ( −0.25 , 0.09 )
−0.01 ( −0.05 , 0.02 )
−0.15 ( −0.35 , 0.05 )
0.02 ( −0.01 , 0.04 )
0.01 ( −0.02 , 0.04 )
0.00 ( −0.02 , 0.02 )
Per allele effect
rs3093077
−0.1 0.0 0.1 0.2
0.17 ( 0.15 , 0.19 )
0.00 ( −0.01 , 0.00 )
0.00 ( −0.01 , 0.01 )
0.00 ( −0.01 , 0.01 )
0.01 ( 0.00 , 0.02 )
0.00 ( −0.01 , 0.01 )
0.00 ( −0.01 , 0.00 )
0.00 ( 0.00 , 0.01 )
0.00 ( −0.01 , 0.01 )
0.00 ( −0.01 , 0.00 )
0.01 ( 0.00 , 0.02 )
0.00 ( −0.01 , 0.01 )
0.01 ( −0.02 , 0.03 )
0.00 ( −0.02 , 0.02 )
0.00 ( −0.02 , 0.02 )
−0.01 ( −0.02 , 0.00 )
−0.01 ( −0.03 , 0.01 )
0.01 ( 0.00 , 0.02 )
−0.02 ( −0.06 , 0.01 )
0.01 ( −0.01 , 0.02 )
0.01 ( 0.00 , 0.02 )
0.01 ( 0.00 , 0.02 )
Variable
log C−reactive protein (mg/l)
Age at survey (yrs)
Body mass index (kg/m²)
Systolic BP (mmHg)
Diastolic BP (mmHg)
To tal cholesterol (mmol/l)
Non−HDL−C (mmol/l)
HDL−C (mmol/l)
log Tr iglycerides (mmol/l)
LDL−C (mmol/l)
Apo A1 (g/l)
Apo B (g/l)
Albumin (g/l)
Lipoprotein(a) (mg/dl)
log Interleukin−6 (mg/l)
Fibrinogen (µmol/l)
log Leukocyte count (× 10^9/l)
Glucose (mmol/l)
Smoking amount (pack yrs)
Weight (kg)
Height (cm)
Waist/Hip ratio
Per allele effect
rs1205
−0.2 0.0 0.1 0.2 0.3
0.26 ( 0.23 , 0.30 )
−0.02 ( −0.04 , 0.01 )
−0.02 ( −0.04 , 0.01 )
0.00 ( −0.03 , 0.03 )
0.00 ( −0.02 , 0.03 )
0.01 ( −0.02 , 0.05 )
0.00 ( −0.04 , 0.04 )
0.02 ( 0.00 , 0.05 )
−0.01 ( −0.06 , 0.03 )
0.01 ( −0.03 , 0.05 )
0.00 ( −0.04 , 0.04 )
0.01 ( −0.03 , 0.05 )
0.00 ( −0.04 , 0.05 )
−0.05 ( −0.11 , 0.01 )
0.00 ( −0.05 , 0.05 )
0.00 ( −0.04 , 0.04 )
0.00 ( −0.05 , 0.06 )
0.00 ( −0.03 , 0.04 )
−0.04 ( −0.11 , 0.04 )
−0.02 ( −0.05 , 0.02 )
0.00 ( −0.03 , 0.03 )
0.00 ( −0.03 , 0.04 )
Variable
log C−reactive protein (mg/l)
Age at survey (yrs)
Body mass index (kg/m²)
Systolic BP (mmHg)
Diastolic BP (mmHg)
To tal cholesterol (mmol/l)
Non−HDL−C (mmol/l)
HDL−C (mmol/l)
log Tr iglycerides (mmol/l)
LDL−C (mmol/l)
Apo A1 (g/l)
Apo B (g/l)
Albumin (g/l)
Lipoprotein(a) (mg/dl)
log Interleukin−6 (mg/l)
Fibrinogen (µmol/l)
log Leukocyte count (× 10^9/l)
Glucose (mmol/l)
Smoking amount (pack yrs)
Weight (kg)
Height (cm)
Waist/Hip ratio
Per allele effect
rs1800947
FIGURE 2. Associations (estimates in standard deviation units and 95% condence intervals) of four genetic variants in the CRP
gene region with a range of covariates per C-reactive protein increasing allele. Adapted from CRP CHD Genetics Collaboration.
16

Frequently asked questions

Q1. What are the contributions in this paper?

In this article, the authors discuss a range of sensitivity analyses that will either support or question the validity of causal inference from a Mendelian randomization analysis with multiple genetic variants. 7 A diagram corresponding to these assumptions is presented in Figure 1. the authors further assume that all valid instrumental variables identify the same causal parameter ; they return to this assumption in the discussion. In this article, the authors assume that the effects of ( i ) the instrumental variables on the risk factor, ( ii ) the instrumental variables on the outcome, ( iii ) the risk factor on the outcome are linear without effect modification ; and ( iv ) the association of the genetic variant with the risk factor is homogeneous in the population. This is an open access article distributed under the creative commons Attribution license 4. 0 ( ccBY ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The authors report no conflicts of interest. Supplemental digital content is available through direct URl citations in the HtMl and PDF versions of this article ( www. epidem. com ). Editor ’ s Note: A commentary on this article appears on p. 43. The authors focus on sensitivity analyses of greatest practical relevance for ensuring robust causal inferences, and those that can be undertaken using summarized data. 

Q2. What is the practical difficulty of determining which variants to include in a mendelian?

A practical difficulty of determining which variants to include in a Mendelian randomization analysis using measured covariates, aside from that of distinguishing between pleiotropy and mediation, is that of multiple testing. 

Q3. what is the role of anakinra in reducing interleukin-1 levels?

For instance, inhibition of interleukin-1 by the drug anakinra has been observed to lead to decreased levels of c-reactive protein and interleukin-6 in clinical trials. 

Q4. what are the main approaches to assess the association of genetic variants with the risk factor?

if there are covariates that by biological considerations should be downstream consequences of the risk factor, then the associations of genetic variants with these covariates can be assessed as positive controls to give confidence that the function of the genetic variants matches the known consequences of the risk factor. 

Q5. What methods allow more general departures from the instrumental variable assumptions for the invalid instruments?

the penalization and median-based methods allow more general departures from the instrumental variable assumptions for the invalid instruments. 

Q6. what are the main approaches to assess the association of genetic variants with a measured covari?

23For instance, if increasing body mass index leads to increased blood pressure, then genetic variants that are instrumental variables for body mass index should also be associated with blood pressure. 

Q7. what is the pleiotropic effect of the egger regression method?

under an assumption that is weaker than standard instrumental variable assumptions, the slope coefficient from the egger regression method provides an estimate of the causal effect that is consistent asymptotically even if all the genetic variants have pleiotropic effects on the outcome. 

Q8. what is the l1 penalization method for cAD?

this approach has been applied for investigating the causal effect of lipid fractions on cAD risk.50 More formal penalizationmethods have been proposed using l1-penalization to downweight the contribution of outlying variants to the analysis in a continuous way.