Statistical issues in Mendelian randomization

284 Pages·2012·7.2 MB·English

Statistical issues in Mendelian
randomization: use of genetic
instrumental variables for assessing
causal associations
Stephen Burgess
MRC Biostatistics Unit
Emmanuel College, University of Cambridge
A thesis submitted for the degree of
Doctor of Philosophy
22nd August 2011
“And furthermore, my son, be admonished: of making many books there is
no end; and much study is a weariness of the flesh.” Ecclesiastes 12:12.
Statement of Collaboration and Acknowledgements
I hereby declare that this thesis is the result of my own work, includes nothing
which is the outcome of work done in collaboration except where specifically
indicated in the text and bibliography, is not substantially the same as any
other work that I have submitted or will be submitting for a degree or diploma
or other qualification at this or any other university, and does not exceed the
prescribed word limit.
I would like to thank my supervisor, Simon G. Thompson, for guidance and
support throughout this period of my life, and my advisors, Julian Higgins,
Jack Bowden and Shaun Seaman, for helpful discussions. In particular, I ac-
knowledge contributions from Shaun Seaman, Debbie Lawlor and Juan Pablo
Casas, who are co-authors of the paper included as Appendix E, which forms
the basis for Chapter 7. This paper was conceived, undertaken and written by
me under the supervision of Simon Thompson, with editorial input from Shaun
Seaman. Debbie Lawlor and Juan Pablo Casas provided data and comments
on the manuscript.
I also acknowledge the contribution of the CRP CHD Genetics Collaboration
(CCGC), specifically of Frances Wensley, who coordinated the collaboration,
Mat Walker and Sarah Watson, who managed the data, and John Danesh, who
oversaw the project. The papers included as Appendices B and C were con-
ceived, undertaken and written by me under the supervision of Simon Thomp-
son; several comments on penultimate versions of these manuscripts were pro-
vided by members of the collaboration. These papers form the basis of work in
Chapters3and5. ThepreliminaryanalysesundertakeninChapter8werecon-
ceived, undertaken and written by me independently of the parallel analyses
in the paper included as Appendix F, for which I contributed the instrumental
variable analysis.
I would also like to thank my wife, Nina, all those with whom I have shared
an office (Aidan, Alex, Ben, Dennis, Emma, Graham, Verena) and those who
have brightened up the journey thus far, my family and friends.
Stephen Burgess: Statistical issues in Mendelian randomization: use of
genetic instrumental variables for assessing causal associations
Mendelian randomization is an epidemiological method for using genetic vari-
ation to estimate the causal effect of the change in a modifiable phenotype on
an outcome from observational data. A genetic variant satisfying the assump-
tions of an instrumental variable for the phenotype of interest can be used
to divide a population into subgroups which differ systematically only in the
phenotype. This gives a causal estimate which is asymptotically free of bias
from confounding and reverse causation. However, the variance of the causal
estimate is large compared to traditional regression methods, requiring large
amounts of data and necessitating methods for efficient data synthesis. Addi-
tionally, if theassociationbetweenthegeneticvariantandthephenotypeisnot
strong, then the causal estimates will be biased due to the “weak instrument”
in finite samples in the direction of the observational association. This bias
may convince a researcher that an observed association is causal. If the causal
parameter estimated is an odds ratio, then the parameter of association will
differ depending on whether viewed as a population-averaged causal effect or
a personal causal effect conditional on covariates.
We introduce a Bayesian framework for instrumental variable analysis, which
is less susceptible to weak instrument bias than traditional two-stage methods,
has correct coverage with weak instruments, and is able to efficiently combine
gene–phenotype–outcome data from multiple heterogeneous sources. Methods
forimputingmissinggeneticdataaredeveloped, allowingmultiplegeneticvari-
ants to be used without reduction in sample size. We focus on the question of
a binary outcome, illustrating how the collapsing of the odds ratio over hetero-
geneous strata in the population means that the two-stage and the Bayesian
methods estimate a population-averaged marginal causal effect similar to that
estimated by a randomized trial, but which typically differs from the condi-
tional effect estimated by standard regression methods. We show how these
methods can be adjusted to give an estimate closer to the conditional effect.
We apply the methods and techniques discussed to data on the causal effect of
C-reactive protein on fibrinogen and coronary heart disease, concluding with
an overall estimate of causal association based on the totality of available data
from 42 studies.
Abbreviations
2SLS two-stage least squares
2SPS two-stage predictor substitution
2SRI two-stage residual inclusion
ACE average causal effect
BMI body mass index
CCGC CRP CHD Genetics Collaboration
CRP C-reactive protein
CHD coronary heart disease
CI /CrI confidence / credible interval
COR causal odds ratio (Chapter 2)
C(L)OR conditional (log) odds ratio (Chapter 4)
CRR causal risk ratio
DIC deviance information criterion
FE / RE fixed-effects / random-effects
GMM generalized method of moments
GWAS genome-wide association study (or studies)
HDL-C high-density lipoprotein cholesterol
HR hazard ratio
HWE Hardy–Weinberg equilibrium
I(L)OR individual (log) odds ratio
IL6 interleukin-6
IPD individual participant data
IV instrumental variable
LIML limited information maximum likelihood
LD linkage disequilibrium
LDL-C low-density lipoprotein cholesterol
lp(a) lipoprotein(a)
MAB median absolute bias
MAF minor allele frequency
MAR missing at random
MCAR missing completely at random
MCMC Monte Carlo Markov chain
MCSE Monte Carlo standard error
MI myocardial infarction
MNAR missing not at random
M(L)OR marginal (log) odds ratio
P(L)OR population (log) odds ratio
RCT randomized controlled trial
SE standard error
(G)SMM (generalized) structural mean model
SNP single nucleotide polymorphism
Abbreviations for the various studies in the CCGC are given in Appendix H.
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Notation
Throughout this dissertation, we use the notation:
X phenotype: the risk factor, or protective factor, or intermediate pheno-
type of interest
Y outcome
U confounder in the X-Y association
V unmeasured confounder (Chapter 3); covariate for Y (Chapters 4 and 6)
G instrumental variable
α parameter of genetic association: regression parameter in the G-X re-
gression
β regression parameter in the X-Y regression
β causal effect of X on Y: the main parameter of interest
1
γ parameter of genetic association for haplotypes: regression parameter in
the G-X regression where G represents a haplotype or diplotype
ρ correlation parameter
σ2 variance parameter
τ2 between-study heterogeneity variance parameter
ψ2 genetic between-study heterogeneity variance parameter
F F statistic from regression of X on G
i subscript indexing individuals
j subscript indexing genotypic subgroups
J total number of genotypic subgroups
k subscript indexing genetic variants (SNPs)
K total number of genetic variants
m subscriptindexingstudiesinameta-analysis, orimputeddatasets(Chap-
ter 7)
M total number of studies, or imputed datasets (Chapter 7)
N total number of individuals
n total number of cases (individuals with a disease event)
t time-to-event
N normal distribution
U uniform distribution
We follow the usual convention of using upper-case letters for random variables and
lower-case letters for data values (except for N and n).
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Contents
1 Introduction to Mendelian randomization 1
1.1 The rise of genetic epidemiology . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Historical background . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.2 Shortcomings of classical epidemiology . . . . . . . . . . . . . . . . 3
1.1.3 The need for an alternative . . . . . . . . . . . . . . . . . . . . . . 4
1.2 What is Mendelian randomization? . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.2 Instrumental variables . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.3 Analogy with randomized controlled trials . . . . . . . . . . . . . . 8
1.2.4 Confounding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.5 Reverse causation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Genetic markers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Examples of Mendelian randomization . . . . . . . . . . . . . . . . . . . . 11
1.5 The CRP CHD Genetic Collaboration dataset . . . . . . . . . . . . . . . . 13
1.5.1 Study design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5.2 Phenotype data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.5.3 Genetic data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.5.4 Outcome data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.5.5 Covariate data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5.6 The need for Mendelian randomization . . . . . . . . . . . . . . . . 18
1.5.7 Statistical issues and difficulties in CCGC . . . . . . . . . . . . . . 18
1.6 Overview of dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.6.1 Chapter structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.6.2 Novelty and publications . . . . . . . . . . . . . . . . . . . . . . . . 21
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CONTENTS
2 Existing statistical methods for Mendelian randomization 22
2.1 Review strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2 Finding a valid instrumental variable . . . . . . . . . . . . . . . . . . . . . 23
2.2.1 Parallel with non-compliance . . . . . . . . . . . . . . . . . . . . . . 24
2.2.2 Violations of the IV assumptions . . . . . . . . . . . . . . . . . . . 25
2.3 Testing for a causal effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4 Estimating the causal effect . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4.1 Additional IV assumptions . . . . . . . . . . . . . . . . . . . . . . . 27
2.4.2 Causal parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4.3 Collapsibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.5 Ratio of coefficients method . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5.1 Confidence intervals . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.6 Two-stage methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.6.1 Continuous outcome - two-stage least squares . . . . . . . . . . . . 31
2.6.2 Binary outcome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.7 Likelihood-based methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.7.1 Limited information maximum likelihood method . . . . . . . . . . 32
2.7.2 Bayesian methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.8 Semi-parametric methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.8.1 Generalized method of moments . . . . . . . . . . . . . . . . . . . . 34
2.8.2 Structural mean models . . . . . . . . . . . . . . . . . . . . . . . . 35
2.9 Method of Greenland and Longnecker . . . . . . . . . . . . . . . . . . . . . 37
2.10 Comparison of methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.11 Efficiency and validity of instruments . . . . . . . . . . . . . . . . . . . . . 38
2.11.1 Use of measured covariates . . . . . . . . . . . . . . . . . . . . . . . 38
2.11.2 Overidentification tests . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.12 Meta-analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.13 Weak instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.14 Computer implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.15 Mendelian randomization in practice . . . . . . . . . . . . . . . . . . . . . 42
2.16 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3 Weak instrument bias for continuous outcomes 46
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2 Demonstrating the bias from IV estimators . . . . . . . . . . . . . . . . . . 47
3.2.1 Bias of IV estimates in small studies . . . . . . . . . . . . . . . . . 47
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3.2.2 Simulation with one IV . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.3 Explaining the bias from IV estimators . . . . . . . . . . . . . . . . . . . . 49
3.3.1 Correlation of associations . . . . . . . . . . . . . . . . . . . . . . . 49
3.3.2 Finite sample violation of IV assumptions . . . . . . . . . . . . . . 51
3.3.3 Sampling variation within genotypic subgroups. . . . . . . . . . . . 52
3.4 Quantifying the bias from IV estimators . . . . . . . . . . . . . . . . . . . 54
3.4.1 Simulation of 2SLS bias with different strengths of 1 and 3 IVs . . . 54
3.4.2 Comparison of bias using different IV methods . . . . . . . . . . . . 55
3.5 Choosing a suitable IV estimator . . . . . . . . . . . . . . . . . . . . . . . 57
3.5.1 Multiple candidate IVs . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.5.2 Overidentification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.5.3 Multiple instruments in the Framingham Heart Study . . . . . . . . 61
3.5.4 Model of genetic association . . . . . . . . . . . . . . . . . . . . . . 62
3.6 Minimizing the bias from IV estimators . . . . . . . . . . . . . . . . . . . . 63
3.6.1 Increasing the F statistic . . . . . . . . . . . . . . . . . . . . . . . . 63
3.6.2 Adjustment for measured covariates . . . . . . . . . . . . . . . . . . 66
3.6.3 Borrowing information across studies . . . . . . . . . . . . . . . . . 67
3.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.7.1 Key points from chapter . . . . . . . . . . . . . . . . . . . . . . . . 72
4 Collapsibility for IV analyses of binary outcomes 73
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2 Collapsibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.2.1 Collapsibility across a covariate . . . . . . . . . . . . . . . . . . . . 75
4.2.2 Collapsibility across the risk factor distribution . . . . . . . . . . . 76
4.3 Exploring differences in odds ratios . . . . . . . . . . . . . . . . . . . . . . 77
4.3.1 Individual and population odds ratios . . . . . . . . . . . . . . . . . 77
4.3.2 Marginal and conditional estimates . . . . . . . . . . . . . . . . . . 80
4.3.3 Population and individual odds ratios in simulated data . . . . . . . 80
4.3.4 Population and individual odds ratios in five studies . . . . . . . . . 82
4.3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.4 Instrumental variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.4.1 Relation of the two-stage IV estimator and population odds ratio . 85
4.4.2 IV estimation in simplistic simulated scenarios . . . . . . . . . . . . 87
4.4.3 IV estimation in more realistic simulated scenarios . . . . . . . . . . 89
4.4.4 Interpretation of the adjusted two-stage estimand . . . . . . . . . . 91
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4.4.5 IV estimation in five studies . . . . . . . . . . . . . . . . . . . . . . 92
4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.5.1 Connection to existing literature and novelty . . . . . . . . . . . . . 94
4.5.2 Choice of target effect estimate . . . . . . . . . . . . . . . . . . . . 95
4.5.3 “Forbidden” regressions . . . . . . . . . . . . . . . . . . . . . . . . 96
4.5.4 Different designs, different parameters . . . . . . . . . . . . . . . . 96
4.5.5 Key points from chapter . . . . . . . . . . . . . . . . . . . . . . . . 97
5 A Bayesian framework for instrumental variable analysis 99
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.2 Continuous outcome — A single genetic marker in one study . . . . . . . . 100
5.2.1 Conventional methods . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.2.2 A Bayesian method . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.3 Continuous outcome — Multiple genetic markers in one study . . . . . . . 104
5.3.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.3.2 Application to C-reactive protein and fibrinogen . . . . . . . . . . . 105
5.4 Continuous outcome — Multiple genetic markers in multiple studies . . . . 108
5.4.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.4.2 Application to C-reactive protein and fibrinogen . . . . . . . . . . . 110
5.5 Binary outcome — Genetic markers in one study . . . . . . . . . . . . . . 112
5.5.1 Conventional methods . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.5.2 A Bayesian method . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.6 Dealing with issues of evidence synthesis in meta-analysis . . . . . . . . . . 117
5.6.1 Cohort studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.6.2 Common SNPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.6.3 Common haplotypes . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.6.4 Lack of phenotype data . . . . . . . . . . . . . . . . . . . . . . . . 120
5.6.5 Tabular data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.7.1 Bayesian methods in IV analysis . . . . . . . . . . . . . . . . . . . . 120
5.7.2 Bayesian analysis as a likelihood-based method . . . . . . . . . . . 121
5.7.3 Meta-analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.7.5 Key points from chapter . . . . . . . . . . . . . . . . . . . . . . . . 123
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