Causal Risk Ratio Research Articles

Using instrumental variables to estimate the attributable fraction.

In order to design efficient interventions aimed to improve public health, policy makers need to be provided with reliable information of the health burden of different risk factors. For this purpose, we are interested in the proportion of cases that could be prevented had some harmful exposure been eliminated from the population, i.e. the attributable fraction. The attributable fraction is a causal measure; thus, to estimate the attributable fraction from observational data, we have to make appropriate adjustment for confounding. However, some confounders may be unobserved, or even unknown to the investigator. A possible solution to this problem is to use instrumental variable analysis. In this work, we present how the attributable fraction can be estimated with instrumental variable methods based on the two-stage estimator or the G-estimator. One situation when the problem of unmeasuredconfounding may be particularly severe is when assessing the effect of low educational qualifications on coronary heart disease. By using Mendelian randomization, a special case of instrumental variable analysis, it has been claimed that low educational qualifications is a causal risk factor for coronary heart disease. We use Mendelian randomization to estimate the causal risk ratio and causal odds ratio of low educational qualifications as a risk factor for coronary heart disease with data from the UK Biobank. We compare the two-stage and G-estimator as well as the attributable fraction based on the two estimators. The plausibility of drawing causal conclusion in this analysis is thoroughly discussed and alternative genetic instrumental variables are tested.

Obesity As A Causal Risk Factor For Aortic Valve Stenosis

Abstract Background Causal risk factors for aortic valve stenosis are poorly understood, limiting the possibility of preventing the most common heart valve disease. Objectives The hypothesis was tested that genetically based obesity measured by body mass index is causally associated with risk of aortic valve stenosis and replacement. Methods The authors included 108,211 individuals from the Copenhagen General Population Study. Participants had measurements of body mass index, waist-hip ratio, and waist circumference, and information on 5 genetic variants associated with obesity. A Mendelian randomization design was used to investigate genetic and observational associations of obesity with incident aortic valve stenosis (n = 1,215) and replacement (n = 467) for a median follow-up time of 8.7 years. Results Genetically increased body mass index was causally associated with increased risk of aortic valve stenosis. Compared with an unweighted allele score of 0 to 3, individuals with an allele score 7 to 10 had a mean increase in body mass index of 0.87 kg/m2, and the age and sex–adjusted hazard ratio for aortic valve stenosis was 1.3 (95% confidence interval [CI]: 1.0 to 1.7) for allele score 4, 1.4 (95% CI: 1.1 to 1.8) for allele score 5 to 6, and 1.6 (95% CI: 1.3 to 2.1) for allele score 7 to 10 (p for trend: 9 × 10−5). A 1-kg/m2 increase in body mass index was associated with causal risk ratios for aortic valve stenosis and replacement, respectively, of 1.52 (95% CI: 1.23 to 1.87) and 1.49 (95% CI: 1.07 to 2.08) genetically, and with corresponding hazard ratios of 1.06 (95% CI: 1.05 to 1.08) and 1.06 (95% CI: 1.03 to 1.08) observationally. Conclusions Obesity from human genetics was causally associated with higher risk of aortic valve stenosis and replacement.

Elevated Lipoprotein(a) and Risk of Ischemic Stroke

BackgroundHigh lipoprotein(a) is associated with increased risk of myocardial infarction and aortic valve stenosis. Previous studies have examined the association of lipoprotein(a) and risk of stroke; however, the results are conflicting. ObjectivesThe purpose of this study was to test if high lipoprotein(a) is associated with high risk of ischemic stroke observationally and causally from human genetics. MethodsThe study included 49,699 individuals from the Copenhagen General Population Study and 10,813 individuals from the Copenhagen City Heart Study with measurements of plasma lipoprotein(a), LPA kringle-IV type 2 number of repeats, and LPA rs10455872. The endpoint of ischemic stroke was ascertained from Danish national health registries and validated by medical doctors. ResultsCompared with individuals with lipoprotein(a) levels <10 mg/dl (<18 nmol/l: first to 50th percentile), the multivariable-adjusted hazard ratio for ischemic stroke was 1.60 (95% confidence interval [CI]:1.24 to 2.05) for individuals with lipoprotein(a) levels >93mg/dl (>199 nmol/L: 96th to 100th percentile). In observational analyses for a 50 mg/dl (105 nmol/l) higher lipoprotein(a) level the age- and sex-adjusted hazard ratio for ischemic stroke was 1.20 (95% CI: 1.13 to 1.28), while the corresponding age- and sex-adjusted genetic causal risk ratio for KIV-2 number of repeats was 1.20 (95% CI: 1.02 to 1.43) and for rs10455872 was 1.27 (95% CI: 1.06 to 1.51). The highest absolute 10-year risk of ischemic stroke was 17% in active smoking individuals >70 years of age with hypertension and lipoprotein(a) levels >93 mg/dl (>199 nmol/l: 96th to 100th percentile). In the Copenhagen City Heart Study, risk estimates for high levels of lipoprotein(a) were in the same direction but did not reach statistical significance. ConclusionsIn a large contemporary general population study, high plasma levels of lipoprotein(a) were associated with increased risk of ischemic stroke both observationally and causally from human genetics.

257-OR: High Glucose Levels and Risk of Vascular Diseases—Observational Studies and Mendelian Randomization Studies of the General Population

Aim: To investigate the observational and causal associations between glucose levels in the nondiabetic range and risk of vascular diseases in the general population. Methods: We included 112,457 nondiabetic individuals from the Copenhagen City Heart Study and the Copenhagen General Population Study and assessed their prospective risk of retinopathy, neuropathy, diabetic nephropathy, peripheral arterial disease (PAD), and myocardial infarction (MI) as a function of increasing glucose levels. We used Mendelian randomization (MR) to assess a potential causal effect of high glucose levels on the vascular endpoints using genetic variants known to be associated with elevated glucose levels. We replicated our analyses in a 2-sample MR design using glucose data from the MAGIC consortium and vascular endpoint data from the UK Biobank. Results: Observationally, glucose levels in the nondiabetic range and higher were associated with increased risks of retinopathy, neuropathy, diabetic nephropathy, PAD, and MI (p trend for all &amp;lt;0.001). In genetic, causal analyses risk ratios for a 1 mmol/L higher glucose were 2.01 (95% confidence interval: 1.18-3.41) for retinopathy, 2.15 (1.38-3.35) for neuropathy, 1.58 (1.04-2.40) for diabetic nephropathy, 1.19 (0.90-1.58) for PAD, and 1.49 (1.02-2.17) for MI. Summary level data from the MAGIC consortium and the UK Biobank gave similar results. Conclusion: In this general population, glucose levels in the nondiabetic range and higher were prospectively associated with a high risk of retinopathy, neuropathy, diabetic nephropathy, PAD, and MI. These associations were causal for retinopathy, neuropathy, diabetic nephropathy, and MI, but causality could not be confirmed for PAD. These findings suggest that elevated glucose levels in the nondiabetic range is an important risk factor for microvascular diseases. Disclosure F. Emanuelsson: None. S. Marott: None. A. Tybjaerg-Hansen: None. B.G. Nordestgaard: None. M. Benn: None. Funding Danish Council for Independent Research

G-estimation of structural nested mean models for competing risks data using pseudo-observations.

This article provides methods of causal inference for competing risks data. The methods are formulated as structural nested mean models of causal effects directly related to the cumulative incidence function or subdistribution hazard, which reflect the survival experience of a subject in the presence of competing risks. The effect measures include causal risk differences, causal risk ratios, causal subdistribution hazard ratios, and causal effects of time-varying exposures. Inference is implemented by g-estimation using pseudo-observations, a technique to handle censoring. The finite-sample performance of the proposed estimators in simulated datasets and application to time-varying exposures in a cohort study of type 2 diabetes are also presented.

Bayesian Modelling for Binary Outcomes in the Regression Discontinuity Design

SummaryThe regression discontinuity (RD) design is a quasi-experimental design which emulates a randomized study by exploiting situations where treatment is assigned according to a continuous variable as is common in many drug treatment guidelines. The RD design literature focuses principally on continuous outcomes. We exploit the link between the RD design and instrumental variables to obtain an estimate for the causal risk ratio for the treated when the outcome is binary. Occasionally this risk ratio for the treated estimator can give negative lower confidence bounds. In the Bayesian framework we impose prior constraints that prevent this from happening. This is novel and cannot be easily reproduced in a frequentist framework. We compare our estimators with those based on estimating equation and generalized methods-of-moments methods. On the basis of extensive simulations our methods compare favourably with both methods and we apply our method to a real example to estimate the effect of statins on the probability of low density lipoprotein cholesterol levels reaching recommended levels.

Body composition and atrial fibrillation: a Mendelian randomization study.

Increases in fat-free mass and fat mass have been associated with higher risk of atrial fibrillation (AF) in observational studies. It is not known whether these associations reflect independent causal processes. Our aim was to evaluate independent causal roles of fat-free mass and fat mass on AF. We conducted a large observational study to estimate the associations between fat-free mass and fat mass on incident AF in the UK Biobank (N = 487404, N events = 10365). Genome-wide association analysis was performed to obtain genetic instruments for Mendelian randomization (MR). We evaluated the causal effects of fat-free mass and fat mass on AF with two-sample method by using genetic associations from AFGen consortium as outcome. Finally, we evaluated independent causal effects of fat-free mass and fat mass with multivariate MR. Both fat-free mass and fat mass had observational associations with incident AF [hazard ratio (HR) = 1.77, 95% confidence interval (CI) 1.72-1.83; HR = 1.40, 95% CI 1.37-1.43 per standard deviation increase in fat-free and fat mass, respectively]. The causal effects using the inverse-variance weighted method were 1.55 (95% CI 1.38-1.75) for fat-free mass and 1.30 (95% CI 1.17-1.45) for fat mass. Weighted median, Egger regression, and penalized methods showed similar estimates. The multivariate MR analysis suggested that the causal effects of fat-free and fat mass were independent of each other (causal risk ratios: 1.37, 95% CI 1.06-1.75; 1.28, 95% CI 1.03-1.58). Genetically programmed increases in fat-free mass and fat mass independently cause an increased risk of AF.

On the collapsibility of measures of effect in the counterfactual causal framework

The relationship between collapsibility and confounding has been subject to an extensive and ongoing discussion in the methodological literature. We discuss two subtly different definitions of collapsibility, and show that by considering causal effect measures based on counterfactual variables (rather than measures of association based on observed variables) it is possible to separate out the component of non-collapsibility which is due to the mathematical properties of the effect measure, from the components that are due to structural bias such as confounding. We provide new weights such that the causal risk ratio is collapsible over arbitrary baseline covariates. In the absence of confounding, these weights may be used for standardization of the risk ratio.

Relationship of Familial Hypercholesterolemia and High Low-Density Lipoprotein Cholesterol to Ischemic Stroke: Copenhagen General Population Study.

Familial hypercholesterolemia (FH) is a condition with very high concentrations of low-density lipoprotein (LDL) cholesterol and high risk of ischemic heart disease including myocardial infarction. However, there is limited and contradictory information on whether FH and high LDL cholesterol per se confer high risk of ischemic stroke. We tested the hypotheses that individuals in the general population with FH and/or high LDL cholesterol have higher risk of ischemic stroke. The associations of FH and high LDL cholesterol with ischemic stroke risk were tested in both causal, genetic, and observational analyses using 106 412 individuals from the CGPS (Copenhagen General Population Study; 2823 ischemic strokes and 3792 myocardial infarctions) and/or 10 372 individuals from the CCHS (Copenhagen City Heart Study; 945 ischemic strokes and 1142 myocardial infarctions). FH causative mutations were LDLR W23X(rs267607213), W66G(rs121908025) and W556S, and APOB R3500Q(rs5742904). A Mendelian randomization design tested whether high LDL cholesterol per se has a causal effect on ischemic stroke risk, using a combination of the FH causative mutations and common genetic variants associated with high LDL cholesterol. The cumulative incidences in individuals in the CGPS with and without FH causative mutations were similar for ischemic stroke ( P=0.50) but not for myocardial infarction ( P<0.001): at age 80 years, 4% and 7% of these individuals developed ischemic stroke and 20% and 8% myocardial infarction, with similar results in the CCHS. There was no association between clinical FH and ischemic stroke, except if personal premature ischemic heart disease was included in the clinical FH criteria. Ischemic heart disease at baseline was associated with higher ischemic stroke risk, explaining the higher ischemic stroke risk in those with high LDL cholesterol. For a 1 mmol/L higher LDL cholesterol, the genetic causal risk ratio was 1.11 (0.62-2.02) for ischemic stroke and 1.45 (1.08-1.93) for myocardial infarction. FH and high LDL cholesterol did not confer an increased risk of ischemic stroke. A positive association with ischemic stroke observed for some clinical FH criteria and high LDL cholesterol appears to be due to previous ischemic heart disease, rather than to high LDL cholesterol per se.

A Note on G-Estimation of Causal Risk Ratios.

G-estimation is a flexible, semiparametric approach for estimating exposure effects in epidemiologic studies. It has several underappreciated advantages over other propensity score-based methods popular in epidemiology, which we review in this article. However, it is rarely used in practice, due to a lack of off-the-shelf software. To rectify this, we show a simple trick for obtaining G-estimators of causal risk ratios using existing generalized estimating equations software. We extend the procedure to more complex settings with time-varying confounders.

Estimation of Causal Effect Measures in the Presence of Measurement Error in Confounders

The odds ratio, risk ratio, and the risk difference are important measures for assessing comparative effectiveness of available treatment plans in epidemiological studies. Estimation of these measures, however, is often challenged by the presence of error-contaminated confounders. In this article, by adapting two correction methods for measurement error effects applicable to the noncausal context, we propose valid methods which consistently estimate the causal odds ratio, causal risk ratio, and the causal risk difference for settings with error-prone confounders. Furthermore, we develop a bootstrap-based procedure to construct estimators with improved asymptotic efficiency. Numerical studies are conducted to assess the performance of the proposed methods.

Plasma apolipoprotein E levels and risk of dementia: A Mendelian randomization study of 106,562 individuals

IntroductionIn recent prospective studies, low plasma levels of apolipoprotein E (apoE) are associated with high risk of dementia. Whether this reflects a causal association remains to be established. MethodsUsing a Mendelian randomization approach, we studied 106,562 and 75,260 individuals from the general population in observational and genetic analyses, respectively. ResultsIn observational analyses risk of Alzheimer's disease and all dementia increased stepwise as a function of stepwise lower apoE levels (P for trend, 2 × 10−17 and 9 × 10−21). APOE-weighted allele scores were associated with stepwise decreases in apoE (P for trend, <1 × 10−300). In instrumental variable analysis, the causal risk ratios for a 1 mg/dL genetically determined lower apoE were 1.41 (1.27–1.57) for Alzheimer's disease and 1.33 (1.25–1.43) for all dementia (F-statistics = 3821). DiscussionGenetic and hence lifelong low apoE is associated with high risk of dementia in the general population. The concordance between observational and genetic estimates suggests a potential causal relationship.

Randomization Inference and Sensitivity Analysis for Composite Null Hypotheses With Binary Outcomes in Matched Observational Studies

ABSTRACTWe present methods for conducting hypothesis testing and sensitivity analyses for composite null hypotheses in matched observational studies when outcomes are binary. Causal estimands discussed include the causal risk difference, causal risk ratio, and the effect ratio. We show that inference under the assumption of no unmeasured confounding can be performed by solving an integer linear program, while inference allowing for unmeasured confounding of a given strength requires solving an integer quadratic program. Through simulation studies and data examples, we demonstrate that our formulation allows these problems to be solved in an expedient manner even for large datasets and for large strata. We further exhibit that through our formulation, one can assess the impact of various assumptions about the potential outcomes on the performed inference. R scripts are provided that implement our methods. Supplementary materials for this article are available online.

Ltmle: An R Package Implementing Targeted Minimum Loss-Based Estimation for Longitudinal Data

In recent years, targeted minimum loss-based estimation methodology has been used to develop estimators of parameters in longitudinal data structures (Gruber and van der Laan 2012; Petersen, Schwab, Gruber, Blaser, Schomaker, and van der Laan 2014; Schnitzer, Moodie, van der Laan, Platt, and Klein 2013). These methods are implemented in the ltmle package for R. The ltmle package provides methods to estimate intervention-specific means and measures of association including the average treatment effect, causal odds ratio and causal risk ratio and parameters of a longitudinal working marginal structural model. The package allows for multiple time point treatments, time-varying covariates and right censoring of the outcome. In this paper we described the usage of the ltmle package and provide examples.

Genetic invalidation of Lp-PLA2 as a therapeutic target: Large-scale study of five functional Lp-PLA2-lowering alleles.

AimsDarapladib, a potent inhibitor of lipoprotein-associated phospholipase A2 (Lp-PLA2), has not reduced risk of cardiovascular disease outcomes in recent randomized trials. We aimed to test whether Lp-PLA2 enzyme activity is causally relevant to coronary heart disease.MethodsIn 72,657 patients with coronary heart disease and 110,218 controls in 23 epidemiological studies, we genotyped five functional variants: four rare loss-of-function mutations (c.109+2T > C (rs142974898), Arg82His (rs144983904), Val279Phe (rs76863441), Gln287Ter (rs140020965)) and one common modest-impact variant (Val379Ala (rs1051931)) in PLA2G7, the gene encoding Lp-PLA2. We supplemented de-novo genotyping with information on a further 45,823 coronary heart disease patients and 88,680 controls in publicly available databases and other previous studies. We conducted a systematic review of randomized trials to compare effects of darapladib treatment on soluble Lp-PLA2 activity, conventional cardiovascular risk factors, and coronary heart disease risk with corresponding effects of Lp-PLA2-lowering alleles.ResultsLp-PLA2 activity was decreased by 64% (p = 2.4 × 10–25) with carriage of any of the four loss-of-function variants, by 45% (p < 10–300) for every allele inherited at Val279Phe, and by 2.7% (p = 1.9 × 10–12) for every allele inherited at Val379Ala. Darapladib 160 mg once-daily reduced Lp-PLA2 activity by 65% (p < 10–300). Causal risk ratios for coronary heart disease per 65% lower Lp-PLA2 activity were: 0.95 (0.88–1.03) with Val279Phe; 0.92 (0.74–1.16) with carriage of any loss-of-function variant; 1.01 (0.68–1.51) with Val379Ala; and 0.95 (0.89–1.02) with darapladib treatment.ConclusionsIn a large-scale human genetic study, none of a series of Lp-PLA2-lowering alleles was related to coronary heart disease risk, suggesting that Lp-PLA2 is unlikely to be a causal risk factor.

A general, multivariate definition of causal effects in epidemiology.

Population causal effects are often defined as contrasts of average individual-level counterfactual outcomes, comparing different exposure levels. Common examples include causal risk difference and risk ratios. These and most other examples emphasize effects on disease onset, a reflection of the usual epidemiological interest in disease occurrence. Exposure effects on other health characteristics, such as prevalence or conditional risk of a particular disability, can be important as well, but contrasts involving these other measures may often be dismissed as non-causal. For example, an observed prevalence ratio might often viewed as an estimator of a causal incidence ratio and hence subject to bias. In this manuscript, we provide and evaluate a definition of causal effects that generalizes those previously available. A key part of the generalization is that contrasts used in the definition can involve multivariate, counterfactual outcomes, rather than only univariate outcomes. An important consequence of our generalization is that, using it, one can properly define causal effects based on a wide variety of additional measures. Examples include causal prevalence ratios and differences and causal conditional risk ratios and differences. We illustrate how these additional measures can be useful, natural, easily estimated, and of public health importance. Furthermore, we discuss conditions for valid estimation of each type of causal effect, and how improper interpretation or inferences for the wrong target population can be sources of bias.

Elevated Lipoprotein(a) Does Not Cause Low-Grade Inflammation Despite Causal Association With Aortic Valve Stenosis and Myocardial Infarction: A Study of 100,578 Individuals from the General Population.

It is unknown whether elevated lipoprotein(a) is causally associated with low-grade inflammation. We tested the hypothesis that elevated lipoprotein(a) is observationally and causally associated with low-grade inflammation together with aortic valve stenosis and myocardial infarction. Using a multidirectional Mendelian randomization approach, we studied 100,578 individuals from the Danish general population with plasma levels of and/or genotypes known to affect levels of lipoprotein(a) and C-reactive protein (CRP), and using information regarding diagnosis of aortic valve stenosis and of myocardial infarction (MI) from registries. Observationally, CRP increased by 29% (95% confidence interval [CI], 23-34) per 50-mg/dL increase in lipoprotein(a). However, two LPA single nucleotide polymorphisms (SNPs) and the kringle IV type 2 (KIV-2) genotype that were associated with 98, 95, and 68 mg/dL higher lipoprotein(a) levels were not causally associated with increased CRP levels. For aortic valve stenosis, a 1-SD increase in lipoprotein(a) levels was associated observationally with a multifactorially adjusted hazard ratio of 1.23 (95% CI, 1.06-1.41), with corresponding causal risk ratios of 1.38 (1.23-1.55) based on LPA SNPs and of 1.21 (1.06-1.40) based on LPA KIV-2 genotype. For myocardial infarction, corresponding values were 1.20 (1.10;1.31) observationally, and 1.18 (1.11;1.26) and 1.31 (1.22;1.42) causally, respectively. Observational hazard ratios for aortic valve stenosis and MI were similar after further adjustment for CRP levels. Elevated levels of lipoprotein(a) were not causally associated with increased low-grade inflammation as measured through CRP despite a causal association with increased risk of aortic valve stenosis and MI.

Effects of BMI, fat mass, and lean mass on asthma in childhood: a Mendelian randomization study.

Observational studies have reported associations between body mass index (BMI) and asthma, but confounding and reverse causality remain plausible explanations. We aim to investigate evidence for a causal effect of BMI on asthma using a Mendelian randomization approach. We used Mendelian randomization to investigate causal effects of BMI, fat mass, and lean mass on current asthma at age 7½ y in the Avon Longitudinal Study of Parents and Children (ALSPAC). A weighted allele score based on 32 independent BMI-related single nucleotide polymorphisms (SNPs) was derived from external data, and associations with BMI, fat mass, lean mass, and asthma were estimated. We derived instrumental variable (IV) estimates of causal risk ratios (RRs). 4,835 children had available data on BMI-associated SNPs, asthma, and BMI. The weighted allele score was strongly associated with BMI, fat mass, and lean mass (all p-values<0.001) and with childhood asthma (RR 2.56, 95% CI 1.38-4.76 per unit score, p = 0.003). The estimated causal RR for the effect of BMI on asthma was 1.55 (95% CI 1.16-2.07) per kg/m2, p = 0.003. This effect appeared stronger for non-atopic (1.90, 95% CI 1.19-3.03) than for atopic asthma (1.37, 95% CI 0.89-2.11) though there was little evidence of heterogeneity (p = 0.31). The estimated causal RRs for the effects of fat mass and lean mass on asthma were 1.41 (95% CI 1.11-1.79) per 0.5 kg and 2.25 (95% CI 1.23-4.11) per kg, respectively. The possibility of genetic pleiotropy could not be discounted completely; however, additional IV analyses using FTO variant rs1558902 and the other BMI-related SNPs separately provided similar causal effects with wider confidence intervals. Loss of follow-up was unlikely to bias the estimated effects. Higher BMI increases the risk of asthma in mid-childhood. Higher BMI may have contributed to the increase in asthma risk toward the end of the 20th century. Please see later in the article for the Editors' Summary.

Testing concordance of instrumental variable effects in generalized linear models with application to Mendelian randomization.

Instrumental variable regression is one way to overcome unmeasured confounding and estimate causal effect in observational studies. Built on structural mean models, there has been considerable work recently developed for consistent estimation of causal relative risk and causal odds ratio. Such models can sometimes suffer from identification issues for weak instruments. This hampered the applicability of Mendelian randomization analysis in genetic epidemiology. When there are multiple genetic variants available as instrumental variables, and causal effect is defined in a generalized linear model in the presence of unmeasured confounders, we propose to test concordance between instrumental variable effects on the intermediate exposure and instrumental variable effects on the disease outcome, as a means to test the causal effect. We show that a class of generalized least squares estimators provide valid and consistent tests of causality. For causal effect of a continuous exposure on a dichotomous outcome in logistic models, the proposed estimators are shown to be asymptotically conservative. When the disease outcome is rare, such estimators are consistent because of the log-linear approximation of the logistic function. Optimality of such estimators relative to the well-known two-stage least squares estimator and the double-logistic structural mean model is further discussed.

Lack of Identification in Semiparametric Instrumental Variable Models With Binary Outcomes

A parameter in a statistical model is identified if its value can be uniquely determined from the distribution of the observable data. We consider the context of an instrumental variable analysis with a binary outcome for estimating a causal risk ratio. The semiparametric generalized method of moments and structural mean model frameworks use estimating equations for parameter estimation. In this paper, we demonstrate that lack of identification can occur in either of these frameworks, especially if the instrument is weak. In particular, the estimating equations may have no solution or multiple solutions. We investigate the relationship between the strength of the instrument and the proportion of simulated data sets for which there is a unique solution of the estimating equations. We see that this proportion does not appear to depend greatly on the sample size, particularly for weak instruments (ρ2 ≤ 0.01). Poor identification was observed in a considerable proportion of simulated data sets for instruments explaining up to 10% of the variance in the exposure with sample sizes up to 1 million. In an applied example considering the causal effect of body mass index (weight (kg)/height (m)2) on the probability of early menarche, estimates and standard errors from an automated optimization routine were misleading.