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Background At the individual level, it is well known that pregnancies have a short-term effect on a woman’s cardiovascular system and blood pressure. The long-term effect of having children on maternal blood pressure, however, is unknown. We thus estimated the causal effect of having children on blood pressure among mothers in India, a country with a history of high fertility rates. Methods We used nationally representative cross-sectional data from the 2015–16 India National Family and Health Survey (NFHS-4). The study population comprised 444 611 mothers aged 15–49 years. We used the sex of the first-born child as an instrumental variable (IV) for the total number of a woman’s children. We estimated the effect of an additional child on systolic and diastolic blood pressure in IV (two-stage least squares) regressions. In additional analyses, we stratified the IV regressions by time since a mother last gave birth. Furthermore, we repeated our analyses using mothers' husbands and partners as the regression sample. Results On average, mothers had 2.7 children [standard deviation (SD): 1.5], a systolic blood pressure of 116.4 mmHg (SD: 14.4) and diastolic blood pressure of 78.5 mmHg (SD: 9.4). One in seven mothers was hypertensive. In conventional ordinary least squares regression, each child was associated with 0.42 mmHg lower systolic [95% confidence interval (CI): –0.46 to –0.39, P Conclusions Bearing and rearing a child decreases blood pressure among mothers in India.

Background We developed a method to make Inference about Causation from Examination of FAmiliaL CONfounding (ICE FALCON) using observational data for related individuals and considering changes in a pair of regression coefficients. ICE FALCON has some similarities to Mendelian randomization (MR) but uses in effect all the familial determinants of the exposure, not just those captured by measured genetic variants, and does not require genetic data nor make strong assumptions. ICE FALCON can assess tracking of a measure over time, an issue often difficult to assess using MR due to lack of a valid instrumental variable. Methods We describe ICE FALCON and present two empirical applications with simulations. Results We found evidence consistent with body mass index (BMI) having a causal effect on DNA methylation at the ABCG1 locus, the same conclusion as from MR analyses but providing about 2.5 times more information per subject. We found evidence that tracking of BMI is consistent with longitudinal causation, as well as familial confounding. The simulations supported the validity of ICE FALCON. Conclusions There are conceptual similarities between ICE FALCON and MR, but empirically they are giving similar conclusions with possibly more information per subject from ICE FALCON. ICE FALCON can be applied to circumstances in which MR cannot be applied, such as when there is no a priori genetic knowledge and/or data available to create a valid instrumental variable, or when the assumptions underlying MR analysis are suspect. ICE FALCON could provide insights into causality for a wide range of public health questions.

Background Conventional epidemiologic studies have evaluated associations between circulating lipid levels and breast cancer risk, but results have been inconsistent. As Mendelian randomization analyses may provide evidence for causal inference, we sought to evaluate potentially unbiased associations between breast cancer risk and four genetically predicted lipid traits. Methods Previous genome-wide association studies (GWAS) have identified 164 discrete variants associated with high density lipoprotein-cholesterol (HDL-C), low density lipoprotein-cholesterol (LDL-C), triglycerides and total cholesterol. We used 162 of these unique variants to construct weighted genetic scores (wGSs) for a total of 101 424 breast cancer cases and 80 253 controls of European ancestry from the Breast Cancer Association Consortium (BCAC). Unconditional logistic regression was used to estimate odds ratios (OR) and 95% confidence intervals (CI) for associations between per standard deviation increase in genetically predicted lipid traits and breast cancer risk. Additional Mendelian randomization analysis approaches and sensitivity analyses were conducted to assess pleiotropy and instrument validity. Results Corresponding to approximately 15 mg/dL, one standard deviation increase in genetically predicted HDL-C was associated with a 12% increased breast cancer risk (OR: 1.12, 95% CI: 1.08–1.16). Findings were consistent after adjustment for breast cancer risk factors and were robust in several sensitivity analyses. Associations with genetically predicted triglycerides and total cholesterol were inconsistent, and no association for genetically predicted LDL-C was observed. Conclusions This study provides strong evidence that circulating HDL-C may be associated with an increased risk of breast cancer, whereas LDL-C may not be related to breast cancer risk.

Focus of Presentation Causation is critical if epidemiology is to be relevant to public health. Mendelian Randomisation makes causal inference about from simply observing an association between the outcome and a genetic variable that has been conferred the title of “instrumental” because the proponents consider it satisfies some assumptions perfectly. We take this association as the starting point of a Popperian approach that tries to falsify causal hypotheses by relaxing assumptions and considering alternate models. Findings We developed methods to calculate test-of-fit statistics for different causal scenarios based on the joint changes in regression coefficients, using simulations and bootstrapping methods. Let Y be the outcome, X a putative cause, and G a potential instrumental variable associated with X and Y. We regress Y against X and G alone, and with X and G together. We predicted the changes to regression coefficients that should occur under three scenarios; (i) X causes Y, (ii) there is a factor C associated with Y, X and G. and (ii) Y causes X. We compared goodness-of-fit statistics across scenarios, and for combinations of scenarios (given multiple causal processes might co-exist). We present findings from application to data on body mass index and DNA methylation and compare with Mendelian Randomisation analyses. Conclusions/Implications Robust inference can be made but the sample sizes and strengths of associations need to be substantive. Key messages Causation is a fundamentally important issue that should, and can, be addressed by trying to disprove it, rather than by finding evidence for it.

Background Mendelian randomisation (MR) uses measured genetic variants and presumes they satisfy the requirements of being instrumental variables for exposures so as to make inference about causation. The application of MR is becoming popular. The validity of MR, however, depends on three strong assumptions (see Table 3), some of which are not easy to test. Although there are methods for addressing the validity of these assumptions, and for assessing the robustness of findings under failed assumptions, MR analysis still requires a priori genetic knowledge and genetic data in order to create such presumed instrumental variables. We have developed a new method, Inference about Causation from Examination FAmiliaL CONfounding (ICE FALCON), which applies to data for related individuals and enables the assessment of evidence for causality between measured factors. ICE FALCON in effect uses all the familial determinants of the exposure, not just those captured by measured genetic variants, and also does not assume they satisfy all the requirements of being instrumental variables. ICE FALCON allows for departures from those requirements and has statistical methods to compare fits under different causal models. To illustrate ICE FALCON, we applied it to two examples and compared ICE FALCON analysis with MR analysis in terms of conclusion, power and flexibility. In Example 1, we considered the cross-sectional association between body mass index (BMI) and blood DNA methylation level at the ABCG1 locus. Two MR analyses involving 4034 and 2170 individuals, respectively, have suggested that this association is due to a causal effect of BMI on methylation. In Example 2, we considered the tracking of BMI over time. For MR to assess the causality between BMI measures over time there would need to be genetic variants associated with BMI at the earlier time that are not associated with BMI at the later time. Genetic variants for adulthood BMI found to date by genome-wide association studies (GWAS) appear to apply to BMI across the whole of adulthood. Therefore, there are as yet no validated genetic variants associated with BMI at an earlier age that are not associated with BMI at a later age; even genetic variants found to be associated with childhood BMI are associated with adulthood BMI. Consequently, MR cannot be applied to this issue. Methods In both examples, we used the data from the Australian Mammographic Density Twins and Sisters Study, a twin and family cohort with baseline and follow-up. We used the BMI and methylation data (both assessed at follow-up) for 65 monozygotic twin pairs in Example 1, and the baseline and follow-up data for 250 monozygotic twin pairs who aged 50 years and 57 years on average at the two time points in Example 2. In both examples, we performed ICE FALCON analysis, which involved three models. For the purposes of explanation, let ‘self’ refer to an individual and ‘co-twin’ refer to the individual’s twin, but recognise that these labels can be swapped and both twins are included in the analysis (the method can also be applied to other related individuals, such as sibling and parent-offspring pairs). Let X = methylation in Example 1 and baseline BMI in Example 2, and Y = BMI in Example 1 and follow-up BMI in Example 2. Therefore, for a twin pair, there are four variables: Xself, Xco-twin, Yself, Yco-twin. Model 1 regressed Yself on Xself to estimate the marginal within-individual association (□self). Model 2 regressed Yself on Xco-twin to estimate the marginal cross-twin association (□co-twin). Model 3 regressed Yself on both Xself and Xco-twin to estimate the conditional within-individual and cross-twin associations (□'self, □'co-twin). The changes in regression coefficients between models (□self vs □'self, □co-twin vs □'co-twin) were investigated. For different causal scenarios, ICE FALCON analysis will give different results (Table 1). Therefore, we can use statistical inference methods to try to find the causal scenario(s), or their combinations, that are most consistent with the observed results. The regression analyses were conducted using the generalised estimating equations model with an exchangeable correlation structure. Statistical inference for the changes in regression coefficients was conducted using standard errors estimated from applying a non-parametric bootstrap method. Results In Example 1, a woman’s methylation level was associated with her own BMI (Model 1; □self=0.13, 95% CI: 0.05, 0.22) and with her co-twin’s BMI (Model 2; □co-twin=0.09, 95% CI: 0.01, 0.17). Conditioning on her co-twin’s BMI (Model 3), the association between her methylation level and her own BMI remained unchanged (P = 0.49); conditioning on her own BMI (Model 3), the association between her methylation level and her co-twin’s BMI attenuated almost completely, by 97%, to become null (□'co-twin=0.003, 95% CI: -0.11, 0.12; P = 0.08). These findings are consistent with the expectations under the ‘X causes Y’ scenario of Table 1. Therefore, the data are consistent with BMI having a causal effect on methylation. In Example 2, a woman’s follow-up BMI was associated with her own baseline BMI (Model 1; □self=0.81, 95% CI: 0.72, 0.90), and with her co-twin’s baseline BMI (Model 2; □co-twin=0.73, 95% CI: 0.65, 0.81). In Model 3, there remained a strong association of a woman’s follow-up BMI with her own baseline BMI (□'self=0.73, 95% CI: 0.63, 0.83), and a weak association with her co-twin’s baseline BMI (□'co-twin=0.15, 95% CI: 0.06, 0.23). Both the associations attenuated compared with the estimates from Models 1 and 2, but to different extents: 10% (P = 0.02) and 80% (P = 10-30), respectively. These findings are not consistent with any single causal scenario of Table 1, but are with both the ‘X causes Y’ and ‘Familial confounders cause the association’ scenarios. Therefore, we interpret these results as being consistent with a longitudinal causation, as well as a small amount of familial confounding, underlying the association between the two BMI measures. Conclusions We found from Example 1 that the ICE FALCON approach gave the same conclusion as from previous MR analyses, i.e., BMI has a causal effect on DNA methylation at the ABCG1 locus. One measure of the amount of information on causality assessment from MR could be from consideration of the test statistic (ZMR) for the association between genetic variants and outcome, in proportion to the square root of the sample size (n). Similar for ICE FALCON, a measure of the amount of information comes from the test statistic (ZIF) for the change in the cross-twin regression coefficient. The two MR analyses had n = 4034 and ZMR=4.00, and n = 2170 and ZMR=2.69, respectively, while our ICE FALCON analysis had n = 130 and ZIF=1.75. Therefore, ZMR/n1/2=0.063 and 0.058, respectively, when using MR, whereas ZIF/n1/2=0.153 when using ICE FALCON. That is, in this example ICE FALCON appears to be extracting about 2.5 times more information on causality per subject than MR. We found from Example 2 that the longitudinal tracking in BMI is consistent with a causal effect of BMI on its future values, as well as a smaller component of familial confounding. As mentioned in the Methods, MR cannot be used to address this question, given there is no valid instrumental variables. ICE FALCON outperforms MR completely for assessing the causality of tracking in a trait over time. ICE FALCON is analogous in some ways to MR. Consider the scenario in which X has a causal effect on Y, ICE FALCON essentially uses all the familial determinants of X as instrumental variables for Xself. The familial determinants of X are not measured, but in this scenario a proxy measure is Xco-twin. ICE FALCON studies the association between the proxy instrumental variable and the outcome, Yself. MR uses measured genetic variants, only a proportion of the familial determinants of X, as an instrumental variable. The validity of MR is subject to three key assumptions. Table 3 summarises a comparison between MR and ICE FALCON for each of these assumptions and shows that ICE FALCON could have some advantages over MR. Example 1 shows that ICE FALCON uses a stronger instrumental variable, and thus is more powerful, regarding to the relevance assumption, and Example 2 shows the validity of ICE FALCON even if the independence and/or exclusion restriction assumptions do not hold. In conclusion, we have developed ICE FALCON, a statistical modelling approach to observational data for related individuals to assess causality between measured variables of interest. There are some conceptual similarities and differences between ICE FALCON and MR and empirically they are giving similar conclusions with possibly more information per subject from ICE FALCON. ICE FALCON can be applied to circumstances in which MR cannot be applied, such as when there is no a priori genetic knowledge or genetic data to create a potential instrumental variable, or when the assumptions underlying MR analysis are suspect. Given ICE FALCON does not rely on genetic knowledge, or measurement of genetic data, but instead utilises the almost universal fact that related individuals (and especially twins) are correlated in exposures, it could provide insights into causality for a wide range of public health questions. Key messages ICE FALCON is similar to, but more powerful and flexible than, MR in assessing causation.

Background Mendelian randomisation uses genetic variants randomly allocated at conception as instrumental variables for an exposure. Methodological advances allow for mediation analysis to be carried out using Mendelian randomisation using either multivariable Mendelian randomisation or two-step Mendelian randomisation. Methods We use simulations and an applied example to demonstrate when multivariable Mendelian randomisation and two-step Mendelian randomisation methods are valid and how they relate to traditional phenotypic regression-based approaches to mediation. We demonstrate how Mendelian randomisation methods can relax assumptions required for causal inference in phenotypic mediation, as well as which Mendelian randomisation specific assumptions are required. We illustrate our methods in data from UK Biobank, estimating the role of body mass index mediating the association between education and cardiovascular outcomes. Results Both multivariable Mendelian randomization and two-step Mendelian randomization are unbiased when estimating the total effect, direct effect, indirect effect and proportion mediated when both confounding, and measurement error are present. Multivariable Mendelian Randomization can be used when multiple mediators are to be investigated in a single model. Conclusions Mendelian randomisation provides an opportunity to improve causal inference in mediation analysis. Although Mendelian randomisation specific assumptions apply, such as no weak instrument bias and no pleiotropic pathways, strong phenotypic assumptions of no confounding and no measurement error can be relaxed. Key messages Mendelian randomisation offers an opportunity to address bias by unmeasured confounding, measurement error and reverse causality in mediation analysis.

markdownabstractBackground Instrumental variable (IV) methods are often used to identify ‘local’ causal effects in a subgroup of the population of interest. Such ‘local’ effects may not be ideal for informing clinical or policy decision making. When the instrument is non-causal, additional difficulties arise for interpreting ‘local’ effects. Little attention has been paid to these difficulties, even though commonly proposed instruments in epidemiology are non-causal (e.g. proxies for physician’s preference; genetic variants in some Mendelian randomization studies). Methods For IV estimates obtained from both causal and non-causal instruments under monotonicity, we present results to help investigators pose four questions about the local effect estimates obtained in their studies. (1) To what subgroup of the population does the effect pertain? Can we (2) estimate the size of or (3) describe the characteristics of this subgroup relative to the study population? (4) Can the sensitivity of the effect estimate to deviations from monotonicity be quantified? Results We show that the common interpretations and approaches for answering these four questions are generally only appropriate in the case of causal instruments. Conclusions Appropriate interpretation of an IV estimate under monotonicity as a ‘local’ effect critically depends on whether the proposed instrument is causal or non-causal. The results and formal proofs presented here can help in the transparent reporting of IV results and in enhancing the use of IV estimates in informing decision-making efforts.

Background Findings in the observational retail food environment and obesity literature are inconsistent, potentially due to a lack of adjustment for residual confounding. Methods Using data from the CARDIA study (n = 12 174 person-observations; 6 examinations; 1985-2011) across four US cities (Birmingham, AL; Chicago, IL; Minneapolis, MN; Oakland, CA), we used instrumental-variables (IV) regression to obtain causal estimates of the longitudinal associations between the percentage of neighbourhood food stores or restaurants (per total food outlets within 1 km network distance of respondent residence) with body mass index (BMI), adjusting for individual-level socio-demographics, health behaviours, city, year, total food outlets and market-level prices. To determine the presence and extent of bias, we compared the magnitude and direction of results with ordinary least squares (OLS) and random effects (RE) regression, which do not control for residual confounding, and with fixed effects (FE) regression, which does not control for time-varying residual confounding. Results Relative to neighbourhood supermarkets (which tend to be larger and have healthier options than grocery stores), a higher percentage of grocery stores [mean = 53.4%; standard deviation (SD) = 31.8%] was positively associated with BMI [β = 0.05; 95% confidence interval (CI) = 0.01, 0.10] using IV regression. However, associations were negligible or null using OLS (β = -0.001; 95% CI = -0.01, 0.01), RE (β = -0.003; 95% CI = -0.01, 0.0001) and FE (β = -0.003; 95% CI = -0.01, 0.0002) regression. Neighbourhood convenience stores and fast-food restaurants were not associated with BMI in any model. Conclusions Longitudinal associations between neighbourhood food outlets and BMI were greater in magnitude using a causal model, suggesting that weak findings in the literature may be due to residual confounding.

A critical assumption of causal inference is that of no unmeasured confounding: for estimated exposure effects to have valid causal interpretations, a sufficient set of predictors of exposure and outcome must be adequately measured and correctly included in the respective inference model(s). In an observational study setting, this assumption will often be unsatisfied, and the potential impact of unmeasured confounding on effect estimates should be investigated. The confounding function approach allows the impact of unmeasured confounding on estimates to be assessed, where unmeasured confounding may be due to unmeasured confounders and/or biases such as collider bias or information bias. Although this approach is easy to implement and pertains to the sum of all bias, its use has not been widespread, and discussion has typically been limited to continuous outcomes. In this paper, we consider confounding functions for use with binary outcomes and illustrate the approach with an example. We note that confounding function choice encodes assumptions about effect modification: some choices encode the belief that the true causal effect differs across exposure groups, whereas others imply that any difference between the true causal parameter and the estimate is entirely due to imbalanced risks between exposure groups. The confounding function approach is a useful method for assessing the impact of unmeasured confounding, in particular when alternative approaches, e.g. external adjustment or instrumental variable approaches, cannot be applied. We provide Stata and R code for the implementation of this approach when the causal estimand of interest is an odds or risk ratio.

Instrumental-variable (IV) methods were invented over 70 years ago, but remain uncommon in epidemiology. Over the past decade or so, non-parametric versions of IV methods have appeared that connect IV methods to causal and measurement-error models important in epidemiological applications. This paper provides an introduction to those developments, illustrated by an application of IV methods to non-parametric adjustment for non-compliance in randomized trials.

BACKGROUND: The number of Mendelian randomization analyses including large numbers of genetic variants is rapidly increasing. This is due to the proliferation of genome-wide association studies, and the desire to obtain more precise estimates of causal effects. However, some genetic variants may not be valid instrumental variables, in particular due to them having more than one proximal phenotypic correlate (pleiotropy).METHODS: We view Mendelian randomization with multiple instruments as a meta-analysis, and show that bias caused by pleiotropy can be regarded as analogous to small study bias. Causal estimates using each instrument can be displayed visually by a funnel plot to assess potential asymmetry. Egger regression, a tool to detect small study bias in meta-analysis, can be adapted to test for bias from pleiotropy, and the slope coefficient from Egger regression provides an estimate of the causal effect. Under the assumption that the association of each genetic variant with the exposure is independent of the pleiotropic effect of the variant (not via the exposure), Egger's test gives a valid test of the null causal hypothesis and a consistent causal effect estimate even when all the genetic variants are invalid instrumental variables.RESULTS: We illustrate the use of this approach by re-analysing two published Mendelian randomization studies of the causal effect of height on lung function, and the causal effect of blood pressure on coronary artery disease risk. The conservative nature of this approach is illustrated with these examples.CONCLUSIONS: An adaption of Egger regression (which we call MR-Egger) can detect some violations of the standard instrumental variable assumptions, and provide an effect estimate which is not subject to these violations. The approach provides a sensitivity analysis for the robustness of the findings from a Mendelian randomization investigation.

A large number of epidemiological studies use genetic variants as instrumental variables to infer causal relationships. For a genetic variant to be a valid instrument in these so-called Mendelian randomization (MR) studies, three assumptions need to hold: __(i)__ the genetic variant is associated with the exposure of interest (relevance assumption); __(ii)__ the genetic variants should be independent of all confounders (independence assumption); __(iii)__ the genetic variants only effect the outcome through the exposure of interest (exclusion restriction). Without specific knowledge about the biological mechanisms affected by genetic variants, it is virtually impossible to prove that the exclusion restriction holds for a specific genetic variant. For example, genetic variants may have pleiotropic effects on both the exposure and the outcome through different biological pathways. [...]

There is increasing interest in the use of instrumental variable analysis to overcome unmeasured confounding in observational pharmacoepidemiological studies. This is partly because instrumental variable analyses are potentially less biased than conventional regression analyses. However, instrumental variable analyses are less precise, and regulators and clinicians find it difficult to interpret conflicting evidence from instrumental variable compared with conventional regression analyses. In this paper, we describe three techniques to assess which approach (instrumental variable versus conventional regression analyses) is least biased. These techniques are negative control outcomes, negative control populations and tests of covariate balance. We illustrate these methods using an analysis of the effects of smoking cessation therapies (varenicline) prescribed in primary care.

Background: Mendelian randomization studies have so far restricted attention to linear associations relating the genetic instrument to the exposure, and the exposure to the outcome. In some cases, however, observational data suggest a non-linear association between exposure and outcome. For example, alcohol consumption is consistently reported as having a U-shaped association with cardiovascular events. In principle, Mendelian randomization could address concerns that the apparent protective effect of light-to-moderate drinking might reflect ‘sick-quitters’ and confounding.\ud \ud Methods: The Alcohol-ADH1B Consortium was established to study the causal effects of alcohol consumption on cardiovascular events and biomarkers, using the single nucleotide polymorphism rs1229984 in ADH1B as a genetic instrument. To assess non-linear causal effects in this study, we propose a novel method based on estimating local average treatment effects for discrete levels of the exposure range, then testing for a linear trend in those effects. Our method requires an assumption that the instrument has the same effect on exposure in all individuals. We conduct simulations examining the robustness of the method to violations of this assumption, and apply the method to the Alcohol-ADH1B Consortium data.\ud \ud Results: Our method gave a conservative test for non-linearity under realistic violations of the key assumption. We found evidence for a non-linear causal effect of alcohol intake on several cardiovascular traits.\ud \ud Conclusions: We believe our method is useful for inferring departure from linearity when only a binary instrument is available. We estimated non-linear causal effects of alcohol intake which could not have been estimated through standard instrumental variable approaches.

In Mendelian randomization (MR) studies, where genetic variants are used as proxy measures for an exposure trait of interest, obtaining adequate statistical power is frequently a concern due to the small amount of variation in a phenotypic trait that is typically explained by genetic variants. A range of power estimates based on simulations and specific parameters for two-stage least squares (2SLS) MR analyses based on continuous variables has previously been published. However there are presently no specific equations or software tools one can implement for calculating power of a given MR study. Using asymptotic theory, we show that in the case of continuous variables and a single instrument, for example a single-nucleotide polymorphism (SNP) or multiple SNP predictor, statistical power for a fixed sample size is a function of two parameters: the proportion of variation in the exposure variable explained by the genetic predictor and the true causal association between the exposure and outcome variable. We demonstrate that power for 2SLS MR can be derived using the non-centrality parameter (NCP) of the statistical test that is employed to test whether the 2SLS regression coefficient is zero. We show that the previously published power estimates from simulations can be represented theoretically using this NCP-based approach, with similar estimates observed when the simulation-based estimates are compared with our NCP-based approach. General equations for calculating statistical power for 2SLS MR using the NCP are provided in this note, and we implement the calculations in a web-based application.

BACKGROUND: Mendelian randomization uses a carefully selected gene as an instrumental-variable (IV) to test or estimate an association between a phenotype and a disease. Classical IV analysis assumes linear relationships between the variables, but disease status is often binary and modelled by a logistic regression. When the linearity assumption between the variables does not hold the IV estimates will be biased. The extent of this bias in the phenotype-disease log odds ratio of a Mendelian randomization study is investigated. METHODS: Three estimators termed direct, standard IV and adjusted IV, of the phenotype-disease log odds ratio are compared through a simulation study which incorporates unmeasured confounding. The simulations are verified using formulae relating marginal and conditional estimates given in the Appendix. RESULTS: The simulations show that the direct estimator is biased by unmeasured confounding factors and the standard IV estimator is attenuated towards the null. Under most circumstances the adjusted IV estimator has the smallest bias, although it has inflated type I error when the unmeasured confounders have a large effect. CONCLUSIONS: In a Mendelian randomization study with a binary disease outcome the bias associated with estimating the phenotype-disease log odds ratio may be of practical importance and so estimates should be subject to a sensitivity analysis against different amounts of hypothesized confounding.

This paper provides a brief overview to four major types of causal models for health-sciences research: Graphical models (causal diagrams), potential-outcome (counterfactual) models, sufficient-component cause models, and structural-equations models. The paper focuses on the logical connections among the different types of models and on the different strengths of each approach. Graphical models can illustrate qualitative population assumptions and sources of bias not easily seen with other approaches; sufficient-component cause models can illustrate specific hypotheses about mechanisms of action; and potential-outcome and structural-equations models provide a basis for quantitative analysis of effects. The different approaches provide complementary perspectives, and can be employed together to improve causal interpretations of conventional statistical results.

Background An allele score is a single variable summarizing multiple genetic variants associated with a risk factor. It is calculated as the total number of risk factor-increasing alleles for an individual (unweighted score), or the sum of weights for each allele corresponding to estimated genetic effect sizes (weighted score). An allele score can be used in a Mendelian randomization analysis to estimate the causal effect of the risk factor on an outcome. Methods Data were simulated to investigate the use of allele scores in Mendelian randomization where conventional instrumental variable techniques using multiple genetic variants demonstrate ‘weak instrument’ bias. The robustness of estimates using the allele score to misspecification (for example non-linearity, effect modification) and to violations of the instrumental variable assumptions was assessed. Results Causal estimates using a correctly specified allele score were unbiased with appropriate coverage levels. The estimates were generally robust to misspecification of the allele score, but not to instrumental variable violations, even if the majority of variants in the allele score were valid instruments. Using a weighted rather than an unweighted allele score increased power, but the increase was small when genetic variants had similar effect sizes. Naive use of the data under analysis to choose which variants to include in an allele score, or for deriving weights, resulted in substantial biases. Conclusions Allele scores enable valid causal estimates with large numbers of genetic variants. The stringency of criteria for genetic variants in Mendelian randomization should be maintained for all variants in an allele score.

Mendelian randomization, which is instrumental variable analysis using genetic variants as instruments, is an increasingly popular method of making causal inferences from observational studies. In order to design efficient Mendelian randomization studies, it is essential to calculate the sample sizes required. We present formulas for calculating the power of a Mendelian randomization study using one genetic instrument to detect an effect of a given size, and the minimum sample size required to detect effects for given levels of significance and power, using asymptotic statistical theory. We apply the formulas to some example data and compare the results with those from simulation methods. Power and sample size calculations using these formulas should be more straightforward to carry out than simulation approaches. These formulas make explicit that the sample size needed for Mendelian randomization study is inversely proportional to the square of the correlation between the genetic instrument and the exposure and proportional to the residual variance of the outcome after removing the effect of the exposure, as well as inversely proportional to the square of the effect size.

The remarkable growth in the popularity of mendelian randomization (MR) studies—analyses using genetic polymorphisms as instrumental variables (IVs)—reflects both the importance of the causal inference problems addressed by MR and the potential power of the approach.1 Although IV analysis is a well-established econometric tool, applications in health research using genetic information as IVs present novel challenges and opportunities. Papers such as that of Au Yeung et al., providing evidence on the characteristics of a proposed genetic IV for a specific phenotype-outcome association,2 are important because they help establish a foundation of evidence relevant for many future MR studies. In this commentary, we discuss the value of integrating evidence from external sources, such as the Au Yeung et al. results in MR studies, with specific focus on using separate-sample IV (SSIV, sometimes called split-sample IV) in data from case-control studies.

Background: MR-Egger regression has recently been proposed as a method for Mendelian randomization (MR) analyses incorporating summary data estimates of causal effect from multiple individual variants, which is robust to invalid instruments. It can be used to test for directional pleiotropy and provides an estimate of the causal effect adjusted for its presence. MR-Egger regression provides a useful additional sensitivity analysis to the standard inverse variance weighted (IVW) approach that assumes all variants are valid instruments. Both methods use weights that consider the single nucleotide polymorphism (SNP)-exposure associations to be known, rather than estimated. We call this the `NO Measurement Error' (NOME) assumption. Causal effect estimates from the IVW approach exhibit weak instrument bias whenever the genetic variants utilized violate the NOME assumption, which can be reliably measured using the F-statistic. The effect of NOME violation on MR-Egger regression has yet to be studied. Methods: An adaptation of the I2 statistic from the field of meta-analysis is proposed to quantify the strength of NOME violation for MR-Egger. It lies between 0 and 1, and indicates the expected relative bias (or dilution) of the MR-Egger causal estimate in the two-sample MR context. We call it IGX2. The method of simulation extrapolation is also explored to counteract the dilution. Their joint utility is evaluated using simulated data and applied to a real MR example. Results: In simulated two-sample MR analyses we show that, when a causal effect exists, the MR-Egger estimate of causal effect is biased towards the null when NOME is violated, and the stronger the violation (as indicated by lower values of IGX2), the stronger the dilution. When additionally all genetic variants are valid instruments, the type I error rate of the MR-Egger test for pleiotropy is inflated and the causal effect underestimated. Simulation extrapolation is shown to substantially mitigate these adverse effects. We demonstrate our proposed approach for a two-sample summary data MR analysis to estimate the causal effect of low-density lipoprotein on heart disease risk. A high value of IGX2 close to 1 indicates that dilution does not materially affect the standard MR-Egger analyses for these data. Conclusions: Care must be taken to assess the NOME assumption via the IGX2 statistic before implementing standard MR-Egger regression in the two-sample summary data context. If IGX2 is sufficiently low (less than 90%), inferences from the method should be interpreted with caution and adjustment methods considered.

Background Mendelian randomization is used to test and estimate the magnitude of a causal effect of a phenotype on an outcome by using genetic variants as instrumental variables (IVs). Estimates of association from IV analysis are biased in the direction of the confounded, observational association between phenotype and outcome. The magnitude of the bias depends on the F-statistic for the strength of relationship between IVs and phenotype. We seek to develop guidelines for the design and analysis of Mendelian randomization studies to minimize bias. Methods IV analysis was performed on simulated and real data to investigate the effect on bias of size of study, number and choice of instruments and method of analysis. Results Bias is shown to increase as the expected F-statistic decreases, and can be reduced by using parsimonious models of genetic association (i.e. not over-parameterized) and by adjusting for measured covariates. Using data from a single study, the causal estimate of a unit increase in log-transformed C-reactive protein on fibrinogen (μmol/l) is shown to increase from -0.005 (P = 0.99) to 0.792 (P = 0.00003) due to injudicious choice of instrument. Moreover, when the observed F-statistic is larger than expected in a particular study, the causal estimate is more biased towards the observational association and its standard error is smaller. This correlation between causal estimate and standard error introduces a second source of bias into meta-analysis of Mendelian randomization studies. Bias can be alleviated in meta-analyses by using individual level data and by pooling genetic effects across studies. Conclusions Weak instrument bias is of practical importance for the design and analysis of Mendelian randomization studies. Post hoc choice of instruments, genetic models or data based on measured F-statistics can exacerbate bias. In particular, the commonly cited rule of thumb that F > 10 avoids bias in IV analysis is misleading.

Investigations into the aetiology of common complex diseases based on observational data should make use of any opportunity to reduce bias due to unobserved confounding. In this context, it has become popular to exploit instrumental variable (IV) methods via Mendelian randomization but the key to success lies in finding suitable genetic instruments. Genome-wide association studies are increasingly yielding large numbers of biomarkers and the understanding of the functionality of these variants is continually improving. However, genetic instruments typically explain only a small proportion of the overall variation in a given exposure and are therefore loosely regarded as ‘weak’ instruments. Combining several instruments intuitively seems like a plausible approach to improving overall instrument strength. Given the likely availability of ever more genetic instruments in the foreseeable future, an investigation into the power and instrument strength requirements of Mendelian randomization analyses with multiple instruments, as proposed by Pierce et al. 1 , is both relevant and timely. In a Mendelian randomization study, the typical target of inference is the effect of an exposure X on a disease outcome Y in the presence of unmeasured confounding factors, U, using one or a combination of several genetic variant(s), G, as an IV. It is often assumed that X and Y are continuous and that all relationships are linear with no interactions, as in

In this issue of the International Journal of Epidemiology (IJE), Day et al.1 conduct a sensitivity analysis of the effects of correlated errors of mis-measured covariates in linear regression models, when the data to permit direct adjustment for the effects of correlated errors are not available. They show yet again what has been documented many times before, but not cited by Day et al.,2–6 that measurement error in more than one model covariate in which both the underlying true values and the errors may be correlated leads to a conflation of the well-known effects of confounding with the well-known effects of measurement error. When the correlations between the underlying variables, the error correlations, and the measurement errors are large, these results can be misleading and unpredictable. It is not entirely clear what is new here, although the detailed example is certainly useful to further emphasize this point to those who didn’t get it the first few times around. There are many instances in nutritional epidemiology when it is of interest to evaluate the independent effects of several nutrients on a health endpoint. The sensitivity analysis approach given by Day et al. is of limited use, since as they and others before them have shown, the regression coefficients relating the nutrients to the health endpoint could take on virtually any value. With the standard study design, it is impossible to validly estimate or even put bounds of any informative width on the regression coefficients of interest. So how can we remove bias in multiple regression coefficients for mis-measured nutrients which both are correlated and may have correlated error? Are there study designs and then methods of data analysis available to correct for these biases? Of course! Day et al.’s revisitation of the problem without reviewing the proposed solutions does not highlight the potential for progress in nutritional epidemiology. Very briefly, Carroll & Stefanski,7 Rosner et al.,8,9 Spiegelman et al.,10,11 and Xie et al.12 presented the regression calibration method for main study/validation study designs, which permit approximately unbiased point and interval estimates of effect to be estimated from linear, Cox, and logistic regression models when the usual methods of exposure assessment can be validated against an unbiased, perhaps ‘alloyed’ gold standard in a sub-sample, possibly outside of the original main study. The method is inherently multivariate and naturally accommodates multiple correlated nutrients with correlated errors as discussed in Day et al.’s paper in this issue of the IJE. Fraser & Stram2 discuss study design in this setting, and show that validation studies of 1000 participants or more are needed when correlations of either the errors or the underlying true values of nutrients are high. Concerns have been raised about the validity of the assumptions made in the original regression calibration method. First, it was suggested that when multiple nutrients are being validated simultaneously by ‘alloyed’ gold standards, the errors in the reference instruments may be correlated with the errors in the surrogates. When this is the case, regression calibration will fail to remove all of the bias due to measurement error— the surrogates will appear to be better measured than they are. Spiegelman et al.13 developed an augmented validation study design which includes a third method of exposure assessment (e.g. a biomarker), the errors of which can reasonably be assumed to be uncorrelated with the errors in the reference instrument (e.g. diet records) and surrogate (e.g. food frequency questionnaires). With these measurements in hand, they extended the regression calibration procedure to allow for correlated errors between the reference and surrogate measures. Error correlations were small in examples of vitamin E and physical activity. Next, Kipnis et al.14 suggested that in addition to the presence of correlated random within-person error of the reference and surrogate exposure measures, there may also be systematic correlated within-person error between these two instruments. In response, Spiegelman et al.15 developed a class of augmented main study/validation study designs which permit valid regression calibration estimation of relative risks and other regression slopes, requiring either replicated biomarkers or an additional ‘instrumental variable’ which has some correlation with the underlying exposure of interest with error jointly uncorrelated with all other errors. It should be pointed out that these extended regression calibration methods have not been expanded to handle multiple regression covariates measured with error, as Day et al. focus upon here.1 Should there be sufficient interest, it is straightforward to derive these further generalizations and produce user-friendly SAS macros to facilitate use, as has already been done with standard multivariate regression calibration (www.hsph.harvard.edu/faculty/spiegelman/software.html). The truly limiting issue in practice is the one underscored by Fraser & Stram—we nutritional epidemiologists have not been conducting validation studies of sufficient size to make the necessary bias corrections with meaningful precision. To my knowledge, there is not yet a single validation study available for which it would be informative to fully investigate the independent effects of several nutrients together, allowing for correlated random and possibly systematic errors of the reference instrument (diaries) with the surrogate (food frequency questionnaire or diet recall). Let’s do it, and do the job right!

In epidemiological research, the unavailability of possible confounders may lead to inconsistent estimates when a randomized experimental design is not feasible. A related challenge can be possible reverse causal orders between the dependent and the independent variables. This is a so-called endogeneity problem, which is well known in econometrics. 1 However, if possible confounders are unavailable, or we suspect reverse causal orders, the appropriate use of instrumental variables (IVs) can generate consistent estimates. IVs have the properties that they are correlated with the explanatory variables and uncorrelated

Economists have much to learn from social epidemiologists. But perhaps economists can teach epidemiologists some useful things. My profession is not alone in being obsessed with cause and effect. Economists teach students from their first university term about identification theory, instrumental variables, and why classical ordinary least squares regression equations will produce the wrong answers if some of the independent variables are endogenously determined. An oft-used example is in equations where the dependent variable is people's rates of pay and one of the independent variables is their level of education. Education is endogenous––chosen by the person. So we teach our students not to read off the coefficient on education in a wage equation and claim that that measures the size of the causal effect.

MendelianRandomization is a software package for the R open-source software environment that performs Mendelian randomization analyses using summarized data. The core functionality is to implement the inverse-variance weighted, MR-Egger and weighted median methods for multiple genetic variants. Several options are available to the user, such as the use of robust regression, fixed- or random-effects models and the penalization of weights for genetic variants with heterogeneous causal estimates. Extensions to these methods, such as allowing for variants to be correlated, can be chosen if appropriate. Graphical commands allow summarized data to be displayed in an interactive graph, or the plotting of causal estimates from multiple methods, for comparison. Although the main method of data entry is directly by the user, there is also an option for allowing summarized data to be incorporated from the PhenoScanner database of genotype-phenotype associations. We hope to develop this feature in future versions of the package. The R software environment is available for download from [https://www.r-project.org/]. The MendelianRandomization package can be downloaded from the Comprehensive R Archive Network (CRAN) within R, or directly from [https://cran.r-project.org/web/packages/MendelianRandomization/]. Both R and the MendelianRandomization package are released under GNU General Public Licenses (GPL-2|GPL-3).

Background A robust method for Mendelian randomization does not require all genetic variants to be valid instruments to give consistent estimates of a causal parameter. Several such methods have been developed, including a mode-based estimation method giving consistent estimates if a plurality of genetic variants are valid instruments; i.e. there is no larger subset of invalid instruments estimating the same causal parameter than the subset of valid instruments. Methods We here develop a model-averaging method that gives consistent estimates under the same 'plurality of valid instruments' assumption. The method considers a mixture distribution of estimates derived from each subset of genetic variants. The estimates are weighted such that subsets with more genetic variants receive more weight, unless variants in the subset have heterogeneous causal estimates, in which case that subset is severely down-weighted. The mode of this mixture distribution is the causal estimate. This heterogeneity-penalized model-averaging method has several technical advantages over the previously proposed mode-based estimation method. Results The heterogeneity-penalized model-averaging method outperformed the mode-based estimation in terms of efficiency and outperformed other robust methods in terms of Type 1 error rate in an extensive simulation analysis. The proposed method suggests two distinct mechanisms by which inflammation affects coronary heart disease risk, with subsets of variants suggesting both positive and negative causal effects. Conclusions The heterogeneity-penalized model-averaging method is an additional robust method for Mendelian randomization with excellent theoretical and practical properties, and can reveal features in the data such as the presence of multiple causal mechanisms.

Mendelian randomization studies perform instrumental variable (IV) analysis using genetic IVs. Results of individual Mendelian randomization studies can be pooled through meta-analysis. We explored how different variance estimators influence the meta-analysed IV estimate. Two versions of the delta method (IV before or after pooling), four bootstrap estimators, a jack-knife estimator and a heteroscedasticity-consistent (HC) variance estimator were compared using simulation. Two types of meta-analyses were compared, a two-stage meta-analysis pooling results, and a one-stage meta-analysis pooling datasets. Using a two-stage meta-analysis, coverage of the point estimate using bootstrapped estimators deviated from nominal levels at weak instrument settings and/or outcome probabilities ≤ 0.10. The jack-knife estimator was the least biased resampling method, the HC estimator often failed at outcome probabilities ≤ 0.50 and overall the delta method estimators were the least biased. In the presence of between-study heterogeneity, the delta method before meta-analysis performed best. Using a one-stage meta-analysis all methods performed equally well and better than two-stage meta-analysis of greater or equal size. In the presence of between-study heterogeneity, two-stage meta-analyses should preferentially use the delta method before meta-analysis. Weak instrument bias can be reduced by performing a one-stage meta-analysis.

Background: Sample size calculations are an important tool for planning epidemiological studies. Large sample sizes are often required in Mendelian randomization investigations. Methods and results: Resources are provided for investigators to perform sample size and power calculations for Mendelian randomization with a binary outcome. We initially provide formulae for the continuous outcome case, and then analogous formulae for the binary outcome case. The formulae are valid for a single instrumental variable, which may be a single genetic variant or an allele score comprising multiple variants. Graphs are provided to give the required sample size for 80% power for given values of the causal effect of the risk factor on the outcome and of the squared correlation between the risk factor and instrumental variable. R code and an online calculator tool are made available for calculating the sample size needed for a chosen power level given these parameters, as well as the power given the chosen sample size and these parameters. Conclusions: The sample size required for a given power of Mendelian randomization investigation depends greatly on the proportion of variance in the risk factor explained by the instrumental variable. The inclusion of multiple variants into an allele score to explain more of the variance in the risk factor will improve power, however care must be taken not to introduce bias by the inclusion of invalid variants.