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Mendelian randomization (MR) is a method of exploiting genetic variation to unbiasedly estimate a causal effect in presence of unmeasured confounding. MR is being widely used in epidemiology and other related areas of population science. In this paper, we study statistical inference in the increasingly popular two-sample summary-data MR design. We show a linear model for the observed associations approximately holds in a wide variety of settings when all the genetic variants satisfy the exclusion restriction assumption, or in genetic terms, when there is no pleiotropy. In this scenario, we derive a maximum profile likelihood estimator with provable consistency and asymptotic normality. However, through analyzing real datasets, we find strong evidence of both systematic and idiosyncratic pleiotropy in MR, echoing the omnigenic model of complex traits that is recently proposed in genetics. We model the systematic pleiotropy by a random effects model, where no genetic variant satisfies the exclusion restriction condition exactly. In this case, we propose a consistent and asymptotically normal estimator by adjusting the profile score.We then tackle the idiosyncratic pleiotropy by robustifying the adjusted profile score. We demonstrate the robustness and efficiency of the proposed methods using several simulated and real datasets.

Mendelian randomization (MR) is a statistical approach using genetic variants as instrumental variables to estimate causal effects of a single exposure on an outcome. Multivariable MR (MVMR) extends this to estimate the direct effect of multiple exposures simulatiously. MR and MVMR can be biased by the presence of pleiotropic genetic variants in the set used as instrumental variables, violating one of the core IV assumptions. Genetic variants that give outlying estimates are often considered to be potentially pleiotropic variants. Radial plots can be used in MR to help identify these variants. Analogous plots for MVMR have so far been unavailable due to the multidimensional nature of the analysis. We propose a radial formulation of MVMR, and an adapted Galbraith radial plot, which allows for the estimated effect of each exposure within an MVMR analysis to be visualised. Radial MVMR additionally includes an option for removal of outlying SNPs which may violate one or more assumptions of MVMR. A RMVMR R package is presented as accompanying software for implementing the methods described. We demonstrate the effectiveness of the radial MVMR approach through simulations and applied analyses. We highlight how outliers with respect to all exposures can be visualised and removed through Radial MVMR. We present simulations that illustrate how outlier removal decreases the bias in estimated effects under various forms of pleiotropy. We apply Radial MVMR to estimate the effect of lipid fractions on coronary heart disease (CHD). In combination with simulated examples, we highlight how important features of MVMR analyses can be explored using a range of tools incorporated within the RMVMR R package. Radial MVMR effectively visualises causal effect estimates, and provides valuable diagnostic information with respect to the underlying assumptions of MVMR.

Background Mental health conditions represent one of the major groups of non-transmissible diseases. Physical activity (PA) and sedentary time (ST) have been shown to affect mental health outcomes in opposite directions. In this study, we use accelerometery-derived measures of PA and ST from the UK Biobank (UKB) and depression, anxiety and well-being data from the UKB mental health questionnaire as well as published summary statistics to explore the causal associations between these phenotypes. Methods We used MRlap to test if objectively measured PA and ST associate with mental health outcomes using UKB data and summary statistics from published genome-wide association studies. We also tested for bidirectional associations. We performed sex stratified as well as sensitivity analyses. Results Genetically instrumented higher PA was associated with lower odds of depression (OR = 0.92; 95% CI: 0.88, 0.97) and depression severity (beta =  − 0.11; 95% CI: − 0.18, − 0.04), Genetically instrumented higher ST was associated higher odds of anxiety (OR = 2.59; 95% CI: 1.10, 4.60). PA was associated with higher well-being (beta = 0.11, 95% CI: 0.04; 0.18) and ST with lower well-being (beta =  − 0.18; 95% CI: − 0.32, − 0.03). Similar findings were observed when stratifying by sex. There was evidence for a bidirectional relationship, with higher genetic liability to depression associated with lower PA (beta =  − 0.25, 95% CI: − 0.42; − 0.08) and higher well-being associated with higher PA (beta = 0.15; 95% CI: 0.05, 0.25). Conclusions We have demonstrated the bidirectional effects of both PA and ST on a range of mental health outcomes using objectively measured predictors and MR methods for causal inference. Our findings support a causal role for PA and ST in the development of mental health problems and in affecting well-being.

Background Major depressive disorder (MDD) has a significant impact on global burden of disease. Complications in clinical management can occur when response to pharmacological modalities is considered inadequate and symptoms persist (treatment-resistant depression (TRD)). We aim to investigate inflammation, proxied by C-reactive protein (CRP) levels, and body mass index (BMI) as putative causal risk factors for depression and subsequent treatment resistance, leveraging genetic information to avoid confounding via Mendelian randomisation (MR). Methods We used the European UK Biobank subcohort ( n = 451 , 025 ), the mental health questionnaire (MHQ) and clinical records. For treatment resistance, a previously curated phenotype based on general practitioner (GP) records and prescription data was employed. We applied univariable and multivariable MR models to genetically predict the exposures and assess their causal contribution to a range of depression outcomes. We used a range of univariable, multivariable and mediation MR models techniques to address our research question with maximum rigour. In addition, we developed a novel statistical procedure to apply pleiotropy-robust multivariable MR to one sample data and employed a Bayesian bootstrap procedure to accurately quantify estimate uncertainty in mediation analysis which outperforms standard approaches in sparse binary outcomes. Given the flexibility of the one-sample design, we evaluated age and sex as moderators of the effects. Results In univariable MR models, genetically predicted BMI was positively associated with depression outcomes, including MDD ( β ( 95 % CI): 0.133(0.072, 0.205)) and TRD (0.347(0.002, 0.682)), with a larger magnitude in females and with age acting as a moderator of the effect of BMI on severity of depression (0.22(0.050, 0.389)). Multivariable MR analyses suggested an independent causal effect of BMI on TRD not through CRP (0.395(0.004, 0.732)). Our mediation analyses suggested that the effect of CRP on severity of depression was partly mediated by BMI. Individuals with TRD ( n = 2199 ) observationally had higher CRP and BMI compared with individuals with MDD alone and healthy controls. Discussion Our work supports the assertion that BMI exerts a causal effect on a range of clinical and questionnaire-based depression phenotypes, with the effect being stronger in females and in younger individuals. We show that this effect is independent of inflammation proxied by CRP levels as the effects of CRP do not persist when jointly estimated with BMI. This is consistent with previous evidence suggesting that overweight contributed to depression even in the absence of any metabolic consequences. It appears that BMI exerts an effect on TRD that persists when we account for BMI influencing MDD.

Results from genome-wide association studies (GWAS) can be used to infer causal relationships between phenotypes, using a strategy known as 2-sample Mendelian randomization (2SMR) and bypassing the need for individual-level data. However, 2SMR methods are evolving rapidly and GWAS results are often insufficiently curated, undermining efficient implementation of the approach. We therefore developed MR-Base ( http://www.mrbase.org ): a platform that integrates a curated database of complete GWAS results (no restrictions according to statistical significance) with an application programming interface, web app and R packages that automate 2SMR. The software includes several sensitivity analyses for assessing the impact of horizontal pleiotropy and other violations of assumptions. The database currently comprises 11 billion single nucleotide polymorphism-trait associations from 1673 GWAS and is updated on a regular basis. Integrating data with software ensures more rigorous application of hypothesis-driven analyses and allows millions of potential causal relationships to be efficiently evaluated in phenome-wide association studies. Our health is affected by many exposures and risk factors, including aspects of our lifestyles, our environments, and our biology. It can, however, be hard to work out the causes of health outcomes because ill-health can influence risk factors and risk factors tend to influence each other. To work out whether particular interventions influence health outcomes, scientists will ideally conduct a so-called randomized controlled trial, where some randomly-chosen participants are given an intervention that modifies the risk factor and others are not. But this type of experiment can be expensive or impractical to conduct. Alternatively, scientists can also use genetics to mimic a randomized controlled trial. This technique – known as Mendelian randomization – is possible for two reasons. First, because it is essentially random whether a person has one version of a gene or another. Second, because our genes influence different risk factors. For example, people with one version of a gene might be more likely to drink alcohol than people with another version. Researchers can compare people with different versions of the gene to infer what effect alcohol drinking has on their health. Every day, new studies investigate the role of genetic variants in human health, which scientists can draw on for research using Mendelian randomization. But until now, complete results from these studies have not been organized in one place. At the same time, statistical methods for Mendelian randomization are continually being developed and improved. To take advantage of these advances, Hemani, Zheng, Elsworth et al. produced a computer programme and online platform called “MR-Base”, combining up-to-date genetic data with the latest statistical methods. MR-Base automates the process of Mendelian randomization, making research much faster: analyses that previously could have taken months can now be done in minutes. It also makes studies more reliable, reducing the risk of human error and ensuring scientists use the latest methods. MR-Base contains over 11 billion associations between people’s genes and health-related outcomes. This will allow researchers to investigate many potential causes of poor health. As new statistical methods and new findings from genetic studies are added to MR-Base, its value to researchers will grow.

Clinical researchers have often preferred to use a fixed effects model for the primary interpretation of a meta-analysis. Heterogeneity is usually assessed via the well known Q and I2 statistics, along with the random effects estimate they imply. In recent years, alternative methods for quantifying heterogeneity have been proposed, that are based on a 'generalised' Q statistic. We review 18 IPD meta-analyses of RCTs into treatments for cancer, in order to quantify the amount of heterogeneity present and also to discuss practical methods for explaining heterogeneity. Differing results were obtained when the standard Q and I2 statistics were used to test for the presence of heterogeneity. The two meta-analyses with the largest amount of heterogeneity were investigated further, and on inspection the straightforward application of a random effects model was not deemed appropriate. Compared to the standard Q statistic, the generalised Q statistic provided a more accurate platform for estimating the amount of heterogeneity in the 18 meta-analyses. Explaining heterogeneity via the pre-specification of trial subgroups, graphical diagnostic tools and sensitivity analyses produced a more desirable outcome than an automatic application of the random effects model. Generalised Q statistic methods for quantifying and adjusting for heterogeneity should be incorporated as standard into statistical software. Software is provided to help achieve this aim.

Background Confidence intervals for the between study variance are useful in random-effects meta-analyses because they quantify the uncertainty in the corresponding point estimates. Methods for calculating these confidence intervals have been developed that are based on inverting hypothesis tests using generalised heterogeneity statistics. Whilst, under the random effects model, these new methods furnish confidence intervals with the correct coverage, the resulting intervals are usually very wide, making them uninformative. Methods We discuss a simple strategy for obtaining 95 % confidence intervals for the between-study variance with a markedly reduced width, whilst retaining the nominal coverage probability. Specifically, we consider the possibility of using methods based on generalised heterogeneity statistics with unequal tail probabilities, where the tail probability used to compute the upper bound is greater than 2.5 %. This idea is assessed using four real examples and a variety of simulation studies. Supporting analytical results are also obtained. Results Our results provide evidence that using unequal tail probabilities can result in shorter 95 % confidence intervals for the between-study variance. We also show some further results for a real example that illustrates how shorter confidence intervals for the between-study variance can be useful when performing sensitivity analyses for the average effect, which is usually the parameter of primary interest. Conclusions We conclude that using unequal tail probabilities when computing 95 % confidence intervals for the between-study variance, when using methods based on generalised heterogeneity statistics, can result in shorter confidence intervals. We suggest that those who find the case for using unequal tail probabilities convincing should use the ‘1–4 % split’, where greater tail probability is allocated to the upper confidence bound. The ‘width-optimal’ interval that we present deserves further investigation.

Over a decade of genome-wide association studies (GWAS) have led to the finding of extreme polygenicity of complex traits. The phenomenon that “all genes affect every complex trait” complicates Mendelian Randomization (MR) studies, where natural genetic variations are used as instruments to infer the causal effect of heritable risk factors. We reexamine the assumptions of existing MR methods and show how they need to be clarified to allow for pervasive horizontal pleiotropy and heterogeneous effect sizes. We propose a comprehensive framework GRAPPLE to analyze the causal effect of target risk factors with heterogeneous genetic instruments and identify possible pleiotropic patterns from data. By using GWAS summary statistics, GRAPPLE can efficiently use both strong and weak genetic instruments, detect the existence of multiple pleiotropic pathways, determine the causal direction and perform multivariable MR to adjust for confounding risk factors. With GRAPPLE, we analyze the effect of blood lipids, body mass index, and systolic blood pressure on 25 disease outcomes, gaining new information on their causal relationships and potential pleiotropic pathways involved.

Recent analyses have shown educational attainment to be associated with a number of health outcomes. This association may, in part, be due to an effect of educational attainment on smoking behaviour. In this study, we apply a multivariable Mendelian randomisation design to determine whether the effect of educational attainment on smoking behaviour is due to educational attainment or general cognitive ability. We use individual data from the UK Biobank study ( N  = 120,050) and summary data from large GWA studies of educational attainment, cognitive ability and smoking behaviour. Our results show that more years of education are associated with a reduced likelihood of smoking that is not due to an effect of general cognitive ability on smoking behaviour. Given the considerable physical harms associated with smoking, the effect of educational attainment on smoking is likely to contribute to the health inequalities associated with differences in educational attainment. Higher educational attainment is positively associated with a number of health outcomes. Here, Sanderson et al. use multivariable Mendelian randomisation analysis to test whether the association of educational attainment with smoking behaviour is direct or indirectly mediated via general cognitive ability.