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J Obes Metab Syndr 2025; 34(1): 14-26

Published online January 30, 2025 https://doi.org/10.7570/jomes24031

Copyright © Korean Society for the Study of Obesity.

A Review of Mendelian Randomization: Assumptions, Methods, and Application to Obesity-Related Diseases

Seungjae Lee1, Woojoo Lee1,2,*

1Institute of Health and Environment, Seoul National University, Seoul; 2Department of Public Health Sciences, Graduate School of Public Health, Seoul National University, Seoul, Korea

Correspondence to:
Woojoo Lee
https://orcid.org/0000-0001-7447-7045
Department of Public Health Sciences, Graduate School of Public Health, Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul 08826, Korea
Tel: +82-2-880-2899
Fax: +82-2-762-9105
E-mail: lwj221@gmail.com

Received: September 19, 2024; Reviewed : December 2, 2024; Accepted: January 2, 2025

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Mendelian randomization (MR) is a statistical method that uses genetic variants as instrumental variables to estimate the causal effect of exposure on an outcome in the presence of unmeasured confounding. In this review, we argue that it is crucial to acknowledge the instrumental variable assumptions in MR analysis. We describe widely used MR methods, using an example from obesity-related metabolic disorders. We describe situations in which instrumental variable assumptions are violated and explain how to evaluate these violations and employ robust methods for accommodating such violations.

Keywords: Causality, Confounding factors, epidemiologic, Genetic pleiotropy, Genetic variation, Genome-wide association study, Human genetics, Mendelian randomization analysis

The instrumental variable (IV) method is a method of causal inference in the presence of unmeasured confounding. Mendelian randomization (MR) is a special form of IV analysis using genetic variants as IVs to estimate the causal effect of exposure on the outcome of interest.1 Single-nucleotide polymorphisms (SNPs) are widely used IVs in MR analysis. Owing to the ease of use of genome-wide association studies (GWAS),2 many MR studies have emerged in the medical and public health sciences.3,4 The scope of MR,5-8 associated statistical methods,9 and analytical guidelines10 have been considered in the literature. Burgess and Thompson11 provides a book-length overview of MR implementation.

This study aims to provide a review of MR with a focus on current practical considerations. We first outline standard IV assumptions underlying MR. The advantage of IV methods in avoiding bias due to unmeasured confounding is not obtained for free but involves specific assumptions, some of which cannot be verified from observed data. Subject domain knowledge is necessary to justify the use of IV methods.

Next, we categorize MR analysis to help researchers understand which MR methodologies are suitable. After explaining the methods for estimating causal effects using one-sample individual-level data, we describe estimation methods for causal effects using two-sample summary-level data. Finally, we discuss some challenges for MR.

To test whether an exposure has a causal effect on an outcome using IV methods, the following IV conditions are assumed:12

IV1. Non-null association assumption: The genetic variant is associated with the exposure.

IV2. Independence assumption: The genetic variant is not associated with the outcome through confounding pathways.

IV3. Exclusion restriction assumption: The genetic variant affects the outcome only through the exposure.

The directed acyclic graph is an intuitive way of expressing the three IV assumptions (Fig. 1). Assumption IV1 is represented by an arrow from the genetic variant (G) to the exposure (X), although a non-zero association between G and X is sufficient to satisfy IV1. Assumption IV2 is illustrated by the absence of an arrow between the genetic variant and the unmeasured confounder (U), indicating that G and U are independent. Assumption IV3 is depicted as absence of a pathway from the genetic variant to the outcome (Y), other than that passing through the exposure. Of the three IV assumptions, IV1 can be tested using observed data, whereas IV2 cannot be verified using empirical data alone, and IV3 can only be partially verified using some models.13

It is important to acknowledge when these IV assumptions are violated. First, genetic variants may affect the outcome through pathways other than the exposure. This may result in bias of the MR estimate. Fig. 2A and B are examples of (horizontal) pleiotropy. Fig. 2A shows that IV2 would be violated because the genetic variant is associated with the outcome via an unmeasured confounding variable, which is depicted by the existence of a bidirectional arrow between G and U. Fig. 2B shows that IV3 would be violated because of the presence of an alternative causal pathway (not via the exposure) from the genetic variant to the outcome, as illustrated by a direct arrow from G to Y. Second, as illustrated in Fig. 2C both by the bidirectional arrow between G1 and G2 and by the pathway of G2 to the outcome not via the exposure, two (or more) genetic variants in linkage disequilibrium (LD) may result in confounding between X and Y, which can violate IV3. Third, population stratification, assortative mating, and dynastic effects may affect the distribution of genetic variants of specific traits within populations, violating IV2.14 Confounding by population stratification (illustrated by arrows from Z to both G and Y in Fig. 2D), in which ancestry is correlated with both phenotypes and genetic variants, is a major concern during the early development of MR.15 Assortative mating occurs due to non-random matching between spouses based on particular characteristics.16 Dynastic effects can occur when the expression of a parental genetic variant that affects the parental phenotype directly influences the offspring’s phenotype.17 In practice, the principal components of genetic variants are often adjusted to avoid bias owing to population stratification.18 Alternatively, family-based study designs could be considered to solve these issues.14,19

Assumptions IV1–IV3 are insufficient to estimate the magnitude of the causal effect. For estimating the causal effect, the following IV assumption is often employed:20

IV4. Monotonicity: Any change in the exposure from varying the IV should be in the same direction for all individuals in the population.

Under the monotonicity assumption, an IV estimate is often referred to as a local average treatment effect or complier average causal effect. For more details about causal interpretation in MR studies, see Swanson and Hernán.21

Two important factors for categorizing MR analysis are (1) the number of data sources (a single sample or two samples) and (2) the type of dataset (individual-level data or summary-level data). However, this categorization is not flawless and contains grey areas. One-sample MR is often conducted using a single sample, whereas two-sample MR uses two samples drawn from either one or two different populations. When some samples overlap in the two-sample MR, this categorization is not clear. Individual-level data include genetic variants, exposure, outcome, and confounders. By contrast, summary-level data consist of genetic association estimates derived from regressions of the exposure or outcome on a genetic variant.

Most one-sample MR studies have utilized individual-level data,22 although methods based on summary-level data can theoretically be applied to a one-sample MR context.23,24 MR studies are more commonly conducted in a two-sample setting, utilizing published summary statistics from two independent GWAS: one for exposure and the other for outcome.25 Therefore, conveniently, we refer to one-sample individual-level MR as one-sample MR and two-sample summary-level MR as two-sample MR.

One-sample MR versus two-sample MR

One-sample MR offers analytical flexibility, enabling the application of various regression models and inclusion or exclusion of covariates and individuals. This approach allows flexible confounder adjustment and comparison of causal estimates using testing methods such as the Durbin-Wu-Hausman test.26,27 Additionally, onesample MR can model complex interactions between a genetic variant and a covariate,28 between exposures,29,30 consider survival times,31 and assess non-linear effects,32 although these may require additional assumptions and large sample sizes. The key requirement for a one-sample MR is access to individual-level genetic and phenotypic data. However, such results may suffer weak instrument bias or the winner’s curse when genetic variants used in the MR analysis are initially discovered in the data under analysis.33

In many situations, the results from one-sample MR have low statistical power owing to the limited sample size and number of genetic variants.34 The increasing availability of publicly available summary statistics from large-scale GWAS enhances the statistical power, which is a strength of two-sample MR.23 In addition, weak IVs will bias causal estimates toward the null in the two-sample MR (with non-overlapping samples), whereas weak instrument bias will favor the confounded estimate in one-sample MR. However, twosample MR can produce biased results if the genetic associations with the exposure or with variables on pleiotropic pathways differ between the two samples, which could affect the validity of the IV assumptions. Reliance on summary data from original GWAS models limits the flexibility of this approach, and harmonizing the SNPexposure and SNP-outcome associations can be complex. Also, more sophisticated analyses of interactions or survival times are often not feasible in two-sample MR studies. Despite these difficulties, the expansion of GWAS has significantly enhanced the popularity of two-sample MR by providing datasets for estimating genetic associations with exposures and outcomes.35

Multiple genetic variants in MR

Baiocchi et al.36 provides a comprehensive review of IV methods using a single IV in medical applications. Smith and Ebrahim37 acknowledged that an individual genetic variant explains only a small proportion of variance in exposure, and sufficient statistical power in an MR analysis with a single IV requires a huge sample size. This problem can be resolved partially using multiple genetic variants.38 If each genetic variant explains an additional independent variation in the exposure, then a combined causal estimate using all genetic variants will be more precise than the estimate from any of the individual genetic variants.39

In MR studies involving multiple genetic variants, the simplest approach is to combine genetic variants into a single allele score.40,41 This score with K multiple genetic variants can be calculated as follows:

G*= j=1KwjGj

where weight wj reflects the effect of the jth genetic variant on exposure and can be obtained from the data under analysis or from independent data. Combining multiple genetic variants also can reduce the risk of bias from many weak IVs.42 However, use of multiple genetic variants often increases the risk of violating the IV assumptions. If any genetic variant that constitutes a single allele score violates the assumptions of IV2 or IV3, the resulting IV estimate may be biased. Nevertheless, the use of multiple genetic variants and a combined single allele score is becoming more common in MR studies.38 Therefore, throughout this study, we focus on MR methods that employ multiple independent genetic variants.

To implement one-sample MR, the ratio of coefficients method, or Wald method,43 is the simplest way to estimate the causal effect of exposure X on outcome Y based on a single allele score combining multiple genetic variants G*. The ratio estimate of the causal effect is the ratio of the coefficient of G* in the regression of Y on G* to the coefficient of G* in the regression of X on G*. This ratio estimate can be calculated simply from regression coefficients, which are summary-level data. This ratio of coefficients method also plays an important role in two-sample MR.

An alternative method is the two-stage least squares (2SLS) estimation. When there are K genetic variants, the first-stage regression model is

X=γ0+γ1G1++γKGK+εX

where εX is a random error. The predicted value of the exposure X^ is then used in the following second-stage regression model:

Y=θ0+θX^+εY

where εY is a random error. The estimated value of θ is the 2SLS estimate of the causal effect. However, the standard error should not be computed using the standard formula for the linear regression model and should be corrected for uncertainty resulting from the first-stage regression model. IV software implements this correction as the default method. An example is the ivreg command in the Applied Econometrics with R (AER) package, which implements the 2SLS method in R (R Foundation for Statistical Computing).44 With a single genetic variant, the 2SLS estimate is the same as the ratio estimate. However, with multiple genetic variants, the 2SLS estimate can be regarded as a weighted average of the variant-specific ratio estimates, where the weights are determined by the relative strength of each IV in the first-stage regression.45,46

Assessing strength of IVs

Although IV1–IV4 hold, IV may lack a strong association with exposure and is referred to as a weak IV.47 The causal effect estimate based on weak IVs becomes unbiased as the sample size approaches infinity. However, the performance of the IV estimate can be poor for small sample sizes. In the 2SLS method, if the IVs are weak, the fitted values from the first-stage regression are estimated with a large uncertainty.

Assumption IV1 can be evaluated using the F-statistic in the first stage of the 2SLS method, which is used to test associations between SNPs and exposure. The F-statistic represents the strength of the association between IVs and exposure in the IV estimation.48,49 As a rule of thumb, if the F-statistic is <10, the IVs are weak.

Test for validity of IV assumptions

With individual-level data, the overidentification test (also called the heterogeneity test) assesses whether the causal effects for each IV are the same or whether the IVs have residual associations with the outcome once the main effect of the exposure has been removed. Sargan’s or Hansen’s J test of overidentification determines evidence of a difference between IV estimates based on each genetic variant.50,51 Sargan’s J test becomes a special case of Hansen’s J test under homoskedasticity. Rejection of the overidentification tests indicates heterogeneity in the effect estimates from each IV, suggesting that the IV assumptions may be violated for one or more genetic variants. Higgins’ I2 statistic also can be used to quantify the degree of heterogeneity.52 Alternatively, Hausman’s endogeneity test (also called Durbin-Wu-Hausman’s test) evaluates whether there is any evidence that the IV estimate differs from the ordinary least squares estimate.53

Methods with some robustness to violations of IV assumptions

In the context of individual-level MR design, we consider robust methods that do not require all genetic variants to be valid IVs to provide consistent estimates of causal effects. The validity of MR investigations becomes less reliable in the presence of outlying causal effects, especially in the presence of substantial heterogeneity, because outliers may represent pleiotropic variants. There are MR methods that are robust to outliers, such as weighted median54 and weighted mode.55 These methods derive their estimates from the median or mode of ratio estimates, which naturally withstand the influence of extreme values. Additionally, approaches such as Some Invalid Some Valid Instrumental Variables Estimator (sisVIVE)56 and adaptive Lasso57 offer MR estimates when the proportion of invalid IVs is <50%. Using penalized regression, these methods enable valid causal inferences. Another approach is to adjust for the pleiotropic effects of genetic variants. Methods implementing this strategy include multivariable 2SLS58 and constrained IV.59 Multi-variable 2SLS controls for pleiotropic phenotypes as covariates in both first- and second-stage regression models. Constrained IV constructs a new IV by maximizing the genetic association with exposure and minimizing its association with pleiotropic phenotypes. Finally, Tchetgen Tchetgen et al.60 proposed MR G-Estimation under No Interaction with Unmeasured Selection (MR-GENIUS), which improves G-estimation to be robust to both additive unmeasured confounding and violation of the IV3 assumption. Ye et al.61 proposed MR-GENIUS MAny Weak Invalid IV (MR-GENIUS-MAWII), an extended version of MR-GENIUS that is additionally robust to many weak IVs.

Now, we explore two-sample MR methods using GWAS summary statistics only. This consists of the following three steps.

1. Select genetic variants from the exposure GWAS database. Based on the selected genetic variants, extract the summary statistics from the exposure and outcome GWAS databases, respectively.

2. Harmonize the effect sizes for genetic variants on exposure and outcome to be each for the same reference allele.

3. Perform MR analysis for estimating effect size and applying robust MR methods.

In the ‘Practical application of MR’ section, we will consider Gormley et al.62 as an example of practical application of these three steps.

Selecting genetic variants and extracting summary statistics

Genetic variants in MR should be associated with exposure to satisfy assumption IV1. Typically, genetic variants are often required to be associated with exposure at a specific genome-wide significance level, such as P-value < 5 × 10–8. Next, genetic variants are required to be uncorrelated. The inclusion of genetic variants in high LD might not contribute additional information for estimating causal effects and could lead to biased standard errors if this correlation structure is not considered.25 Therefore, genetic variants in strong LD can be excluded using LD-based clumping or pruning, which can be performed using the PLINK software tool63 or the clump_ data function of the TwoSampleMR package in R.64 In the LD-based clumping procedure, within a certain physical distance between pairs of loci, the square of the LD correlation coefficient (denoted by r2) is calculated. Among SNPs with r2 greater than the specified threshold, only those with the lowest P-values are retained.

Even though we select genetic variants that are significantly associated with exposure using P-value < 5 × 10–8, it is important to check the strength of genetic variants. This strength can be assessed using the proportion of variance in exposure explained by genetic variants (denoted by R2). For a biallelic SNP, the proportion of variance explained by the jth SNP (denoted by Rj2) is calculated by Rj2=2×MAFj×1MAFj×γ^j2, where MAFj denotes the minor allele frequency and γ^j denotes the SNP-exposure association estimate.65 For uncorrelated genetic variants, the total proportion of variance explained becomes R2=j=1KRj2. Alternatively, to assess the strength, we can use the F-statistic, defined as F=NK1KR21R2, where N is the sample size and K is the number of genetic variants.39 In practice, as a rule of thumb, an F-statistic < 10 indicates the presence of weak IVs.

Harmonizing two-sample summary-level data

In two-sample MR, summary statistics for the SNP-exposure and SNP-outcome associations are used to estimate the causal effect of exposure on the outcome. When combining two or more independently generated datasets,66 it is crucial to confirm consistent orientation.23 If the effect allele of a particular SNP differs between datasets, both the SNP-outcome association estimate and the effect allele frequency in the outcome data need to be harmonized to reflect the same effect allele as in the exposure data. This can be achieved by reversing the sign of the SNP-outcome association estimate (i.e., multiplying by –1) and adjusting the effect allele frequency in the outcome data to its complement (i.e., 1–current effect allele frequency). Using knowledge of the effect alleles (with their frequencies), one can automatically harmonize the exposure and outcome datasets using the MR-base web application or the harmonise_data function of the TwoSampleMR package in R.64

With palindromic SNPs, the effect allele frequency can be used to infer the strand and determine whether it is consistent across datasets. When the effect allele frequency is not available, it may be necessary to exclude palindromic SNPs, for which it is not possible to infer the direction.23

To overcome potential errors in harmonization, it is advised to assess the correlation between effect allele frequencies before and after harmonization. Providing results for both the pre- and post-harmonization datasets helps assess the quality of harmonization. Furthermore, conducting sensitivity analyses with and without difficult-to-harmonize variants can effectively evaluate their influence on the harmonization process.

Effect size estimation

In summary-level data, assume that we only observe (γ^j, σx,j, Γ^j, σy,j), where γ^j is the jth SNP-exposure association estimate, σx,j is the standard error of γ^j , Γ^j is the jth SNP-outcome association estimate, and σy,j is the standard error of Γ^j. These values are obtained from regressions of the exposure or outcome on each genetic variant.

The inverse variance weighting (IVW) method, as a meta-analysis technique, was designed to combine several independent estimates and has been used widely in two-sample summary-level MR.67 This method assumes that the IV assumptions hold for all genetic variants selected in MR studies. The IVW estimate is obtained by regressing the SNP-outcome association Γ^j on the SNP-exposure association γ^j, weighted by the inverse variance of the SNP-outcome association σy,j2, without an intercept. The assumed model is

Γ^j=θγ^j+εj,  εjN0,σy,j2.

The slope estimate is equal to the IVW estimate68 and is expressed as follows:

θ^=j=1K γ ^ j Γ ^ jσy,j2j=1K γ ^ jσy,j2.

One important assumption for IVW estimation is that genetic variants are independent.69 When genetic variants are uncorrelated, the IVW estimate is equivalent to the 2SLS estimate.58 Some methods have been developed to consider LD between genetic variants in summary-level MR.68,70

Heterogeneity in variant-specific causal estimates greater than that expected by chance may occur because one or more genetic variants are invalid IVs. In particular, testing for heterogeneity is a statistical assessment of pleiotropy. Cochran’s Q statistic can be used to test for between-variant heterogeneity.71,72 Under the null hypothesis of homogeneity, Cochran’s Q statistic approximately follows a χ2 distribution with K–1 degrees of freedom, where K denotes the number of genetic variants. A large value of Q means the variant-specific ratio estimates differ more than expected by chance. Higgins’ I2 statistic also can be used to quantify the degree of heterogeneity.52 Outliers and influential points, which may represent invalid IVs, can be identified using studentized residuals and Cook’s distances when implementing the IVW method using weighted linear regression.73 Additionally, scatter plots,54 funnel plots,74 forest plots, and radial plots75 are useful tools for exploring heterogeneity and identifying outliers. If evidence of heterogeneity exists, care is necessary when interpreting MR results.

Presence of heterogeneity due to pleiotropy does not necessarily invalidate MR studies. Across all genetic variants, if (1) the amount of pleiotropy is independent of the effect of genetic variant on the exposure (known as the Instrument Strength Independent of Direct Effect [InSIDE] assumption)76 and (2) the average pleiotropic effect is zero (known as directional pleiotropy), then a standard additive or multiplicative random-effects meta-analysis still yields reliable inferences for the causal effect.77 The resulting random-effects IVW estimate is unbiased only when the InSIDE assumption is satisfied and the average pleiotropy is zero.

Robust methods using invalid IVs

Robust methods have been developed for situations where some genetic variants are invalid IVs. These methods are often classified into four categories: (1) consensus, (2) outlier removal, (3) outlier adjustment, and (4) pleiotropy adjustment.

Consensus methods include the weighted median54 and weighted mode55 methods, which exploit the median or mode of the ratio estimate distribution as the causal estimate. Consequently, they are naturally robust to outliers, as the median and mode are robust to extreme values. Outlier removal methods include the MR Pleiotropy RESidual Sum and Outlier (MR-PRESSO)78 and MR-Lasso79 methods, which involve the identification and removal of individual IVs, and a causal estimate is obtained by the IVW method (and other robust methods) using the remaining IVs. Outlier adjustment methods include MR-Robust,79 MR-Mixture,80 and contamination mixture81 and assume that only some of the IVs are valid. The MR-Robust method downweighs IVs with heterogeneous causal estimates, whereas the other methods divide IVs into valid and invalid IVs (two categories for the contamination mixture method and four categories for the MR-Mixture method). Finally, pleiotropy adjustment methods include the MR-Egger,76 multivariable MR,82 generalized summary MR (GSMR),70 MR Causal Analysis Using Summary Effect estimates (MR-CAUSE),83 MR-Robust Adjusted Profile Score (MR-RAPS),84 Genome-wide mR Analysis under Pervasive PLEiotropy (GRAPPLE),85 and debiased IVW86 methods, which allow most or all IVs included in the estimation to have pleiotropic effects on the outcome. The MR-RAPS,84 GRAPPLE,85 and debiased IVW86 methods are also robust to many weak IVs. However, these categories are not mutually exclusive, and the classification of some methods can be ambiguous. Table 1 lists the functions and packages that implement commonly used methods in R. See Slob and Burgess87 for a comparison of the performances of robust MR methods across several simulation studies.

We focus on the MR-Egger method, which is currently the most popular robust MR method and provides a consistent causal estimate under the InSIDE assumption.76,88 An MR-Egger estimate θ^ can be obtained by fitting the following regression model with an intercept term θ0:

Γ^j=θ0+θγ^j+εj,  εjN0,σy,j2.

Unlike the IVW method, the average pleiotropic effect does not need to equal zero, and this effect is absorbed into the intercept term under the InSIDE assumption. The intercept can differ from zero when (1) the average pleiotropic effect is non-zero or (2) the In-SIDE assumption is violated. Therefore, testing whether the intercept differs from zero can serve as a test for IV assumptions. After fitting the MR-Egger regression and adjusting for the mean pleiotropic effect, it is possible to use Rücker’s Q statistic to test for any residual heterogeneity due to pleiotropy.77,89 Under the null hypothesis that the pleiotropic effect is the same across all variants, Rücker’s Q statistic approximately follows a χ2 distribution with K–2 degrees of freedom. Using weak IVs, MR approaches can violate the NO Measurement Error (NOME) assumption, which assumes that the variance of the SNP-exposure associations is negligible. For the MR-Egger estimate, the degree of violation of the NOME assumption can be quantified using the I2 statistic (denoted by IGX2), which ranges from 0 to 1.88 Lower values of IGX2 indicate a stronger violation of the NOME assumption, resulting in a more biased MR-Egger estimate because of weak IVs.

Gormley et al.62 conducted two-sample MR using genetic variants associated with adiposity, glycemic, and blood pressure traits in GWAS and evaluated the effects of these metabolic traits on oral and oropharyngeal cancer risk. In this section, we examine the details of the three steps described in the ‘two-sample MR using summary-level data’ section, using a concrete example from Gormley et al.62

The first step is to select genetic variants and extract their summary statistics from the exposure and outcome GWAS databases. Gormley et al.62 identified genome-wide significant genetic variants (P <5× 10–8) for metabolic traits from previously conducted GWAS. They performed LD clumping to extract independent sets of SNP-exposure associations based on a specified clumping threshold of r2 at 0.001. After LD clumping, 312 genetic variants for body mass index (BMI) were identified from a GWAS meta-analysis of 806,834 individuals of European ancestry, including data from the Genetic Investigation of ANthropometric Traits (GIANT) consortium and the UK Biobank. The SNP-outcome association estimates were obtained from a GWAS of 6,034 oral and oropharyngeal cases and 6,585 controls from 12 studies that were part of the Genetic Associations and Mechanisms in Oncology (GAME-ON) network. The F-statistic of genetic variants for BMI was 89.5, indicating sufficient strength for MR analyses. The proportion of variance in BMI explained by the selected genetic variants (i.e., r2) was 4.0%.

The second step is to harmonize the effect sizes for genetic variants on exposure and outcome to be each for the same reference allele. In Gormley et al.,62 after extracting summary-level data for the SNP-exposure and SNP-outcome associations, the exposure and outcome summary statistics were harmonized such that genetic association estimates corresponded to the same effect allele. Additionally, palindromic SNPs were identified and corrected using effect allele frequencies, aligning alleles when minor allele frequencies were < 0.3 and excluding them otherwise.

The third step is to perform MR analysis for estimating effect size and applying robust MR methods. Following harmonization, Gormley et al.62 performed IVW analysis and heterogeneity tests. The IVW-based odds ratio (OR) estimate for oral and oropharyngeal cancer was 0.89 (95% confidence interval [CI], 0.72 to 1.09) per 1 standard deviation increase in BMI. There was no clear evidence of heterogeneity between the variant-specific causal estimates for BMI (Cochran’s Q= 311.47, degrees of freedom= 271, P = 0.05); however, a small number of outliers was present upon visual inspection of the scatter plots. Also, they employed the MR-Egger, weighted median, and weighted mode methods to evaluate the potential for unbalanced horizontal pleiotropy. The MR-PRESSO was also used to detect and correct potential outliers. The results were similar to the IVW results (MR-Egger OR, 0.66 [95% CI, 0.40 to 1.10]; weighted median OR, 0.71 [95% CI, 0.50 to 1.00]; and weighted mode OR, 0.63 [95% CI, 0.37 to 1.04]). The MR-Egger intercepts were not strongly indicative of directional pleiotropy (θ^ = 0.006, P = 0.21). There was no clear evidence of heterogeneity in the SNP effect estimates in the MR-Egger regression (Rücker’s Q= 309.67, degrees of freedom = 270, P = 0.05). The NOME assumption employed in the MR-Egger method appeared acceptable, as both the unweighted and weighted IGX2s for BMI were > 0.90. When excluding 19 outliers identified using the MR-PRESSO method, the analysis yielded effects consistent with the primary IVW analysis, which demonstrated a protective effect on oral and oropharyngeal cancer risk (IVW OR, 0.77 [95% CI, 0.62 to 0.94]; MR-Egger OR, 0.68 [95% CI, 0.41 to 1.13]; weighted median OR, 0.67 [95% CI, 0.48 to 0.93]; and weighted mode OR, 0.61 [95% CI, 0.37 to 1.00]).

With the availability of summary-level data, a two-sample MR analysis has become more straightforward. Table 2 in Davies et al.5 and Table 2 in Richmond and Davey Smith7 provide lists of publicly available data sources for two-sample MR studies. The OpenGWAS project is a large database of GWAS results that can be used in MR analysis.64 More information is available here: https://gwas.mrcieu.ac.uk/. However, the primary challenge is not merely executing the analysis but ensuring its reliability and reproducibility. In this review, we delineate the fundamental principles of MR, elucidate the IV conditions integral to MR estimation, and explore various estimation methodologies. We emphasize that the underlying assumptions of an MR study should be evaluated based on observed data and domain knowledge. In particular, because not all assumptions can be verified using observed data, it is often crucial to understand how these unverifiable assumptions can be supported by domain knowledge. Furthermore, for secure MR analysis, estimation methods that are robust to violations of assumptions are often recommended as sensitivity analysis methods.

To help in reporting MR studies transparently, a set of guidelines entitled ‘Strengthening the Reporting of Observational Studies in Epidemiology using MR’ (STROBE-MR) has been developed.90,91 As a result, many articles published in high-impact journals have employed the STROBE-MR guidelines.62,92-96 The STROBE-MR guidelines help to ensure that the necessary content is included in an MR study.

There are important topics not covered in this review. One of these is power calculations.34 The STROBE-MR guidelines recommend reporting whether power or sample size calculations were performed before the main analysis. In a given sample size, power calculations provide information on reliably detecting a particular magnitude of causal effect. Online tools for performing power calculations in MR analysis are available at https://shiny.cnsgenomics.com/mRnd/ and https://sb452.shinyapps.io/power/. Additional topics that were not covered in this review include network MR,97 which evaluates intermediates in the causal pathway between exposures and outcomes; non-linear MR,98 which estimates the nonlinear (e.g., dose-response) effect of exposure on an outcome; and bidirectional MR,99 which considers the prevailing direction of causality between two traits. Regarding these novel methods, it is important to clarify the assumptions needed for the advanced MR method and the untestable assumptions required in MR and to recognize whether different results arise from these differences in assumptions. Determining whether such findings are truly scientific cannot rely solely on P-values but requires the accumulation of evidence from various perspectives.

Fig. 1. Directed acyclic graph for the instrumental variable assumptions. The arrow indicates a causal effect. G=genetic variant(s); X=exposure; U=unmeasured confounder(s); and Y=outcome.
Fig. 2. Directed acyclic graph when there is a genetic variant that violates the instrumental variable assumptions. (A) Horizontal pleiotropy (1); (B) horizontal pleiotropy (2); (C) confounding by linkage disequilibrium; and (D) confounding by population stratification. The arrow indicates a causal effect. The bidirectional arrow indicates an association between the variables. G=genetic variant(s); X=exposure; U=unmeasured confounder(s); Y=outcome; C=(potentially unmeasured) phenotype; and Z=(residual) population stratification.

List of MR estimation methods

Category Relaxation of IV assumptions Method Function (package)
Standard MR None IVW mr_ivw (MendelianRandomization)
mr_ivw (TwoSampleMR)
Consensus Balanced pleiotropy Weighted median mr_median (MendelianRandomization)
mr_median (TwoSampleMR)
Directional pleiotropy Weighted mode mr_mbe (MendelianRandomization)
mr_mode (TwoSampleMR)
Outlier removal Directional pleiotropy MR-PRESSO mr_presso (MRPRESSO)
Directional pleiotropy MR-Lasso mr_lasso (MendelianRandomization)
Outlier adjustment Directional pleiotropy MR-Robust mr_ivw using robust=TRUE option (MendelianRandomization)
Directional pleiotropy MR-Mixture MRMix (MRMix)
Directional pleiotropy Contamination mixture mr_conmix (MendelianRandomization)
Pleiotropy adjustment Directional pleiotropy MR-Egger mr_egger (MendelianRandomization)
mr_egger_regression (TwoSampleMR)
Directional pleiotropy MVMR IVW mr_egger (MendelianRandomization)
mr_egger_regression (TwoSampleMR)
Directional pleiotropy GSMR gsmr (gsmr2)
Correlated and uncorrelated pleiotropy MR-CAUSE cause (cause)
Weak IVs; Balanced pleiotropy MR-RAPS mr_raps (TwoSampleMR)
Weak IVs; Directional pleiotropy GRAPPLE grappleRobustEst (GRAPPLE)
Weak IVs; Balanced pleiotropy Debiased IVW mr_divw (MendelianRandomization)

These methods are most frequently used in MR studies. Directional pleiotropy is the pleiotropy when the average pleiotropic effect is not zero. Balanced pleiotropy is the pleiotropy when the average pleiotropic effect is not zero and the Instrument Strength Independent of Direct Effect (InSIDE) assumption is satisfied. Correlated pleiotropy is the pleiotropy when the genetic variants are associated with a confounder of the exposure and outcome.

MR, Mendelian randomization; IV, instrumental variable; IVW, inverse variance weighting estimation method; MR-PRESSO, MR Pleiotropy RESidual Sum and Outlier; MVMR, multivariable Mendelian randomization; GSMR, generalized summary Mendelian randomization; MR-CAUSE, MR Causal Analysis Using Summary Effect estimates; MR-RAPS, MR-Robust Adjusted Profile Score; GRAPPLE, Genome-wide mR Analysis under Pervasive PLEiotropy.

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