P = 0.007; unadjusted for multiple testing). Mediation analysis suggested that CE (38%) and RTV (44%) substantially mediated the association between inattention and the T‐allele of SNP rs3785157 in the norepinephrine transporter gene (SLC6A2) and the T‐allele of SNP rs7984966 in HTR2A, respectively. The SNPs tag risk‐haplotypes but are not thought to be functionally significant. While these exploratory findings are preliminary, requiring replication, this study demonstrates the value of this approach that can be adapted to the investigation of multiple genetic markers and polygenic risk scores. © 2016 The Authors. American Journal of Medical Genetics Part B: Neuropsychiatric Genetics Published by Wiley Periodicals, Inc.Family and twin studies have identified endophenotypes that capture familial and genetic risk in attention‐deficit/hyperactivity disorder (ADHD), but it remains unclear if they lie on the causal pathway. Here, we illustrate a stepwise approach to identifying intermediate phenotypes. First, we use previous quantitative genetic findings to delineate the expected pattern of genetically correlated phenotypes. Second, we identify overlapping genetic associations with ADHD‐related quantitative traits. Finally, we test for the mediating role of associated endophenotypes. We applied this approach to a sample of 1,312 twins aged 7–10. Based on previous twin model‐fitting analyses, we selected hyperactivity–impulsivity, inattention, reading difficulties (RD), reaction time variability (RTV) and commission errors (CE), and tested for association with selected ADHD risk alleles. For nominally significant associations with both a symptom and a cognitive variable, matching the expected pattern based on previous genetic correlations, we performed mediation analysis to distinguish pleiotropic from mediating effects. The strongest association was observed for the rs7984966 SNP in the serotonin receptor gene (HTR2A), and RTV ( Attention‐deficit/hyperactivity disorder (ADHD) is characterized by developmentally inappropriate levels of inattention and hyperactivity–impulsivity. In addition to the core behavioral symptoms, ADHD is associated with multiple cognitive impairments [Kuntsi et al., Using a population‐based twin sample, we recently replicated the etiological separation of reaction time variability (RTV) and commission errors (CE) [Kuntsi et al., The comparability of these findings across both diagnostic and dimensional approaches to the ADHD phenotype is consistent with evidence from family and twin studies that the clinical diagnosis of ADHD reflects the extreme and impairing tail end of one or more continuous dimensions of psychopathology [Levy et al., The cognitive phenotypes identified in these studies (such as RTV and CE) are often assumed to be intermediate phenotypes that lie on the causal pathway from genes to ADHD symptoms, since they are identified by the same genes that increase risk for ADHD. However, genetically correlated phenotypes may also reflect pleiotropy, whereby overlapping sets of genes give rise to multiple phenotypes that index genetic risk, but do not mediate between genes and the clinical phenotype [Kendler and Neale, As yet, only a few studies have investigated the mediating role of phenotypes in pathways from genetic variants to ADHD symptoms. One study found the association between the high‐activity catechol‐O‐methyltransferase gene (COMT) valine/valine genotype and antisocial behavior in ADHD was partially mediated by impaired social cognition, whereas impairments in executive control represented pleiotropic effects  [Langley et al., Overall, our knowledge of the causal links from genetic risk markers to ADHD remains very limited, in particular for the two ADHD symptom dimensions of inattention and hyperactivity–impulsivity considered separately. The aim of this study was, therefore, to address this gap by using genetic variants reported to be associated with ADHD in previous studies, to look for overlapping associations with both ADHD symptoms and cognitive performance. Then in a second step, we explicitly test for the mediating role of phenotypes in the association with ADHD genetic risk variants.The selection of genetic variants for this study was based on candidate gene association findings with the dopaminergic, noradrenergic and serotoninergic neurotransmitter systems that had previously been reported to be associated with ADHD. Pharmacological, neuroimaging, and animal studies have established a role of these neurotransmitter systems in ADHD. For example, the main pharmacological action of stimulant mediation used to treat ADHD, such as methylphenidate and dexamphetamine, is to increase synaptic dopamine, generating much interest in genes involved in dopaminergic pathways. Adrenergic and serotonergic neurotransmission is also affected by ADHD medications. Moreover, there is evidence for a role of serotoninergic dysfunction in impulsive behavior in both human and animal studies [Koskinen et al., We selected SNPs and variable number tandem repeat (VNTR) polymorphisms that had previously provided evidence of association with ADHD in two or more studies . Sampling methods and data collection procedures are described in detail elsewhere [Kuntsi et al., Parents and teachers were asked to complete the Long Versions of Conners' Parent (CPRS‐R:L) and Teacher (CTRS:R:L) Rating Scales [Conners et al., The vocabulary, similarities, picture completion and block design subtests from the WISC‐III were used to obtain an estimate of the child's IQ (prorated following procedures described by Sattler ).Reading Difficulties Questionnaire (RDQ) is a subscale of the Colorado Learning Difficulties Questionnaire [Willcutt et al., On each trial, one of two possible stimuli appeared for 300 ms in the middle of the computer screen. The child was instructed to respond only to the “go” stimuli and to react as quickly as possible, but to maintain a high level of accuracy. The proportion of “go” stimuli to “no‐go” stimuli was 4:1. The participants performed the task under three conditions , matched for length of time on task. Herein, we present data from the slow condition, which had an inter‐stimulus interval of 8 sec and consisting of 72 trials, and the fast condition, with an inter‐stimulus interval of 1 sec and consisting of 462 trials. The order of presentation of the slow and fast conditions varied randomly across participants. We focus here on two variables obtained from the task: CE and RTV.The baseline condition, with a foreperiod of 8 sec and consisting of 72 trials, followed a standard warned four‐choice RT task. A warning signal  first appeared on the screen. At the end of the foreperiod , the circle designated as the target signal for that trial was filled (colored) in. The participant was asked to make a compatible choice by pressing the response key that directly corresponded in position to the location of the target stimulus. Following a response, the stimuli disappeared from the screen and a fixed inter‐trial interval of 2.5 sec followed. Speed and accuracy were emphasized equally. If the child did not respond within 10 sec, the trial terminated. A comparison condition with a fast event rate (1 sec) and incentives followed the baseline condition [Andreou et al., Tests of allelic association were performed on the exact same final variables used in corresponding quantitative genetic analysis in the same twin sample [Paloyelis et al., Nineteen polymorphisms were selected on the basis of previous reports of association with ADHD [Bobb et al., DNA was extracted from buccal swabs .Genotype error rates were estimated from genotype discordance rates within MZ twin pairs using PEDSTATS, a feature of the Quantitative Transmission Disequilibrium Test (QTDT) program [Abecasis et al., P > 0.05) and results are interpreted from the AT. Conversely, results are interpreted from the AW if significant stratification effects are detected [Mill et al., Tests of allelic association were performed using the QTDT program [Abecasis et al., P < 0.003 to attain study‐wise statistical significance. We did not correct for the number of phenotypes tested as the majority of the quantitative traits are significantly correlated . Furthermore, we set out to establish whether there was any evidence that the pattern of findings from the molecular genetic analysis would match that predicted by the pattern of genetic correlations in twin studies.To correct for multiple testing, a Bonferroni correction for the 16 independent markers analyzed was applied, requiring P‐value for tests of alleles with an allele frequency > 0.05. UNPHASED  [Dudbridge, VNTR markers were tested using the “multi‐allelic” function in QTDT. This provides a single c′ represents the direct effect of X  on Y , after controlling for the effect of M . The coefficient a represents the effect of the SNP on the mediator, and the coefficient b represents the effect of the mediator on the outcome. Therefore, the mediated (or indirect) effect of the SNP on the outcome via the mediator is represented as ab. The total effect of the SNP on the outcome (including the indirect and direct effect) is represented by c, and is estimated as ab + c′.Nominal genetic associations that overlapped across behavioral ratings and cognitive measures, and matched previous expectation from quantitative genetic findings were taken forward to test whether associations reflected pleiotropic or mediating genetic effects. Mediation analyses were conducted using a series of regression analyses performed in STATA. The mediation model is presented in Figure ab) by its standard error, and comparing this value to a standard normal distribution to test for significance. There are multiple formulas to estimate the standard error (SE) of the mediated (ab) effect [see MacKinnon et al., a2 × SEb2) + (b2 × SEa2) (where SEa refers to the standard error of a and SEb refers to the standard error of b) [MacKinnon et al., ab is then divided by its standard error, and tested for significance using MacKinnon's z distribution. For these analyses, we adopted an alpha value of 0.05, for which a z value that does not lie between 1.96 and −1.96 is considered significant.Approaches to test mediation have been subjected to extensive research over the last two decades [Fairchild and MacKinnon, P = 0.007), although this did not withstand correction for multiple testing. The T‐allele of the same SNP in HTR2S was also nominally associated with RD (P = 0.02). The T‐allele of another HTR2A SNP (rs7322347) showed nominal association with inattention (P = 0.01). Increased RTV was also nominally associated with the G‐allele of SNP rs11646411 in CDH13 (P = 0.03). The T‐allele of one of the SNPs (rs3785143) in SLC6A2 showed a nominal AT association with CE (P = 0.03).Association findings are listed in Table P = 0.04) and CE (P = 0.05). In addition, nominal associations emerged between the G‐allele of SNP rs6296 in the HTR1B gene and increased CE (P = 0.02). Although we found an AT nominal association between the TPH2 SNP rs1843809 and inattention, since there was also evidence for significant population stratification, the AT association was ignored in favor of the AW estimate, which was non‐significant.Taking into account the presence of population stratification effects, and therefore using the AW results, nominal AW associations were found for the T‐allele of the other SLC6A2 SNP (rs3785157) and both hyperactivity–impulsivity , and the T‐allele in rs3785157 and hyperactivity–impulsivity (P = 0.04) and CE (P = 0.05). In addition further, though weaker, trends emerged with hyperactivity–impulsivity  and inattention . In addition, HTR2A is of potential interest because of overlapping associations of the T‐allele in rs7984966 SNP with RTV (P = 0.007) and RD (P = 0.02), with a further trend for association with inattention (P = 0.09). Although all the overlapping associations are nominal and non‐significant when adjusted for multiple testing, the observed pattern of findings is consistent with previous findings derived from our previous genetic model‐fitting in the same sample. Therefore, we tested meditation models for these overlapping associations with both SLC6A2 SNPs and the HTR2A SNP rs7984966.None of the associations withstood correction for the number of SNPs examined, requiring P = 0.05) and both hyperactivity–impulsivity (P = 0.04), and to a lesser extent inattention (P = 0.09). Therefore, we tested two candidate pathways using two separate models with hyperactivity–impulsivity/inattention modeled as the outcome in alternative models. As the QTDT associations with rs3785157 were found using the AW test, within‐pair differences for phenotype and genotype  were used in the regression tests for mediation. Although we found overlapping associations with the other SLC6A2 SNP (rs3785143) and both hyperactivity–impulsivity and CE, these could not be tested in a mediation model, as associations in QTDT were mixed . Associations were also observed for the HTR2A SNP rs7984996 with RTV (P = 0.007), RD (P = 0.02) and, to a lesser extent, inattention (0.09), and as these associations were found in the AT test, difference scores were not used for the regression analysis. Instead, individual genotype and phenotype data were used, and the “cluster” command employed to account for the genetic relationship within twin pairs. Only models where significant mediation is observed are presented  with CE  = 0.044, P = 0.073). In the second regression step, we found that the SLC6A2 SNP rs3785157 significantly predicted CE  = 3.53, P = 0.006). In the next step of the regression analysis, a hierarchical regression analysis was performed to test pathway b  and c′ . The indirect effect (ab) is estimated at −0.0097, and the SE of ab (SEab) is estimated at 0.0065 (√(b2 × SEa2) + (a2 × SEb2)). The z statistic was estimated at −1.48 (ab/SEab), P = 0.14. Therefore, the indirect (mediated) effect of CE on the association between SLC6A2 and hyperactivity–impulsivity was not significant.In the first step of the regression analysis, we tested pathway see Fig. , and fouP = 0.33). The second step in this mediation model is a repeat of the second step in the previous model (as the SNP and mediator are the same), and so indicated that the SNP is significantly related to the mediator (CE) (P = 0.006). In the hierarchical regression analysis, CE significantly predicted inattention  and when controlling for the effects of CE on inattention, the effect of rs3785157 on inattention was β = −0.028, SE = 0.047, P = 0.55). The indirect effect (ab) is estimated at −0.017, and the SE of ab is estimated at 0.0086. The z statistic was estimated at −2.03, P = 0.04. Therefore, the indirect (mediated) effect of CE on the association between SLC6A2 and inattention was significant , and RTV . In the hierarchical regression analysis, RTV significantly predicted inattention  and when controlling for the effects of RTV on inattention, the effect of rs7984966 on inattention was β = 0.017, SE = 0.024, P = 0.47). The mediated effect of RTV on the association between the HTR2A SNP rs7984966 and inattention was significant   = 0.05, P = 0.03). In the second regression step, we found that the SNP predicted inattention  = 0.024, P = 0.13). In the next step of the regression analysis, a hierarchical regression analysis was performed to test pathway b and c′. Inattention significantly predicted RD , and when controlling for the effects of inattention on RD, the effect of the HTR2A SNP rs7984966 on RD was β = 0.079, SE = 0.045, P = 0.08. The mediated effect of inattention on the association between the HTR2A SNP rs7984966 and RD was not significant .In the first step of the regression analysis, we tested pathway Informed by our previous genetic model‐fitting results [Paloyelis et al., From the mediation tests we obtained preliminary evidence that RTV mediated the association between the T‐allele of rs7984966 in HTR2A and inattention, and CE mediated the association between the T‐allele of rs3785157 in SLC6A2 and inattention. In contrast, there was no evidence for the mediating role of inattention in the association between rs7984966 in HTR2A and RD, or for CE in the association between rs3785158 in SLC6A2 and hyperactivity–impulsivity, suggesting pleiotropic (non‐mediating) genetic effects.The strongest association was between the T‐allele of SNP rs7984966 in HTR2A and RTV, which was also associated with RD, and at trend level with inattention. A further SNP in HTR2A (rs7322347) was also associated with inattention. These SNPs have been associated both in single marker and haplotype (multiple markers) analyses with ADHD (DSM‐IV combined type) in both children and adults [Ribases et al., Overlapping nominal associations were also found for two SNPs in SLC6A2: both SNPs were associated with CE but not associated with either RTV or RD, and one SNP (rs3785157) was also further associated with both ADHD behavioral dimensions . In the analysis, we identified the T allele of the SLC6A2 SNP, rs3785157, as the risk allele, in line with some studies [Bobb et al., Overall, the pattern of findings we observed  is in line with the results from our quantitative genetic analyses [Paloyelis et al., While these findings are preliminary, based on weak nominal significance with individual genetic variants, the similar pattern of findings between the molecular and quantitative genetic findings is of interest and met our criteria for conducting mediation tests. We therefore performed mediation analyses to formally distinguish if overlapping associations reflect pleiotropic or mediating effects [Kendler and Neale, Although overlapping associations suggested that rs3785157 in SLC6A2 may be a potential genetic candidate contributing to the association of CE with both ADHD behavioral dimensions, our further analyses suggested that the overlapping nominal genetic associations with hyperactivity–impulsivity reflect pleiotropic genetic effects. In contrast, CE mediated 38% of the effect of SLC6A2 on inattention. Another overlapping set of associations was observed for rs7984966 in HTR2A with RTV, RD and inattention, implicating it as a potential candidate marker, in line with our previous quantitative genetic finding that the association of RTV and RD with ADHD largely reflects genetic influences shared with inattention that are distinct from those underlying hyperactivity–impulsivity [Paloyelis et al., P = 0.05) for genetic loci explaining around 0.5% of the phenotypic variance. The genetic associations, therefore, need to be treated with considerable caution, as none of the individual associations reported withstood correction for multiple testing, and did not account for the multiple phenotypes also targeted. Furthermore, some of the allelic specific associations we identified were not in the same direction as predicted by the previous literature. Despite these limitations, we found nominal associations with previously implicated ADHD susceptibility genes and psychometrically robust ADHD‐related phenotypes, selected on the basis of quantitative genetic model‐fitting analyses. Since the pattern of findings was consistent with those from the previous quantitative genetic analyses these are less likely to have arisen by chance.In common with previous studies incorporating quantitative assessments of ADHD within molecular genetic investigations, power to detect genetic associations was limited due to a small sample size. Power for this sample was estimated using the genetic power calculator from Purcell et al.  and founIt is essential that these findings are extended and replicated; candidate gene associations are notoriously fickle and this could potentially be the case with the present findings. The next stage of this approach is already underway with collaboration between international investigators generating large samples, with clinical phenotypic data (ADHD symptom counts), cognitive measures and genome‐wide association data. The findings presented here are nevertheless encouraging and outline a general strategy that can easily be adapted to the investigation of multiple genetic markers and polygenic risk scores. Matching genetic association findings to the predicted patterns of genetically correlated traits, from twin model fitting and other quantitative genetic designs, may help to delineate the role of specific gene or gene pathways, and provides a framework for testing for the causal (mediating) role of cognitive endophenotypes on ADHD.


By means of simulation studies, we show that under a wide range of genetic architectures, the statistical power and predictive accuracy provided by this calculator are accurate. We compare the predictions from the MetaGAP calculator with actual results obtained in the GWAS literature. Specifically, we use genomic-relatedness-matrix restricted maximum likelihood to estimate the SNP heritability and cross-study genetic correlation of height, BMI, years of education, and self-rated health in three large samples. These estimates are used as input parameters for the MetaGAP calculator. Results from the calculator suggest that cross-study heterogeneity has led to attenuation of statistical power and predictive accuracy in recent large-scale GWAS efforts on these traits . Hence, cross-study heterogeneity contributes to the missing heritability.Large-scale genome-wide association results are typically obtained from a fixed-effects meta-analysis of GWAS summary statistics from multiple studies spanning different regions and/or time periods. This approach averages the estimated effects of genetic variants across studies. In case genetic effects are heterogeneous across studies, the statistical power of a GWAS and the predictive accuracy of polygenic scores are attenuated, contributing to the so-called ‘missing heritability’. Here, we describe the online Meta-GWAS Accuracy and Power (MetaGAP) calculator (available at  www.devlaming.eu, which accounts for these effects. By means of this calculator, we show that imperfect genetic correlations between studies substantially decrease statistical power and predictive accuracy and, thereby, contribute to the missing heritability. The MetaGAP calculator helps researchers to gauge how sensitive their results will be to heterogeneity in genetic effects across studies. If strong heterogeneity is expected, random-effects meta-analysis methods should be used instead of fixed-effects methods.Large-scale genome-wide association studies are uncovering the genetic architecture of traits which are affected by many genetic variants. In such efforts, one typically meta-analyzes association results from multiple studies spanning different regions and/or time periods. Results from such efforts do not yet capture a large share of the heritability. The origins of this so-called ‘missing heritability’ have been strongly debated. One factor exacerbating the missing heritability is heterogeneity in the effects of genetic variants across studies. The effect of this type of heterogeneity on statistical power to detect associated genetic variants and the accuracy of polygenic predictions is poorly understood. In the current study, we derive the precise effects of heterogeneity in genetic effects across studies on both the statistical power to detect associated genetic variants as well as the accuracy of polygenic predictions. We present an online calculator, available at  Large-scale GWAS efforts are rapidly elucidating the genetic architecture of polygenic traits, including anthropometrics , 2 and dDifferences across strata , in genetic effects, phenotype measurement, and phenotype accuracy, lead to loss of signal –20. Hencwww.devlaming.eu) that accounts for the cross-study genetic correlation (CGR). This calculator infers the statistical power to detect associated SNPs and the predictive accuracy of the PGS in a meta-analysis of GWAS results from genetically and phenotypically heterogeneous studies, and quantifies the loss in power and predictive accuracy incurred by this cross-study heterogeneity. Using simulations, we show that the MetaGAP calculator is accurate under a wide range of genetic architectures, even when the assumptions of the calculator are violated.Despite empirical evidence of transethnic genetic heterogeneity in diseases  and the METAL )TN = 19,184 for self-rated health. Details per phenotype on sample size, univariate estimates of SNP heritability, and bivariate estimates of genetic correlation, stratified across studies, and cross-study averages, are provided in In The univariate estimates of SNP heritability based on the pooled data assume perfect CGRs. Therefore, such estimates of SNP heritability are downwards biased when based on data from multiple studies with imperfect CGRs. To circumvent this bias, we estimated SNP heritability in each study separately, and focused on the sample-size-weighted cross-study average estimate of SNP heritability.EduYears) we found a CGR around 0.8 when averaged across pairs of studies. Similarly, the genetic correlation for EduYears in females and males lies around 0.8. The CGR of self-rated health is substantially below one across the pairs of studies, whilst the genetic correlation between females and males seems to lie around one. The reason for this difference in the genetic correlation of self-rated health between pairs of studies and between females and males may be due to the difference in the questionnaire across studies, discussed in CurrCigt and CurrDrinkFreq, the estimates of CGR and of genetic correlation between females and males are non-informative. For these two traits the standard errors of the genetic correlations estimates are large, mostly greater than 0.5. In addition, for CurrDrinkFreq there is strong volatility in the CGR estimate across pairs of studies.For both height and BMI, we observed genetic correlations close to one across pairs of studies and between females and males. For years of schooling , we used the MetaGAP calculator to predict the number of hits in a set of discovery samples and the PGS EduYears, and self-rated health, we focused on these four phenotypes. For these traits, we computed sample-size-weighted average CGR estimates across the pairs of studies. R2 reported in the most comprehensive GWAS efforts to date for the traits of interest, together with predictions from the MetaGAP calculator. We tried several values for the number of independent haplotype blocks  and for the number of trait-associated blocks . Overall, 250k blocks of which 20k trait-affecting yielded theoretical predictions in best agreement with the empirical observations; we acknowledge the potential for some overfitting .Since we only had accurate estimates for height, BMI, R2 for the two largest GWAS efforts are much in line with theoretical predictions. For the smaller GWAS of 13,665 individuals [R2. Provided there are some differences in R2 between causal SNPs, the first SNPs that are likely to reach genome-wide significance in relatively small samples, are the ones with a comparatively large R2. This view is supported by the fact that a PGS based on merely 20 SNPs already explains 2.9% of the variation in height. Hence, for relatively small samples our theoretical predictions of power and R2 may be somewhat conservative. In addition, the 10k SNPs with the lowest meta-analysis p-values can explain about 60% of the SNP heritability [For height—the trait with the lowest standard error in the estimates of ividuals , our estR2 is also supported by the predicted and observed number of hits for the reported self-rated-health GWAS [R2 are present. However, there is uncertainty in the number of truly associated loci. More accurate estimates of this number may improve the accuracy of our theoretical predictions.The notion of a GWAS first picking up the SNPs with a relatively high lth GWAS ; given alth GWAS  and 16% R2 were quite in line with empirical results. However, for the number of hits, our predictions for the largest efforts seemed overly optimistic. We therefore suspect that the number of independent SNPs associated with BMI is higher than 20k; a higher number of associated SNPs would reduce the GWAS power, while preserving PGS R2, yielding good agreement with empirical observation. Nevertheless, given the limited number of data points, this strategy of setting the number of causal SNPs would increase the chance of overfitting.For BMI our predictions of PGS EduYears we observed that the reported number of hits is in between the expected number of hits when the CGR is set to the averaged GREML estimate of 0.783 and when the CGR is set to one. Given the standard errors in the CGR estimates for EduYears, the CGR might very well be somewhat greater than 0.783, which would yield a good fit with the reported number of hits. However, as with the number of truly associated SNPs for BMI, in light of the risk of overfitting, we can make no strong claims about a slightly higher CGR of EduYears.For R2 are in moderate agreement with empirical observations, especially when bearing in mind that we are looking at a limited number of data points, making chance perturbations from expectation likely. In addition, regarding the number of hits, the listed studies are not identical in terms of the procedure to obtain the independent hits. Therefore, the numbers could have been slightly different, had the same pruning procedure been used across all reported studies.Overall, our theoretical predictions of the number of hits and PGS R2 when assuming either a CGR equal to one, or a CGR in accordance with empirical estimates, with traits with lower CGR suffering from stronger attenuation in power and predictive accuracy. In line with theory, R2 falls approximately quadratically with CGR. For instance, for self-rated health, the estimated CGR of about 0.5, would yield a PGS that retains approximately 0.52 = 25% of the R2 it would have had under a CGR of one. Hence the approximated attenuation is 75%. This approximation is corroborated by the theoretical relative attenuation of 78%.Regarding attenuation, we observed a substantial spread in the predicted number of hits and PGS R2 is 6% for height, 14% for BMI, 36% for EduYears, and 78% for self-rated health, when compared to the R2 of PGSs under perfect CGRs  in a hold-out sample to the number of studies, sample size and SNP heritability per study, and the CGR. The underlying theory shows that there is a quadratic response of the PGS R2 to CGR. Moreover, we showed that the power per associated SNP is also affected by CGR.We have shown that imperfect cross-study genetic correlations (CGRs) are likely to contribute to the gap between the phenotypic variation accounted for by all SNPs jointly and by the leading GWAS efforts to date. We arrived at this conclusion in five steps. First, we developed a Meta-GWAS Accuracy and Power (MetaGAP) calculator that accounts for the CGR. This online calculator relates the statistical power to detect associated SNPs and the Second, we used simulations to demonstrate that our theory is robust to several violations of the assumptions about the underlying data-generating process, regarding the relation between allele frequency and effect size, the distribution of allele frequencies, and the factors contributing to CGR. Further research needs to assess whether our theoretical predictions are also accurate under an even broader set of scenarios .Third, we used a sample of unrelated individuals from the Rotterdam Study, the Swedish Twin Registry, and the Health and Retirement Study, to estimate SNP-based heritability as well as the CGR for traits such as height and BMI. Although our CGR estimates have considerable standard errors, the estimates make it likely that for many polygenic traits the CGR is positive, albeit smaller than one.R2 for the most prominent studies to date for these traits. We found that our predictions are in moderate agreement with empirical observations. Our theory seems slightly conservative for smaller GWAS samples. For large-scale GWAS efforts our predictions were in line with the outcomes of these efforts. More accurate estimates of the number of truly associated loci may further improve the accuracy of our theoretical predictions.Fourth, based on these empirical estimates of SNP heritability and CGR for height, BMI, years of education, and self-rated health, we used the MetaGAP calculator to predict the number of expected hits and the expected PGS R2 and the expected number of hits, due to imperfect CGRs. Though these observations are in line with expectation for predictive accuracy, for statistical power the effect was larger than we anticipated. This finding can be explained, however, by the fact that though the absolute decrease in power per SNP is small, the relative decrease is large, since the statistical power per associated SNP is often low to begin with.Fifth, we used our theoretical model to assess statistical power and predictive accuracy for these GWAS efforts, had the CGR been equal to one for the traits under consideration. Our estimates of power and predictive accuracy in this scenario indicated a strong decrease in the PGS R2 will continue to fall short of the full SNP-based heritability. Hence, this study contributes to the understanding of the hiding heritability reported in the GWAS literature.Overall, our study affirms that although PGS accuracy improves substantially with further increasing sample sizes, in the end PGS Regarding the etiology of imperfect CGRs, the likely reasons are heterogeneous phenotype measures across studies, gene–environment interactions with underlying environmental factors differing across studies, and gene–gene interactions where the average effects differ across studies due to differences in allele frequencies. Our study is not able to disentangle these different causes; by estimating the CGR for different traits we merely quantify the joint effect these three candidates have on the respective traits.However, in certain situations it may be possible to disentangle the etiology of imperfect CGRs to some extent. For instance, in case one considers a specific phenotype that is usually studied by means of a commonly available but relatively heterogeneous and/or noisy measure, while there also exists a less readily available but more accurate and homogeneous measure. If one has access to both these measures in several studies, one can compare the CGR estimates for the more accurate measure and the CGR estimates for the less accurate but more commonly available measure. Such a comparison would help to disentangle the contribution of phenotypic heterogeneity and genetic heterogeneity to the CGR of the more commonly available measure.In considering how to properly address imperfect CGRs, it is important to note that having a small set of large studies, rather than a large set of small studies, does not necessarily abate the problem of imperfect genetic correlations. Despite the fact that having fewer studies can help to reduce the effects of heterogeneous phenotype measures, larger studies are more likely to sample individuals from different environments. If gene–environment interactions do play a role, strong differences in environment between subsets of individuals in a study can lead to imperfect genetic correlations within that study. The attenuation in power and accuracy resulting from such within-study heterogeneity may be harder to address than cross-study heterogeneity.Our findings stress the importance of considering the use of more sophisticated meta-analysis methods that account for cross-study heterogeneity –31. We bS1 Derivations(PDF)Click here for additional data file.S2 Derivations(PDF)Click here for additional data file.S1 Note(PDF)Click here for additional data file.S1 Simulations(PDF)Click here for additional data file.S1 Data(PDF)Click here for additional data file.S1 Estimation(PDF)Click here for additional data file.S1 Table(PDF)Click here for additional data file.S2 Table(PDF)Click here for additional data file.S3 Table(PDF)Click here for additional data file.S4 Table(PDF)Click here for additional data file.


Currently, the analyses of most genome-wide association studies (GWAS) have been performed on a single phenotype. There is increasing evidence showing that pleiotropy is a widespread phenomenon in complex diseases. Therefore, using only one single phenotype may lose statistical power to identify the underlying genetic mechanism. There is an increasing need to develop and apply powerful statistical tests to detect association between multiple phenotypes and a genetic variant. In this paper, we develop an Adaptive Fisher’s Combination (AFC) method for joint analysis of multiple phenotypes in association studies. The AFC method combines p-values obtained in standard univariate GWAS by using the optimal number of p-values which is determined by the data. We perform extensive simulations to evaluate the performance of the AFC method and compare the power of our method with the powers of TATES, Tippett’s method, Fisher’s combination test, MANOVA, MultiPhen, and SUMSCORE. Our simulation studies show that the proposed method has correct type I error rates and is either the most powerful test or comparable with the most powerful test. Finally, we illustrate our proposed methodology by analyzing whole-genome genotyping data from a lung function study. To date, genome-wide association studies (GWAS) have become a tool of choice for the identification of genetic variants associated with complex human diseases. Currently, the analyses of most GWAS have been performed on a single phenotype. There is increasing evidence showing that pleiotropy, the effect of one variant on multiple phenotypes, is a widespread phenomenon in complex diseases1356Recently, several statistical methods for detecting association using multivariate phenotypes have been developed89101112915161711221124Motivated by TATES, in this article, we propose an Adaptive Fisher’s Combination (AFC) method for joint analysis of multiple phenotypes in genetic association studies. We first test the association between each of the phenotypes and a genetic variant of interest using standard GWAS software. Then, AFC uses the optimal number of p-values which is determined by the data to test the association. Using extensive simulation studies, we evaluate the performance of the proposed method and compare the power of the proposed method with the powers of TATES, Tippett’s methodn unrelated individuals. Each individual has K phenotypes. Denote Yk = T as the kth phenotype of n individuals. Denote X = T as the genotypic score of n individuals at a genetic variant of interest, where xi ∈ {0, 1, 2} is the number of minor alleles that the ith individual carries at the genetic variant. We propose a new method to test the null hypothesis H0: none of the K phenotypes are associated with the genetic variant.Consider a sample of Yk  and the genotypic score X using a standard GWAS software 303132333435p1, p2, …, pK denote the p-values obtained by the standard univariate GWAS. Based on these p-values, we propose an Adaptive Fisher’s Combination (AFC) method to test the association between multiple phenotypes and the genetic variant. Let pk), …, pK) denote the value of Tk based on the bth. permuted data, where b = 0 represents the original data.1. In each permutation, we randomly shuffle the genotypes and recalculate 2. We transfer Tall is given by3. Let p1, p2, …, pK and thus of Tall do not depend on the genetic variant being tested. Thus, the permutation procedure described above to generate an empirical null distribution of Tall needs to be done only once for a GWAS.As shown in Appendix, the null distributions of http://www.math.mtu.edu/~shuzhang/software.html.The R code of AFC is available at Shuanglin Zhang’s homepage We compare the performance of our method with those of TATESK phenotype-specific p-values obtained in standard univariate GWAS to acquire one overall p-value, me denotes the effective number of independent p-values of all K phenotypes, and me(k) denotes the effective number of p-values among the top k p-values.Combine the Y = XβT + ε, where Y denotes the n × K matrix of phenotypes, βT =  is a vector of coefficients corresponding to the K phenotypes, and ε. is the n × K matrix of random errors with each row of ε to be i.i.d. MVN , where Σ is the covariance matrix of ε. To test H0 : β = 0, the likelihood ratio test is equivalent to the Wilk’s Lambda test statistic of MANOVAH0 to the likelihood function under H1, l is the log-likelihood function, β, and |·| denotes the determinant of a matrix. Then the test statistic has an asymptotic Consider a multivariate multiple linear regression model: X = T as a response variable, and the correlated phenotypes Yk = T as predictors.By performing ordinaregression, MultiPhen develops a reversed analysis for joint analysis of multiple phenotypes by considering a genetic variant of interest kth phenotype and the genetic variant. The test statistic of SUMSCORE is given by TSUMSCORE is estimated using a permutation procedure.Let TTippett is estimated using a permutation procedure.The test statistic of Tippett is given by et al.The Fisher’s combination test statistic is defined as AFC, FC, and Tippett are closely related. Intuitively, when only one phenotype or very few phenotypes are associated with the variant, Tippett is more powerful than FC because in this case FC contains a lot of noises. When all phenotypes or a large proportion of the phenotypes are associated with the variant, FC is more powerful than Tippett because in this case Tippett only uses the minimum p-value and loses information. AFC can be adaptive to the number of phenotypes associated with the variant.et al.Yk = T as the kth phenotype of n individuals and X = T as the genotypic score of the n individuals at the genetic variant.We generate genotype data at a genetic variant according to a minor allele frequency (MAF) under Hardy-Weinberg equilibrium. Phenotypes are generated similarly to that of van der Sluis f = βX, where f is the n by 1 common factor and β is the effect size. Then we simulate K phenotypes byconsidering one factor model with genetic variant effect on the factor. We first generate a common factor, a is a factor loading, εk = T ~ MVN, and In is the identity matrix.where K)/4 (K is a multiple of 4) phenotypes. We generate 4 correlated factors using T ~ MVN, where Σ = (1−ρfa)I + ρfa A, A is a matrix with elements of 1, I is the identity matrix, and ρfa is the correlation between any two factors. Then, we transform the fourth factor f4 to K phenotypes usingconsidering 4 factor model with the genetic variant effect on the fourth factor, each factor has T ~ MVN for j = 1, …, 4, and β is the effect size.where K)/(2) (K is a multiple of 2) phenotypes. We generate two correlated factors using T ~ MVN, where Σ = (1−ρfa)I + ρfaA, A is a matrix with elements of 1, I is the identity matrix, and ρfa is the correlation between any two factors. Then, we transform the second factor f2 to K phenotypes usingconsidering two factor model with the genetic variant effect on the second factor, each factor has T ~ MVN for j = 1, 2, and β is the effect size.where Kth phenotype, each factor has (K)/4 (K is a multiple of 4) phenotypes. By using the original factors f1, f2, f3, f4 in scenario 2, we simulate K phenotypes usingconsidering 4 factor model with genetic variant effect specific to the a is a factor loading, εj = T ~ MVN for j = 1, …, 4, and β is the effect size.where Kth phenotype. We simulate K phenotypes byconsidering one factor model with the genetic variant effect specific to the f~MVN, a is a factor loading, εk = T ~ MVN, and β is the effect size.where K phenotypes are correlated and the correlation structure is not due to one or multiple underlying common factors. We generate K phenotypes independent of genotypes for each individual by using Σ = (1 − ρphe)I + ρpheA, A is a matrix with elements of 1, I is the identity matrix, and ρphe is the correlation between any two phenotypes. After generatingεk = T ~ MVN.considering a network model, where the a2 and between-factor correlaiton is a2ρfa. To evaluate type I error of the proposed method, we generate phenotypic values independent of genotypes by assigning β = 0. To evaluate power, we generate phenotypic values according to the six scenarios described above.In scenarios 2–5, the within-factor correlation is We use two sets of simulations to evaluate the type I error rates of the proposed method. The first set of simulations is normal simulation studies and includes 10,000 replicated samples for each sample size under each scenario. The p-values are estimated using 10,000 permutations. For 10,000 replicated samples, the 95% confidence intervals (CIs) for type I error rates at the nominal levels 0.01 and 0.001 are  and , respectively. The estimated type I error rates of the proposed test (AFC) are summarized in 6 variant sites under each scenario. The phenotypes are independent of genotypes. The MAF at each variant is a random number between 0.05 and 0.5. The null distributions of T1, …, TK and Tall are generated by 106 permutations using the genotypes at the first variant. We consider genotypes at 106 variants as 106 replicated samples. For 106 replicated samples, the 95% confidence intervals (CIs) for the type I error rates at the nominal levels 10−3, 10−4, and 10−5 are ,  and , respectively. The estimated type I error rates of the proposed test (AFC) are summarized in The second set of simulations mimics GWAS. To be comparable with the real data analysis, we generate 6,000 unrelated individuals with 8 phenotypes at 10ρfa) under scenarios 2–4 and power as a function of phenotypic correlation (ρphe) under scenario 6 because scenarios 1 and 5 are one factor model and thus have no ρfa and ρphe involved. For 7 permutations. The powers of all tests are evaluated using 1,000 replicated samples at 10−6 significance level.For power comparisons, we consider (1) power as a function of the effect size under all six scenarios, and 2) power as a function of factorial correlation ( power asρfa) under scenarios 2–4 and as a function of the phenotypic correlation (ρphe) under scenario 6 are given by et al.−6 with 107 permutations and 1,000 replicates for the power as a function of effect size (β) under scenario 2  is a multi-center genetic and epidemiologic investigation to study COPD. This study is sufficiently large and appropriately designed for genome-wide association analysis of COPD. In this study, a total of more than 10,000 subjects have been recruited including non-Hispanic Whites (NHW) and African-Americans (AA). The participants in this study have completed a detailed protocol, including questionnaires, pre- and post-bronchodilator spirometry, high-resolution CT scanning of the chest, exercise capacity , and blood samples for genotyping. The participants were genotyped using the Illumina OmniExpress platform. The genotype data have gone through standard quality-control procedures for genome-wide association analysis detailed at http://www.copdgene.org/sites/default/files/GWAS_QC_Methodology_20121115.pdf. Variants with MAF <1% were excluded in the data set.Chronic obstructive pulmonary disease (COPD) is one of the most common lung diseases characterized by long term poor airflow and is a major public health problemet al.To evaluate the performance of our proposed method on a real data set, we applied all of the 7 methods to the COPDGene of NHW population to carry out GWAS of COPD-related phenotypes. Based on the literature studies of COPD42−8. The results were summarized in 44454647484950515253545556To identify SNPs associated with the phenotypes, we adopted the commonly used genome-wide significance level 5 × 10GWAS have identified many variants with each variant affecting multiple phenotypes, which suggests that pleiotropic effects on human complex phenotypes may be widespread. Therefore, statistical methods that can jointly analyze multiple phenotypes in GWAS may have advantages over analyzing each phenotype individually. In this article, we developed a new method AFC to jointly analyze multivariate phenotypes in genetic association studies. We used extensive simulation studies as well as real data application to compare the performance of AFC with TATES, Tippett, FC, MANOVA, MultiPhen, and SUMSCORE. Our simulation results showed that AFC has correct type I error rates. With respect to power, AFC is either the most powerful test or has similar power with the most powerful test under a variety of simulation scenarios. Additionally, the real data analysis results demonstrated that the proposed method has great potential in GWAS on complex diseases with multiple phenotypes such as COPD.AFC has several important advantages. First, it allows researchers to test genetic associations using standard GWAS software. Second, phenotypes of different types  can be easily analyzed simultaneously. Third, since AFC is based on p-values obtained from standard univariate GWAS, it can not only test the association between multiple phenotypes and one genetic variant of interest, but also can test the association between multiple phenotypes and multiple genetic variants. For common variants, multiple-variant AFC can be applied based on p-values obtained in standard univariate GWAS for each variant and each phenotype. For rare variants, we can first combine genotypes of rare variants by giving different weights, hoping that we give big weights to the variants having strong associations with the phenotypes. Then, we can apply AFC to test the association between the combined genotypes and multiple phenotypes. In conclusion, we showed that our proposed method provides a useful framework for joint analysis of multiple phenotypes in association studies.L independent studies containing the variant of interest and each study has K phenotypes. Let Tl1, …, TKl denote the summary statistics from the lth study. Suppose that Tl = T~N under the null hypothesis, where Σl can be estimated from the values of Tl for all independent SNPs in the GWAS from the lth studydiag . From T, we can calculate the corresponding p-values Pl = T. The AFC test statistic is based on the p-values P. In the permutation procedure, we can generate T according to the distribution N and then we can calculate the p-values P in each permutation.It is well known that the effect sizes of identified variants are often small and that a large sample size is necessary to ensure sufficient power to detect such variants. A common strategy to increase sample size is to perform a meta-analysis by combining summary statistics from a series of studies. The proposed AFC method can be applied to meta-analysis. Suppose that there are How to cite this article: Liang, X. et al. An Adaptive Fisher’s Combination Method for Joint Analysis of Multiple Phenotypes in Association Studies. Sci. Rep.6, 34323; doi: 10.1038/srep34323 (2016).


Mendelian randomization (MR) is being increasingly used to strengthen causal inference in observational studies. Availability of summary data of genetic associations for a variety of phenotypes from large genome-wide association studies (GWAS) allows straightforward application of MR using summary data methods, typically in a two-sample design. In addition to the conventional inverse variance weighting (IVW) method, recently developed summary data MR methods, such as the MR-Egger and weighted median approaches, allow a relaxation of the instrumental variable assumptions.Here, a new method - the mode-based estimate (MBE) - is proposed to obtain a single causal effect estimate from multiple genetic instruments. The MBE is consistent when the largest number of similar  individual-instrument causal effect estimates comes from valid instruments, even if the majority of instruments are invalid. We evaluate the performance of the method in simulations designed to mimic the two-sample summary data setting, and demonstrate its use by investigating the causal effect of plasma lipid fractions and urate levels on coronary heart disease risk.The MBE presented less bias and lower type-I error rates than other methods under the null in many situations. Its power to detect a causal effect was smaller compared with the IVW and weighted median methods, but was larger than that of MR-Egger regression, with sample size requirements typically smaller than those available from GWAS consortia.The MBE relaxes the instrumental variable assumptions, and should be used in combination with other approaches in sensitivity analyses. Summary data Mendelian randomization, typically in a two-sample setting, is being increasingly used due to the availability of summary association results from large genome-wide association studies.Mendelian randomization analyses using multiple genetic instruments are prone to bias due to horizontal pleiotropy, especially when genetic instruments are selected based solely on statistical criteria.A causal effect estimate robust to horizontal pleiotropy can be obtained using the mode-based estimate (MBE).The MBE requires that the most common causal effect estimate is a consistent estimate of the true causal effect, even if the majority of instruments are invalid .Plotting the smoothed empirical density function is useful to explore the distribution of causal effect estimates, and to understand how the MBE is determined. ,Using germline genetic variants as instrumental variables of modifiable exposure phenotypes can strengthen causal inference in observational studies by applying the principles of Mendelian randomization (MR).,Recent MR methods allow performing MR with multiple genetic instruments, typically single nucleotide polymorphisms (SNPs), using summary data estimates from genome-wide association studies (GWAS).,,However, using many instruments in an MR analysis increases the probability of including at least one invalid instrument, which could easily bias the estimate. For example, the inverse variance weighting (IVW) method requires that either all variants are valid instruments or that there is balanced horizontal pleiotropy  and that such pleiotropic effects are independent of instrument strength across all variants (i.e. the Instrument Strength Independent of Direct Effects – InSIDE – assumption).In this paper, we describe a new summary data MR method – the mode-based estimate (MBE). We clarify when this will be a consistent estimate of the causal effect, compare it with established summary data MR methods using simulations and illustrate its application using real data examples.In order to motivate the summary data methods discussed in this paper, we assume the following data-generating model linking genetic variant ,Let The ratio estimate IJE online), we briefly review three such summary data methods – IVW,Under models (1) and (2), variant In this paper we propose a new causal effect estimator – the MBE – that offers robustness to horizontal pleiotropy in a different manner to that of the IVW, MR-Egger or weighted median methods. Its ability to consistently estimate the true causal effect relies on the following fundamental assumption termed the ZEro Modal Pleiotropy Assumption (ZEMPA): across all instruments, the most frequent value  of In order to formalize this, let The breakdown level  of the simple (i.e. unweighted) MBE ranges from 100For example, in The breakdown level of the weighted MBE can be similarly defined as ranging from 50% (exclusive) to 100% (exclusive). In other words, the weighted MBE is biased if To calculate the MBE, we propose using the mode of the smoothed empirical density function of all IJE online).Consider the normal kernel density function of the The simulations were performed using the following model to generate individual 2 = 1% of IJE online).Constraining the variances in this way enables easy interpretation of the parameters in models (8)–(10). For example, Although the consistency property of an estimator provides a formal justification of the approach, it is equally important to understand how well it works in practice for realistically sized datasets in comparison with other methods. Therefore, we evaluated our proposed estimator in four different simulation scenarios. In all simulations, the number of variants Simulations 1 and 2 were aimed at evaluating the performance of the MBE under the causal null  or fully (100%) overlapping samples . It was implemented as for simulation 3, except P-values). Genetic variants were classified as instruments for each lipid fraction using a statistical criterion (P < 1 × 10−8), resulting in 73 instruments for LDL-C, 85 for HDL-C and 31 for triglycerides.Do and colleaguesP < 5 × 10−7) were used as genetic instruments, and the required summary statistics were obtained from the GWAS catalogue [https://www.ebi.ac.uk/gwas/].White and colleaguesIn all simulation scenarios, causal effect estimates were obtained using established MR methods .MR methods were also applied to estimate the causal effect of plasma lipid fractions and urate levels on CHD risk. The magnitude of regression dilution bias in IVW and MR-Egger regression was assessed by the IJE online).The results of simulation 1 – where directional horizontal pleiotropy (if any) occurs only under the InSIDE assumption – are shown in IJE online). Moreover, in this scenario, the simple MBE was generally less biased than the weighted counterparts, and setting When InSIDE is violated , again tIJE online).IJE online) displays the performance of the methods under the causal null when the samples used to estimate instrument-exposure and instrument-outcome associations overlap. MR-Egger regression presented the largest bias, followed by the weighted MBE assuming NOME, the weighted MBE not assuming NOME, weighted median, simple MBE and IVW. Setting IJE online). Importantly, the precision of the MBE was very low, suggesting that the method may be prohibitively underpowered in small samples, thus being best suited for the two-sample setting using precise summary association results. Gains in precision by making the NOME assumption were more noticeable than in the other simulations with larger sample sizes.We used real datasets of summary association results to further explore the influence of the stimates . In the P-values of the Cochran’s Q test ranged from 0.0003 (urate) to 1.7 × 10−21 (HDL-C), thus providing strong statistical evidence for heterogeneity between the ratio estimates. Nevertheless, results for LDL-C and triglycerides consistently suggested risk-increasing causal effects. In the case of HDL-C, the IVW method suggested a protective effect, with one standard deviation increase in HDL-C being associated with a 0.254  decrease in CHD ln(odds). However, the other methods did not confirm this result, suggesting that it was due to negative horizontal pleiotropy . Finally, the IVW method suggested a 0.163  increase in CHD ln(odds) per standard deviation increase in urate levels. Other methods did not confirm this finding, suggesting that it could be a result of positive horizontal pleiotropy .Results of the MR analysis are shown in We have proposed a new MR method – the MBE – for causal effect estimation using summary data of multiple genetic instruments. Its performance was evaluated in a simulation study and its application illustrated in real data examples. An overview of the summary data MR methods that we evaluated (as well as the simple median) is provided in Consistent causal effect estimation using the MBE requires that ZEMPA holds. ZEMPA is an assumption that relates to the underlying bias parameters  when deriving weights for the MBE.Although the simple MBE was less precise than the weighted MBE, it was less prone to bias due to violations of the InSIDE assumption. However, it was more prone to bias when InSIDE held. Indeed, a similar pattern has been previously shown for the simple and weighted median.IJE online), but over-coverage in the latter was even greater when there were no or few invalid instruments. Moreover, after a certain proportion of invalid instruments (around 50%), coverage of the percentile method reduced markedly, whereas this occurred gradually in the normal approximation method. We therefore proposed the latter method to compute confidence intervals, but there might be better alternatives.In our simulations, the 95% confidence intervals of the MBE computed using the normal approximation presented over-coverage (i.e. coverage larger than 95%). This may be due to the MBE being less influenced by outlying instruments (which is indeed the basis of the method), which correspond to the most imprecise ones when all instruments are valid. Therefore, the causal effect estimate fluctuates less around the true causal effect Another aspect of the MBE method (and of the weighted median) that requires further research is regression dilution bias in the two-sample setting. Understanding how regression dilution bias operates in IVW and MR-Egger contributed to developing correction methods,et al.et al.’s approach also relies on the assumption that the most common causal effect estimate is a consistent estimate of the true causal effect (i.e. ZEMPA). In fact, both our approach and Guo et al.’s can be viewed as methods that fully exploit the power of the consistency criterion defined originally by Kang et al.et al.’s method and the MBE (which was developed independently from their work) are very different in their implementation. Ours is designed to be simple to understand and implement, does not require selecting instruments, and is easy to extend to any weighting scheme one desires. Moreover, plotting the empirical density function using different bandwidths may be a useful tool to visually explore the distribution of the et al.’s method after translating it to the two-sample context, and suggest some simple modifications to improve its performance.Although this is the first description of using the MBE as a causal effect estimate in MR, other closely related methods have already been published. For example, Guo It is also important to consider that there are other strategies to compute the mode of continuous data. In preliminary simulations, the modified Silverman’s rule was both generally more robust against horizontal pleiotropy than the original Silverman’s rule,We propose the MBE as an additional MR method that should be used in combination with other approaches in a sensitivity analysis framework. Using several methods that make different assumptions, rather than a single method, is a useful strategy to assess the robustness of the results against violations of the instrumental variable assumptions.IJE online.The Medical Research Council (MRC) and the University of Bristol support the MRC Integrative Epidemiology Unit . J.B. is additionally supported by an MRC Methodology Research Fellowship (grant MR/N501906/1).Conflict of interest: None declared.Supplementary Table S1Click here for additional data file.Supplementary Table S2Click here for additional data file.Supplementary Table S3Click here for additional data file.Supplementary Table S4Click here for additional data file.Supplementary Table S5Click here for additional data file.Supplementary Table S6Click here for additional data file.Supplementary Table S7Click here for additional data file.Supplementary Figure S1Click here for additional data file.Supplementary MethodsClick here for additional data file.