Hyperopia Is Not Causally Associated With a Major Deficit in Educational Attainment.

Purpose: Hyperopia (farsightedness) has been associated with a deficit in children's educational attainment in some studies. We aimed to investigate the causality of the relationship between refractive error and educational attainment. Methods: Mendelian randomization (MR) analysis in 74,463 UK Biobank participants was used to estimate the causal effect of refractive error on years spent in full-time education, which was taken as a measure of educational attainment. A polygenic score for refractive error derived from 129 genetic variants was used as the instrumental variable. Both linear and nonlinear (allowing for a nonlinear relationship between refractive error and educational attainment) MR analyses were performed. Results: Assuming a linear relationship between refractive error and educational attainment, the causal effect of refractive error on years spent in full-time education was estimated as -0.01 yr/D (95% confidence interval, -0.04 to +0.02; P = 0.52), suggesting minimal evidence for a non-zero causal effect. Nonlinear MR supported the hypothesis of the nonlinearity of the relationship (I2 = 80.3%; Cochran's Q = 28.2; P = 8.8e-05) but did not suggest that hyperopia was associated with a major deficit in years spent in education. Conclusions: This work suggested that the causal relationship between refractive error and educational attainment was nonlinear but found no evidence that moderate hyperopia caused a major deficit in educational attainment. Importantly, however, because statistical power was limited and some participants with moderate hyperopia would have worn spectacles as children, modest adverse effects may have gone undetected. Translational Relevance: These findings suggest that moderate hyperopia does not cause a major deficit in educational attainment.

Strategies for pairwise searches in forensic kinship analysis.

Testing kinship between pairs of individuals is central to a wide range of applications. We focus on cases where many tests are done jointly. Typical examples include cases where DNA profiles are available from a burial site, a plane crash or a database of convicted offenders. The task is to determine the relationships between DNA profiles or individuals. Our approach generalises previous methods and implementations in several respects. We model general, possibly inbred, pairwise relationships which is important for non-human applications and in archaeological studies of ancient inbred populations. Furthermore, we do not restrict attention to autosomal markers. Some cases, such as distinguishing between maternal and paternal half siblings, can be solved using X-chromosomal markers. When many tests are done, the risk of errors increases. We address this problem by building on the theory of multiple testing and show how optimal thresholds for tests can be determined. We point out that the likelihood ratios in a blind search may be dependent so multiple testing methods and interpretation need to account for this. In addition, we show how a Bayesian approach can be helpful. Our examples, using simulated and real data, demonstrate the practical importance of the methods and implementation is based on freely available software.

The use of two-sample methods for Mendelian randomization analyses on single large datasets.

BACKGROUND: With genome-wide association data for many exposures and outcomes now available from large biobanks, one-sample Mendelian randomization (MR) is increasingly used to investigate causal relationships. Many robust MR methods are available to address pleiotropy, but these assume independence between the gene-exposure and gene-outcome association estimates. Unlike in two-sample MR, in one-sample MR the two estimates are obtained from the same individuals, and the assumption of independence does not hold in the presence of confounding. METHODS: With simulations mimicking a typical study in UK Biobank, we assessed the performance, in terms of bias and precision of the MR estimate, of the fixed-effect and (multiplicative) random-effects meta-analysis method, weighted median estimator, weighted mode estimator and MR-Egger regression, used in both one-sample and two-sample data. We considered scenarios differing by the: presence/absence of a true causal effect; amount of confounding; and presence and type of pleiotropy (none, balanced or directional). RESULTS: Even in the presence of substantial correlation due to confounding, all two-sample methods used in one-sample MR performed similarly to when used in two-sample MR, except for MR-Egger which resulted in bias reflecting direction and magnitude of the confounding. Such bias was much reduced in the presence of very high variability in instrument strength across variants (IGX2 of 97%). CONCLUSIONS: Two-sample MR methods can be safely used for one-sample MR performed within large biobanks, expect for MR-Egger. MR-Egger is not recommended for one-sample MR unless the correlation between the gene-exposure and gene-outcome estimates due to confounding can be kept low, or the variability in instrument strength is very high.

Epidemiology, genetic epidemiology and Mendelian randomisation: more need than ever to attend to detail.

In the current era, with increasing availability of results from genetic association studies, finding genetic instruments for inferring causality in observational epidemiology has become apparently simple. Mendelian randomisation (MR) analyses are hence growing in popularity and, in particular, methods that can incorporate multiple instruments are being rapidly developed for these applications. Such analyses have enormous potential, but they all rely on strong, different, and inherently untestable assumptions. These have to be clearly stated and carefully justified for every application in order to avoid conclusions that cannot be replicated. In this article, we review the instrumental variable assumptions and discuss the popular linear additive structural model. We advocate the use of tests for the null hypothesis of 'no causal effect' and calculation of the bounds for a causal effect, whenever possible, as these do not rely on parametric modelling assumptions. We clarify the difference between a randomised trial and an MR study and we comment on the importance of validating instruments, especially when considering them for joint use in an analysis. We urge researchers to stand by their convictions, if satisfied that the relevant assumptions hold, and to interpret their results causally since that is the only reason for performing an MR analysis in the first place.

Adjustment for index event bias in genome-wide association studies of subsequent events.

Following numerous genome-wide association studies of disease susceptibility, there is increasing interest in genetic associations with prognosis, survival or other subsequent events. Such associations are vulnerable to index event bias, by which selection of subjects according to disease status creates biased associations if common causes of incidence and prognosis are not accounted for. We propose an adjustment for index event bias using the residuals from the regression of genetic effects on prognosis on genetic effects on incidence. Our approach eliminates this bias when direct genetic effects on incidence and prognosis are independent, and otherwise reduces bias in realistic situations. In a study of idiopathic pulmonary fibrosis, we reverse a paradoxical association of the strong susceptibility gene MUC5B with increased survival, suggesting instead a significant association with decreased survival. In re-analysis of a study of Crohn's disease prognosis, four regions remain associated at genome-wide significance but with increased standard errors.

A Bayesian approach to Mendelian randomisation with dependent instruments.

Mendelian randomisation (MR) is a method for establishing causality between a risk factor and an outcome by using genetic variants as instrumental variables. In practice, the association between individual genetic variants and the risk factor is often weak, which may lead to a lack of precision in the MR and even biased MR estimates. Usually, the most significant variant within a genetic region is selected to represent the association with the risk factor, but there is no guarantee that this variant will be causal or that it will capture all of the genetic association within the region. It may be advantageous to use extra variants selected from the same region in the MR. The problem is to decide which variants to select. Rather than selecting a specific set of variants, we investigate the use of Bayesian model averaging (BMA) to average the MR over all possible combinations of genetic variants. Our simulations demonstrate that the BMA version of MR outperforms classical estimation with many dependent variants and performs much better than an MR based on variants selected by penalised regression. In further simulations, we investigate robustness to violations in the model assumptions and demonstrate sensitivity to the inclusion of invalid instruments. The method is illustrated by applying it to an MR of the effect of body mass index on blood pressure using SNPs in the FTO gene.

Potential causal associations of serum 25-hydroxyvitamin D with lipids: a Mendelian randomization approach of the HUNT study.

Observational studies have shown consistent associations between higher circulating 25-hydroxyvitamin D [25(OH)D] levels and favorable serum lipids. We sought to investigate if such associations were causal. A Mendelian randomization (MR) study was conducted on a population-based cohort comprising 56,435 adults in Norway. A weighted 25(OH)D allele score was generated based on vitamin D-increasing alleles of rs2282679, rs12785878 and rs10741657. Linear regression analyses of serum lipid levels on the allele score were performed to assess the presence of causal associations of serum 25(OH)D with the lipids. To quantify the causal effects, the inverse-variance weighted method was used for calculating MR estimates based on summarized data of individual single-nucleotide polymorphisms. The MR estimate with 95% confidence interval (CI) represents percentage difference in the lipid level per genetically determined 25 nmol/L increase in 25(OH)D. The 25(OH)D allele score demonstrated a clear association with high-density lipoprotein (HDL) cholesterol (p = 0.007) but no association with total or non-HDL cholesterol or triglycerides (p ≥ 0.27). The MR estimate showed 2.52% (95% CI 0.79-4.25%) increase in HDL cholesterol per genetically determined 25 nmol/L increase in 25(OH)D, which was stronger than the corresponding estimate of 1.83% (95% CI 0.85-2.81%) from the observational analysis. The MR estimates for total cholesterol (0.60%, 95% CI - 0.73 to 1.94%), non-HDL cholesterol (0.04%, 95% CI - 1.79 to 1.88%) and triglycerides (- 2.74%, 95% CI - 6.16 to 0.67%) showed no associations. MR analysis of data from a population-based cohort suggested a causal and positive association between serum 25(OH)D and HDL cholesterol.

Improving the accuracy of two-sample summary-data Mendelian randomization: moving beyond the NOME assumption.

BACKGROUND: Two-sample summary-data Mendelian randomization (MR) incorporating multiple genetic variants within a meta-analysis framework is a popular technique for assessing causality in epidemiology. If all genetic variants satisfy the instrumental variable (IV) and necessary modelling assumptions, then their individual ratio estimates of causal effect should be homogeneous. Observed heterogeneity signals that one or more of these assumptions could have been violated. METHODS: Causal estimation and heterogeneity assessment in MR require an approximation for the variance, or equivalently the inverse-variance weight, of each ratio estimate. We show that the most popular 'first-order' weights can lead to an inflation in the chances of detecting heterogeneity when in fact it is not present. Conversely, ostensibly more accurate 'second-order' weights can dramatically increase the chances of failing to detect heterogeneity when it is truly present. We derive modified weights to mitigate both of these adverse effects. RESULTS: Using Monte Carlo simulations, we show that the modified weights outperform first- and second-order weights in terms of heterogeneity quantification. Modified weights are also shown to remove the phenomenon of regression dilution bias in MR estimates obtained from weak instruments, unlike those obtained using first- and second-order weights. However, with small numbers of weak instruments, this comes at the cost of a reduction in estimate precision and power to detect a causal effect compared with first-order weighting. Moreover, first-order weights always furnish unbiased estimates and preserve the type I error rate under the causal null. We illustrate the utility of the new method using data from a recent two-sample summary-data MR analysis to assess the causal role of systolic blood pressure on coronary heart disease risk. CONCLUSIONS: We propose the use of modified weights within two-sample summary-data MR studies for accurately quantifying heterogeneity and detecting outliers in the presence of weak instruments. Modified weights also have an important role to play in terms of causal estimation (in tandem with first-order weights) but further research is required to understand their strengths and weaknesses in specific settings.

Age at puberty and risk of asthma: A Mendelian randomisation study.

BACKGROUND: Observational studies on pubertal timing and asthma, mainly performed in females, have provided conflicting results about a possible association of early puberty with higher risk of adult asthma, possibly due to residual confounding. To overcome issues of confounding, we used Mendelian randomisation (MR), i.e., genetic variants were used as instrumental variables to estimate causal effects of early puberty on post-pubertal asthma in both females and males. METHODS AND FINDINGS: MR analyses were performed in UK Biobank on 243,316 women using 254 genetic variants for age at menarche, and on 192,067 men using 46 variants for age at voice breaking. Age at menarche, recorded in years, was categorised as early (<12), normal (12-14), or late (>14); age at voice breaking was recorded and analysed as early (younger than average), normal (about average age), or late (older than average). In females, we found evidence for a causal effect of pubertal timing on asthma, with an 8% increase in asthma risk for early menarche (odds ratio [OR] 1.08; 95% CI 1.04 to 1.12; p = 8.7 × 10(-5)) and an 8% decrease for late menarche (OR 0.92; 95% CI 0.89 to 0.97; p = 3.4 × 10(-4)), suggesting a continuous protective effect of increasing age at puberty. In males, we found very similar estimates of causal effects, although with wider confidence intervals (early voice breaking: OR 1.07; 95% CI 1.00 to 1.16; p = 0.06; late voice breaking: OR 0.93; 95% CI 0.87 to 0.99; p = 0.03). We detected only modest pleiotropy, and our findings showed robustness when different methods to account for pleiotropy were applied. BMI may either introduce pleiotropy or lie on the causal pathway; secondary analyses excluding variants associated with BMI yielded similar results to those of the main analyses. Our study relies on self-reported exposures and outcomes, which may have particularly affected the power of the analyses on age at voice breaking. CONCLUSIONS: This large MR study provides evidence for a causal detrimental effect of early puberty on asthma, and does not support previous observational findings of a U-shaped relationship between pubertal timing and asthma. Common biological or psychological mechanisms associated with early puberty might explain the similarity of our results in females and males, but further research is needed to investigate this. Taken together with evidence for other detrimental effects of early puberty on health, our study emphasises the need to further investigate and address the causes of the secular shift towards earlier puberty observed worldwide.

Age at menarche and adult body mass index: a Mendelian randomization study.

BACKGROUND: Pubertal timing has psychological and physical sequelae. While observational studies have demonstrated an association between age at menarche and adult body mass index (BMI), confounding makes it difficult to infer causality. METHODS: The Mendelian randomization (MR) technique is not limited by traditional confounding and was used to investigate the presence of a causal effect of age at menarche on adult BMI. MR uses genetic variants as instruments under the assumption that they act on BMI only through age at menarche (no pleiotropy). Using a two-sample MR approach, heterogeneity between the MR estimates from individual instruments was used as a proxy for pleiotropy, with sensitivity analyses performed if detected. Genetic instruments and estimates of their association with age at menarche were obtained from a genome-wide association meta-analysis on 182,416 women. The genetic effects on adult BMI were estimated using data on 80,465 women from the UK Biobank. The presence of a causal effect of age at menarche on adult BMI was further investigated using data on 70,692 women from the GIANT Consortium. RESULTS: There was evidence of pleiotropy among instruments. Using the UK Biobank data, after removing instruments associated with childhood BMI that were likely exerting pleiotropy, fixed-effect meta-analysis across instruments demonstrated that a 1 year increase in age at menarche reduces adult BMI by 0.38 kg/m2 (95% CI 0.25-0.51 kg/m2). However, evidence of pleiotropy remained. MR-Egger regression did not suggest directional bias, and similar estimates to the fixed-effect meta-analysis were obtained in sensitivity analyses when using a random-effect model, multivariable MR, MR-Egger regression, a weighted median estimator and a weighted mode-based estimator. The direction and significance of the causal effect were replicated using GIANT Consortium data. CONCLUSION: MR provides evidence to support the hypothesis that earlier age at menarche causes higher adult BMI. Complex hormonal and psychological factors may be responsible.

Correcting the Standard Errors of 2-Stage Residual Inclusion Estimators for Mendelian Randomization Studies.

Mendelian randomization studies use genotypes as instrumental variables to test for and estimate the causal effects of modifiable risk factors on outcomes. Two-stage residual inclusion (TSRI) estimators have been used when researchers are willing to make parametric assumptions. However, researchers are currently reporting uncorrected or heteroscedasticity-robust standard errors for these estimates. We compared several different forms of the standard error for linear and logistic TSRI estimates in simulations and in real-data examples. Among others, we consider standard errors modified from the approach of Newey (1987), Terza (2016), and bootstrapping. In our simulations Newey, Terza, bootstrap, and corrected 2-stage least squares (in the linear case) standard errors gave the best results in terms of coverage and type I error. In the real-data examples, the Newey standard errors were 0.5% and 2% larger than the unadjusted standard errors for the linear and logistic TSRI estimators, respectively. We show that TSRI estimators with modified standard errors have correct type I error under the null. Researchers should report TSRI estimates with modified standard errors instead of reporting unadjusted or heteroscedasticity-robust standard errors.

Mendelian randomization incorporating uncertainty about pleiotropy.

Mendelian randomization (MR) requires strong assumptions about the genetic instruments, of which the most difficult to justify relate to pleiotropy. In a two-sample MR, different methods of analysis are available if we are able to assume, M1 : no pleiotropy (fixed effects meta-analysis), M2 : that there may be pleiotropy but that the average pleiotropic effect is zero (random effects meta-analysis), and M3 : that the average pleiotropic effect is nonzero (MR-Egger). In the latter 2 cases, we also require that the size of the pleiotropy is independent of the size of the effect on the exposure. Selecting one of these models without good reason would run the risk of misrepresenting the evidence for causality. The most conservative strategy would be to use M3 in all analyses as this makes the weakest assumptions, but such an analysis gives much less precise estimates and so should be avoided whenever stronger assumptions are credible. We consider the situation of a two-sample design when we are unsure which of these 3 pleiotropy models is appropriate. The analysis is placed within a Bayesian framework and Bayesian model averaging is used. We demonstrate that even large samples of the scale used in genome-wide meta-analysis may be insufficient to distinguish the pleiotropy models based on the data alone. Our simulations show that Bayesian model averaging provides a reasonable trade-off between bias and precision. Bayesian model averaging is recommended whenever there is uncertainty about the nature of the pleiotropy.

A Mendelian Randomization Study of Metabolite Profiles, Fasting Glucose, and Type 2 Diabetes.

Mendelian randomization (MR) provides us the opportunity to investigate the causal paths of metabolites in type 2 diabetes and glucose homeostasis. We developed and tested an MR approach based on genetic risk scoring for plasma metabolite levels, utilizing a pathway-based sensitivity analysis to control for nonspecific effects. We focused on 124 circulating metabolites that correlate with fasting glucose in the Erasmus Rucphen Family (ERF) study (n = 2,564) and tested the possible causal effect of each metabolite with glucose and type 2 diabetes and vice versa. We detected 14 paths with potential causal effects by MR, following pathway-based sensitivity analysis. Our results suggest that elevated plasma triglycerides might be partially responsible for increased glucose levels and type 2 diabetes risk, which is consistent with previous reports. Additionally, elevated HDL components, i.e., small HDL triglycerides, might have a causal role of elevating glucose levels. In contrast, large (L) and extra large (XL) HDL lipid components, i.e., XL-HDL cholesterol, XL-HDL-free cholesterol, XL-HDL phospholipids, L-HDL cholesterol, and L-HDL-free cholesterol, as well as HDL cholesterol seem to be protective against increasing fasting glucose but not against type 2 diabetes. Finally, we demonstrate that genetic predisposition to type 2 diabetes associates with increased levels of alanine and decreased levels of phosphatidylcholine alkyl-acyl C42:5 and phosphatidylcholine alkyl-acyl C44:4. Our MR results provide novel insight into promising causal paths to and from glucose and type 2 diabetes and underline the value of additional information from high-resolution metabolomics over classic biochemistry.

Age at menarche and lung function: a Mendelian randomization study.

A trend towards earlier menarche in women has been associated with childhood factors (e.g. obesity) and hypothesised environmental exposures (e.g. endocrine disruptors present in household products). Observational evidence has shown detrimental effects of early menarche on various health outcomes including adult lung function, but these might represent spurious associations due to confounding. To address this we used Mendelian randomization where genetic variants are used as proxies for age at menarche, since genetic associations are not affected by classical confounding. We estimated the effects of age at menarche on forced vital capacity (FVC), a proxy for restrictive lung impairment, and ratio of forced expiratory volume in one second to FVC (FEV1/FVC), a measure of airway obstruction, in both adulthood and adolescence. We derived SNP-age at menarche association estimates for 122 variants from a published genome-wide meta-analysis (N = 182,416), with SNP-lung function estimates obtained by meta-analysing three studies of adult women (N = 46,944) and two of adolescent girls (N = 3025). We investigated the impact of departures from the assumption of no pleiotropy through sensitivity analyses. In adult women, in line with previous evidence, we found an effect on restrictive lung impairment with a 24.8 mL increase in FVC per year increase in age at menarche (95% CI 1.8-47.9; p = 0.035); evidence was stronger after excluding potential pleiotropic variants (43.6 mL; 17.2-69.9; p = 0.001). In adolescent girls we found an opposite effect (-56.5 mL; -108.3 to -4.7; p = 0.033), suggesting that the detrimental effect in adulthood may be preceded by a short-term post-pubertal benefit. Our secondary analyses showing results in the same direction in men and boys, in whom age at menarche SNPs have also shown association with sexual development, suggest a role for pubertal timing in general rather than menarche specifically. We found no effect on airway obstruction (FEV1/FVC).

Assessing the suitability of summary data for two-sample Mendelian randomization analyses using MR-Egger regression: the role of the I2 statistic.

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

Family tree and ancestry inference: is there a need for a 'generational' consent?

BACKGROUND: Genealogical research and ancestry testing are popular recreational activities but little is known about the impact of the use of these services on clients' biological and social families. Ancestry databases are being enriched with self-reported data and data from deoxyribonucleic acid (DNA) analyses, but also are being linked to other direct-to-consumer genetic testing and research databases. As both family history data and DNA can provide information on more than just the individual, we asked whether companies, as a part of the consent process, were informing clients, and through them clients' relatives, of the potential implications of the use and linkage of their personal data. METHODS: We used content analysis to analyse publically-available consent and informational materials provided to potential clients of ancestry and direct-to-consumer genetic testing companies to determine what consent is required, what risks associated with participation were highlighted, and whether the consent or notification of third parties was suggested or required. RESULTS: We identified four categories of companies providing: 1) services based only on self-reported data, such as personal or family history; 2) services based only on DNA provided by the client; 3) services using both; and 4) services using both that also have a research component. The amount of information provided on the potential issues varied significantly across the categories of companies. 'Traditional' ancestry companies showed the greatest awareness of the implications for family members, while companies only asking for DNA focused solely on the client. While in some cases companies included text recommending clients inform their relatives, showing they recognised the issues, often it was located within lengthy terms and conditions or privacy statements that may not be read by potential clients. CONCLUSIONS: We recommend that companies should make it clearer that clients should inform third parties about their plans to participate, that third parties' data will be provided to companies, and that that data will be linked to other databases, thus raising privacy and issues on use of data. We also suggest investigating whether a 'generational consent' should be created that would include more than just the individual in decisions about participating in genetic investigations.

Detecting pleiotropy in Mendelian randomisation studies with summary data and a continuous outcome.

Mendelian randomisation (MR) estimates causal effects of modifiable phenotypes on an outcome by using genetic variants as instrumental variables, but its validity relies on the assumption of no pleiotropy, that is, genes influence the outcome only through the given phenotype. Excluding pleiotropy is difficult, but the use of multiple instruments can indirectly address the issue: if all genes represent valid instruments, their MR estimates should vary only by chance. The Sargan test detects pleiotropy when individual phenotype, outcome and genotype data are measured in the same subjects. We propose an alternative approach to be used when only summary genetic data are available or data on gene-phenotype and gene-outcome come from different subjects. The presence of pleiotropy is investigated using the between-instrument heterogeneity Q test (together with the I(2) index) in a meta-analysis of MR Wald estimates, derived separately from each instrument. For a continuous outcome, we evaluate the approach through simulations and illustrate it using published data. For the scenario where all data come from the same subjects, we compare it with the Sargan test. The Q test tends to be conservative in small samples. Its power increases with the degree of pleiotropy and the sample size, as does the precision of the I(2) index, in which case results are similar to those of the Sargan test. In MR studies with large sample sizes based on summary data, the between-instrument Q test represents a useful tool to explore the presence of heterogeneity due to pleiotropy or other causes.

Improved maximum likelihood reconstruction of complex multi-generational pedigrees.

The reconstruction of pedigrees from genetic marker data is relevant to a wide range of applications. Likelihood-based approaches aim to find the pedigree structure that gives the highest probability to the observed data. Existing methods either entail an exhaustive search and are hence restricted to small numbers of individuals, or they take a more heuristic approach and deliver a solution that will probably have high likelihood but is not guaranteed to be optimal. By encoding the pedigree learning problem as an integer linear program we can exploit efficient optimisation algorithms to construct pedigrees guaranteed to have maximal likelihood for the standard situation where we have complete marker data at unlinked loci and segregation of genes from parents to offspring is Mendelian. Previous work demonstrated efficient reconstruction of pedigrees of up to about 100 individuals. The modified method that we present here is not so restricted: we demonstrate its applicability with simulated data on a real human pedigree structure of over 1600 individuals. It also compares well with a very competitive approximate approach in terms of solving time and accuracy. In addition to identifying a maximum likelihood pedigree, we can obtain any number of pedigrees in decreasing order of likelihood. This is useful for assessing the uncertainty of a maximum likelihood solution and permits model averaging over high likelihood pedigrees when this would be appropriate. More importantly, when the solution is not unique, as will often be the case for large pedigrees, it enables investigation into the properties of maximum likelihood pedigree estimates which has not been possible up to now. Crucially, we also have a means of assessing the behaviour of other approximate approaches which all aim to find a maximum likelihood solution. Our approach hence allows us to properly address the question of whether a reasonably high likelihood solution that is easy to obtain is practically as useful as a guaranteed maximum likelihood solution. The efficiency of our method on such large problems bodes well for extensions beyond the standard setting where some pedigree members may be latent, genotypes may be measured with error and markers may be linked.

Mixtures with relatives: a pedigree perspective.

DNA mixture evidence pertains to cases where several individuals may have contributed to a biological stain. Statistical methods and software for such problems are available and a large number of cases can be handled adequately. However, one class of mixture problems remains untreated in full generality in the literature, namely when the contributors may be related. Disregarding a plausible close relative of the perpetrator as an alternative contributor (identical twin is the most extreme case) may lead to overestimating the evidence against a suspect. Existing methods only accommodate pairwise relationships such as the case where the suspect and the victim are siblings, for example. In this paper we consider relationships in full generality, conveniently represented by pedigrees. In particular, these pedigrees may involve inbreeding, for instance when the parents of an individual of interest are first cousins. Furthermore our framework handles situations where the opposing parties in a court case (prosecution and defence) propose different family relationships. Consequently, our approach combines classical mixture and kinship problems. The basic idea of this paper is to formulate the problem in a way that allows for the exploitation of currently available methods and software designed originally for linkage applications. We have developed a freely available R package, euroMix based on another package, paramlink, and we illustrate the ideas and methods on real and simulated data.

Maximum likelihood pedigree reconstruction using integer linear programming.

Large population biobanks of unrelated individuals have been highly successful in detecting common genetic variants affecting diseases of public health concern. However, they lack the statistical power to detect more modest gene-gene and gene-environment interaction effects or the effects of rare variants for which related individuals are ideally required. In reality, most large population studies will undoubtedly contain sets of undeclared relatives, or pedigrees. Although a crude measure of relatedness might sometimes suffice, having a good estimate of the true pedigree would be much more informative if this could be obtained efficiently. Relatives are more likely to share longer haplotypes around disease susceptibility loci and are hence biologically more informative for rare variants than unrelated cases and controls. Distant relatives are arguably more useful for detecting variants with small effects because they are less likely to share masking environmental effects. Moreover, the identification of relatives enables appropriate adjustments of statistical analyses that typically assume unrelatedness. We propose to exploit an integer linear programming optimisation approach to pedigree learning, which is adapted to find valid pedigrees by imposing appropriate constraints. Our method is not restricted to small pedigrees and is guaranteed to return a maximum likelihood pedigree. With additional constraints, we can also search for multiple high-probability pedigrees and thus account for the inherent uncertainty in any particular pedigree reconstruction. The true pedigree is found very quickly by comparison with other methods when all individuals are observed. Extensions to more complex problems seem feasible.