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Structural equation models (SEMs) involving feedback loops may offer advantages over standard instrumental variables estimators in terms of modelling causal effects in the presence of bidirectional relationships. In the following note, we show that in the case of a single "exposure" and "outcome" variable, modelling relationships using a SEM with a simple bidirectional linear feedback loop offers no advantage over traditional instrumental variables estimators in terms of consistency (i.e. both approaches yield consistent estimates of the causal effect, provided that causal estimates are obtained in both directions). In the case of finite samples, traditional IV estimators and SEM exhibited similar power across many of the conditions we examined, although which method performed best depended on the residual correlation between variables and the strength of the instruments. In particular, the power of SEM was insensitive to the residual correlation between variables, whereas the power of the Wald estimator/2SLS improved (deteriorated) relative to SEM as the magnitude of the residual correlation increased (decreased) assuming a positive causal effect of the exposure on the outcome. The power of SEM improved relative to the Wald estimator/2SLS as the instruments explained more residual variance in the "outcome" variable., (© 2024. The Author(s).)

Disaggregation and estimation of genetic effects from offspring and parents has long been of interest to statistical geneticists. Recently, technical and methodological advances have made the genome-wide and loci-specific estimation of direct offspring and parental genetic nurture effects more possible. However, unbiased estimation using these methods requires datasets where both parents and at least one child have been genotyped, which are relatively scarce. Our group has recently developed a method and accompanying software (IMPISH; Hwang et al. in PLoS Genet 16:e1009154, 2020) which is able to impute missing parental genotypes from observed data on sibships and estimate their effects on an offspring phenotype conditional on the effects of genetic transmission. However, this method is unable to disentangle maternal and paternal effects, which may differ in magnitude and direction. Here, we introduce an extension to the original IMPISH routine which takes advantage of all available nuclear families to impute parent-specific missing genotypes and obtain asymptotically unbiased estimates of genetic effects on offspring phenotypes. We apply this this method to data from related individuals in the UK Biobank, showing concordance with previous estimates of maternal genetic effects on offspring birthweight. We also conduct the first GWAS jointly estimating offspring-, maternal-, and paternal-specific genetic effects on body-mass index.

The Classical Twin Method (CTM) compares the similarity of monozygotic (MZ) twins with that of dizygotic (DZ) twins to make inferences about the relative importance of genes and environment in the etiology of individual differences. The design has been applied to thousands of traits across the biomedical, behavioral and social sciences and is arguably the most widely used natural experiment known to science. The fundamental assumption of the CTM is that trait relevant environmental covariation within MZ pairs is the same as that found within DZ pairs, so that zygosity differences in within-pair variance must be due to genetic factors uncontaminated by the environment. This equal environments assumption (EEA) has been, and still is hotly contested, and has been mentioned as a possible contributing factor to the missing heritability conundrum. In this manuscript, we introduce a new model for testing the EEA, which we call the Augmented Classical Twin Design which uses identity by descent (IBD) sharing between DZ twin pairs to estimate separate environmental variance components for MZ and DZ twin pairs, and provides a test of whether these are equal. We show through simulation that given large samples of DZ twin pairs, the model provides unbiased estimates of variance components and valid tests of the EEA under strong assumptions (e.g. no epistatic variance, IBD sharing in DZ twins estimated accurately etc.) which may not hold in reality. Sample sizes in excess of 50,000 DZ twin pairs with genome-wide genetic data are likely to be required in order to detect substantial violations of the EEA with moderate power. Consequently, we recommend that the Augmented Classical Twin Design only be applied to datasets with very large numbers of DZ twin pairs (> 50,000 DZ twin pairs), and given the strong assumptions relating to the absence of epistatic variance, appropriate caution be exercised regarding interpretation of the results.