A long-standing challenge in epidemiology has been to infer causality from correlational data in observational studies. For many scientists, correlational studies are starting points for exploring the causal associations between variables. However, by themselves, correlations are insufficient to identify causality, due to the potential existence of background confounding and ambiguous direction of causality.
The primary alternative to observational studies is the randomised controlled trial (RCT), in which study participants are randomly allocated to treatment/experimental and control groups. This approach averages the effects of any confounders equally among the groups, so that any difference in the outcome can be attributed to the intervention. However, RCTs are not always feasible. Difficulties may arise due to ethical considerations, e.g., when the aim is to estimate the causal effects of putative risk factors on disease outcomes, as it is fundamental to not harm participants. This is particularly problematic in psychiatry, for example, when working with samples of children, which hinders intervention evaluation.
Mendelian randomization (MR) can be used to investigate causality in cases where RCTs are not feasible or ethical. It is based on Mendel’s laws of inheritance of segregation and independent assortment. Specifically, it uses the randomization that happens in meiosis, when genetic information is shuffled between chromosomes (crossing-over) and these chromosomes then form the gametes (Madole and Harden 2022). Genetic variants associated with phenotypic exposures are identified in large scale genome-wide association studies (GWAS) and meta-analyses. These genetic variants, or weighted combinations thereof, are potentially useful instrumental variables. Instrumental variables are variables correlated with the predictor, but only indirectly affect the outcome, and are commonly used in econometrics (Evans and Davey Smith 2015; Sanderson et al. 2022).
Several key assumptions are involved in causal inference based on MR (Sanderson et al. 2022). First, the strength of the association between the genetic variant(s) and the exposure must be sufficient, which is known as the relevance assumption. Second, MR is based on the assumption that the instrument’s effect on the outcome is completely mediated by the exposure (known as the exclusion restriction assumption).. In genetic studies this assumption is known as no horizontal pleiotropy. This assumption may not hold, given that GWAS have shown that the same variant may influence multiple traits, sometimes affecting both exposure and outcome, i.e., if the exposure acts as a mediator between the variant and the outcome then it will not be a case of horizontal pleiotropy (Verbanck et al. 2018; Jordan et al. 2019). Horizontal pleiotropy, the direct effect of the instrument on both the exposure and the outcome, violates the assumption of exclusion restriction in MR.
Several solutions have been proposed to detect and/or accommodate horizontal pleiotropy in causal inference based on MR (Sanderson et al. 2022). One can use methods that relax this assumption, and only require the instrument strength to be independent of the direct effect of the exposure on the outcome (Bowden et al. 2015), or one can triangulate results from different MR methods to confirm that the strength and direction of the causal signal are consistent over tests (Burgess et al. 2020). Alternative methods that integrate MR in the twin-design were proposed to address the restriction assumption (Hwang et al. 2021), notably MR-DoC (Minica et al. 2018), which combines MR with the Direction of Causation (DoC) twin design (Heath et al. 1993; Duffy and Martin 1994). The MR-DoC model includes a (horizontal) pleiotropic path, accommodating a direct relationship between instrumental variable and the outcome, and thus allowing for a test of directional horizontal pleiotropy (path b2, Fig. 1B). MR-DoC was extended to accommodate bidirectional causation (Castro-de-Araujo et al. 2023; MR-DoC2, Fig. 1) by adding an additional polygenic score that acts as an instrumental variable for the outcome, thus allowing for estimation of the potential effects of reverse causation (paths b1 and b3, Fig. 1C). Henceforth we refer to the Minica et al. (2018) method as MR-DoC, and the Castro-de-Araujo et al. (2023) method as MR-DoC2.
The DoC model uses cross-twin cross-trait correlations to extract information on possible causal paths between two phenotypes. However, it is known to have the following limitations. First, differences in reliability of the variables in the model may bias causal inference estimates (Heath et al. 1993; Gillespie et al. 2003). Specifically, the more reliable variable is more likely to be identified as the cause of the less reliable variable (Heath et al. 1993; Duffy and Martin 1994). Castro-de-Araujo et al. (2023) reported that this is not an issue in MR-DoC2, but it is unknown whether this is an issue in MR-DoC. Second, both the DoC and MR-DOC models require the assumption that the unshared environmental correlation, the parameter re in Fig. 1, is zero in order to estimate the causal path between the exposure and the outcome. This constraint implies the assumption that unshared environmental influences are not a source of confounding. Violation of this assumption biases the causal estimates in DoC models (Rasmussen et al. 2019), but it is not a problem for MR-DoC2, which explicitly models this type of confounding. For MR-DoC it is unknown if re ≠ 0 introduces biases to the estimates of interest, in particular the causal path (g1), or other paths (Fig. 1, A).
The statistical power of MR-DoC and MR-DoC2 models has been explored in Minica et al. (2018) and in Castro-de-Araujo et al. (2023), respectively. However, a comparison of the power profiles of the three models (i.e., DoC, MR-Doc, and MR-Doc2) is lacking. While all three models focus on causal inference, the models differ with respect to their assumptions. First, the DoC model can accommodate both unidirectional and bidirectional causation, provided that some of the parameters are fixed. That is, in addition to the two causal paths, only one of three possible sources (A, C, E) of confounding can be accommodated. This implies that of the three correlations, ra, rc, and re, two have to be constrained (to zero). In general, a bivariate ACE model is identified with any three of the five possible paths coefficients (parameters ra, rc, re, g1 and g2; Fig. 1), which connect the two phenotypes, freely estimated (Maes et al. 2021). Second, the MR-DoC model is usually specified with unidirectional causation (as bidirectional causation requires further constraints to the background ACE confounding), and assumes no unshared environmental confounding (re = 0, Fig. 1). Third, MR-DoC2 is bidirectional and assumes no direct horizontal pleiotropy (b2 path in MR-DoC is fixed to zero in MR-DoC2). However, MR-DoC2 does accommodate two sources of indirect horizontal pleiotropy (rf*b1 and rf*b3 in Fig. 1). Of note, this type of pleiotropy is effectively equivalent to direct horizontal pleiotropy.
In this paper, we compared the power profiles within and between the three models. We estimated the effect of phenotypic measurement error, and the effect of environmental confounding in DoC and MR-DoC (re ≠ 0). We finally identify situations in which each model performs optimally in terms of power. The outline of this paper is as follows. First, the model specifications will be presented; Second, the simulation designs will be presented; Third, bias due to measurement error will be tested by introducing unreliable phenotypes; Fourth, results from simulations where exclusive environmental confounding (re, in Fig. 1) is fixed to zero, when in fact it is present in the data generation process will be presented. Finally, the models' power profiles will be reported.