The Multivariate SEM-PGS Model: Using Polygenic Scores to Investigate Cross-Trait Genetic Nurture and Assortative Mating

Reviewer #1 (Public review):

Summary:

The authors develop a multivariate extension of SEM models incorporating transmitted and non-transmitted polygenic scores to disentangle genetic and environmental intergenerational effects across multiple traits. Their goal is to enable unbiased estimation of cross-trait vertical transmission, genetic nurture, gene-environment covariance, and assortative mating within a single coherent framework. By formally deriving multivariate path-tracing rules and validating the model through simulation, they show that ignoring cross-trait structure can severely bias both cross- and within-trait estimates. The proposed method provides a principled tool for studying complex gene-environment interplay in family genomic data.

Strengths:

It has become apparent in recent years that multivariate processes play an important role in genetic effects that are studied (e.g., Border et al., 2022), and these processes can affect the interpretation of these studies. This paper develops a comprehensive framework for polygenic score studies using trio data. Their model allows for assortative mating, vertical transmission, gene-environment correlation, and genetic nurture. Their study makes it clear that within-trait and cross-trait influences are important considerations. While their exposition and simulation focus on a bivariate model, the authors point out that their approach can be easily extended to higher-dimensional applications.

Weaknesses:

(1) My primary concern is that the paper is very difficult to follow. Perhaps this is inevitable for a model as complicated as this one. Admittedly, I have limited experience working with SEMs, so that might be partly why I really struggled with this paper, but I ultimately still have many questions about how to interpret many aspects of the path diagram, even after spending a considerable amount of time with it. Below, I will try to point out the areas where I got confused (and some where I still am confused). If the authors choose to revise the paper, clarifying some of these points would substantially broaden the paper's accessibility and impact.

(1a) Figure 1 contains a large number of paths and variable names, and it is not always apparent which variables correspond to which paths. For example, at a first glance, the "k + g_c" term next to the "T_m" box could arguably correspond to any of the four paths near it. Disentangling this requires finding other, more reasonable variables for the other lines and sifting through the 3 pages of tables describing the elements of the figure.

(1b) More hand-holding, describing the different parameters in the model, would help readers who don't have experience with SEMs. For example, many parameters show up several times (e.g., delta, a, g_c, i_c, w) and describing what these parameters are and why they show up several times would help. Some of this information is found in the tables (e.g., "Note: [N]T denotes either NT or T, as both share the same matrix content"), though I don't believe it is explained what it means to "share the same matrix content."

(1c) Relatedly, descriptions of the path tracing were very confusing to me. I was relieved to see the example on the bottom of page 10 and top of page 11, but then as I tried to follow the example, I was again confused. Because multiple paths have the same labels, I was not able to follow along which exact path from Figure 1 corresponded to the elements of the sum that made up Theta_{Tm}. Also, based on my understanding of the path-tracing rules described, some paths seemed to be missing. After a while, I think I decided that these paths were captured by the (1/2)*w term since that term didn't seem to be represented by any particular path in the figure, but I'm still not confident I'm right. In this example, rather than referring to things like "four paths through the increased genetic covariance from AM", it might be useful to identify the exact paths represented by indicating the nodes those paths go through. If there aren't space constraints, the authors might even consider adding a figure which just contains the relevant paths for the example

(1d) The paper has many acronyms and variable names that are defined early in the paper and used throughout. Generally, I would limit acronyms wherever possible in a setting like this, where readers are not necessarily specialists. For the variables, while the definitions are technically found in the paper, it would be useful to readers if they were reminded what the variables stood for when they are referred to later, especially if that particular variable hasn't been mentioned for a while. As I read, I found myself constantly having to scroll back up to the several pages of figures and tables to remind myself of what certain variables meant. Then I would have to find where I was again. It really made a dense paper even harder to follow.

(1e) Relatedly, on page 13, the authors make reference to a parameter eta, and I don't see it in Figure 1 or any of the tables. What is that parameter?

(2) This point may be related to me misunderstanding the model, but if LT_p represent the actual genetic factors for the two traits for variants that are transmitted to the child, and T_p represents the PGS of for transmitted variants, shouldn't their be a unidirectional arrow from LT_p to T_p (since the genetic factor affects the PGS and not the other way around) and shouldn't there be no arrow from T_p to Y_0 (since the entire effect of the transmitted SNPs is represented by the arrow from LT_p to Y_0)? If I'm mistaken here, it would be useful to explain why these arrows are necessary.

(3) Some explanation of how the interpretation of the coefficients differs in a univariate model versus a bivariate model would be useful. For example, in a univariate model, the delta parameter represents the "direct effect" of the PGI on the offspring's outcome (roughly corresponding to a regression of the offspring's outcome onto the offspring's PGI and each parent's PGI). Does it have the same interpretation in the bivariate case, or is it more closely related to a regression of one of the outcomes onto the PGIs for both traits?

(4) It appears from the model that the authors are assuming away population stratification since the path coefficient between T_m and T_m is delta (the same as the path coefficient between T_m and Y_0). Similarly, I believe the effect of NT_m on Y_0 only has a genetic nurture interpretation if there is no population stratification. Some discussion of this would be valuable.

References:

Border, R., Athanasiadis, G., Buil, A., Schork, AJ, Cai, N., Young, AI, ... & Zaitlen, N.A. (2022). Cross-trait assortative mating is widespread and inflates genetic correlation estimates. Science , 378 (6621), 754-761.

Reviewer #2 (Public review):

(1) Summary and overall comments:

This is an impressive and carefully executed methodological paper developing an SEM framework with substantial potential. The manuscript is generally very well written, and I particularly appreciated the pedagogical approach: the authors guide the reader step by step through a highly complex model, with detailed explanations of the structure and the use of path tracing rules. While this comes at the cost of length, I think the effort is largely justified given the technical audience and the novelty of the contribution.

The proposed SEM aims to estimate cross-trait indirect genetic effects and assortative mating, using genotype and phenotype data from both parents and one offspring, and builds on the framework introduced by Balbona et al. While I see the potential interest of the model, it is still a bit unclear in which conditions I could use it in practice. However, this paper made a clear argument for the need for cross-traits models, which changed my mind on the topic (I would have accommodated myself with univariate models and only interpreted in the light of likely pleiotropy, but I am now excited by the potential to actually disentangle cross-traits effects).

The paper is written in a way that makes me trust the authors' thoroughness and care, even when I do not fully understand every step of the model. I want to stress that I am probably not well-positioned to identify technical errors in the implementation. My comments should therefore be interpreted primarily from the perspective of a potential user of the method: I focus on what I understand, what I do not, and where I see (or fail to see) the practical benefits.

For transparency, here is some context on my background. I have strong familiarity with the theoretical concepts involved (e.g., genetic nurture, gene-environment covariance, dynastic effects), and I have worked on those with PGS regressions and family-based comparison designs. My experience with SEM is limited to relatively simple models, and I have never used OpenMx. Reading this paper was therefore quite demanding for me, although still a better experience than many similarly technical papers, precisely because of the authors' clear effort to explain the model in detail. That said, keeping track of all moving parts in such a complex framework was difficult, and some components remain obscure to me.

(2) Length, structure, and clarity:

I do not object in principle to the length of the paper. This is specialized work, aimed at a relatively narrow audience, and the pedagogical effort is valuable. However, I think the manuscript would benefit from a clearer and earlier high-level overview of the model and its requirements. I doubt that most readers can realistically "just skim" the paper, and without an early hook clearly stating what is estimated and what data are required, some readers may disengage.

In particular, I would suggest clarifying early on:

• What exactly is estimated?

For example, in the Discussion, the first two paragraphs seem to suggest slightly different sets of estimands: "estimate the effects of both within- and cross-trait AM, genetic nurture, VT, G-E covariance, and direct genetic effects." versus "model provides unbiased estimates of direct genetic effects (a and δ), VT effects (f), genetic nurture effects (ϕ and ρ), G-E covariance w and v, AM effects (μ), and other parameters when its assumptions are met." A concise and consistent summary of parameters would be helpful.

• What data are strictly required?

At several points, I thought that phenotypes for both parents were required, but later in the Discussion, the authors consider scenarios where parental phenotypes are unavailable. I found this confusing and would appreciate a clearer statement of what is required, what is optional, and what changes when data are missing.

• Which parameters must be fixed by assumption, rather than estimated from the data?

Relatedly, in the Discussion, the authors mention the possibility of adding an additional latent shared environmental factor across generations. It would help to clearly distinguish:

• the baseline model,
• the model actually tested in the paper, and
• possible extensions.

Making these distinctions explicit would improve accessibility.

This connects to a broader concern I had when reading Balbona et al. (2021): at first glance, the model seemed readily applicable to commonly available data, but in practice, this was not the case. I wondered whether something similar applies here. A clear statement of what data structures realistically allow the model to be fitted would be very useful.

I found the "Suggested approach for fitting the multivariate SEM-PGS model" in the Supplementary Information particularly helpful and interesting. I strongly encourage highlighting this more explicitly in the main manuscript. If the authors want the method to be widely used, a tutorial or at least a detailed README in the GitHub repository would greatly improve accessibility.

Finally, while the pedagogical repetition can be helpful, there were moments where it felt counterproductive. Some concepts are reintroduced several times with slightly different terminology, which occasionally made me question whether I had misunderstood something earlier. Streamlining some explanations and moving more material to the SI could improve clarity without sacrificing rigor.

(3) Latent genetic score (LGS) and the a parameter

I struggled to understand the role of the latent genetic score (LGS), and I think this aspect could be explained more clearly. In particular, why is this latent genetic factor necessary? Is it possible to run the model without it?

My initial intuition was that the LGS represents the "true" underlying genetic liability, with the PGS being a noisy proxy. Under that interpretation, I expected the i matrix to function as an attenuation factor. However, i is interpreted as assortative-mating-induced correlation, which suggests that my intuition is incorrect. Or should the parameter be interpreted as an attenuation factor?

Relatedly, in the simulation section, the authors mention simulating both PGS and LGS, which confused me because the LGS is not a measured variable. I did not fully understand the logic behind this simulation setup.

Finally, I was unsure whether the values simulated for parameter a in Figures 8-9 are higher than what would typically be expected given the current literature, though this uncertainty may reflect my incomplete understanding of a itself. I appreciated the Model assumptions section of the discussion, and I wonder if this should not be discussed earlier.

(4) Vertical transmission versus genetic nurture

I am not sure I fully understand the distinction between vertical transmission (VT) and genetic nurture as defined in this paper. From the Introduction, I initially had the impression that these concepts were used almost interchangeably, but Table 3 suggests they are distinct.

Relatedly:

• Why are ϕ and ρ not represented in the path diagram?

• Are these parameters estimated in the model?

The authors also mention that these parameters target different estimands compared to other approaches. It would be helpful to elaborate on this point. Relatedly, where would the authors expect dynastic effects to appear in this framework?

(5) Univariate model and misspecification

In the simulations where a univariate model is fitted to data generated under a true bivariate scenario, I have a few clarification questions.

What is the univariate model used (e.g., Table 5)? Is it the same as the model described in Balbona et al. (2025)? Does it include an LGS?

If the genetic correlation in the founder generation is set to zero, does this imply that all pleiotropy arises through assortative mating? If so, is this a realistic mechanism, and does it meaningfully affect the interpretation of the results?

(6) Simulations

Overall, I found the simulations satisfying to read; they largely test exactly the kinds of issues I would want them to test, and the rationale for these tests is clear.

That said, I was confused by the notation Σ and did not fully understand what it represents.

In the Discussion, the authors mention testing the misspecification of social versus genetic homogamy, but I do not recall this being explicitly described in the simulation section. They also mention this issue in the SI ("Suggested approach for fitting..."). I think it would be very helpful to include an example illustrating this form of misspecification.

(7) Cross-trait specific limitations

I am wondering - and I don't think this is addressed - what is the impact of the difference in the noisiness and the heritability of the traits used for this multivariate analysis?

Using the example, the authors mention of BMI and EA, one could think that these two traits have different levels of noise (maybe BMI is self-reported and EA comes from a registry), and similarly for the GWAS of these traits, let's say one GWAS is less powered than the other ones. Does it matter? Should I select the traits I look at carefully in function of these criteria? Should I interpret the estimates differently if one GWAS is more powered than the other one?