Dear Lecturers and Tutors,
Thank you for your excellent lectures and tutorings. I still have three quick questions. Though workshops ended, I just wonder if I can ask these questions:

In slide 13 (Day 8, pdf on Estimating the Importance of Maternal Genetic Effects on Offspring Phenotypes with “MGCTA), why Xij and Xik follow a binomial distribution (though each of each has a value of 0,1 or 2). I asked this question in the tutorial session, but still not clear).

In slide 17 (Day 8, pdf on Estimating Parental Effects using Polygenic Scores Part II: Model Extensions), why only interested in Cov(Yo, NTp), not Cov(Yo, NTm)?

In slide 5253 (Day 8, pdf on Estimating Parental Effects using Polygenic Scores Part II: Model Extensions), why this model is bivariate version of SEM given this model has one observed factor (Yo) and 3 latent dependent variables (Fo, Yp & Ym)?
I look forward to receiving your attention and response.
Hi Huy,
Those are excellent questions! While I cannot personally address the first one (regarding MGCTA), I can help out with #2 and #3:

You are absolutely correct that both cov(Yo, NTp) and cov(Yo, NTm) are important. Both are used to provide us with estimates of vertical transmission and genetic nurture, and neither one is more important than the other. Cov(Yo, NTp) was just arbitrarily chosen as the path tracing example in my presentation, but I could just as easily have chosen any other covariance for this.

The only difference between the bivariate and univariate versions of SEMPGS is that, in the univariate, each path coefficient is equal to a single scalar value. For example, µ could be equal to .5. However, in the bivariate version, each of those path coefficients are instead equal to 2x2 matrices, which allow for the different relationships between different variables to be accounted for. Beyond this, the univariate and bivariate versions of SEMPGS are more or less the same as one another They have the same number of observed variables ( 5 or more, depending on whether Yp and Ym are observed), and the same number of latent variables.
With that said, please do not worry too much about the specifics of the bivariate model! I had included it in the presentation to showcase an example of how the model can be extended, rather than to fully explain how bivariate modeling works.
Thank you again for your questions, and please feel free to reach out (either on this forum or via email) if you have any others!
Best,
Jared
Thank you so much Jared for useful explanations. It clear my understanding.