# Twin Model Identification Matrix Algebra

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Benny
Joined: 05/15/2020

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Hello,

I was reading the book by Neale and Maes (2004) and they have a nice part about how to check the identification of a univariate twin model using matrix algebra (p. 104).

It is possible to check the identification of the model by representing the expected (co-)variances as a system of equation in matrix algebra:

Ax=b

where x is the vector of parameters, b is the vector of observed statistics and A is the matrix containing the weights of elements in x.

In the univariate case (ACE model), x=(a^2,c^2,e^2).

I was reading the book by Neale and Maes (2004) and they have a nice part about how to check the identification of a univariate twin model using matrix algebra (p. 104).

It is possible to check the identification of the model by representing the expected (co-)variances as a system of equation in matrix algebra:

Ax=b

where x is the vector of parameters, b is the vector of observed statistics and A is the matrix containing the weights of elements in x.

In the univariate case (ACE model), x=(a^2,c^2,e^2).

My first question is: Is it possible to express x=(a,c,e) instead of x=(a^2,c^2,e^2)= How would Ax=b look like in this case?

My second question is: How would like the Ax=b system of equation for a bivariate Cholesky model? Here my parameters would be x=(a11,a21,a22,c11,c21,c22,e11,e21,e22). However, as some of the covariances are a function of the squared parameters, I don't know how to construct A in this case.

I hope you understand the point of my questions.

Thank you,

Benny

## A paper and a function

Hunter, M.D., Garrison, S.M., Burt, S.A. et al. The Analytic Identification of Variance Component Models Common to Behavior Genetics. Behav Genet 51, 425–437 (2021). https://doi.org/10.1007/s10519-021-10055-x

Also, the `mxCheckIdentification()` function takes a more general approach by checking the dimension (i.e., rank) of the mapping from the free parameters to the summary statistics. See its help page for more details.

Finally, for multi-phenotype models the same process of identification applies. It's just a little more complicated. Once you read the paper and check out the function, it should seem straightforward.

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