# Moderation of OSDZ intrapair correlation (ra, rc) in a sex-lim model

I want to study if the similarity (through A and C) between female twin and male twin, within pair, changes due to a moderator (as age) using the GxE approach of Purcell (2002) in a sex-limitation model.

I thought to put it in a very simple way such as:

rados <- mxMatrix(type="Full", nrow=1, ncol=1, free=F, values=0.5, label="rg", lbound=-1, ubound=1, name="ra")

rcdos <- mxMatrix(type="Full", nrow=1, ncol=1, free=T, values=.01, label="re", lbound=-1, ubound=1, name="rc")

```
```#I am currenty interested in the moderation of rc

rBados <- mxMatrix(type="Full", nrow=1, ncol=1, free=F, values=0, label="rBg", name="rBa")

rBcdos <- mxMatrix(type="Full", nrow=1, ncol=1, free=T, values=.01, label="rBe", name="rBc")

regRa <- mxAlgebra(expression= ra+DefVar*rBa, name = "regRg") #being DefVar any continuous definition variable

regRc <- mxAlgebra(expression= rc+DefVar*rBc, name = "regRe")

`covDZo <- mxAlgebra( expression= regRg%x%(am%*%t(af))+regRe%x%(cm%*%t(cf)), name="covDZmf" )`

But I'm not sure if this is both statistically and theoretically correct, and I can't find this information in the papers or textbooks available to me. Is it possible to do it like this?

Also, I want to implement this MxAlgebra along with a moderation of the variance components.

Thank you very much in advance for any insight you could give me about this.

Best regards,

Jose.

## Interesting idea. I have a

First, I'll grant that it might be an interesting research question to ask "do opposite-sex DZ twins have less-similar environments than same-sex twins by virtue of being opposite-sex?" But, the common environment, by definition, comprises all environmental factors that are shared by the two twins in a pair. That is, by definition, the twin correlation of common-environmental factors is 1.0. So, I don't think it's theoretically coherent to talk about moderating the shared-environmental correlation. In fact, I'm not sure that the shared-environmental correlation would be an identified parameter in the model. I could be mistaken about all this, though.

Second, in modeling qualitative sex-limitation, it's typical to parameterize the additive-genetic contribution to opposite-sex covariance as the expected DZ-twin genetic resemblance times the opposite-sex genetic correlation times the additive-genetic covariance matrix--say,

mxAlgebra( expression= 0.5%*%rg%x%(af%*%t(am))+cf%*%t(cm), name="covDZfm" )

Note that, in the conventional parameterization,

`rg`

refers to the correlation of trait-relevant genetic factors in a male DZ twin with the trait-relevant genetic factors in his female co-twin. There's nothing wrong with your parameterization, but I just wanted to point out the convention.Third, you'll need to deal with the fact that the coefficient (named "regRg" in your syntax) in front of the genetic covariance matrix needs to be bounded. An inelegant way would be to use MxConstraints. A better way would be to use a different link function between the linear composite involving the definition variable and "regRg". The hyperbolic tangent,

`tanh()`

, maps the real line to the interval (-1,1), so you could instead doregRa <- mxAlgebra(expression= tanh(ra+DefVar*rBa), name = "regRg") #being DefVar any continuous definition variable

Even better: due to the intraclass nature of twin data, we can reasonably restrict "regRg" to the unit interval, so we can use a logit link, e.g.

regRa <- mxAlgebra(expression= exp(ra+DefVar*rBa) / (1+exp(ra+DefVar*rBa)), name = "regRg") #being DefVar any continuous definition variable

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