# Model-averaged point estimates

I am trying to calculate the heritability for some fat-related traits. I have fitted the ACE model and some submodels (CE and AE) and then learned that it was recommended to use ` mxModelAverage()`

to gain the model-averaged point estimates.

Here is part of the script I am using to fit ACE model:

modelbase <- mxModel("base",

mxMatrix(type="Full",nrow=1,ncol=ncv,free=TRUE, values=0,labels=c('Bage','BSUBTOT_MASS'), name="B" ),

mxMatrix( type="Full", nrow=1, ncol=ntv, free=TRUE, labels="mean", name="meanG" ),

# Matrices declared to store a, c, and e Path Coefficients

mxMatrix( type="Lower", nrow=nv, ncol=nv, free=T, values=sqrt(svace), label="a11",lbound=0.0001, name="a" ),

mxMatrix( type="Lower", nrow=nv, ncol=nv, free=T, values=sqrt(svace), label="c11",lbound=0.0001, name="c" ),

mxMatrix( type="Lower", nrow=nv, ncol=nv, free=T, values=sqrt(svace), label="e11",lbound=0.0001, name="e" ),

# Matrices generated to hold A, C, and E components and total Variance

mxAlgebra( expression=a %*% t(a), name="A" ),

mxAlgebra( expression=c %*% t(c), name="C" ),

mxAlgebra( expression=e %*% t(e), name="E" ),

mxAlgebra( expression=A+C+E, name="V" ),

mxAlgebra( expression=A/V, name="h2" ),

mxAlgebra( expression=C/V, name="c2" ),

mxAlgebra( expression=E/V, name="e2" ))

And the code I'm using to calculate the model-averaged point estimates is as follows:

mxModelAverage(reference=c("base.h2[1,1]","base.c2[1,1]","base.e2[1,1]"), include="onlyFree",

type="AICc", SE=T, models=list(AceFit,AEfit, CEfit))

I learned that, for the purpose of statistical inferences, it's recommended to set `include="onlyFree"`

. However, in this way, the sum of model-averaged point estimates of h2, c2 and e2 is not equal to 1. The result is showed as follows:

$`Model-Average Estimates`

Estimate SE

base.h2[1,1] 0.6400199 0.14768808

base.c2[1,1] 0.2225559 0.11887596

base.e2[1,1] 0.2259699 0.02830278

```
```$`Model-wise Estimates`

ACE AE CE

base.h2[1,1] 0.5502440 0.7759045 NA

base.c2[1,1] 0.2225497 NA 0.6412221

base.e2[1,1] 0.2272063 0.2240955 0.3587779

$`Model-wise Sampling Variances`

ACE AE CE

base.h2[1,1] 0.0154727014 0.000742769 NA

base.c2[1,1] 0.0141290903 NA 0.001150016

base.e2[1,1] 0.0008354342 0.000742769 0.001150016

`$`Akaike-Weights Table``

model AICc delta AkaikeWeight inConfidenceSet

1 ACE 995.5385 0.0000000 6.021586e-01 *

2 AE 996.3675 0.8289793 3.978325e-01 *

3 CE 1017.7799 22.2413237 8.913921e-06

Thus, what do you think can be done to fix it? Using `include="all"`

at the cost of violating normal sampling distribution?

Additionally, according to the code of the ACE model, it seems also feasible to calculate model-averaged point estimates of path coefficients **a, c, e** and variance of **A, C, E**. However, this will also result in different estimates of heritability. In this case, what parameters do you recommend to be calculated model-averaged point estimates?

Many thanks in advance!

## raw variance components

I think it makes the most sense to report model-averaged estimates for the raw variance components--that is, the unstandardized variances of _A_, _C_, and _E_.

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In reply to raw variance components by AdminRobK

## The CI of heritability

Just to check, if I calculate the model-averaged point estimates of A, C, and E, then I can only get the point estimate of heritability, rather than the CI of it. Is it right? Thank you.

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In reply to The CI of heritability by diana

## heritability

Yes, that's true, *if* you use `include="onlyFree"` and `refAsBlock=FALSE`. If you instead use `include="all"` and `refAsBlock=TRUE`, you'll get the joint sampling covariance matrix of the model-averaged _A_, _C_, and _E_ variance components. You can then use that joint covariance matrix to calculate the standard error of the heritability coefficient, using the delta method.

Per Chebyshev's Inequality, ±5SEs will give you a confidence interval with coverage probability of at least 96%.

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