Difference between Cholesky decomposition and variance-based ACE model

May I ask what's the difference between multi-variate ACE models based on paths (Cholesky decomposition) and models based on variance, such as advantages and disadvantages of the two methods? I was told that the models based on variance and covariance will not suffer from the type I error issues, by compared to the Cholesky decomposition, why?

The two are equivalent parameterizations. Under the "Cholesky" parameterization, the free parameters that define the model-expected covariance matrix are the elements of the Cholesky factors of the _A_, _C_, and _E_ covariance matrices. Under the "direct-symmetric" parameterization, the free parameters that define the model-expected covariance matrix are the elements of the _A_, _C_, and _E_ covariance matrices themselves. Much of the time--probably, most of the time--results will not differ between the two parameterizations. But, sometimes results _will_ differ, because the Cholesky parameterization imposes an implicit constraint that the direct-symmetric parameterization does not, namely, that the _A_, _C_, and _E_ covariance matrices must be positive-definite. But, the model-expected covariance matrix remains within the parameter space of the multivariate-normal distribution as long as it itself is positive-definite, whether or not the _A_, _C_, and _E_ covariance matrices are positive-definite.

Thus, the Cholesky parameterization introduces a "boundary condition," which can interfere with the usual asymptotic maximum-likelihood theory. One of the first (perhaps first) researchers to recognize how the Cholesky parameterization can interfere with Type I error rates was Greg Carey, in a 2005 paper, "Cholesky Problems". This year's Boulder Workshop was the first year we taught the direct-symmetric parameterization.

It is true that the direct-symmetric parameterization can result in negative estimates of variance components, but that is not necessarily a bad thing. I'd say the only real advantage the Cholesky parameterization has is its interpretability.

Could I model a multi-variate ACE model based on variance in OpenMx? Any demo codes would be appreciated.

See here for a collection of twin-modeling scripts for one phenotype and one covariate, maintained by Hermine Maes. The scripts that are described as "estimating variance components" use the direct-symmetric parameterization, and those described as "estimating path coefficients" use the Cholesky parameterization.

Thanks Rob. The parameter estimations were the results of the full ACE model. However, if the sub-models were introduced the AE model would survived. In this condition, which one would be more trustable, the CIs of the full ACE or the model comparison between the full genetic model and its sub-models?

Assuming that the full ACE model and the submodels converged, and that all of the CIs succeeded, then I would say both are informative. The model-comparison chi-square for ACE versus (say) AE is an omnibus test of the null hypothesis that there is no _C_, that is, that the 6(?) _C_-related parameters are all zero. The CIs instead represent inference about one quantity at a time.

How could we get the heritability from the direct-symmetric parameterization and explain the results? The parameters estimation were higher than 1 and the sum of variance of A, C and E seemed not equal to 1.

I'd like to see those results.

Hi! My similar bivariate ACE model results were as below:
confidence intervals:
CholCE.corC[2,1] NA 1.000000e+00 NA !!!
CholCE.corC[1,2] NA 1.000000e+00 NA !!!
CholCE.corC[2,2] NA 1.000000e+00 NA !!!
CholCE.corE[1,1] NA 1.000000e+00 NA !!!
CholCE.corE[2,1] NA -8.420869e-02 NA !!!
CholCE.corE[1,2] NA -8.420869e-02 NA !!!
CholCE.corE[2,2] NA 1.000000e+00 NA !!!
CI details:
58 CholCE.corC[1,1] 1.00000000 upper 1156.910 5.010251 0.10257746 -4.200825e-02
59 CholCE.corC[2,1] 0.96243518 lower 1153.141 5.238043 0.09977091 -2.814626e-02
60 CholCE.corC[2,1] 1.00000000 upper 1153.069 5.238043 0.09978666 -5.515351e-07
61 CholCE.corC[1,2] 1.00000000 lower 1153.069 5.238043 0.09978666 -5.515351e-07
62 CholCE.corC[1,2] 1.00000000 upper 1153.069 5.238043 0.09978666 -5.515351e-07
63 CholCE.corC[2,2] 1.00000000 lower 1153.069 5.238043 0.09978666 -5.515351e-07
64 CholCE.corC[2,2] 1.00000000 upper 1153.069 5.238043 0.09978666 -5.515351e-07
65 CholCE.corE[1,1] 1.00000000 lower 1153.069 5.238043 0.09978666 -5.515351e-07
66 CholCE.corE[1,1] 1.00000000 upper 1153.069 5.238043 0.09978666 -5.515351e-07

I am confused why there is no CholCE.corE[1,2] & CholCE.corE[2,2] results in CI details. How can I get it ?

Here is my scripts:

# Run Cholesky CE model
CholCeModel <- mxModel( CholAceFit, name="CholCE" )
pathA <- mxMatrix( type="Lower", nrow=nv, ncol=nv, free=FALSE, values=paZero,
labels=aLabs, lbound=paLBoD, name="a" )#pathA is 2X2 matrix
CholCeModel$MZ$a <-pathA
CholCeModel$DZ$a <-pathA
CholCeModel$a <-pathA
rA <- mxAlgebra(I*A, name="corA")
CholCeModel$corA <- rA
CholCeModel$MZ$corA <- rA
CholCeModel$DZ$corA <- rA

CholCeFit <- mxRun(CholCeModel, intervals = TRUE)
CholCeSumm <- summary(CholCeFit,verbose=T)
CholCeSumm
round(CholCeFit@output$estimate,4)
expectedMeansCovariances(CholCeFit)
tableFitStatistics(CholAceFit,CholCeFit)

*****Here is another part of the results:
58 success infeasible non-linear constraint
59 active box constraint infeasible non-linear constraint
60 alpha level not reached iteration limit/blue
61 alpha level not reached iteration limit/blue
62 alpha level not reached iteration limit/blue
63 alpha level not reached iteration limit/blue
64 alpha level not reached iteration limit/blue
65 alpha level not reached iteration limit/blue
66 alpha level not reached iteration limit/blue
[ reached getOption("max.print") -- omitted 14 rows ]

I don't know how to understand this part?
Wish for your help！
Thank you!

Very relevant to the topic of this thread: Type I Error Rates and Parameter Bias in Multivariate Behavioral Genetic Models.

Abstract:

For many multivariate twin models, the numerical Type I error rates are lower than theoretically expected rates using a likelihood ratio test (LRT), which implies that the significance threshold for statistical hypothesis tests is more conservative than most twin researchers realize. This makes the numerical Type II error rates higher than theoretically expected. Furthermore, the discrepancy between the observed and expected error rates increases as more variables are included in the analysis and can have profound implications for hypothesis testing and statistical inference. In two simulation studies, we examine the Type I error rates for the Cholesky decomposition and Correlated Factors models. Both show markedly lower than nominal Type I error rates under the null hypothesis, a discrepancy that increases with the number of variables in the model. In addition, we observe slightly biased parameter estimates for the Cholesky decomposition and Correlated Factors models. By contrast, if the variance–covariance matrices for variance components are estimated directly (without constraints), the numerical Type I error rates are consistent with theoretical expectations and there is no bias in the parameter estimates regardless of the number of variables analyzed. We call this the direct symmetric approach. It appears that each model-implied boundary, whether explicit or implicit, increases the discrepancy between the numerical and theoretical Type I error rates by truncating the sampling distributions of the variance components and inducing bias in the parameters. The direct symmetric approach has several advantages over other multivariate twin models as it corrects the Type I error rate and parameter bias issues, is easy to implement in current software, and has fewer optimization problems. Implications for past and future research, and potential limitations associated with direct estimation of genetic and environmental covariance matrices are discussed.