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Genetic correlation: two ways, two different estimates

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lemire's picture
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Genetic correlation: two ways, two different estimates
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Hi,

I implemented two ways of calculating the genetic correlation between two phenotypes, estimated from twins (direct model of rg between A1 and A2, and Cholesky).

My two implementations are based on the path diagrams from H.H. Maes' slides 18 and 53 found at

https://vipbg.vcu.edu/media/course/HGEN619_2015/BivariateGeneticAnalysis2.pdf

I do not get the same estimates between the two methods (-0.26 vs 0.51). Below is the code with the reproducible fake data and results. The code is also attached.

If someone could help figuring out why the two methods differ, of help me find a bug in my code, I would appreciate. Many thanks for looking into this.

In the notation below, indices 1 and 2 refer to phenotype 1 and phenotype 2; indices x and y refer to twin 1 and twin 2.

Mathieu

####
 
library(OpenMx)
 
#######
# creating a reproducible data set
library(  mvtnorm )
set.seed(12345)
 
twincov<- 0.30
trcov<- 0.1
N<-200
sigmz<-matrix( c( 1, twincov, trcov,  trcov, 
                twincov, 1,  trcov,  trcov, 
                trcov,   trcov,   1,  twincov, 
                trcov,   trcov, twincov,1 ), ncol=4 ) 
 
sigdz <-matrix( c( 1,  twincov/2, trcov,  trcov, 
                twincov/2, 1,   trcov,  trcov, 
                trcov,   trcov,   1,   twincov/2, 
                trcov,   trcov,   twincov/2,1 ), ncol=4 ) 
 
tmz <- as.data.frame( rmvnorm(N , c(0,0,0,0) , sigmz ) )
names(tmz)<- c("phen1x", "phen1y", "phen2x", "phen2y")
 
tdz <-  as.data.frame(rmvnorm(N , c(0,0,0,0) , sigdz ))
names(tdz)<- c("phen1x", "phen1y", "phen2x", "phen2y")
 
dataMZ       <- mxData( observed=tmz , type="raw" )
dataDZ       <- mxData( observed=tdz , type="raw" )
 
 
##############
# defining variables and path
# the following are common to the two methods of calculating rg
# and will be re-used 
 
phenVars<- c("phen1x", "phen1y" ,"phen2x", "phen2y" )
# pheno 1 
aceVars1<- c("A1x","C1x","E1x","A1y","C1y","E1y")
# pheno 2 
aceVars2<- c("A2x","C2x","E2x","A2y","C2y","E2y")
 
latentVariances <- mxPath( from=c( aceVars1, aceVars2 ), arrows=2, 
                        free=FALSE, values=1 )
 
# means of latent variables
latentMeans     <- mxPath( from="one", to=c( aceVars1, aceVars2 ) , arrows=1, 
                        free=FALSE, values=0 )
 
# means of observed variables
obsMeans1     <- mxPath( from="one", to=c("phen1x", "phen1y" )  , arrows=1, 
                        free=TRUE, values=0.01 , labels="mean1" )
obsMeans2    <- mxPath( from="one", to=c("phen2x", "phen2y" )  , arrows=1, 
                        free=TRUE, values=0.02 , labels="mean2" )
 
# path coefficients for twin 1, phen 1
path1x    <- mxPath( from=c( "A1x","C1x","E1x"  ), to="phen1x", arrows=1, 
                        free=TRUE, values=c( .12,.22,.32) ,  label=c( "a1","c1","e1") )
# path coefficients for twin 2, phen 1 
path1y    <- mxPath( from=c( "A1y","C1y","E1y"  ), to="phen1y", arrows=1, 
                        free=TRUE, values=c( .12,.22,.32) ,  label=c( "a1","c1","e1") )
 
# path coefficients for twin 1, phen 2
path2x    <- mxPath( from=c( "A2x","C2x","E2x"  ), to="phen2x", arrows=1, 
                        free=TRUE, values=c( .18,.28,.38) ,  label=c( "a2","c2","e2") )
# path coefficients for twin 2, phen 2 
path2y    <- mxPath( from=c( "A2y","C2y","E2y"  ), to="phen2y", arrows=1, 
                        free=TRUE, values= c( .18,.28,.38) ,  label=c( "a2","c2","e2") )
 
# covariance between C1x and C2y, etc 
covC1      <- mxPath( from="C1x", to="C1y", arrows=2, 
                        free=FALSE , values=1 )
covC2      <- mxPath( from="C2x", to="C2y", arrows=2, 
                        free=FALSE , values=1 )
 
# cov between A1x, A1y etc . index x is twin 1 , y in twin 2
covA1mz <- mxPath( from="A1x", to="A1y", arrows=2, 
                        free=FALSE, values=1 )
covA2mz <- mxPath( from="A2x", to="A2y", arrows=2, 
                        free=FALSE, values=1 )
covA1dz <- mxPath( from="A1x", to="A1y", arrows=2, 
                        free=FALSE, values=.5 )
covA2dz <- mxPath( from="A2x", to="A2y", arrows=2, 
                        free=FALSE, values=.5 )
 
commonPaths  <-  list( latentVariances,  latentMeans,obsMeans1, obsMeans2,  
                       path1x, path1y, path2x, path2y ,  covC1, covC2  )
 
 
# END of paths common to both methods
######################################
 
 
 
########
# method one: directly modeling the genetic correlation between A1 and A2 
# index 1 refers to phenotype 1, 2 to phenotype 2
# x refers to twin 1, y to twin 2
 
pathRGx <- mxPath( from="A1x", to="A2x" , arrows=2 , free=TRUE , value= .5, label="rg" )
pathRGy <- mxPath( from="A1y", to="A2y" , arrows=2 , free=TRUE , value= .5, label="rg" )
 
# solution seems highly dependent on starting values (see below).
# 0.5 was chosen as starting value because
# that's close to the estimate of method 2 below 
 
# confidence intervals 
ci.rg <- mxCI("rg")
 
 
# Defining the models the paths 
paths.rg        <- list( pathRGx, pathRGy  )
 
modelMZ.rg      <- mxModel(model="MZrg", type="RAM", manifestVars=phenVars, 
                        latentVars=c(aceVars1, aceVars2 ) , commonPaths, paths.rg , covA1mz, covA2mz, dataMZ ,  ci.rg  )
 
modelDZ.rg      <- mxModel(model="DZrg", type="RAM", manifestVars=phenVars, 
                        latentVars=c(aceVars1, aceVars2 ) , commonPaths, paths.rg , covA1dz, covA2dz,  dataDZ   )
 
multi.rg <- mxFitFunctionMultigroup(c("MZrg","DZrg" ) )
model.rg      <- mxModel(model="ACErg", modelMZ.rg, modelDZ.rg, multi.rg )
 
 
 
# check identification 
print( mxCheckIdentification( model.rg  ) )
# the model is identified
 
# Run Model
 
fit.rg    <- mxRun( model.rg , intervals=T )
 
print(fit.rg$output$confidenceIntervals )
 
#  lbound   estimate ubound
#rg     NA -0.2639032     NA
 
# Note that the estimate is far from method 2 (cholesky, below). The sign is also different. 
# Also, why can't the bounds of the CI be calculated? 
 
# If "rg" gets a starting value of 0.25 instead of .5, we get
# an estimate out of the intuitive range: (???)
#   lbound estimate ubound
#rg     NA 12.27812     NA
 
 
 
########
# method 2 : cholesky
# from A1 to phen2 
path21x<- mxPath( from="A1x", to="phen2x" , arrows=1 , free=TRUE , value=.15, label="a21" )
path21y<- mxPath( from="A1y", to="phen2y" , arrows=1 , free=TRUE , value=.15, label="a21" )
 
# confidence intervals 
cimat.cho <- mxMatrix(type="Full",nrow=3,ncol=1,free=T ,values=c( .12,.18,.15   ) ,
    labels=c( "a1","a2","a21" ),name="aaa") 
cialg.cho <- mxAlgebra( aaa[1,1]*aaa[3,1]/sqrt( aaa[1,1]^2 * ( aaa[3,1]^2+aaa[2,1]^2 ) ), name="rg",
            dimnames=list( NULL   ,NULL ) )
ci.cho <- mxCI("rg")
 
 
# Defining the models the paths 
paths.cho        <- list( path21x, path21y  )
 
modelMZ.cho      <- mxModel(model="MZcho", type="RAM", manifestVars=phenVars, 
                        latentVars=c(aceVars1, aceVars2 ) , commonPaths, paths.cho , covA1mz, covA2mz, dataMZ ,
            cimat.cho,cialg.cho, ci.cho  )
 
modelDZ.cho      <- mxModel(model="DZcho", type="RAM", manifestVars=phenVars, 
                        latentVars=c(aceVars1, aceVars2 ) , commonPaths, paths.cho , covA1dz, covA2dz,  dataDZ   )
 
multi.cho <- mxFitFunctionMultigroup(c("MZcho","DZcho" ) )
model.cho      <- mxModel(model="ACEcho", modelMZ.cho, modelDZ.cho, multi.cho )
 
 
 
# check identification 
print( mxCheckIdentification( model.cho  ) )
# the model is indentified
 
# Run Model
 
fit.cho    <- mxRun( model.cho  , intervals=T )
 
print(fit.cho$output$confidenceIntervals )
 
#                 lbound  estimate ubound
#MZcho.rg[1,1] 0.1515944 0.5169708      1
#
# Different estimate from above.
AdminRobK's picture
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Joined: 01/24/2014 - 12:15
bound the correlation

First off, if you know a two-headed path coefficient is a correlation, you probably ought to bound it to the parameter space of a correlation, for instance:

pathRGx <- mxPath( from="A1x", to="A2x" , arrows=2 , free=TRUE , value= .25, label="rg", lbound=-1, ubound=1 )
pathRGy <- mxPath( from="A1y", to="A2y" , arrows=2 , free=TRUE , value= .25, label="rg", lbound=-1, ubound=1 )

If I do that, I get the same point estimate with start values of 0.25 as with 0.5.

As for the rest of your post, I'll have to give it some more thought. I'm rather surprised that fit.rg has a worse -2logL than fit.cho. If it were the other way around, I'd have a clearer suspicion about what's going on.

BTW, you can see more information about the estimated confidence limits (such as why they're being reported as NA) if you pass verbose=TRUE to summary(), e.g. summary(fit.rg,verbose=TRUE).

EDIT: Hang on. You're using the symbol "rg" to refer to two different things. In model.rg, 'rg' is the correlation between A1 and A2. But in model.cho, 'rg' is the genetic correlation between the two phenotypes. Those aren't the same thing.

lemire's picture
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Joined: 12/13/2019 - 12:35
Thank you for your comments.

Thank you for your comments. Regarding the rg's not being the same things: based on Hermine's course on slide 57, aren't they the same thing, though? I merely implemented the two alternative representations from that slide that can be used to calculate the same genetic correlation (see also slides 17 and 18).

Course found at
https://vipbg.vcu.edu/media/course/HGEN619_2015/BivariateGeneticAnalysis2.pdf

lemire's picture
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Joined: 12/13/2019 - 12:35
slide 53

I meant slide 53, not 57

AdminRobK's picture
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Joined: 01/24/2014 - 12:15
Hmm, yeah. I think you're

Hmm, yeah. I think you're right, and I was mistaken.

AdminRobK's picture
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Joined: 01/24/2014 - 12:15
This is an interesting case.

This is an interesting case. A few remarks...

First, I should point out that in model.rg, the sign of 'rg' is not identified. Assuming you've bounded 'rg' to [-1,1], you can see what I mean if you do summary(fit.rg,verbose=T), specifically:

CI details:
  parameter  side value      fit            diagnostic statusCode            method         a1           c1        e1         a2        c2        e2 rg       mean1      mean2
1        rg lower    -1 4507.172 active box constraint         OK neale-miller-1997 -0.5443192 2.241322e-06 0.8267003  0.1700324 0.4037318 0.9087478 -1 -0.01174902 0.03194610
2        rg upper     1 4507.172               success         OK neale-miller-1997 -0.5029672 2.557402e-01 0.8482156 -0.2021780 0.4658332 0.8661036  1 -0.02967843 0.04124539

The point estimate of 'rg' is negative, but at the solution for the upper confidence limit, 'rg' is positive but the sign of 'a2' has flipped to negative, thus keeping the genetic covariance positive. In model.cho, 'rg' doesn't have this problem. You can get saner CI results for the genetic correlation if you define the MZ model as:

modelMZ.rg      <- mxModel(
    model="MZrg", type="RAM", manifestVars=phenVars, 
    mxAlgebra(rbind(
        cbind(a1^2,a1*rg*a2),
        cbind(a1*rg*a2,a2^2)
    ),name="covA"),
    mxAlgebra(cov2cor(covA),name="corA"),
    mxCI("corA[1,2]"),
    latentVars=c(aceVars1, aceVars2 ) , commonPaths, paths.rg , covA1mz, covA2mz, dataMZ ,  ci.rg  )

Compare:

confidence intervals:
                   lbound   estimate    ubound note
MZrg.corA[1,2] -0.2228341  0.2639025 0.9999999     
rg                     NA -0.2639025 1.0000000  !!!

Second, the Cholesky parameterization ensures that the covariance matrix of the latent variables remains positive definite. You can see that with eigen(fit.cho$MZcho$expectation$UnfilteredExpCov[5:16,5:16]). There are no negative eigenvalues, though the matrix is nearly singular. model.rg is not subject to that implicit constraint; see eigen(fit.rg$MZrg$expectation$UnfilteredExpCov[5:16,5:16]). The fact that model.rg lacks an implicit constraint which model.cho has is why I was surprised that fit.rg has the worse -2logL.

lemire's picture
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.26 or .51?

Thanks a lot, I am too much of a novice to figure out all the hidden gems in openmx for diagnostic purposes. As much as I would like to dig deeper, I am only doing all this for a one-liner in a paper!

It now makes intuitive sense to me that the sign of rg for correlated latent variables is not well defined.

So back to my original question, which estimate should I report? You pointed that the log-likelihood is smaller for cholesky, does that mean the other method got stuck in a local minimum? Are the log-lik comparable anyway? (at least because the number of parameters is the same)? If so, I can base my decision on that (hence cholesky)?

AdminRobK's picture
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Joined: 01/24/2014 - 12:15
report Cholesky
You pointed that the log-likelihood is smaller for cholesky, does that mean the other method got stuck in a local minimum?

Perhaps, but I tried running it with mxTryHard() with exhaustive=TRUE yesterday, and didn't find a better solution.

Are the log-lik comparable anyway? (at least because the number of parameters is the same)? If so, I can base my decision on that (hence cholesky)?

Yeah, on those grounds, I would report the result from the Cholesky-parameterized model.

tbates's picture
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Joined: 07/31/2009 - 14:25
impossible covariation

I think the issue is in the data you're simulating. I'm guessing you've transposed what should be the two phenotypes (x and y) with what should be twins (1 and 2), and in the process generated a dataset where trait1 correlates more with trait2 that it does with itself.

In addition the DZ correlations are only modified in two cells, which generates a weird all-C no-C pattern between the cells. Guessing this is not a positive definite target expected covariance matrix.

Try the built-in twinData (say wt and ht) and you'll see the equivalence of correlated factors with a correlation via cholesky paths.

A great paper on this topic of equivalent model transformations is the inimitable John Loehlin (1996).

Loehlin, J. C. (1996). The Cholesky approach: A cautionary note. Behavior Genetics, 26(1), 65-69.

http://ibg.colorado.edu/cdrom2012/demoor/MultivariateCholesky/Papers/Loehlin_BG_1996.pdf

AdminRobK's picture
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Joined: 01/24/2014 - 12:15
data generation
I think the issue is in the data you're simulating. I'm guessing you've transposed what should be the two phenotypes (x and y) with what should be twins (1 and 2), and in the process generated a dataset where trait1 correlates more with trait2 that it does with itself.

Good catch, Tim.

lemire's picture
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my problem was based on real data

Thanks a lot for the reference, much appreciated. I'll take a look.

I first saw the problem of the different estimates using our real data that we are not willing to share publicly. I merely simulated data to get a reproducible, testable framework for my post.

Also, my code seems consistent with my choice of indices (1 and 2 for phenotypes, x and y for twins). So does the correlation matrices, unless there's something I am missing.

cor(tmz)
phen1x phen1y phen2x phen2y
phen1x 1.00000000 0.35936636 0.07024392 0.06686935
phen1y 0.35936636 1.00000000 0.10982890 0.07460252
phen2x 0.07024392 0.10982890 1.00000000 0.24500391
phen2y 0.06686935 0.07460252 0.24500391 1.00000000

cor(tdz)
phen1x phen1y phen2x phen2y
phen1x 1.0000000 0.15550815 0.1511272 0.14248904
phen1y 0.1555081 1.00000000 0.1119594 0.06931447
phen2x 0.1511272 0.11195939 1.0000000 0.22150121
phen2y 0.1424890 0.06931447 0.2215012 1.00000000

AdminRobK's picture
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Joined: 01/24/2014 - 12:15
Hmm

Hmm. Then maybe Tim was mistaken.

lemire's picture
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enlightening

Guys, I can't thank you enough for all of your comments and for being there for users. It was super enlightening and useful!

Until next time,

Mathieu

AdminRobK's picture
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Joined: 01/24/2014 - 12:15
You're welcome. Glad to be

You're welcome. Glad to be of assistance.

I still think there's something weird going on in your dataset, though. How large is the sample?

lemire's picture
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sample size

Oh the sample size is quite small, in this case 46 MZ pairs and 125 DZ pairs without missing data. For my mean-centered data, I'm getting an estimate of rg=0.234 for the correlated A's model and 0.309 with cholesky.

> cor( mzData )
phen1x phen1y phen2x phen2y
phen1x 1.0000000 0.7675690 0.177232 0.2401964
phen1y 0.7675690 1.0000000 0.156969 0.2297162
phen2x 0.1772320 0.1569690 1.000000 0.4758010
phen2y 0.2401964 0.2297162 0.475801 1.0000000
>
> cor( dzData )
phen1x phen1y phen2x phen2y
phen1x 1.00000000 0.1336467 0.2357775 -0.01569437
phen1y 0.13364668 1.0000000 0.0265778 0.11114247
phen2x 0.23577754 0.0265778 1.0000000 0.27126851
phen2y -0.01569437 0.1111425 0.2712685 1.00000000