Analysis of bivariate binomial data: Twin analysis
Table of Contents
Overview
When looking at bivariate binomial data with the aim of learning about the dependence that is present, possibly after correcting for some covariates many models are available.
- Random-effects models logistic regression covered elsewhere (glmer in lme4).
in the mets package you can fit the
- Pairwise odds ratio model
- Bivariate Probit model
- With random effects
- Special functionality for polygenic random effects modelling such as ACE, ADE ,AE and so forth.
- Additive gamma random effects model
- Special functionality for polygenic random effects modelling such as ACE, ADE ,AE and so forth.
Typically it can be hard or impossible to specify random effects models with special structure among the parameters of the random effects. This is possible in our models.
To be concrete about the model structure assume that we have paired binomial data \( Y_1, Y_2, X_1, X_2 \) where the responses are \( Y_1, Y_2 \) and we have covariates \( X_1, X_2 \).
We start by giving a brief description of these different models. First we for bivariate data one can specify the marginal probability using logistic regression models \[ logit(P(Y_i=1|X_i)) = \alpha_i + X_i^T \beta i=1,2. \] These model can be estimated under working independence \cite{zeger-liang-86}.
A typical twin analysis will typically consist of looking at both
- Pairwise odds ratio model
- Bivariate Probit model
The additive gamma can be used for the same as the bivariate probit model but is more restrictive in terms of dependence structure, but is nevertheless still valuable to have also as a check of results of the bivariate probit model.
Biprobit with random effects
For these model we assume that given random effects \(Z\) and a covariate vector \( V_{12} \) we have independent logistic regression models \[ probit(P(Y_i=1|X_i, Z)) = \alpha_i + X_i^T \beta + V_{12}^T Z i=1,2. \] where \( Z \) is a bivariate normal distribution with some covariance \( \Sigma \). The general covariance structure \( \Sigma \) makes the model very flexible.
We note that
- Paramters \( \beta \) are subject specific
- The \( \Sigma \) will reflect dependence
The more standard link function \( logit \) rather than the \( probit \) link is often used and implemented in for example \cite{mm}. The advantage is that one now gets an odds-ratio interpretation of the subject specific effects, but one then needs numerical integration to fit the model.
#We note that
Pairwise odds ratio model
Now the pairwise odds ratio model the specifies that given \( X_1, X_2 \) the marginal models are \[ logit(P(Y_i=1|X_i)) = \alpha_i + X_i^T \beta i=1,2 \]
The primary object of interest are the odds ratio between \(Y_{1}\) and \(Y_{2}\) \[ \gamma_{12} = \frac{ P( Y_{ki} =1 , Y_{kj} =1) P( Y_{ki} =0 , Y_{kj} =0) }{ P( Y_{ki} =1 , Y_{kj} =0) P( Y_{ki} =0 , Y_{kj} =1) } \] given \(X_{ki}\), \(X_{kj}\), and \(Z_{kji}\).
We model the odds ratio with the regression \[ \gamma_{12} = \exp( Z_{12}^T \lambda) \] Where \( Z_{12} \) are some covarites that may influence the odds-ratio between between \(Y_{1}\) and \(Y_{2}\) and contains the marginal covariates, \cite{carey-1993,dale1986global,palmgren1989,molenberghs1994marginal}. This odds-ratio is given covariates as well as marginal covariates. The odds-ratio and marginals specify the joint bivariate distribution via the so-called Placckett-distribution.
One way of fitting this model is the ALR algoritm, the alternating logistic regression ahd this has been described in several papers \cite{kuk2004permutation,kuk2007hybrid,qaqish2012orthogonalized}. We here simply estimate the parameters in a two stage-procedure
- Estimating the marginal parameters via GEE
- Using marginal estimates, estimate dependence parameters
This gives efficient estimates of the dependence parameters because of orthogonality, but some efficiency may be gained for the marginal parameters by using the full likelihood or iterative fitting such as for the ALR.
The pairwise odds-ratio model is very useful, but one do not have a random effects model.
Additive gamma model
Again we operate under marginal logistic regression models are \[ logit(P(Y_i=1|X_i)) = \alpha_i + X_i^T \beta i=1,2 \]
First with just one random effect \( Z \) we assume that conditional on \( Z \) the responses are independent and follow the model \[ logit(P(Y_i=1|X_i,Z)) = exp( -Z \cdot \Psi^{-1}(\lambda_{\bullet},\lambda_{\bullet},P(Y_i=1|X_i)) ) \] where \( \Psi \) is the laplace transform of \( Z \) where we assume that \( Z \) is gamma distributed with variance \( \lambda_{\bullet}^{-1} \) and mean 1. In general \( \Psi(\lambda_1,\lambda_2) \) is the laplace transform of a Gamma distributed random effect with \( Z \) with mean \( \lambda_1/\lambda_2 \) and variance \( \lambda_1/\lambda_2^2 \).
We fit this model by
- Estimating the marginal parameters via GEE
- Using marginal estimates, estimate dependence parameters
To deal with multiple random effects we consider random effects \( Z_i i=1,...,d \) such that \( Z_i \) is gamma distributed with mean \( \lambda_j/\lambda_{\bullet} \) and variance \( \lambda_j/\lambda_{\bullet}^2 \), where we define the scalar \( \lambda_{\bullet} \) below.
Now given a cluster-specific design vector \( V_{12} \) we assume that \[ V_{12}^T Z \] is gamma distributed with mean 1 and variance \( \lambda_{\bullet}^{-1} \) such that critically the random effect variance is the same for all clusters. That is \[ \lambda_{\bullet} = V_{12}^T (\lambda_1,...,\lambda_d)^T \] We return to some specific models below, and show how to fit the ACE and AE model using this set-up.
One last option in the model-specification is to specify how the parameters \( \lambda_1,...,\lambda_d \) are related. We thus can specify a matrix \( M \) of dimension \( p \times d \) such that \[ (\lambda_1,...,\lambda_d)^T = M \theta \] where \( \theta \) is d-dimensional. If \( M \) is diagonal we have no restrictions on parameters.
This parametrization is obtained with the var.par=0 option that thus estimates \( \theta \).
The DEFAULT parametrization instead estimates the variances of the random effecs (var.par=1) via the parameters \( \nu \) \[ M \nu = ( \lambda_1/\lambda_{\bullet}^2, ...,\lambda_d/\lambda_{\bullet}^2)^T \]
The basic modelling assumption is now that given random effects \(Z=(Z_1,...,Z_d)\) we have independent probabilites \[ logit(P(Y_i=1|X_i,Z)) = exp( -V_{12,i}^T Z \cdot \Psi^{-1}(\lambda_{\bullet},\lambda_{\bullet},P(Y_i=1|X_i)) ) i=1,2 \]
We fit this model by
- Estimating the marginal parameters via GEE
- Using marginal estimates, estimate dependence parameters
Even though the model not formaly in this formulation allows negative correlation in practice the paramters can be negative and this reflects negative correlation. An advanatage is that no numerical integration is needed.
The twin-stutter data
We consider the twin-stutter where for pairs of twins that are either dizygotic or monozygotic we have recorded whether the twins are stuttering \cite{twinstut-ref}
We here consider MZ and same sex DZ twins.
Looking at the data
library(mets) data(twinstut) twinstut$binstut <- 1*(twinstut$stutter=="yes") twinsall <- twinstut twinstut <- subset(twinstut,zyg%in%c("mz","dz")) head(twinstut)
Loading required package: timereg Loading required package: survival Loading required package: lava lava version 1.4.7.1 mets version 1.2.1 Attaching package: ‘mets’ The following object is masked _by_ ‘.GlobalEnv’: object.defined tvparnr zyg stutter sex age nr binstut 1 2001005 mz no female 71 1 0 2 2001005 mz no female 71 2 0 3 2001006 dz no female 71 1 0 8 2001012 mz no female 71 1 0 9 2001012 mz no female 71 2 0 11 2001015 dz no male 71 1 0
Pairwise odds ratio model
We start by fitting an overall dependence OR for both MZ and DZ even though the dependence is expected to be different across zygosity.
The first step is to fit the marginal model adjusting for marginal covariates. We here note that there is a rather strong gender effect in the risk of stuttering.
margbin <- glm(binstut~factor(sex)+age,data=twinstut,family=binomial())
summary(margbin)
Call: glm(formula = binstut ~ factor(sex) + age, family = binomial(), data = twinstut) Deviance Residuals: Min 1Q Median 3Q Max -0.4419 -0.4078 -0.2842 -0.2672 2.6395 Coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) -3.027625 0.104012 -29.108 < 2e-16 *** factor(sex)male 0.869826 0.062197 13.985 < 2e-16 *** age -0.005983 0.002172 -2.754 0.00588 ** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 (Dispersion parameter for binomial family taken to be 1) Null deviance: 9328.6 on 21287 degrees of freedom Residual deviance: 9117.0 on 21285 degrees of freedom AIC: 9123 Number of Fisher Scoring iterations: 6
Now estimating the OR parameter. We see a strong dependence with an OR at around 8 that is clearly significant.
bina <- binomial.twostage(margbin,data=twinstut,var.link=1,
clusters=twinstut$tvparnr,detail=0)
summary(bina)
Dependence parameter for Odds-Ratio (Plackett) model With log-link $estimates theta se dependence1 2.085347 0.1274536 $or Estimate Std.Err 2.5% 97.5% P-value dependence1 8.05 1.03 6.04 10.1 4.3e-15 $type [1] "plackett" attr(,"class") [1] "summary.mets.twostage"
Now, and more interestingly, we consider an OR that depends on zygosity and note that MZ have a much larger OR than DZ twins. This type of trait is somewhat complicated to interpret, but clearly, one option is that that there is a genetic effect, alternatively there might be a stronger environmental effect for MZ twins.
### design for OR dependence theta.des <- model.matrix( ~-1+factor(zyg),data=twinstut) bin <- binomial.twostage(margbin,data=twinstut,var.link=1, clusters=twinstut$tvparnr,theta.des=theta.des) summary(bin)
Dependence parameter for Odds-Ratio (Plackett) model With log-link $estimates theta se factor(zyg)dz 0.5221651 0.2401355 factor(zyg)mz 3.4853933 0.1866076 $or Estimate Std.Err 2.5% 97.5% P-value factor(zyg)dz 1.69 0.405 0.892 2.48 3.12e-05 factor(zyg)mz 32.64 6.090 20.699 44.57 8.38e-08 $type [1] "plackett" attr(,"class") [1] "summary.mets.twostage"
We now consider further regression modelling of the OR structure by considering possible interactions between sex and zygozsity. We see that MZ has a much higher dependence and that males have a much lower dependence. We tested for interaction in this model and these were not significant.
twinstut$cage <- scale(twinstut$age) theta.des <- model.matrix( ~-1+factor(zyg)+factor(sex),data=twinstut) bina <- binomial.twostage(margbin,data=twinstut,var.link=1, clusters=twinstut$tvparnr,theta.des=theta.des) summary(bina)
Dependence parameter for Odds-Ratio (Plackett) model With log-link $estimates theta se factor(zyg)dz 0.8098841 0.3138423 factor(zyg)mz 3.7318076 0.2632250 factor(sex)male -0.4075409 0.3055349 $or Estimate Std.Err 2.5% 97.5% P-value factor(zyg)dz 2.248 0.705 0.865 3.63 0.001441 factor(zyg)mz 41.755 10.991 20.213 63.30 0.000145 factor(sex)male 0.665 0.203 0.267 1.06 0.001064 $type [1] "plackett" attr(,"class") [1] "summary.mets.twostage"
Alternative syntax
We now demonstrate how the models can fitted jointly and with anohter syntax, that ofcourse just fits the marginal model and subsequently fits the pairwise OR model.
First noticing as before that MZ twins have a much higher dependence.
## refers to zygosity of first subject in eash pair : zyg1 ## could also use zyg2 (since zyg2=zyg1 within twinpair's) out <- easy.binomial.twostage(stutter~factor(sex)+age,data=twinstut, response="binstut",id="tvparnr",var.link=1, theta.formula=~-1+factor(zyg1)) summary(out)
Dependence parameter for Odds-Ratio (Plackett) model With log-link $estimates theta se factor(zyg1)dz 0.5221651 0.2401355 factor(zyg1)mz 3.4853933 0.1866076 $or Estimate Std.Err 2.5% 97.5% P-value factor(zyg1)dz 1.69 0.405 0.892 2.48 3.12e-05 factor(zyg1)mz 32.64 6.090 20.699 44.57 8.38e-08 $type [1] "plackett" attr(,"class") [1] "summary.mets.twostage"
Now considering all data and estimating separate effects for the OR for opposite sex DZ twins and same sex twins. We here find that os twins are not markedly different from the same sex DZ twins.
## refers to zygosity of first subject in eash pair : zyg1 ## could also use zyg2 (since zyg2=zyg1 within twinpair's)) desfs<-function(x,num1="zyg1",num2="zyg2") c(x[num1]=="dz",x[num1]=="mz",x[num1]=="os")*1 margbinall <- glm(binstut~factor(sex)+age,data=twinsall,family=binomial()) out3 <- easy.binomial.twostage(binstut~factor(sex)+age, data=twinsall,response="binstut",id="tvparnr",var.link=1, theta.formula=desfs,desnames=c("dz","mz","os")) summary(out3)
Dependence parameter for Odds-Ratio (Plackett) model With log-link $estimates theta se dz 0.5278527 0.2396796 mz 3.4850037 0.1864190 os 0.7802940 0.2894394 $or Estimate Std.Err 2.5% 97.5% P-value dz 1.70 0.406 0.899 2.49 3.02e-05 mz 32.62 6.081 20.703 44.54 8.13e-08 os 2.18 0.632 0.944 3.42 5.50e-04 $type [1] "plackett" attr(,"class") [1] "summary.mets.twostage"
Bivariate Probit model
library(mets) data(twinstut) twinstut <- subset(twinstut,zyg%in%c("mz","dz")) twinstut$binstut <- 1*(twinstut$stutter=="yes") head(twinstut)
tvparnr zyg stutter sex age nr binstut 1 2001005 mz no female 71 1 0 2 2001005 mz no female 71 2 0 3 2001006 dz no female 71 1 0 8 2001012 mz no female 71 1 0 9 2001012 mz no female 71 2 0 11 2001015 dz no male 71 1 0
First testing for same dependence in MZ and DZ that we recommend doing by comparing the correlations of MZ and DZ twins. Apart from regression correction in the mean this is an un-structured model, and the useful concordance and casewise concordance estimates can be reported from this analysis.
b1 <- bptwin(binstut~sex,data=twinstut,id="tvparnr",zyg="zyg",DZ="dz",type="un") summary(b1)
Estimate Std.Err Z p-value (Intercept) -1.794823 0.023289 -77.066728 0.0000 sexmale 0.401432 0.030179 13.301813 0.0000 atanh(rho) MZ 1.096916 0.073574 14.909087 0.0000 atanh(rho) DZ 0.132458 0.062516 2.118800 0.0341 Total MZ/DZ Complete pairs MZ/DZ 8777/12511 3255/4058 Estimate 2.5% 97.5% Tetrachoric correlation MZ 0.79939 0.74101 0.84577 Tetrachoric correlation DZ 0.13169 0.00993 0.24960 MZ: Estimate 2.5% 97.5% Concordance 0.01698 0.01411 0.02042 Casewise Concordance 0.46730 0.40383 0.53185 Marginal 0.03634 0.03287 0.04016 Rel.Recur.Risk 12.85882 10.87510 14.84253 log(OR) 3.75632 3.37975 4.13289 DZ: Estimate 2.5% 97.5% Concordance 0.00235 0.00140 0.00393 Casewise Concordance 0.06456 0.03937 0.10413 Marginal 0.03634 0.03287 0.04016 Rel.Recur.Risk 1.77662 0.92746 2.62577 log(OR) 0.63527 0.09013 1.18040 Estimate 2.5% 97.5% Broad-sense heritability 1 NaN NaN
Polygenic modelling
We now turn attention to specific polygenic modelling where special random effects are used to specify ACE, AE, ADE models and so forth. This is very easy with the bptwin function. The key parts of the output are the sizes of the genetic component A and the environmental component, and we can compare with the results of the unstructed model above. Also formally we can test if this submodel is acceptable by a likelihood ratio test.
b1 <- bptwin(binstut~sex,data=twinstut,id="tvparnr",zyg="zyg",DZ="dz",type="ace") summary(b1)
Estimate Std.Err Z p-value (Intercept) -3.70371 0.24449 -15.14855 0 sexmale 0.83310 0.08255 10.09201 0 log(var(A)) 1.18278 0.17179 6.88512 0 log(var(C)) -25.34566 0.34507 -73.45055 0 Total MZ/DZ Complete pairs MZ/DZ 8777/12511 3255/4058 Estimate 2.5% 97.5% A 0.76545 0.70500 0.82590 C 0.00000 0.00000 0.00000 E 0.23455 0.17410 0.29500 MZ Tetrachoric Cor 0.76545 0.69793 0.81948 DZ Tetrachoric Cor 0.38272 0.35210 0.41253 MZ: Estimate 2.5% 97.5% Concordance 0.01560 0.01273 0.01912 Casewise Concordance 0.42830 0.36248 0.49677 Marginal 0.03643 0.03294 0.04027 Rel.Recur.Risk 11.75741 9.77237 13.74246 log(OR) 3.52382 3.13466 3.91298 DZ: Estimate 2.5% 97.5% Concordance 0.00558 0.00465 0.00670 Casewise Concordance 0.15327 0.13749 0.17050 Marginal 0.03643 0.03294 0.04027 Rel.Recur.Risk 4.20744 3.78588 4.62900 log(OR) 1.69996 1.57262 1.82730 Estimate 2.5% 97.5% Broad-sense heritability 0.76545 0.70500 0.82590
b0 <- bptwin(binstut~sex,data=twinstut,id="tvparnr",zyg="zyg",DZ="dz",type="ae") summary(b0)
Estimate Std.Err Z p-value (Intercept) -3.70371 0.24449 -15.14855 0 sexmale 0.83310 0.08255 10.09201 0 log(var(A)) 1.18278 0.17179 6.88512 0 Total MZ/DZ Complete pairs MZ/DZ 8777/12511 3255/4058 Estimate 2.5% 97.5% A 0.76545 0.70500 0.82590 E 0.23455 0.17410 0.29500 MZ Tetrachoric Cor 0.76545 0.69793 0.81948 DZ Tetrachoric Cor 0.38272 0.35210 0.41253 MZ: Estimate 2.5% 97.5% Concordance 0.01560 0.01273 0.01912 Casewise Concordance 0.42830 0.36248 0.49677 Marginal 0.03643 0.03294 0.04027 Rel.Recur.Risk 11.75741 9.77237 13.74246 log(OR) 3.52382 3.13466 3.91298 DZ: Estimate 2.5% 97.5% Concordance 0.00558 0.00465 0.00670 Casewise Concordance 0.15327 0.13749 0.17050 Marginal 0.03643 0.03294 0.04027 Rel.Recur.Risk 4.20744 3.78588 4.62900 log(OR) 1.69996 1.57262 1.82730 Estimate 2.5% 97.5% Broad-sense heritability 0.76545 0.70500 0.82590
Additive gamma random effects
Fitting first a model with different size random effects for MZ and DZ. We note that as before in the OR and biprobit model the dependence is much stronger for MZ twins. We also test if these are the same by parametrizing the OR model with an intercept. This clearly shows a significant difference.
theta.des <- model.matrix( ~-1+factor(zyg),data=twinstut) margbin <- glm(binstut~sex,data=twinstut,family=binomial()) bintwin <- binomial.twostage(margbin,data=twinstut,model="gamma", clusters=twinstut$tvparnr,detail=0,theta=c(0.1)/1,var.link=1, theta.des=theta.des) summary(bintwin) ### test for same dependence in MZ and DZ theta.des <- model.matrix( ~factor(zyg),data=twinstut) margbin <- glm(binstut~sex,data=twinstut,family=binomial()) bintwin <- binomial.twostage(margbin,data=twinstut,model="gamma", clusters=twinstut$tvparnr,detail=0,theta=c(0.1)/1,var.link=1, theta.des=theta.des) summary(bintwin)
Dependence parameter for Clayton-Oakes model Variance of Gamma distributed random effects With log-link $estimates theta se factor(zyg)dz -2.61194495 0.4854454 factor(zyg)mz -0.01817181 0.1030735 $vargam Estimate Std.Err 2.5% 97.5% P-value factor(zyg)dz 0.0734 0.0356 0.00356 0.143 3.94e-02 factor(zyg)mz 0.9820 0.1012 0.78361 1.180 2.96e-22 $type [1] "gamma" attr(,"class") [1] "summary.mets.twostage" Dependence parameter for Clayton-Oakes model Variance of Gamma distributed random effects With log-link $estimates theta se (Intercept) -2.611945 0.4854454 factor(zyg)mz 2.593773 0.4962675 $vargam Estimate Std.Err 2.5% 97.5% P-value (Intercept) 0.0734 0.0356 0.00356 0.143 0.0394 factor(zyg)mz 13.3802 6.6401 0.36573 26.395 0.0439 $type [1] "gamma" attr(,"class") [1] "summary.mets.twostage"
Polygenic modelling
First setting up the random effects design for the random effects and the the relationship between variance parameters. We see that the genetic random effect has size one for MZ and 0.5 for DZ subjects, that have shared and non-shared genetic components with variance 0.5 such that the total genetic variance is the same for all subjects. The shared environmental effect is the samme for all. Thus two parameters with these bands.
out <- twin.polygen.design(twinstut,id="tvparnr",zygname="zyg",zyg="dz",type="ace") head(cbind(out$des.rv,twinstut$tvparnr),10) out$pardes
MZ DZ DZns1 DZns2 env 1 1 0 0 0 1 2001005 2 1 0 0 0 1 2001005 3 0 1 1 0 1 2001006 8 1 0 0 0 1 2001012 9 1 0 0 0 1 2001012 11 0 1 1 0 1 2001015 12 0 1 1 0 1 2001016 13 0 1 0 1 1 2001016 15 0 1 1 0 1 2001020 18 0 1 1 0 1 2001022 [,1] [,2] [1,] 1.0 0 [2,] 0.5 0 [3,] 0.5 0 [4,] 0.5 0 [5,] 0.0 1
Now, fitting the ACE model, we see that the variance of the genetic, component, is 1.5 and the environmental variance is -0.5. Thus suggesting that the ACE model does not fit the data. When the random design is given we automatically use the gamma fralty model.
margbin <- glm(binstut~sex,data=twinstut,family=binomial()) bintwin1 <- binomial.twostage(margbin,data=twinstut, clusters=twinstut$tvparnr,detail=0,theta=c(0.1)/1,var.link=0, random.design=out$des.rv,theta.des=out$pardes) summary(bintwin1)
Dependence parameter for Clayton-Oakes model Variance of Gamma distributed random effects $estimates theta se dependence1 1.5261839 0.2475041 dependence2 -0.5447955 0.1942159 $type [1] "clayton.oakes" $h Estimate Std.Err 2.5% 97.5% P-value dependence1 1.555 0.187 1.189 1.922 9.11e-17 dependence2 -0.555 0.187 -0.922 -0.189 2.99e-03 $vare NULL $vartot Estimate Std.Err 2.5% 97.5% P-value p1 0.981 0.102 0.781 1.18 8.29e-22 attr(,"class") [1] "summary.mets.twostage"
For this model we estimate the concordance and casewise concordance as well as the marginal rates of stuttering for females.
concordance.twin.ace(bintwin1,type="ace")
$MZ Estimate Std.Err 2.5% 97.5% P-value concordance 0.0182 0.00147 0.0153 0.0211 2.61e-35 casewise concordance 0.5033 0.03256 0.4395 0.5672 6.49e-54 marginal 0.0362 0.00188 0.0325 0.0399 7.15e-83 $DZ Estimate Std.Err 2.5% 97.5% P-value concordance 0.00235 0.000589 0.0012 0.00351 6.45e-05 casewise concordance 0.06501 0.015836 0.0340 0.09604 4.04e-05 marginal 0.03620 0.001877 0.0325 0.03988 7.15e-83
The E component was not consistent with the fit of the data and we now consider instead the AE model.
out <- twin.polygen.design(twinstut,id="tvparnr",zygname="zyg",zyg="dz",type="ae") bintwin <- binomial.twostage(margbin,data=twinstut, clusters=twinstut$tvparnr,detail=0,theta=c(0.1)/1,var.link=0, random.design=out$des.rv,theta.des=out$pardes) summary(bintwin)
Dependence parameter for Clayton-Oakes model Variance of Gamma distributed random effects $estimates theta se dependence1 0.9094847 0.09536268 $type [1] "clayton.oakes" $h Estimate Std.Err 2.5% 97.5% P-value dependence1 1 0 1 1 0 $vare NULL $vartot Estimate Std.Err 2.5% 97.5% P-value p1 0.909 0.0954 0.723 1.1 1.47e-21 attr(,"class") [1] "summary.mets.twostage"
Again, the concordance can be computed:
concordance.twin.ace(bintwin,type="ae")
$MZ Estimate Std.Err 2.5% 97.5% P-value concordance 0.0174 0.00143 0.0146 0.0202 5.00e-34 casewise concordance 0.4795 0.03272 0.4154 0.5437 1.20e-48 marginal 0.0362 0.00188 0.0325 0.0399 7.15e-83 $DZ Estimate Std.Err 2.5% 97.5% P-value concordance 0.00477 0.000393 0.0040 0.00554 5.94e-34 casewise concordance 0.13175 0.005417 0.1211 0.14237 1.14e-130 marginal 0.03620 0.001877 0.0325 0.03988 7.15e-83