---
title: "A practical guide to Human Genetics with Lifetime Data"
author: Klaus Holst & Thomas Scheike
date: "2026-07-29"
#output: pdf_document
output:
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fig_caption: no
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%\VignetteIndexEntry{A practical guide to Human Genetics with Lifetime Data}
%\VignetteEngine{knitr::rmarkdown}
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---
This vignette demonstrates how to analyze familial resemblance for
twins using the \texttt{mets} {\bf R}-package and is accompanying the
review by Scheike and Holst (2020).
We consider a data-set in that resembles the data of \cite{Hjelmborg2014} that
were based on the NorTwinCan a collaborative research project studying the genetic and environmental components of prostate
cancer. The data comprises around 18,000 DZ twins and 11,000 MZ twins.
It was a population based register study based on the Danish, Finnish, Norwegian, and Swedish twin registries.
We first illustrate a hazards based analysis to show how one would study
dependence in survival data. This needs to be done under assumptions about independent
competing risks
when the outcome of interest is observed subject to competing risks (here death).
This seems reasonable here since the occurrence of cancer prior to death only contains weak
association with the risk of death for the other twin, and vice-versa.
First looking at the data
``` r
library(mets)
set.seed(122)
data(prt)
dtable(prt,~status+cancer)
#>
#> cancer 0 1
#> status
#> 0 21283 0
#> 1 6997 0
#> 2 0 942
dtable(prt,~zyg+country,level=1)
#>
#> zyg
#> DZ MZ
#> 17991 11231
#>
#> country
#> Denmark Finland Norway Sweden
#> 9671 3926 4107 11518
```
we see that there are 21283 censorings and 6997 deaths (prior to cancer) and
a total of 942 prostate cancers.
Approximately half the data consist of DZ twins.
In addition we see that there are around 10000 twins from Denmark and Sweden, and only 4000 from Norway and Finland, respectively.
Survival
==========
Under assumption of random effects acting independently on different cause
specific hazards we can analyse competing risks data considering the
cause-specific hazard. Typically, this can be questionable and the cumulative
incidence modelling below does not rely on this assumption.
We consider the cause specific hazard of cancer in the competing risks
model with death and cancer.
First estimating the marginal hazards for each country.
``` r
# Marginal Cox model here stratified on country without covariates
margph <- phreg(Surv(time,cancer)~strata(country)+cluster(id),data=prt)
plot(margph)
```
We see that the marginal of Denmark in particular is quite different.
Then we fit a two-stage random effects models with country specific
marginals and random-effects variances that differ for MZ and DZ twins.
``` r
# Clayton-Oakes, MLE , overall variance
fitco1<-twostageMLE(margph,data=prt,theta=2.7)
```
``` r
summary(fitco1)
#> Dependence parameter for Clayton-Oakes model
#> Variance of Gamma distributed random effects
#> $estimates
#> Coef. SE z P-val Kendall tau SE
#> dependence1 2.782962 0.4225572 6.586001 4.518319e-11 0.5818491 0.0369421
#>
#> $type
#> NULL
#>
#> attr(,"class")
#> [1] "summary.mets.twostage"
```
``` r
fitco2 <- survival_twostage(margph,data=prt,theta=2.7,clusters=prt$id,var.link=0)
```
``` r
summary(fitco2)
#> Dependence parameter for Clayton-Oakes model
#> Variance of Gamma distributed random effects
#> $estimates
#> Coef. SE z P-val Kendall tau SE
#> dependence1 2.782962 0.4225529 6.586069 4.516254e-11 0.5818491 0.03694172
#>
#> $type
#> [1] "clayton.oakes"
#>
#> attr(,"class")
#> [1] "summary.mets.twostage"
```
With different random effects for MZ and DZ
``` r
mm <- model.matrix(~-1+factor(zyg),prt)
fitco3<-twostageMLE(margph,data=prt,theta=1,theta.des=mm)
```
``` r
summary(fitco3)
#> Dependence parameter for Clayton-Oakes model
#> Variance of Gamma distributed random effects
#> $estimates
#> Coef. SE z P-val Kendall tau SE
#> factor(zyg)DZ 1.318966 0.3861577 3.415614 6.363831e-04 0.3974027 0.07011148
#> factor(zyg)MZ 5.421921 0.9626267 5.632423 1.776956e-08 0.7305280 0.03495065
#>
#> $type
#> NULL
#>
#> attr(,"class")
#> [1] "summary.mets.twostage"
```
``` r
fitco4 <- survival_twostage(margph,data=prt,theta=1,clusters=prt$id,var.link=0,theta.des=mm)
```
``` r
summary(fitco4)
#> Dependence parameter for Clayton-Oakes model
#> Variance of Gamma distributed random effects
#> $estimates
#> Coef. SE z P-val Kendall tau SE
#> factor(zyg)DZ 1.318966 0.3861745 3.415466 6.367306e-04 0.3974027 0.07011454
#> factor(zyg)MZ 5.421920 0.9625027 5.633148 1.769493e-08 0.7305280 0.03494615
#>
#> $type
#> [1] "clayton.oakes"
#>
#> attr(,"class")
#> [1] "summary.mets.twostage"
round(estimate(coef=fitco4$coef,vcov=fitco4$var.theta)$coefmat[,c(1,3:4)],2)
#> Estimate 2.5% 97.5%
#> p1 1.32 0.56 2.08
#> p2 5.42 3.54 7.31
## mz kendalls tau
kendall_ClaytonOakes_twin_ace(fitco4$theta[2],0,K=1000)$mz.kendall
#> [1] 0.7380818
## dz kendalls tau
kendall_ClaytonOakes_twin_ace(fitco4$theta[1],0,K=1000)$mz.kendall
#> [1] 0.4154807
```
The dependence of MZ twins is much stronger, and is summarized by a
variance at $5.42$ in contrast to the $DZ$ variance at $1.32$.
Now we look at the polygenic modelling for survival data, here applied to
the cause specific hazards.
``` r
### setting up design for random effects and parameters of random effects
desace <- twin_polygen_design(prt,type="ace")
### ace model
fitace <- survival_twostage(margph,data=prt,theta=1,
clusters=prt$id,var.link=0,model="clayton.oakes",
numDeriv=1,random.design=desace$des.rv,theta.des=desace$pardes)
```
``` r
summary(fitace)
#> Dependence parameter for Clayton-Oakes model
#> Variance of Gamma distributed random effects
#> $estimates
#> Coef. SE z P-val Kendall tau SE
#> dependence1 7.223855 1.841525 3.922757 8.754161e-05 0.7831709 0.04328951
#> dependence2 -1.582396 1.088423 -1.453843 1.459898e-01 -3.7892257 12.48240540
#>
#> $type
#> [1] "clayton.oakes"
#>
#> $h
#> Estimate Std.Err 2.5% 97.5% P-value
#> [1,] 1.2805 0.1698 0.9477 1.61324 4.615e-14
#> [2,] -0.2805 0.1698 -0.6132 0.05226 9.850e-02
#>
#> $vare
#> NULL
#>
#> $vartot
#> Estimate Std.Err 2.5% 97.5% P-value
#> p1 5.641 0.9894 3.702 7.581 1.184e-08
#>
#> attr(,"class")
#> [1] "summary.mets.twostage"
```
``` r
### ace model with positive random effects variances
# fitacee <- survival.twostage(margph,data=prt,theta=1,
# clusters=prt$id,var.link=1,model="clayton.oakes",
# numDeriv=1,random.design=desace$des.rv,theta.des=desace$pardes)
#summary(fitacee)
### ae model
#desae <- twin_polygen_design(prt,type="ae")
#fitae <- survival.twostage(margph,data=prt,theta=1,
# clusters=prt$id,var.link=0,model="clayton.oakes",
# numDeriv=1,random.design=desae$des.rv,theta.des=desae$pardes)
#summary(fitae)
### de model
desde <- twin_polygen_design(prt,type="de")
fitde <- survival.twostage(margph,data=prt,theta=1,clusters=prt$id,var.link=0,model="clayton.oakes",
numDeriv=1,random.design=desde$des.rv,theta.des=desde$pardes)
```
``` r
summary(fitde)
#> Dependence parameter for Clayton-Oakes model
#> Variance of Gamma distributed random effects
#> $estimates
#> Coef. SE z P-val Kendall tau SE
#> dependence1 5.940643 0.9837336 6.038873 1.551941e-09 0.7481312 0.03120299
#>
#> $type
#> [1] "clayton.oakes"
#>
#> $h
#> Estimate Std.Err 2.5% 97.5% P-value
#> [1,] 1 0 1 1 0
#>
#> $vare
#> NULL
#>
#> $vartot
#> Estimate Std.Err 2.5% 97.5% P-value
#> p1 5.941 0.9837 4.013 7.869 1.552e-09
#>
#> attr(,"class")
#> [1] "summary.mets.twostage"
```
The DE model fits quite well. In summary all shared variance is due to
genes and there is no suggestion of a shared environmental effect.
Concordance and Casewise
========================
First we estimate the concordance of joint prostate cancer. The two-twins are
censored at the same time, otherwise we would enforce this in the data by
artificially censor both twins at the first censoring time. Given, however,
that we have the same-censoring assumption satisfied we can do the stanadar
Aalen-Johansen product-limit estimator of the concordance probabilities for MZ
and DZ twins.
For simplicity we do not do this for each country even though as we show there are big
differences between the countries.
``` r
prt <- force.same.cens(prt,cause="status")
dtable(prt,~status+cancer)
#>
#> cancer 0 1
#> status
#> 0 21580 77
#> 1 6700 0
#> 2 0 865
dtable(prt,~status+country)
#>
#> country Denmark Finland Norway Sweden
#> status
#> 0 7394 2622 3154 8487
#> 1 2140 1138 830 2592
#> 2 137 166 123 439
dtable(prt,~zyg+country)
#>
#> country Denmark Finland Norway Sweden
#> zyg
#> DZ 6191 2833 2393 6574
#> MZ 3480 1093 1714 4944
```
``` r
## cumulative incidence with cluster standard errors.
cif1 <- cif(Event(time,status)~strata(country)+cluster(id),prt,cause=2)
plot(cif1,se=1)
```
``` r
cifa <- cif(Event(time,status)~+1,prt,cause=2)
### concordance estimator, ignoring country differences.
p11 <- bicomprisk(Event(time,status)~strata(zyg)+id(id),data=prt,cause=c(2,2))
#> Strata 'DZ'
#> Strata 'MZ'
p11mz <- p11$model$"MZ"
p11dz <- p11$model$"DZ"
```
``` r
par(mfrow=c(1,2))
## Concordance
plot(p11mz,ylim=c(0,0.1));
plot(p11dz,ylim=c(0,0.1));
```
Now we compare the concordance to the marginals to get a measure
that takes the marginals into account when evaluating the strength
of the association.
``` r
library(prodlim)
outm <- prodlim(Hist(time,status)~+1,data=prt)
cifzyg <- cif(Event(time,status)~+strata(zyg)+cluster(id),data=prt,cause=2)
cifprt <- cif(Event(time,status)~country+cluster(id),data=prt,cause=2)
times <- 70:100
cifmz <- predict(outm,cause=2,time=times,newdata=data.frame(zyg="MZ")) ## cause is 2 (second cause)
cifdz <- predict(outm,cause=2,time=times,newdata=data.frame(zyg="DZ"))
### concordance for MZ and DZ twins<
cc <- bicomprisk(Event(time,status)~strata(zyg)+id(id),data=prt,cause=c(2,2),prodlim=TRUE)
#> Strata 'DZ'
#> Strata 'MZ'
ccdz <- cc$model$"DZ"
ccmz <- cc$model$"MZ"
cdz <- casewise(ccdz,outm,cause.marg=2)
cmz <- casewise(ccmz,outm,cause.marg=2)
dd <- bicompriskData(Event(time,status)~country+strata(zyg)+id(id),data=prt,cause=c(2,2))
conczyg <- cif(Event(time,status)~strata(zyg)+cluster(id),data=dd,cause=1)
par(mfrow=c(1,2))
plot(conczyg,se=TRUE,col=cols[2:1], lty=ltys[2:1], legend=FALSE,xlab="Age",ylab="Concordance")
legend("topleft",c("concordance-MZ","concordance-DZ"),col=cols[1:2],lty=ltys[1:2])
plot(cmz,ci=NULL,ylim=c(0,.8),xlim=c(70,97),legend=FALSE,col=cols[c(1,3,3)],lty=ltys[c(1,3,3)],
ylab="Casewise",xlab="Age")
plot(cdz,ci=NULL,ylim=c(0,.8),xlim=c(70,97),legend=FALSE,ylab="Casewise",xlab="Age",
col=c(cols[2],NA,NA), lty=ltys[c(2,3,3)], add=TRUE)
with(data.frame(cmz$casewise),plotConfRegionSE(time,casewise.conc,se.casewise,col=cols[1]))
with(data.frame(cdz$casewise),plotConfRegionSE(time,casewise.conc,se.casewise,col=cols[2]))
legend("topleft",c("casewise-MZ","casewise-DZ","marginal"),col=cols, lty=ltys, bg="white")
```
``` r
summary(cdz)
#> Casewise concordance and standard errors
#> time casewise conc se casewise
#> [1,] 59.5 0.0866 0.0865
#> [2,] 60.5 0.0659 0.0659
#> [3,] 61.6 0.0593 0.0593
#> [4,] 62.7 0.0483 0.0483
#> [5,] 63.7 0.0358 0.0358
#> [6,] 64.8 0.0279 0.0279
#> [7,] 65.8 0.0223 0.0223
#> [8,] 66.9 0.0197 0.0197
#> [9,] 68.0 0.0415 0.0297
#> [10,] 69.0 0.0335 0.0240
#> [11,] 70.1 0.0452 0.0264
#> [12,] 71.1 0.0855 0.0352
#> [13,] 72.2 0.0728 0.0300
#> [14,] 73.2 0.0888 0.0317
#> [15,] 74.3 0.1010 0.0321
#> [16,] 75.4 0.1020 0.0310
#> [17,] 76.4 0.1130 0.0318
#> [18,] 77.5 0.1230 0.0320
#> [19,] 78.5 0.1400 0.0334
#> [20,] 79.6 0.1470 0.0332
#> [21,] 80.7 0.1530 0.0329
#> [22,] 81.7 0.1460 0.0307
#> [23,] 82.8 0.1470 0.0298
#> [24,] 83.8 0.1600 0.0307
#> [25,] 84.9 0.1470 0.0282
#> [26,] 86.0 0.1620 0.0297
#> [27,] 87.0 0.1680 0.0300
#> [28,] 88.1 0.1820 0.0311
#> [29,] 89.1 0.1760 0.0301
#> [30,] 90.2 0.1950 0.0323
#> [31,] 91.2 0.2040 0.0332
#> [32,] 92.3 0.1970 0.0321
#> [33,] 93.4 0.1940 0.0315
#> [34,] 94.4 0.1970 0.0318
#> [35,] 95.5 0.1940 0.0314
#> [36,] 96.5 0.1930 0.0312
#> [37,] 97.6 0.2040 0.0330
#> [38,] 98.7 0.2010 0.0325
#> [39,] 99.7 0.1990 0.0322
#> [40,] 101.0 0.1980 0.0321
#> [41,] 102.0 0.1950 0.0316
#> [42,] 103.0 0.1940 0.0314
#> [43,] 104.0 0.1940 0.0314
#> [44,] 105.0 0.1930 0.0312
#> [45,] 106.0 0.1920 0.0311
#> [46,] 107.0 0.1920 0.0311
#> [47,] 108.0 0.1910 NaN
summary(cmz)
#> Casewise concordance and standard errors
#> time casewise conc se casewise
#> [1,] 60.6 0.519 0.2590
#> [2,] 61.6 0.466 0.2330
#> [3,] 62.7 0.380 0.1900
#> [4,] 63.7 0.285 0.1420
#> [5,] 64.8 0.286 0.1280
#> [6,] 65.8 0.228 0.1020
#> [7,] 66.9 0.295 0.1120
#> [8,] 67.9 0.306 0.1090
#> [9,] 68.9 0.327 0.1040
#> [10,] 70.0 0.338 0.0981
#> [11,] 71.0 0.345 0.0926
#> [12,] 72.1 0.399 0.0946
#> [13,] 73.1 0.414 0.0909
#> [14,] 74.2 0.426 0.0874
#> [15,] 75.2 0.388 0.0798
#> [16,] 76.3 0.391 0.0773
#> [17,] 77.3 0.410 0.0769
#> [18,] 78.4 0.392 0.0723
#> [19,] 79.4 0.410 0.0721
#> [20,] 80.5 0.423 0.0714
#> [21,] 81.5 0.400 0.0666
#> [22,] 82.6 0.442 0.0685
#> [23,] 83.6 0.446 0.0676
#> [24,] 84.7 0.433 0.0643
#> [25,] 85.7 0.413 0.0612
#> [26,] 86.8 0.389 0.0578
#> [27,] 87.8 0.396 0.0578
#> [28,] 88.9 0.396 0.0573
#> [29,] 89.9 0.399 0.0574
#> [30,] 91.0 0.386 0.0556
#> [31,] 92.0 0.400 0.0570
#> [32,] 93.1 0.393 0.0560
#> [33,] 94.1 0.415 0.0590
#> [34,] 95.2 0.477 0.0669
#> [35,] 96.2 0.493 0.0690
#> [36,] 97.3 0.511 0.0714
#> [37,] 98.3 0.507 0.0708
#> [38,] 99.4 0.500 0.0699
#> [39,] 100.0 0.525 0.0739
#> [40,] 101.0 0.520 0.0731
#> [41,] 103.0 0.514 0.0723
#> [42,] 104.0 0.541 0.0767
#> [43,] 105.0 0.541 0.0767
cpred(cmz$casewise,c(70,80))
#> new.time casewise conc se casewise
#> [1,] 70 0.3381311 0.09811729
#> [2,] 80 0.4096715 0.07211013
cpred(cdz$casewise,c(70,80))
#> new.time casewise conc se casewise
#> [1,] 70 0.03351513 0.02398337
#> [2,] 80 0.14694869 0.03321234
```
``` r
dd <- bicompriskData(Event(time,status)~country+strata(zyg)+id(id),data=prt,cause=c(2,2))
conczyg <- cif(Event(time,status)~strata(zyg)+cluster(id),data=dd,cause=1)
par(mfrow=c(1,2))
plot(conczyg,se=TRUE,legend=FALSE,xlab="Age",ylab="Concordance")
legend("topleft",c("concordance-DZ","concordance-MZ"),col=c(1,2),lty=1)
plot(cmz,ci=NULL,ylim=c(0,0.6),xlim=c(70,100),legend=FALSE,col=c(2,3,3),ylab="Casewise",xlab="Age",lty=c(1,3))
plot(cdz,ci=NULL,ylim=c(0,0.6),xlim=c(70,100),legend=FALSE,ylab="Casewise",xlab="Age",
col=c(1,3,3), add=TRUE, lty=c(2,3))
legend("topleft",c("casewise-MZ","casewise-DZ","marginal"),col=c(2,1,3),lty=1)
with(data.frame(cmz$casewise),plotConfRegionSE(time,casewise.conc,se.casewise,col=2))
with(data.frame(cdz$casewise),plotConfRegionSE(time,casewise.conc,se.casewise,col=1))
```
The standard errors above are slightly off since they only reflect the
uncertainty from the concordance estimation. This can be improved by
doing specific calculations for a specific time-point uisng the
binomial regression function that gives and iid decomposition for the
paramters. We thus apply the binomial regression to estimate the
concordance as well as the marginal, and combine the iid decompositions
when estimating the standard error. We also do this ignoring country differences.
``` r
### new version of Casewise for specific time-point based on binreg
dd <- bicompriskData(Event(time,status)~country+strata(zyg)+id(id),data=prt,cause=c(2,2))
newdata <- data.frame(zyg=c("DZ","MZ"),id=1)
## concordance
bcif1 <- binreg(Event(time,status)~-1+factor(zyg)+cluster(id),dd,time=80,cause=1,cens.model=~strata(zyg))
pconc <- predict(bcif1,newdata)
## marginal estimates
mbcif1 <- binreg(Event(time,status)~cluster(id),prt,time=80,cause=2)
mc <- predict(mbcif1,newdata)
### casewise with improved se's from log-scale
cse <- binregCasewise(bcif1,mbcif1)
```
``` r
cse
#> $coef
#> Estimate 2.5% 97.5%
#> p1 0.1586277 0.1445496 0.1740770
#> p2 0.4041311 0.3682646 0.4434908
#>
#> $logcoef
#> Estimate Std.Err 2.5% 97.5% P-value
#> p1 -1.841 0.04742 -1.934 -1.7483 0.000e+00
#> p2 -0.906 0.04742 -0.999 -0.8131 2.208e-81
```
It can be useful also to simply model the concordance given covariates, and in
this case we might find it important to adjust for country, or to see if the
differences between MZ and DZ are comparable across contries even though
clearly DK has a much lower cumulative incidence of prostate cancer.
``` r
### semi-parametric modelling of concordance
dd <- bicompriskData(Event(time,status)~country+strata(zyg)+id(id),data=prt,cause=c(2,2))
regconc <- cifreg(Event(time,status)~country*zyg,data=dd,prop=NULL)
regconc
#> Call:
#> cifreg(formula = Event(time, status) ~ country * zyg, data = dd,
#> propodds = NULL)
#>
#> n events
#> 14222 106
#>
#> coefficients:
#> countryFinland countryNorway countrySweden
#> 2.207372 2.316572 2.022366
#> zygMZ countryFinland:zygMZ countryNorway:zygMZ
#> 2.308718 -1.267368 -2.146122
#> countrySweden:zygMZ
#> -1.325599
### interaction test
summary(estimate(regconc), contrast=list(5,6,7))
#> Call: estimate.default(x = b)
#> ────────────────────────────────────────────────────────────
#> Estimate Std.Err 2.5% 97.5% P-value
#> countryFinland 2.207 0.7628 0.7123 3.7024 0.003807
#> countryNorway 2.317 0.7899 0.7684 3.8648 0.003361
#> countrySweden 2.022 0.7397 0.5725 3.4722 0.006258
#> zygMZ 2.309 0.7740 0.7918 3.8257 0.002855
#> countryFinland:zygMZ -1.267 0.8706 -2.9737 0.4390 0.145467
#> countryNorway:zygMZ -2.146 0.9435 -3.9953 -0.2969 0.022926
#> countrySweden:zygMZ -1.326 0.8219 -2.9366 0.2854 0.106797
#> ────────────────────────────────────────────────────────────
#> Null Hypothesis:
#> [countryFinland:zygMZ] = 0
#> [countryNorway:zygMZ] = 0
#> [countrySweden:zygMZ] = 0
#>
#> chisq = 5.2661, df = 3, p-value = 0.1533
regconc <- cifreg(Event(time,status)~country+zyg,data=dd,prop=NULL)
regconc
#> Call:
#> cifreg(formula = Event(time, status) ~ country + zyg, data = dd,
#> propodds = NULL)
#>
#> n events
#> 14222 106
#>
#> coefficients:
#> countryFinland countryNorway countrySweden zygMZ
#> 1.2924623 0.9333627 1.0743929 1.0105487
## logistic link
logitregconc <- cifreg(Event(time,status)~country+zyg,data=dd)
slr <- summary(logitregconc)
```
``` r
slr
#>
#> n events
#> 14222 106
#>
#> 14222 clusters
#> coefficients:
#> Estimate S.E. dU^-1/2 P-value
#> countryFinland 1.30427 0.35262 0.35146 0.0002
#> countryNorway 0.94077 0.39999 0.39365 0.0187
#> countrySweden 1.08494 0.32247 0.31871 0.0008
#> zygMZ 1.02335 0.20283 0.19873 0.0000
#>
#> exp(coefficients):
#> Estimate 2.5% 97.5%
#> countryFinland 3.6850 1.8462 7.3552
#> countryNorway 2.5619 1.1698 5.6111
#> countrySweden 2.9592 1.5729 5.5676
#> zygMZ 2.7825 1.8697 4.1408
```
Competing risk using additive Gamma
====================================
Here we do the cumulative incidence random effects modelling (commented out to avoid timereg dependence)
``` r
timereg <- 0
if (timereg==1) {
times <- seq(50,90,length.out=5)
cif1 <- timereg::comp.risk(Event(time,status)~-1+factor(country)+cluster(id),prt,
cause=2,times=times,max.clust=NULL)
mm <- model.matrix(~-1+factor(zyg),prt)
out1<-random.cif(cif1,data=prt,cause1=2,cause2=2,theta=1,
theta.des=mm,same.cens=TRUE,step=0.5)
summary(out1)
round(estimate(coef=out1$theta,vcov=out1$var.theta)$coefmat[,c(1,3:4)],2)
desace <- twin.polygen.design(prt,type="ace")
outacem <- Grandom.cif(cif1,data=prt,cause1=2,cause2=2,
same.cens=TRUE,theta=c(0.45,0.15),var.link=0,
step=0.5,theta.des=desace$pardes,random.design=desace$des.rv)
##outacem$score
}
```
``` r
timereg <- 0
if (timereg==1) {
summary(outacem)
### variances
estimate(coef=outacem$theta,vcov=outacem$var.theta,f=function(p) p/sum(p)^2)
## AE polygenic model
# desae <- twin.polygen.design(prt,type="ae")
# outaem <- Grandom.cif(cif1,data=prt,cause1=2,cause2=2,
# same.cens=TRUE,theta=c(0.45,0.15),var.link=0,
# step=0.5,theta.des=desae$pardes,random.design=desae$des.rv)
# outaem$score
# summary(outaem)
# estimate(coef=outaem$theta,vcov=outaem$var.theta,f=function(p) p/sum(p)^2)
## AE polygenic model
# desde <- twin.polygen.design(prt,type="de")
# outaem <- Grandom.cif(cif1,data=prt,cause1=2,cause2=2,
# same.cens=TRUE,theta=c(0.35),var.link=0,
# step=0.5,theta.des=desde$pardes,random.design=desde$des.rv)
# outaem$score
# summary(outaem)
# estimate(coef=outaem$theta,vcov=outaem$var.theta,f=function(p) p/sum(p)^2)
times <- 90
cif1 <- timereg::comp.risk(Event(time,status)~-1+factor(country)+cluster(id),prt,
cause=2,times=times,max.clust=NULL)
mm <- model.matrix(~-1+factor(zyg),prt)
out1<-random.cif(cif1,data=prt,cause1=2,cause2=2,theta=1,
theta.des=mm,same.cens=TRUE,step=0.5)
summary(out1)
round(estimate(coef=out1$theta,vcov=out1$var.theta)$coefmat[,c(1,3:4)],2)
desde <- twin.polygen.design(prt,type="de")
outaem <- Grandom.cif(cif1,data=prt,cause1=2,cause2=2,
same.cens=TRUE,theta=c(0.35),var.link=0,
step=0.5,theta.des=desde$pardes,random.design=desde$des.rv)
outaem$score
summary(outaem)
estimate(coef=outaem$theta,vcov=outaem$var.theta,f=function(p) p/sum(p)^2)
}
```
Competing risk modeling using the Liabilty Threshold model
===========================================================
First we fit the bivariate probit model (same marginals in MZ and DZ twins but different correlation parameter). Here we evaluate the risk of getting cancer before the last double cancer event (95 years)
``` r
rm(prt)
data(prt)
prt0 <- force.same.cens(prt, cause="status", cens.code=0, time="time", id="id")
prt0$country <- relevel(prt0$country, ref="Sweden")
prt_wide <- fast.reshape(prt0, id="id", num="num", varying=c("time","status","cancer"))
prt_time <- subset(prt_wide, cancer1 & cancer2, select=c(time1, time2, zyg))
tau <- 95
tt <- seq(70, tau, length.out=5) ## Time points to evaluate model in
```
``` r
b0 <- bptwin.time(cancer ~ 1, data=prt0, id="id", zyg="zyg", DZ="DZ", type="cor",
cens.formula=Surv(time,status==0)~zyg, breaks=tau)
```
``` r
summary(b0)
#>
#> Estimate Std.Err Z p-value
#> (Intercept) -1.348188 0.026276 -51.3086 < 2.2e-16 ***
#> atanh(rho) MZ 0.735992 0.087838 8.3789 < 2.2e-16 ***
#> atanh(rho) DZ 0.353022 0.068234 5.1737 2.295e-07 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Total MZ/DZ Complete pairs MZ/DZ
#> 1994/3618 997/1809
#>
#> Estimate 2.5% 97.5%
#> Tetrachoric correlation MZ 0.62672 0.51081 0.72024
#> Tetrachoric correlation DZ 0.33905 0.21584 0.45164
#>
#> MZ:
#> Estimate 2.5% 97.5%
#> Concordance 0.03504 0.02779 0.04409
#> Casewise Concordance 0.39458 0.31876 0.47584
#> Marginal 0.08880 0.08086 0.09743
#> Rel.Recur.Risk 4.44351 3.50521 5.38182
#> log(OR) 2.34131 1.87105 2.81157
#> DZ:
#> Estimate 2.5% 97.5%
#> Concordance 0.01952 0.01449 0.02625
#> Casewise Concordance 0.21983 0.16667 0.28415
#> Marginal 0.08880 0.08086 0.09743
#> Rel.Recur.Risk 2.47556 1.81095 3.14016
#> log(OR) 1.23088 0.81020 1.65156
#>
#> Estimate 2.5% 97.5%
#> Broad-sense heritability 0.57533 0.25790 0.89276
#>
#>
#> Event of interest before time 95
```
Liability threshold model with ACE random effects structure
``` r
b1 <- bptwin.time(cancer ~ 1, data=prt0, id="id", zyg="zyg", DZ="DZ", type="ace",
cens.formula=Surv(time,status==0)~zyg, breaks=tau)
```
``` r
summary(b1)
#>
#> Estimate Std.Err Z p-value
#> (Intercept) -2.20664 0.16463 -13.4034 <2e-16 ***
#> log(var(A)) 0.43261 0.39149 1.1050 0.2691
#> log(var(C)) -1.98291 2.52347 -0.7858 0.4320
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Total MZ/DZ Complete pairs MZ/DZ
#> 1994/3618 997/1809
#>
#> Estimate 2.5% 97.5%
#> A 0.57533 0.25790 0.89276
#> C 0.05139 -0.20836 0.31114
#> E 0.37328 0.26874 0.47782
#> MZ Tetrachoric Cor 0.62672 0.51081 0.72024
#> DZ Tetrachoric Cor 0.33905 0.21584 0.45164
#>
#> MZ:
#> Estimate 2.5% 97.5%
#> Concordance 0.03504 0.02779 0.04409
#> Casewise Concordance 0.39458 0.31876 0.47584
#> Marginal 0.08880 0.08086 0.09743
#> Rel.Recur.Risk 4.44351 3.50520 5.38182
#> log(OR) 2.34131 1.87105 2.81157
#> DZ:
#> Estimate 2.5% 97.5%
#> Concordance 0.01952 0.01449 0.02625
#> Casewise Concordance 0.21983 0.16667 0.28415
#> Marginal 0.08880 0.08086 0.09743
#> Rel.Recur.Risk 2.47556 1.81095 3.14016
#> log(OR) 1.23088 0.81020 1.65156
#>
#> Estimate 2.5% 97.5%
#> Broad-sense heritability 0.57533 0.25790 0.89276
#>
#>
#> Event of interest before time 95
```
In this case the ACE model fits the data well - it is in fact indistinguishable from the flexible bivariate Probit model as seen by the IPCW weighted AIC measure
``` r
AIC(b0, b1)
#> df AIC
#> b0 3 17340.13
#> b1 3 17340.13
```
ACE model with marginal adjusted for country
``` r
b2 <- bptwin.time(cancer ~ country, data=prt0, id="id", zyg="zyg", DZ="DZ", type="ace",
cens.formula=Surv(time,status==0)~zyg+country, breaks=95)
```
``` r
summary(b2)
#>
#> Estimate Std.Err Z p-value
#> (Intercept) -1.97165 0.15371 -12.8267 < 2.2e-16 ***
#> countryDenmark -0.72489 0.11920 -6.0812 1.193e-09 ***
#> countryFinland 0.18968 0.12518 1.5152 0.1297
#> countryNorway -0.11611 0.16621 -0.6986 0.4848
#> log(var(A)) 0.40388 0.40524 0.9966 0.3189
#> log(var(C)) -3.88760 17.56390 -0.2213 0.8248
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Total MZ/DZ Complete pairs MZ/DZ
#> 1994/3618 997/1809
#>
#> Estimate 2.5% 97.5%
#> A 0.59474 0.25169 0.93779
#> C 0.00814 -0.27297 0.28925
#> E 0.39712 0.28435 0.50989
#> MZ Tetrachoric Cor 0.60288 0.47809 0.70381
#> DZ Tetrachoric Cor 0.30551 0.17238 0.42767
#>
#> MZ:
#> Estimate 2.5% 97.5%
#> Concordance 0.04295 0.03307 0.05561
#> Casewise Concordance 0.40128 0.32263 0.48535
#> Marginal 0.10703 0.09453 0.12096
#> Rel.Recur.Risk 3.74923 2.94155 4.55690
#> log(OR) 2.15979 1.67935 2.64023
#> DZ:
#> Estimate 2.5% 97.5%
#> Concordance 0.02439 0.01747 0.03396
#> Casewise Concordance 0.22788 0.17060 0.29749
#> Marginal 0.10703 0.09453 0.12096
#> Rel.Recur.Risk 2.12912 1.54508 2.71315
#> log(OR) 1.06262 0.62460 1.50064
#>
#> Estimate 2.5% 97.5%
#> Broad-sense heritability 0.59474 0.25169 0.93779
#>
#>
#> Event of interest before time 95
```
``` r
bt0 <- bptwin.time(cancer ~ 1, data=prt0, id="id", zyg="zyg", DZ="DZ", type="ace",
cens.formula=Surv(time,status==0)~zyg,
summary.function=function(x) x, breaks=tt)
h2 <- Reduce(rbind, lapply(bt0$coef, function(x) x$heritability))[,c(1,3,4),drop=FALSE]
concMZ <- Reduce(rbind, lapply(bt0$coef, function(x) x$probMZ["Concordance",,drop=TRUE]))
```
``` r
par(mfrow=c(1,2))
plot(tt, h2[,1], type="s", lty=1, col=cols[3], xlab="Age", ylab="Heritability", ylim=c(0,1))
lava::confband(tt, h2[,2], h2[,3],polygon=TRUE, step=TRUE, col=lava::Col(cols[3], 0.1), border=NA)
plot(tt, concMZ[,1], type="s", lty=1, col=cols[1], xlab="Age", ylab="Concordance", ylim=c(0,.1))
lava::confband(tt, concMZ[,2], concMZ[,3],polygon=TRUE, step=TRUE, col=lava::Col(cols[1], 0.1), border=NA)
```
Bivariate probit model at time different time points
``` r
a.mz <- biprobit.time(cancer~1, id="id", data=subset(prt0, zyg=="MZ"),
cens.formula = Surv(time,status==0)~1, pairs.only=TRUE,
breaks=tt)
a.dz <- biprobit.time(cancer~1, id="id", data=subset(prt0, zyg=="DZ"),
cens.formula = Event(time,status==0)~1, pairs.only=TRUE,
breaks=tt)
#system.time(a.zyg <- biprobit.time(cancer~1, rho=~1+zyg, id="id", data=prt,
# cens.formula = Event(time,status==0)~1,
# eqmarg=FALSE, fix.cens.weight
# breaks=seq(75,100,by=10)))
a.mz
#>
#> 1:Concordance
#> 2:Casewise Concordance
#> 3:Marginal
#> 4:Rel.Recur.Risk
#> 5:OR
#> 6:Tetrachoric correlation
#>
#> Time 1:Concor... 2:Casewi... 3:Marginal 4:Rel.Re... 5:OR 6:Tetrac...
#> [1,] 70.00 0.0049 0.2976 0.0166 17.9647 35.3869 0.6973
#> [2,] 76.25 0.0125 0.3468 0.0362 9.5940 21.1451 0.6834
#> [3,] 82.50 0.0247 0.3759 0.0656 5.7307 13.1453 0.6481
#> [4,] 88.75 0.0308 0.3675 0.0839 4.3781 9.4430 0.5993
#> [5,] 95.00 0.0409 0.4144 0.0988 4.1952 10.3180 0.6352
a.dz
#>
#> 1:Concordance
#> 2:Casewise Concordance
#> 3:Marginal
#> 4:Rel.Recur.Risk
#> 5:OR
#> 6:Tetrachoric correlation
#>
#> Time 1:Concor... 2:Casewi... 3:Marginal 4:Rel.Re... 5:OR 6:Tetrac...
#> [1,] 70.00 0.0007 0.0767 0.0088 8.6701 9.9968 0.3855
#> [2,] 76.25 0.0037 0.1612 0.0228 7.0633 9.6181 0.4682
#> [3,] 82.50 0.0074 0.1660 0.0445 3.7329 4.9289 0.3752
#> [4,] 88.75 0.0136 0.2001 0.0680 2.9417 4.0348 0.3614
#> [5,] 95.00 0.0174 0.2091 0.0831 2.5163 3.4243 0.3336
plot(conczyg,se=TRUE,legend=FALSE,xlab="Age",ylab="Concordance", ylim=c(0,0.07))
plot(a.mz, ylim=c(0,.07), col=cols[1], lty=ltys[1], legend=FALSE, add=TRUE)
plot(a.dz, col=cols[2], lty=ltys[2], add=TRUE)
```
Bivariate probit model adjusting for country
``` r
a.mz_country <- biprobit.time(cancer~country, id="id", data=subset(prt0, zyg=="MZ"),
cens.formula = Surv(time,status==0)~country, pairs.only=TRUE,
breaks=tt)
a.dz_country <- biprobit.time(cancer~country, id="id", data=subset(prt0, zyg=="DZ"),
cens.formula = Event(time,status==0)~country, pairs.only=TRUE,
breaks=tt)
s_mz_country <- summary(a.mz_country)
s_dz_country <- summary(a.dz_country)
```
``` r
s_mz_country
#> $Concordance
#> Time Estimate 2.5% 97.5%
#> 1 70.00 0.00592116 0.002986187 0.01170690
#> 2 76.25 0.01432204 0.008753604 0.02334930
#> 3 82.50 0.02921012 0.020110789 0.04224904
#> 4 88.75 0.04037734 0.029107095 0.05576080
#> 5 95.00 0.04970743 0.035562694 0.06907515
#>
#> $`Casewise Concordance`
#> Time Estimate 2.5% 97.5%
#> 1 70.00 0.3061394 0.1804998 0.4691652
#> 2 76.25 0.3502164 0.2491859 0.4667449
#> 3 82.50 0.3867092 0.3017318 0.4791950
#> 4 88.75 0.3906869 0.3129362 0.4744164
#> 5 95.00 0.4190152 0.3369656 0.5058026
#>
#> $Marginal
#> Time Estimate 2.5% 97.5%
#> 1 70.00 0.01934138 0.01265752 0.02944941
#> 2 76.25 0.04089483 0.02990128 0.05569823
#> 3 82.50 0.07553512 0.05950340 0.09544783
#> 4 88.75 0.10334964 0.08432125 0.12608082
#> 5 95.00 0.11862917 0.09702101 0.14428086
#>
#> $Rel.Recur.Risk
#> Time Estimate 2.5% 97.5%
#> 1 70.00 15.828206 6.402412 25.254000
#> 2 76.25 8.563831 5.232353 11.895310
#> 3 82.50 5.119594 3.659965 6.579223
#> 4 88.75 3.780244 2.838404 4.722085
#> 5 95.00 3.532143 2.733073 4.331213
#>
#> $OR
#> Time Estimate 2.5% 97.5%
#> 1 70.00 31.799536 12.251225 82.53954
#> 2 76.25 18.914488 9.701134 36.87794
#> 3 82.50 11.952708 7.033263 20.31308
#> 4 88.75 8.488621 5.238032 13.75644
#> 5 95.00 8.501683 5.275432 13.70099
#>
#> $`Tetrachoric correlation`
#> Time Estimate 2.5% 97.5%
#> 1 70.00 0.6943849 0.4989198 0.8226244
#> 2 76.25 0.6744412 0.5302683 0.7807050
#> 3 82.50 0.6416861 0.5173829 0.7394707
#> 4 88.75 0.5945601 0.4712735 0.6950575
#> 5 95.00 0.6079029 0.4838645 0.7079982
s_dz_country
#> $Concordance
#> Time Estimate 2.5% 97.5%
#> 1 70.00 0.0009355085 0.0003014655 0.002899205
#> 2 76.25 0.0053849065 0.0030067815 0.009625779
#> 3 82.50 0.0090012641 0.0055139065 0.014661739
#> 4 88.75 0.0172556341 0.0117010771 0.025379254
#> 5 95.00 0.0221106772 0.0153703190 0.031711692
#>
#> $`Casewise Concordance`
#> Time Estimate 2.5% 97.5%
#> 1 70.00 0.08242731 0.02825081 0.2172688
#> 2 76.25 0.17830239 0.10976467 0.2763507
#> 3 82.50 0.16491577 0.11033790 0.2392308
#> 4 88.75 0.20763626 0.15174905 0.2773756
#> 5 95.00 0.21841377 0.16270904 0.2866602
#>
#> $Marginal
#> Time Estimate 2.5% 97.5%
#> 1 70.00 0.01134950 0.007572628 0.01697786
#> 2 76.25 0.03020098 0.022824994 0.03986330
#> 3 82.50 0.05458098 0.044180067 0.06725818
#> 4 88.75 0.08310511 0.069910900 0.09852566
#> 5 95.00 0.10123298 0.086500435 0.11815021
#>
#> $Rel.Recur.Risk
#> Time Estimate 2.5% 97.5%
#> 1 70.00 7.262640 -0.6527463 15.178027
#> 2 76.25 5.903861 2.9616380 8.846085
#> 3 82.50 3.021488 1.8476330 4.195343
#> 4 88.75 2.498478 1.7411885 3.255767
#> 5 95.00 2.157536 1.5525687 2.762503
#>
#> $OR
#> Time Estimate 2.5% 97.5%
#> 1 70.00 8.438348 2.374765 29.984323
#> 2 76.25 8.262968 4.176798 16.346645
#> 3 82.50 3.898749 2.289201 6.639977
#> 4 88.75 3.386718 2.150855 5.332698
#> 5 95.00 2.894875 1.876319 4.466352
#>
#> $`Tetrachoric correlation`
#> Time Estimate 2.5% 97.5%
#> 1 70.00 0.3735042 0.1024577 0.5929218
#> 2 76.25 0.4623450 0.2958916 0.6015474
#> 3 82.50 0.3333242 0.1903655 0.4624397
#> 4 88.75 0.3304641 0.1996073 0.4497404
#> 5 95.00 0.3013521 0.1715672 0.4208554
```
``` r
## ACE model (time-varying) with and without adjustment for country
a1 <- bptwin.time(cancer~1, id="id", data=prt0, type="ace",
zyg="zyg", DZ="DZ",
cens.formula=Surv(time,status==0)~zyg,
breaks=tt)
#a2 <- bptwin.time(cancer~country, id="id", data=prt0, #type="ace",
# zyg="zyg", DZ="DZ",
# #cens.formula=Surv(time,status==0)~country+zyg,
# breaks=tt)
```
``` r
plot(a.mz, which=c(6), xlab="Age", ylab="Correlation", ylim=c(0,1), col=cols[1], lty=ltys[1], legend=NULL, alpha=.1)
plot(a.dz, which=c(6), col=cols[2], lty=ltys[2], legend=NULL, add=TRUE, alpha=.1)
legend("topleft", c("MZ tetrachoric correlation", "DZ tetrachoric correlation"),
col=cols, lty=ltys, lwd=2)
```
``` r
plot(a.mz, which=c(4), xlab="Age", ylab="Relative Recurrence Risk",
ylim=c(1,20), col=cols[1], lty=ltys[1], legend=NULL, lwd=2, alpha=.1)
plot(a.dz, which=c(4), col=cols[2], lty=ltys[2], legend=NULL, add=TRUE, lwd=2, alpha=.1)
legend("topright", c("MZ relative recurrence risk", "DZ relative recurrence risk"),
col=cols, lty=ltys, lwd=2)
```
``` r
plot(a1, which=c(5,6), xlab="Age", ylab="Correlation", ylim=c(0,1), col=cols[1:2], lty=ltys[1:2], lwd=2, alpha=0.1,
legend=c("MZ tetrachoric correlation", "DZ tetrachoric correlation"))
```
``` r
plot(a1, which=c(1), xlab="Age", ylim=c(0,1), col="black", lty=1, ylab="Heritability", legend=NULL, alpha=.1)
```
SessionInfo
============
``` r
sessionInfo()
#> R version 4.6.0 (2026-04-24)
#> Platform: x86_64-pc-linux-gnu
#> Running under: Ubuntu 24.04.4 LTS
#>
#> Matrix products: default
#> BLAS: /home/kkzh/.asdf/installs/r/4.6.0/lib/R/lib/libRblas.so
#> LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.12.0 LAPACK version 3.12.0
#>
#> locale:
#> [1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C
#> [3] LC_TIME=en_US.UTF-8 LC_COLLATE=en_US.UTF-8
#> [5] LC_MONETARY=en_US.UTF-8 LC_MESSAGES=en_US.UTF-8
#> [7] LC_PAPER=en_US.UTF-8 LC_NAME=C
#> [9] LC_ADDRESS=C LC_TELEPHONE=C
#> [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C
#>
#> time zone: Europe/Copenhagen
#> tzcode source: system (glibc)
#>
#> attached base packages:
#> [1] splines stats graphics grDevices utils datasets methods
#> [8] base
#>
#> other attached packages:
#> [1] prodlim_2026.03.11 timereg_2.0.7 survival_3.8-6 mets_1.3.12
#>
#> loaded via a namespace (and not attached):
#> [1] cli_3.6.6 knitr_1.51 rlang_1.2.0
#> [4] xfun_0.57 KernSmooth_2.23-26 otel_0.2.0
#> [7] data.table_1.18.4 future.apply_1.20.2 listenv_0.10.1
#> [10] lava_1.9.2.1 stats4_4.6.0 grid_4.6.0
#> [13] evaluate_1.0.5 yaml_2.3.12 mvtnorm_1.3-7
#> [16] numDeriv_2016.8-1.1 compiler_4.6.0 codetools_0.2-20
#> [19] Rcpp_1.1.1-1.1 ucminf_1.2.3 future_1.70.0
#> [22] lattice_0.22-9 digest_0.6.39 parallelly_1.47.0
#> [25] parallel_4.6.0 Matrix_1.7-5 tools_4.6.0
#> [28] RcppArmadillo_15.2.6-1 globals_0.19.1
```