# Twin models In mets: Analysis of Multivariate Event Times

This document provides a brief tutorial to analyzing twin data using the mets package:

options(warn=-1, family="Times")
knitr::opts_chunk$set( collapse = TRUE, comment = "#>", #dev="png", #dpi=72, out.width = "70%") library("mets")  ( \newcommand{\cov}{\mathbb{C}\text{ov}} \newcommand{\cor}{\mathbb{C}\text{or}} \newcommand{\var}{\mathbb{V}\text{ar}} \newcommand{\E}{\mathbb{E}} \newcommand{\unitfrac}[2]{#1/#2} \newcommand{\n}{} ) The development version may be installed from github: # install.packages("remotes") remotes::install_github("kkholst/mets", dependencies="Suggests")  # Twin analysis, continuous traits In the following we examine the heritability of Body Mass Index\n{}korkeila_bmi_1991 hjelmborg_bmi_2008, based on data on self-reported BMI-values from a random sample of 11,411 same-sex twins. First, we will load data library(mets) data("twinbmi") head(twinbmi)  The data is on long format with one subject per row. • tvparnr: twin id • bmi: Body Mass Index ((\mathrm{kg}/{\mathrm{m}^2})) • age: Age (years) • gender: Gender factor (male,female) • zyg: zygosity (MZ, DZ) We transpose the data allowing us to do pairwise analyses twinwide <- fast.reshape(twinbmi, id="tvparnr",varying=c("bmi")) head(twinwide)  Next we plot the association within each zygosity group library("cowplot") scatterdens <- function(x) { require(ggplot2) sp <- ggplot(x, aes_string(colnames(x)[1], colnames(x)[2])) + theme_minimal() + geom_point(alpha=0.3) + geom_density_2d() xdens <- ggplot(x, aes_string(colnames(x)[1],fill=1)) + theme_minimal() + geom_density(alpha=.5)+ theme(axis.text.x = element_blank(), legend.position = "none") + labs(x=NULL) ydens <- ggplot(x, aes_string(colnames(x)[2],fill=1)) + theme_minimal() + geom_density(alpha=.5) + theme(axis.text.y = element_blank(), axis.text.x = element_text(angle=90, vjust=0), legend.position = "none") + labs(x=NULL) + coord_flip() g <- plot_grid(xdens,NULL,sp,ydens, ncol=2,nrow=2, rel_widths=c(4,1.4),rel_heights=c(1.4,4)) return(g) }  We here show the log-transformed data which is slightly more symmetric and more appropiate for the twin analysis (see Figure \@ref(fig:scatter1) and \@ref(fig:scatter2)) mz <- log(subset(twinwide, zyg=="MZ")[,c("bmi1","bmi2")]) scatterdens(mz)  dz <- log(subset(twinwide, zyg=="DZ")[,c("bmi1","bmi2")]) scatterdens(dz)  The plots and raw association measures shows considerable stronger dependence in the MZ twins, thus indicating genetic influence of the trait cor.test(mz[,1],mz[,2], method="spearman")  cor.test(dz[,1],dz[,2], method="spearman")  Ńext we examine the marginal distribution (GEE model with working independence) l0 <- lm(bmi ~ gender + I(age-40), data=twinbmi) estimate(l0, id=twinbmi$tvparnr)

library("splines")
l1 <- lm(bmi ~ gender*ns(age,3), data=twinbmi)
marg1 <- estimate(l1, id=twinbmi\$tvparnr)

dm <- Expand(twinbmi,
bmi=0,
gender=c("male"),
age=seq(33,61,length.out=50))
df <- Expand(twinbmi,
bmi=0,
gender=c("female"),
age=seq(33,61,length.out=50))

plot(marg1, function(p) model.matrix(l1,data=dm)%*%p,
data=dm["age"], ylab="BMI", xlab="Age",
ylim=c(22,26.5))
plot(marg1, function(p) model.matrix(l1,data=df)%*%p,
legend("bottomright", c("Male","Female"),
col=c("black","red"), lty=1, bty="n")


## Polygenic model

We can decompose the trait into the following variance components

\begin{align} Y_i = A_i + D_i + C + E_i, \quad i=1,2 \end{align}

• (A): Additive genetic effects of alleles
• (D): Dominante genetic effects of alleles
• (C): Shared environmental effects
• (E): Unique environmental genetic effects

Dissimilarity of MZ twins arises from unshared environmental effects only, (\cor(E_1,E_2)=0) and

\begin{align} \cor(A_1^{MZ},A_2^{MZ}) = 1, \quad \cor(D_1^{MZ},D_2^{MZ}) = 1, \end{align}

\begin{align} \cor(A_1^{DZ},A_2^{DZ}) = 0.5, \quad \cor(D_1^{DZ},D_2^{DZ}) = 0.25, \end{align}

\begin{align} Y_i = A_i + C_i + D_i + E_i \end{align}

\begin{align} A_i \sim\mathcal{N}(0,\sigma_A^2), C_i \sim\mathcal{N}(0,\sigma_C^2), D_i \sim\mathcal{N}(0,\sigma_D^2), E_i \sim\mathcal{N}(0,\sigma_E^2) \end{align}

\begin{gather} \cov(Y_{1},Y_{2}) = \ \begin{pmatrix} \sigma_A^2 & 2\Phi\sigma_A^2 \ 2\Phi\sigma_A^2 & \sigma_A^2 \end{pmatrix} + \begin{pmatrix} \sigma_C^2 & \sigma_C^2 \ \sigma_C^2 & \sigma_C^2 \end{pmatrix} + \begin{pmatrix} \sigma_D^2 & \Delta_{7}\sigma_D^2 \ \Delta_{7}\sigma_D^2 & \sigma_D^2 \end{pmatrix} + \begin{pmatrix} \sigma_E^2 & 0 \ 0 & \sigma_E^2 \end{pmatrix} \end{gather}

dd <- na.omit(twinbmi)
l0 <- twinlm(bmi ~ age+gender, data=dd, DZ="DZ", zyg="zyg", id="tvparnr", type="sat")

# different marginals (but within pair)
lf <- twinlm(bmi ~ age+gender, data=dd,DZ="DZ", zyg="zyg", id="tvparnr", type="flex")

# same marginals but free correlation with MZ, DZ
lu <- twinlm(bmi ~ age+gender, data=dd, DZ="DZ", zyg="zyg", id="tvparnr", type="u")
estimate(lu,contr(5:6,6))
estimate(lu)

lf <- twinlm(bmi ~ zyg, data=dd, DZ="DZ", zyg="zyg", id="tvparnr", type="flex")
coef(lf)

###sink("lu-est-summary.txt")
lu <- twinlm(bmi ~ zyg, data=dd, DZ="DZ", zyg="zyg", id="tvparnr", type="u")
summary(lu)
estimate(lu)
crossprod(iid(lu))^.5
###sink()

vcov(lu)

estimate(lu)
dim(iid(lu))

estimate(lu,contr(4:5,5))

estimate(coef=coef(lu),vcov=vcov(lu),contr(4:5,5))

wald.test(coef=coef(lu),vcov=vcov(lu),contrast=c(0,0,0,1,-1))

l <- twinlm(bmi ~ ns(age,1)+gender, data=twinbmi,
DZ="DZ", zyg="zyg", id="tvparnr", type="cor", missing=TRUE)
summary(l)


A formal test of genetic effects can be obtained by comparing the MZ and DZ correlation:

estimate(l,contr(5:6,6))

l <- twinlm(bmi ~ ns(age,1)+gender, data=twinbmi,
DZ="DZ", zyg="zyg", id="tvparnr", type="cor", missing=TRUE)
summary(l)


We also consider the ACE model

ace0 <- twinlm(bmi ~ age+gender, data=dd, DZ="DZ", zyg="zyg", id="tvparnr", type="ace")


# Bibliography

[korkeila_bmi_1991] Korkeila, Kaprio, Rissanen & Koskenvuo, Effects of gender and age on the heritability of body mass index, Int J Obes, 15(10), 647-654 (1991).

[hjelmborg_bmi_2008] Hjelmborg, Fagnani, Silventoinen, McGue, Korkeila, Christensen, Rissanen & Kaprio, Genetic influences on growth traits of BMI: a longitudinal study of adult twins, Obesity (Silver Spring), 16(4), 847-852 (2008).

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mets documentation built on Sept. 6, 2021, 9:08 a.m.