Twin models

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.

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,
     data=df["age"], col="red", add=TRUE)
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}

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")

Twin analysis, censored outcomes

Twin analysis, binary traits

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.