# Twin models

#### Klaus Holst & Thomas Scheike

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

\[ \newcommand{\cov}{\mathbb{C}\text{ov}} \newcommand{\cor}{\mathbb{C}\text{or}} \newcommand{\var}{\mathbb{V}\text{ar}} \newcommand{\E}{\mathbb{E}} \newcommand{\unitfrac}[2]{#1/#2} \newcommand{\n}{} \]

# Twin analysis, continuous traits

In the following we examine the heritability of Body Mass Indexkorkeila_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)
    #>   tvparnr      bmi      age gender zyg id num
    #> 1       1 26.33289 57.51212   male  DZ  1   1
    #> 2       1 25.46939 57.51212   male  DZ  1   2
    #> 3       2 28.65014 56.62696   male  MZ  2   1
    #> 5       3 28.40909 57.73097   male  DZ  3   1
    #> 7       4 27.25089 53.68683   male  DZ  4   1
    #> 8       4 28.07504 53.68683   male  DZ  4   2

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)
    #>    tvparnr     bmi1      age gender zyg id num     bmi2
    #> 1        1 26.33289 57.51212   male  DZ  1   1 25.46939
    #> 3        2 28.65014 56.62696   male  MZ  2   1       NA
    #> 5        3 28.40909 57.73097   male  DZ  3   1       NA
    #> 7        4 27.25089 53.68683   male  DZ  4   1 28.07504
    #> 9        5 27.77778 52.55838   male  DZ  5   1       NA
    #> 11       6 28.04282 52.52231   male  DZ  6   1 22.30936

Next we plot the association within each zygosity group

We show the log-transformed data here, which is slightly more symmetric and more appropriate for the twin analysis (see Figure @ref(fig:scatter1) and @ref(fig:scatter2)).

    mz <- log(subset(twinwide, zyg=="MZ")[,c("bmi1","bmi2")])
    plot_twin(mz)

Scatter plot of logarithmic BMI measurements in MZ twins

    dz <- log(subset(twinwide, zyg=="DZ")[,c("bmi1","bmi2")])
    plot_twin(dz)

Scatter plot of logarithmic BMI measurements in DZ twins

The plots and raw association measures show considerably stronger dependence in the MZ twins, indicating genetic influence on the trait.

    cor.test(mz[,1],mz[,2], method="spearman")
    #>
    #>  Spearman's rank correlation rho
    #>
    #> data:  mz[, 1] and mz[, 2]
    #> S = 165457624, p-value < 2.2e-16
    #> alternative hypothesis: true rho is not equal to 0
    #> sample estimates:
    #>       rho
    #> 0.6956209

    cor.test(dz[,1],dz[,2], method="spearman")
    #>
    #>  Spearman's rank correlation rho
    #>
    #> data:  dz[, 1] and dz[, 2]
    #> S = 2162514570, p-value < 2.2e-16
    #> alternative hypothesis: true rho is not equal to 0
    #> sample estimates:
    #>       rho
    #> 0.4012686

Next we examine the marginal distribution (GEE model with working independence):

    l0 <- lm(bmi ~ gender + I(age-40), data=twinbmi)
    estimate(l0, id=twinbmi$tvparnr)
    #>             Estimate  Std.Err    2.5%   97.5%    P-value
    #> (Intercept)  23.3687 0.054534 23.2618 23.4756  0.000e+00
    #> gendermale    1.4077 0.073216  1.2642  1.5512  2.230e-82
    #> I(age - 40)   0.1177 0.004787  0.1083  0.1271 1.499e-133

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

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

Marginal association between BMI and Age for males and females.

## 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
- \(D\): Dominant genetic effects of alleles
- \(C\): Shared environmental effects
- \(E\): Unique environmental 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)

Saturated model (different marginals in MZ and DZ twins and different marginals for twin 1 and twin 2):

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

Different marginals for MZ and DZ twins (but same marginals within a 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="eqmarg")
    estimate(lu)
    #>                      Estimate  Std.Err    2.5%   97.5%    P-value
    #> bmi.1@1               18.6037 0.251036 18.1116 19.0957  0.000e+00
    #> bmi.1~age.1@1          0.1189 0.005635  0.1078  0.1299  9.177e-99
    #> bmi.1~gendermale.1@1   1.3848 0.086573  1.2151  1.5544  1.376e-57
    #> log(var)@1             2.4424 0.022095  2.3991  2.4857  0.000e+00
    #> atanh(rhoMZ)@1         0.7803 0.036249  0.7092  0.8513 9.008e-103
    #> atanh(rhoDZ)@2         0.2987 0.020953  0.2576  0.3397  4.288e-46

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

    estimate(lu,lava::contr(5:6,6))
    #>                           Estimate Std.Err   2.5%  97.5%   P-value
    #> [atanh(rhoMZ)@1] - [a....   0.4816 0.04177 0.3997 0.5635 9.431e-31
    #> ────────────────────────────────────────────────────────────
    #> Null Hypothesis:
    #>   [atanh(rhoMZ)@1] - [atanh(rhoDZ)@2] = 0
    #>
    #> chisq = 132.9162, df = 1, p-value < 2.2e-16

We also consider the ACE model

    ace0 <- twinlm(bmi ~ age+gender, data=dd, DZ="DZ", zyg="zyg", id="tvparnr", type="ace")
    summary(ace0)
    #>                  Estimate Std. Error Z value Pr(>|z|)
    #> bmi            1.8599e+01 2.5576e-01  72.720   <2e-16
    #> sd(A)          2.7270e+00 4.2658e-02  63.927   <2e-16
    #> sd(C)          1.7123e-06 3.1064e-01   0.000        1
    #> sd(E)          2.0276e+00 3.4787e-02  58.286   <2e-16
    #> bmi~age        1.1892e-01 5.6246e-03  21.142   <2e-16
    #> bmi~gendermale 1.3846e+00 8.8748e-02  15.601   <2e-16
    #>
    #> MZ-pairs DZ-pairs
    #>     1483     2788
    #>
    #> Variance decomposition:
    #>   Estimate 2.5%    97.5%
    #> A 0.64399  0.61793 0.67005
    #> C 0.00000  0.00000 0.00000
    #> E 0.35601  0.32995 0.38207
    #>
    #>
    #>                          Estimate 2.5%    97.5%
    #> Broad-sense heritability 0.64399  0.61793 0.67005
    #>
    #>                        Estimate 2.5%    97.5%
    #> Correlation within MZ: 0.64399  0.61718 0.66931
    #> Correlation within DZ: 0.32200  0.30890 0.33497
    #>
    #> 'log Lik.' -22019.66 (df=6)
    #> AIC: 44051.32
    #> BIC: 44089.47

# 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). ↩︎