We identified 503 pairs of female twins born between 1931 and 1952 through the Danish Twin Registry. There are 269 monozygotic twin pairs and 234 dizygotic twin pairs identified by

Skytthe et al. (2003). Before they reach natural menopause, some women may experience surgical menopause as a result of removal of the uterus, cervix or ovaries and others may be treated with hormones that affect the timing of menopause. Surgical menopause and hormonal treatment are therefore competing risks for natural menopause. Of 1006 twins, 416 twins experienced natural menopause, 57 twins had surgical related menopause, 183 twins underwent hormonal treatments before natural menopause, 246 twins had censored event times. There were 104 twins with unknown failure types or censoring status and they were excluded from the analysis. These summaries do not reflect that the probabilities of different events changed considerably over the time cohorts. We partition the data into four cohorts of almost equal size according to birthdate. Cohort 1 from 1931–1936 has 120 twin pairs, cohort 2 from 1937–1942 has 123 twin pairs, cohort 3 from 1943–1947 has 121 twin pairs, and cohort 4 from 1948–1952 has 139 twin pairs. In cohort 1, there was zero censoring, 154 natural menopause, 22 surgical menopause, 40 hormonal treatments, and 24 cases with unknown failure types or censoring status. These numbers were (3, 137, 13, 64, 29) for cohort 2, (65, 89, 11, 54, 23) for cohort 3, and (178, 36, 11, 25, 28) for cohort 4. In the analysis, we allow the censoring distribution to depend on cohorts.

We first estimate the marginal cumulative incidence of natural menopause excluding the cohort structure. Let *x*_{ki} = {1, *I*_{ki} (dizygotic twin)}^{T}, where *I*_{ki} (dizygotic twin) is 1 if the *i* th twin from the *k*th family is dizygotic for *i* = 1, 2 and *k* = 1, . . . , 503. Under model (1) without covariate *z*_{ki}, the marginal cumulative incidence is − log{1 − *P*_{1}(*t* | *x*_{ki})} = *η*(*t*)^{T}*x*_{ki}, which leads to the nonparametric estimates for the distributions of the ages to natural menopause for the monozygotic and dizygotic twins. The estimates, 95% confidence intervals and 95% confidence bands based on the Gaussian multiplier resampling technique are shown in for the monozygotic twins. The plots for the dizygotic twins are similar and omitted to save space.

To examine whether the cumulative incidence of natural menopause changes over the different cohorts, we estimate the cumulative incidence function for natural menopause by adjusting for time cohort effects. Let

*I*_{ki} (cohort 2) be the cohort indicator of whether the twins in the

*k*th family are born between 1937 and 1942. In addition to

*x*_{ki} defined above, we introduce cohort covariates

*z*_{ki} to the random effects model (1), where

*z*_{ki} = {

*I*_{ki} (cohort 2),

*I*_{ki} (cohort 3),

*I*_{ki} (cohort 4)}

^{T}. The marginal cumulative incidence functions then follow the semiparametric additive model − log{1 −

*P*_{1}(

*t* |

*x*_{ki},

*z*_{ki})} =

*η*(

*t*)

^{T}*x*_{ki} + (

*γ*^{T}*z*_{ki})

*t*. This model assumes separate nonparametric baseline functions for monozygotic and dizygotic twins and constant time cohort effects. The goodness-of-fit procedures of

Scheike & Zhang (2008) indicate that the cohorts can well be described as having constant effects. shows the estimated cumulative incidence functions of natural menopause for cohort 1 and 4 for monozygotic twins. There is a clear trend over time in that the later cohort experience their natural menopause later. For cohort 4 we only estimate the cumulative incidence curve for the ages between 40 and 55 due to censoring. The estimated cohort effects

are (−1.03, −3.88, −5.72) × 10

^{−2} with the standard errors of (0.65, 0.58, 0.53) × 10

^{−2}, respectively.

We now examine whether there is an association in the occurrence of natural menopause between twins of the same family under the random effects model (1). To allow for different degrees of association for different types of twins, we let

*ν*_{k} =

*α*_{0} +

*α*_{1}*x*_{k} under (2), where

*x*_{k} is 0 for monozygotic twins and 1 for dizygotic twins. Let

*ρ*_{1} =

*α*_{0} and

*ρ*_{2} =

*α*_{0} +

*α*_{1}. Using the estimation procedure presented in § 2.3, we get

_{1} = 1.04 with the standard error of 0.42, and

_{2} = 0.39 with the standard error of 0.29. Hence, a significant association exists in the monozygotic twins but not for the dizygotic twins.

The joint probability of both twins experiencing natural menopause by age

*t* depends on the degree of association between the twins. This probability is

*v*_{kij}(

*t*) given in (7) under the random effects model (1). For monozygotic twins in cohort 1, this joint probability is plotted in against age. The random effect

*θ*_{k} in model (1) represents family-to-family variability and induces dependence between twins. Let

*θ*_{k}_{,0.95} and

*θ*_{k}_{,0.05} be the 95th percentile and the 5th percentile of the gamma distribution with mean 1 and variance

*ν*_{k}, respectively. The joint probabilities of both twins experiencing natural menopause by age

*t* in a twin pair with

*θ*_{k} =

*θ*_{k}_{,0.95} and

*θ*_{k} =

*θ*_{k}_{,0.05} can be expressed as

and

, respectively. For monozygotic twins in cohort 1, these joint probabilities are given in . The plot of the joint probability of both twins experiencing natural menopause by age

*t*,

*P*_{1}(

*t* |

*x*_{ki},

*z*_{ki})

*P*_{1}(

*t* |

*x*_{kj},

*z*_{kj}), under the marginal model (3) and the presumed independence between twins is also given in . Similar plots for dizygotic twins are given in . Other quantities such as the conditional probabilities of the second twin experiencing natural menopause given that the first twin has already experienced her natural menopause can also be calculated and plotted.

Finally, we calculate the cross-odds ratio to show how the onset of natural menopause for one twin affects the probability of the onset for the other twin. The cross-odds ratios for monozygotic twins are around 2. This indicates that, if one twin has experienced natural menopause, the odds of the second twin experiencing natural menopause by the same age essentially doubles the unconditional odds of natural menopause for the second twin. For the dizygotic twins, we find a cross-odds ratio that lies around 1.4.