Relationship between Fertility Distribution and Sibling Distribution

This note gives the formal relationship between a fertility distribution and the sibling distribution it implies in the next generation. The fertility distribution describes the number of children per woman; the sibling distribution describes the number of siblings per person. Once the fertility distribution is specified, the sibling distribution is a deterministic size-biased transform of that fertility distribution.

1 Fertility Distribution

The number of children per woman in the population is modelled using a fertility distribution, which can be represented by various probability distributions. In this article, we focus on two commonly used distributions for modelling count data: Poisson distribution and zero inflated negative binomial distribution (ZINB).

1.1 Poisson distribution

The Poisson distribution is a popular model for count data, particularly when the mean and variance are equal. However, it may not be very suitable for fertility data, which often exhibit zero-inflation and overdispersion (variance greater than the mean). The Poisson distribution with mean fertility (\(\mu\)) is defined as: for the number of children \(k \geq 0\), \[P(N_{child}=k)=e^{-\mu}\frac{\mu^k}{k!}, \tag{1}\] denoted by \(\text{Poisson}(\mu)\). The mean and variance of \(\text{Poisson}(\mu)\) are both \(\mu\).

1.2 Zero inflated negative binomial distribution

The zero inflated negative binomial distribution with parameters mean fertility (\(\mu\)), dispersion/size (\(\theta\)), and zero‑inflation probability (\(\pi_0\))—denoted as \(\text{ZINB}(\mu, \theta, \pi_0)\)—is defined as: for the number of children \(k \geq 0\), \[P(N_{child}=k)=\begin{cases} \pi_0 + (1-\pi_0)p^{\theta} & \text{if } k=0, \\ (1-\pi_0)\binom{k+\theta-1}{k}(1-p)^k p ^{\theta} & \text{if } k \geq 1, \end{cases} \tag{2}\] where \(p = \frac{\theta}{\mu+\theta}\) is denoted as the success probability. For the negative binomial distribution \(\text{NB}(\mu,\theta)\), the mean and variance are \(\mu\) and \(\mu + \frac{\mu^2}{\theta}\). Therefore, the mean and variance of \(\text{ZINB}(\mu, \theta, \pi_0)\) are given by \(\mu(1-\pi_0)\) and \(\mu(1-\pi_0) + \frac{\mu^2(1-\pi_0)}{\theta}+\mu^2\pi_0(1-\pi_0)\), respectively.

2 Sibling distribution

The number of siblings per person in the next generation is described by a sibling distribution. This distribution is derived directly from the fertility distribution because children are sampled from births, not from mothers. Mothers with more children contribute more individuals to the next generation, so their fertility counts receive weight proportional to family size.

2.1 Deterministic fertility-to-sibling transform

Let \(N_{child}\) be the random variable for the number of children per woman, with probability mass function \(p_j = P(N_{child}=j)\) and mean \(\mu_F = E[N_{child}]\). Let \(N_{sib}\) be the number of siblings for a person in the next generation. A person with \(k\) siblings must come from a mother with \(k+1\) children, and that family contributes \(k+1\) children to the population. Therefore, for any count \(k \geq 0\),

\[P(N_{sib}=k) = \frac{(k+1)P(N_{child}=k+1)}{E[N_{child}]} = \frac{(k+1)p_{k+1}}{\mu_F}. \tag{3}\]

This is a deterministic transform: no additional stochastic model is needed to obtain the sibling distribution once the fertility PMF is known. More distant kin counts can be built from the same fertility and sibling relationships, but this note focuses only on the direct fertility-to-sibling derivation.

2.2 Sibling distribution for Poisson and ZINB fertility models

In this section, we derive the sibling distribution for the two fertility distributions discussed above: Poisson distribution and zero inflated negative binomial distribution, using the relationship established in the previous section.

2.2.1 Fertility distribution: Poisson distribution

If the fertility distribution is \(\text{Poisson}(\mu)\), then the sibling distribution is given by: for a count \(k \geq 0\), \[\begin{equation} \begin{split} P(N_{sib}=k) & = \frac{(k+1)P(N_{child}=k+1)}{\mu} \\ & = \frac{(k+1)e^{-\mu}\frac{\mu^{k+1}}{(k+1)!}}{\mu} \\ & = e^{-\mu}\frac{\mu^k}{k!}, \\ \end{split} \end{equation} \tag{4}\] which is also a Poisson distribution with mean \(\mu\).

2.2.2 Fertility distribution: Zero inflated negative binomial distribution

If the fertility distribution is \(\text{ZINB}(\mu, \theta, \pi_0)\), then \(E[N_{child}] = \mu(1-\pi_0)\). The sibling distribution is given by: for a count \(k \geq 0\), \[\begin{equation} \begin{split} P(N_{sib}=k) & = \frac{(k+1)P(N_{child}=k+1)}{\mu(1-\pi_0)} \\ & = \frac{(1-\pi_0)(k+1)\binom{k+\theta}{k+1}(1-p)^{k+1} p^{\theta}}{\mu(1-\pi_0)} \\ & = \binom{k+\theta}{k}(1-p)^k p^{\theta+1}, \\ \end{split} \end{equation} \tag{5}\] where \(p = \frac{\theta}{\mu+\theta}\). The zero-inflation term cancels because childless women do not contribute sampled children to the next generation. Therefore, the sibling distribution is a negative binomial distribution with parameters mean \(\frac{\mu(\theta+1)}{\theta}\) and dispersion/size \(\theta+1\), denoted as \(\text{NB}(\frac{\mu(\theta+1)}{\theta}, \theta+1)\). The variance is \(\frac{\mu(\theta+1)}{\theta} + \frac{\mu^2(\theta+1)}{\theta^2}\).