The probable numbers of kin in a multi-state population: a branching process approach

This paper introduces a novel analytical framework based on branching process theory and recursive probability generating functions to calculate the joint distributions of kin numbers structured by both age and various life stages, such as parity, health, or location, thereby addressing a key challenge in kinship demography.

The Gist

Imagine you are standing in the middle of a massive, bustling family reunion. You want to know: How many cousins do I have? How many sisters are still alive? How many of my grandchildren are currently taking care of their own kids?

For a long time, demographers (people who study populations) could only give you the average. They could say, "On average, a woman in the UK has 1.2 sisters." But averages are like weather forecasts saying "it will be 70°F." It might be 40°F or 100°F. They don't tell you the chance of having zero sisters, or the chance of having five.

This paper introduces a new, super-powered mathematical tool that doesn't just give you the average. It calculates the exact probability of every possible family scenario. It answers: "What are the odds I have exactly two sisters, both of whom are 40 years old and have no children of their own?"

Here is how the paper works, explained through simple analogies:

1. The "Family Tree" as a Game of Dice

The authors use a concept called a Branching Process. Imagine your family tree isn't a static drawing, but a game of dice being rolled over and over again.

• The Roll: Every time a person has a child, it's like rolling a die. Sometimes you roll a "0" (no kids), sometimes a "1," sometimes a "3."
• The Branch: Each child becomes a new player who rolls their own dice.
• The Complexity: In the past, models only looked at the total number of dice rolls. This new model looks at the color and shape of the dice too.

2. The "Two-Coordinate" Map (Age and Stage)

Most old models only tracked Age (how old you are). This new model tracks two things at once: Age and Stage.

Think of "Stage" as a costume or a badge you wear.

• In the paper's example, the "Stage" is Parity (how many kids you have had).
  • Stage 1: No kids yet.
  • Stage 2: One kid.
  • Stage 3: Two kids.
• The model doesn't just ask, "How many sisters does she have?"
• It asks, "How many sisters does she have who are 30 years old AND have exactly two kids?"

This is crucial because having a sister who is 30 with two kids is a very different social reality than having a sister who is 30 with no kids. One might need help with childcare; the other might be free to help you.

3. The "Magic Recipe Book" (Probability Generating Functions)

How do they calculate all these complex odds without simulating millions of families on a computer? They use something called Probability Generating Functions (PGFs).

Think of a PGF as a Magic Recipe Book.

• Instead of writing down a list of numbers, you write a single, complex algebraic formula (the recipe).
• This recipe contains all the possible outcomes hidden inside it, like ingredients in a sealed jar.
• The Secret Sauce: The authors use a technique called Recursive Nesting. Imagine taking the recipe for "Grandchildren" and stuffing it inside the recipe for "Children," and then stuffing that inside the recipe for "You."
• By nesting these recipes inside each other, the math automatically calculates the odds for every generation at once, keeping track of who is alive, who is dead, and what "costume" (stage) they are wearing.

4. Tracking the "Ghost" Relatives (Kin Loss)

One of the most powerful features of this model is that it counts the dead.

• Usually, we only count living relatives. But this model has a special "Ghost Counter."
• It can tell you the probability that you have zero living sisters because they all passed away, or that you have one living sister and two who died.
• This helps answer heartbreaking but important questions: "What are the odds I will be an orphan?" or "What are the odds my grandchild will lose their mother before they turn 18?"

5. The Real-World Test: The UK Family

To prove it works, the authors applied this "Magic Recipe" to real data from the United Kingdom, focusing on Parity (number of children).

• The Finding: They looked at women born in the 1960s.
• The Insight: They found a specific, high-probability scenario: A woman who is childless but has multiple sisters.
• Why it matters: In the 1960s, many women had fewer children but larger families of siblings. This creates a specific demographic: a woman with no kids of her own, but a whole network of sisters who might need care or who might be her only source of family support. The model calculates exactly how common this "childless but surrounded by sisters" scenario is.

The Big Picture

Think of this paper as upgrading from a black-and-white photo of a family to a 3D, interactive hologram.

• Old Way: "You have 1.5 sisters on average." (Flat, blurry, and misleading).
• New Way: "There is a 25% chance you have no sisters, a 40% chance you have one sister who is 50 and childless, and a 10% chance you have two sisters, one of whom has passed away."

This allows governments, social workers, and researchers to plan for the future. If they know the odds of people being "kinless" (having no living relatives) are rising, they can build better support systems for the elderly. If they know the odds of "orphaned grandchildren" are dropping, they can adjust healthcare policies.

It turns the chaotic, random game of family life into a predictable map of probabilities, helping us understand not just how many people we are related to, but who they are and what state they are in.

Technical Summary

1. Problem Statement

Kinship demography has traditionally focused on predicting the expected numbers (means) of kin for a typical population member, often structured by age, sex, or stage (e.g., parity, health status). While recent advances (e.g., by Caswell) have successfully modeled these expectations using matrix projections, they fail to capture the full probability distribution of kin numbers.

In reality, kin numbers are non-negative integers, and knowing only the mean obscures critical demographic uncertainties, such as:

• The probability of having zero kin of a specific type (kinlessness).
• The variance, skew, and kurtosis of kin distributions (important for understanding reproductive success and care networks).
• The joint probabilities of having specific combinations of kin (e.g., one sister at home and one away, versus two sisters).
• The probabilities of kin loss (bereavement) and orphanhood.

Existing probabilistic models were largely limited to single-sex, time-invariant, age-structured populations. There was no analytical framework to derive joint distributions of kin numbers structured simultaneously by age and stage (multi-state) in a time-invariant setting.

2. Methodology

The author proposes a novel analytical framework grounded in Branching Process Theory, specifically utilizing Probability Generating Functions (PGFs) rather than Probability Mass Functions (PMFs).

Core Mathematical Framework

• Multi-State Population: The population is structured by age classes ($a = 0, \dots, \omega$) and discrete stages ($s = 1, \dots, k$). Transitions between stages and survival are governed by matrices $U_a$ (survival and stage transition) and $F_a$ (fertility).
• Recursive PGF Composition: The model treats kinship as a branching process. Instead of calculating means, it recursively composes PGFs to track the distribution of offspring and descendants.
  • State Distribution PGF ($S_t$): Tracks the probability of an individual being alive in a specific stage or dead at time $t$.
  • Lifetime Reproduction PGF ($L$): Encodes the distribution of offspring produced over an individual's life, accounting for the stage transitions of both the parent and the offspring.
  • Recursive Nesting: To find the distribution of the $g$-th generation, the PGF of the $(g-1)$-th generation is nested inside the fertility function of the current generation.
• Genealogical Markov Chains: To model collateral kin (e.g., siblings, cousins) and ancestors, the model employs a genealogical Markov chain derived from the stable population structure (Perron-Frobenius eigenvectors). This allows for the probabilistic inference of ancestral ages and stages at the time of reproduction.
• Size-Biasing: To calculate the distribution of collateral kin (like sisters), the model uses size-biasing techniques. This conditions the reproduction PGF on the event that a specific offspring (the "Focal" individual) was born, effectively removing that offspring from the count to analyze the remaining siblings.

Key Extensions

• Kin Loss and Bereavement: The framework introduces a dummy variable $d$ to explicitly track deaths. This allows for the calculation of joint distributions of living kin and deceased kin, enabling the analysis of bereavement and orphanhood.
• Joint Focal-Kin States: The model can calculate the joint probability of the "Focal" individual being in a specific state (e.g., childless) while having a specific number of kin in other states.

3. Key Contributions

1. First Analytical Multi-State Kin Distribution: This is the first method to derive complete joint probability distributions for kin numbers structured by both age and stage simultaneously.
2. Recursive PGF Approach: It demonstrates how recursive composition of PGFs in a Galton-Watson process can be adapted to complex, overlapping-generation, multi-state demographies.
3. Comprehensive Kin Metrics: The framework provides analytical formulas for:
   • Joint distributions of kin by age and stage.
   • Marginal distributions (by age only or stage only).
   • Joint distributions of the Focal's state and kin's state.
   • Probabilities of kin loss (total and by generation).
   • Probabilities of being kinless or orphaned.
4. Handling Reducible Matrices: The paper addresses the mathematical challenge of reducible projection matrices (common in parity-structured models where parity only increases) by restricting the genealogical chain to reproductive states or using numerical perturbation.

4. Results and Application

The framework was applied to UK parity-specific fertility and mortality data (1964–2023, projected to 2025). Parity (number of children born) was used as the "stage" variable ($s=0, 1, 2, 3+$).

Key Findings:

• Joint Distributions: The model successfully extracted the joint probabilities of having specific numbers of daughters and sisters across different parity levels. For example, it showed how the probability of having a sister with parity 2 changes as the Focal individual ages.
• Childless Cohorts: The model quantified the probability that a Focal individual is childless (parity 0) but has at least one sister. This is relevant for the UK 1960s cohort, where many women were childless but had siblings, impacting informal care networks.
• Bereavement and Orphanhood:
  • The model calculated the probability of losing specific numbers of kin (daughters/sisters) by age.
  • It highlighted a significant decline in the probability of having an orphaned granddaughter between the 1965 and 2025 periods (dropping from ~13% to ~4% by age 95), reflecting improvements in maternal survival.
• Validation: The model's expected values (means) were validated against existing matrix projection models (Caswell [18]), confirming consistency while providing the added value of full distributional data.

5. Significance

• Demographic Uncertainty: By moving beyond expected values to full probability distributions, the model allows researchers to assess the risk of kinlessness or the likelihood of specific care configurations, which is crucial for social policy and planning.
• Informal Care and Support: The ability to model joint states (e.g., a childless parent with living siblings) provides a fast, analytical alternative to computationally expensive micro-simulations (like Soc-Sim) for estimating the size of vulnerable populations.
• Evolutionary Biology: The framework is applicable to animal populations, helping to understand how kin structure (dispersal, death, relatedness) influences cooperative behavior and inclusive fitness.
• Methodological Advance: It bridges the gap between classical matrix demography and stochastic branching processes, offering a flexible tool that can incorporate empirical data or parametric distributions for mortality and fertility.

In conclusion, Butterick's work provides a rigorous, analytical toolkit for understanding the stochastic nature of kinship networks, offering unprecedented detail on the probable numbers, states, and survival of relatives in structured populations.