Relational Gompertz Fertility Model
Also known as: Brass Relational Gompertz Model, Gompertz Relational Fertility Model, Relational Gompertz Function
The relational Gompertz model expresses any population's cumulative fertility schedule as a simple linear transformation of a fixed standard schedule, after both are mapped through a double-logarithm (gompit) transform. Developed by William Brass and given its widely used standard by Heather Booth, it characterizes the entire age pattern of fertility with just two parameters — α, which shifts the schedule earlier or later, and β, which controls how concentrated or spread out childbearing is. This makes it a robust tool for smoothing, fitting, and especially for correcting and estimating fertility from the limited and error-prone data common in developing countries.
Key highlights
- Summarizes a full age schedule of fertility with two interpretable parameters, α for level/timing and β for spread.
- Robust to defective data: the linear gompit relationship exposes implausible points so they can be excluded, and it can reconstruct missing ages.
- Pairs naturally with the Brass P/F ratio method to estimate and correct fertility from limited census and survey data.
- Easy to fit by ordinary least squares and easy to compare across populations and over time.
Intuition
This section is available to Pro members. Upgrade to Pro
How it works
This section is available to Pro members. Upgrade to Pro
When to use it
Use the relational Gompertz model to smooth, complete, or correct an age schedule of fertility, and especially to estimate fertility from the deficient data typical of censuses and surveys in low-income settings, where it is often paired with the P/F ratio method. Its two parameters make fertility schedules easy to compare and project. Assumptions: the population's fertility shape is a Gompertz relational transform of the chosen standard, the standard is appropriate to the fertility regime, and at least the central ages have usable data. Do NOT use it when fertility has an unusual shape the standard cannot match, when both tails of the schedule are needed but the gompit linearity holds only in the centre, or when high-quality complete fertility data already exist and no relational smoothing is required.
Strengths & limitations
- Summarizes a full age schedule of fertility with two interpretable parameters, α for level/timing and β for spread.
- Robust to defective data: the linear gompit relationship exposes implausible points so they can be excluded, and it can reconstruct missing ages.
- Pairs naturally with the Brass P/F ratio method to estimate and correct fertility from limited census and survey data.
- Easy to fit by ordinary least squares and easy to compare across populations and over time.
- The fit and reconstruction depend on choosing an appropriate standard schedule; a mismatched standard biases the estimates.
- The gompit transform is most nearly linear only in the central reproductive ages, so estimates in the youngest and oldest tails are less reliable.
- It assumes the target schedule is a simple two-parameter transform of the standard, which can fail for unusual fertility patterns.
- Reliable application still requires reasonably accurate data at some ages; it cannot rescue a schedule that is corrupt everywhere.
Common pitfalls
This section is available to Pro members. Upgrade to Pro
Applications
This section is available to Pro members. Upgrade to Pro
Frequently asked
What do the parameters α and β represent?
α shifts the fertility schedule in age — a higher α moves childbearing to younger ages relative to the standard — while β controls the spread, with values above one concentrating fertility into a narrower age range and values below one dispersing it. Together they reposition and reshape the standard schedule to match the observed data.
Why use the gompit transform rather than modeling fertility directly?
The cumulative fertility curve is bounded between zero and one and S-shaped, which is hard to model linearly. The gompit (negative double-log) transform maps it onto a scale where its relationship to the standard is nearly a straight line, so a simple two-parameter linear regression captures the whole schedule and defective points stand out as outliers from the line.
How does the relational Gompertz model relate to the Coale-Trussell model?
Both are parametric descriptions of fertility, but Coale-Trussell separates natural fertility from a parametric departure due to fertility control, whereas the relational Gompertz model represents the whole schedule as a transform of a single standard. Booth's improved standard for the relational Gompertz model was in fact derived from Coale and Trussell's work, linking the two approaches.
Sources
- 1.Booth, H. (1984). Transforming Gompertz's function for fertility analysis: The development of a standard for the relational Gompertz function. Population Studies, 38(3), 495–506.
- 2.Preston, S. H., Heuveline, P., & Guillot, M. (2001). Demography: Measuring and Modeling Population Processes. Blackwell.ISBN 9781557864512
You have read it. What now?
Cite this page
ScholarGate. (2026, June 22). Relational Gompertz Fertility Model. ScholarGate. https://scholargate.app/demography/relational-gompertz-fertility-model