# Overview of fertility estimation methods based on the P/F ratio

## Introduction

Almost all methods of estimating fertility indirectly have their origins in the P/F ratio method first proposed by Brass (1964). In addition, the interpretation of the results from other methods (for example, cohort-period fertility rates) and some of the diagnostic tools used to assess the quality of the data when estimating child mortality also rely on the intrinsic logic of the P/F ratio approach. Thus, while the method in its original and modified forms has been superseded by the relational Gompertz model and its variants, it is useful to present the essential logic of the method here. The interested reader is referred to Manual X (UN Population Division 1983) for a full exposition of the approach.

## The Brass P/F ratio method

The foundation of the method rests on the observation that if fertility has been constant for an extended period of time, cohort and period measures of fertility will be identical. In other words, under conditions of constant fertility, the cumulated fertility of a cohort of women up to any given age will be the same as the cumulated fertility up to that same age in any given period.

If we assume that there are no appreciable mortality differentials by the fertility of mother, so that surviving women do not have materially different levels of childbearing from deceased women, the cumulated fertility of a cohort of women up to any given age is the same as the average parity in that cohort. (This assumption is not very important as even if there are differentials in the fertility of living and deceased women, in most populations the magnitude of female mortality in the reproductive ages is very small and the effect of differential survival will therefore be small.)

Brass defined P to be the average parity (cumulated lifetime fertility) of a cohort of women up to a given age, and F to be closely related to the cumulated current (period) fertility up to that same age. The P/F ratio method expresses these two quantities in relation to each other in the form of a ratio for each age group.

The derivation of F is a little more complicated than suggested above for two reasons. First, any comparison of cohort and period fertility has to deal with the probable shifting of the data on recent fertility brought about by the question being based on the age of the mother at the time of the inquiry rather than her age at the time of her most recent birth. Second, while the cumulation of period fertility to any given age will reflect the fertility experience of all women up until that age, the average parities typically calculated reflect those of women in 5-year age groups and hence reflect (approximately) the average parity of women aged at the midpoint of that age group. The method formulated by Brass addresses both these aspects.

It follows that if fertility has been constant in a population for an extended period of time, and if the data are free of error, the P/F ratio would equal 1 in every age group. If fertility has been falling, however, cumulated life time fertility would be greater than cumulated current fertility. In this case (in the absence of errors in the data) the P/F ratio would depart from unity systematically with increasing age of mother.

The corollary to this observation is that one would expect the P/F ratio to be fairly close to unity at the youngest ages because even by women’s mid-twenties one would not expect significant deviation of cumulated period fertility from cumulated lifetime cohort fertility as most of the births to women in that cohort would have happened fairly recently. It is from this observation that the P/F ratio derived from women aged 20-24 at the time of a survey is held to be the most reliable indicator of the quality of the fertility data collected. Conveniently, the supposition is that the average parities of younger women are usually fairly accurately reported, at least relative to those of older women.

It is this characteristic pattern of departure from unity with age of mother that forms the basis for many diagnostic investigations into the nature and quality of data drawn from questions based on recent and lifetime fertility.

## Diagnostics based on the P/F ratio

In reality the data are never free from error, and so the hypothetical pattern of departure of the P/F ratio from unity is confounded and obfuscated by underlying errors in the data.

As discussed on the sections on evaluation of recent fertility data and evaluation of lifetime fertility data, two errors typically affect these data. The first is that reports on lifetime fertility – that is, cumulated cohort fertility – become increasingly inaccurate with age of the respondent, with older women tending to under-report their lifetime fertility. Errors of this kind will therefore tend to depress the numerator of the P/F ratio, particularly at the older ages. If such errors occur in the data, the ratio will tend to be closer to unity than it might truly be.

The second kind of error frequently encountered is that women tend to under-report recent births, regardless of their age. Errors of this type will result in the reported level of recent fertility being somewhat lower than anticipated, thereby causing the P/F ratio to be inflated.

The P/F ratio method seeks to correct the second problem by applying the P/F ratio applicable to younger women (for the reasons set out above) to the directly observed fertility schedule as a scaling factor.

## Summary of methods based on the P/F ratio method

A number of methods described here were originally presented in Manual X as extensions of the P/F ratio method. The relational Gompertz model can be thought of as an improved and more versatile version of the Brass P/F ratio method. The model uses the same input data (and makes the same assumptions about errors that affect fertility data) as its precursor. Importantly, however, the method does not require an assumption that fertility has been constant in the past. Nonetheless, the comparison of lifetime and period fertility lies at the heart of the method.

Most of the extensions to the Brass P/F ratio method presented in Manual X have been recast as extensions to the relational Gompertz model. These extensions include those methods that make use of the data on parity increments from two censuses to estimate fertility; methods that use parity increments in conjunction with a schedule of intercensal fertility rates (the synthetic relational Gompertz model); and indirect methods that make use of data from vital registration systems. Cohort-period fertility rates derived from survey data also rely on the logic of the P/F ratio method to shed light on longer-term trends and dynamics in fertility.

## References

Brass W. 1964. Uses of census or survey data for the estimation of vital rates. Paper prepared for the African Seminar on Vital Statistics, Addis Ababa 14-19 December 1964. Document No. E/CN.14/CAS.4/V57. New York: United Nations.

UN Population Division. 1983. Manual X: Indirect Techniques for Demographic Estimation. New York: United Nations, Department of Economic and Social Affairs, ST/ESA/SER.A/81. http://www.un.org/esa/population/techcoop/DemEst/manual10/manual10.html