# 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been superseded by the relational Gompertz model and its variants, it is useful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

*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

*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

*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

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