# Relational standards used in this project

## Introduction

A number of methods presented here rely on relational models of fertility or mortality for their implementation. To enable others to reproduce the standards used in these methods, we describe briefly the standards used, and their construction.

## Mortality standards

In order to harmonise and standardise the mortality bases used in the preparation of this manual, all methods presented that rely on tabulated values from standard life tables draw on a common subset of these life tables.

In particular, the standard life tables used are those from the Princeton Regional Model Life Tables (North, South, East and West) and the UN Model Life Tables for Developing Countries (General, Latin America, Chile, South Asia, Far East), by sex, all with an expectation of life at birth of 60 years. The original life tables have been modified, extended and enhanced over time to extend them to older ages. We make use of the updated tables provided by the United Nations Population Division, and used by them in their population projections.

Standard
life tables based on the 18 June 2010 revision, by sex with an expectation
of life at birth of 60 years, for each of the nine standards above were used.
These life tables provide values of *l** _{x}*
and

*L*(amongst other quantities) for ages 0, 1, 5, 10, ..., 130.

_{x}Values of* l** _{x}* for ages 2, 3
and 4 were derived as follows:

- For the Princeton Regional Model Life Tables, the
proportionality factors presented by Coale, Demeny and Vaughan (1983: 21) were applied to
*l*and_{1}*l*to generate_{5}*l*,_{2}*l*and_{3}*l*. For the UN Model Life Tables for Developing Countries, deaths between the ages of 1 and 5 were distributed by single years of age in the same proportion as those deaths in the original sex- and region-specific life tables._{4}

Joint-sex life tables (that is, for males and females combined) are required by some methods of child and adult mortality estimation. As these life tables (or their implementation) is not sensitive to the sex ratio at birth, a sex ratio at birth of 105 (boys per 100 girls) was used. Joint-sex life tables were then derived by appropriate weighting of the sex-specific life tables:

where

represents the number of survivors at age x in the joint-sex life table and

and

are the equivalent life table values for men and women respectively.

As these life tables are used almost
exclusively in a relational context (as originally set out by Brass (1971)), standard logits of the *l** _{x}* were calculated for all ages above zero by means of
the formula

Values of these logits are presented in the downloadable spreadsheet at the bottom of this page.

## Fertility standard

Methods based on the relational
Gompertz model draw on a slightly modified version of the standard produced by Zaba (1981). Zaba's standard is the same at all ages older than 15 to
that produced by Booth (1984). At younger ages, however, Zaba observed that the rate of
change in the fertility schedule did not match that found in empirical
observations. The standard applied here uses Zaba's tabulated values, but makes
a small correction for how cumulated fertility, *F,* is derived at half-ages, which are required to accommodate the
usual shift in tabulations of fertility data by age of mother. Zaba's approach
averaged values of *F*(*x*) and *F*(*x+1*) to approximate *F*(*x*+½).
However, since the purpose and the effect of the gompit transform, *Y*(*x*)
= -ln(-ln(*F*(*x*)), is to linearise a curvilinear function, it makes greater sense
to interpolate *Y*(*x*+½) from successive values of *Y*(*x*) and *Y*(*x*+1) and to then take
the antigompit of *Y*(*x*+½) to derive *F*(*x*+½).

Values of the modified Zaba standard are presented in the downloadable spreadsheet at the bottom of this page.

## References

Booth H. 1984. "Transforming Gompertz' function for fertility
analysis: The development of a standard for the relational Gompertz
function", *Population Studies* **38**(3):495-506.

Brass W. 1971. "On the
scale of mortality," in Brass, W (ed). *Biological
Aspects of Demography*. London: Taylor and Francis, pp. 69-110.

Coale AJ, P Demeny and B
Vaughan. 1983. *Regional Model Life Tables
and Stable Populations. *New York: Academic Press.

Zaba B. 1981. *Use of the Relational Gompertz Model in
Analysing Fertility Data Collected in Retrospective Surveys*.* *Centre for Population Studies Research
Paper 81-2. London: Centre for Population Studies, London School of Hygiene
& Tropical Medicine.

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