Cohort parity comparison with vital registration data

Description of method

The synthetic Relational Gompertz model presents a method for comparing average parities with average parity equivalents derived from period fertility rates without having to assume constant fertility. The most important aspect of that method is that average parities are calculated for a period rather than for a series of cohorts. It requires, however, that data on children ever born be available for two points in time, five or ten years apart. If only one source of data on children ever born exists, or if the inter-survey period is not (at least approximately) five or ten years, an alternative procedure that does not require the assump­tion of constant fertility must be used.

The method outlined here is such a procedure. However, while it has distinct theoretical benefits, in practical application the method requires that a fairly long series of annual data on registered births classified by age of mother is available. Such data may not be readily accessible or may be deemed to be so unreliable and defective that the method may not produce sensible results. The method is mainly of use with data on births from a vital registration system, which is normally the only source of information about births by calendar year over a 20-year period. However, if fertility schedules are available from other sources for regular five-year intervals (e.g. complete birth history data collected at a series of time points), such schedules could be used. The method is described here in terms of data from a vital registration system. In general, if parities up to age x are to be used, current fertility data on at least the previous x-15 years are required in order to make reasonable comparisons.

The method makes use of the cohort nature of reported average parities and compares them with parity equivalents obtained from the recorded fertility rates pertaining to the relevant cohorts. If one considers women aged 30-34 at some census, then ten years before the census they were aged 20-24, and 20 years before the census they were aged 10-14. Therefore, on the assumption that childbearing begins at age 15, the children ever born reported by women aged up to 35 at the time of the census reflect the cumulated fertility experi­ence of the women over the preceding 20 years. If mortality and migration are assumed to be unrelated to the fertility experience of women, and fertility rates can be calculated for those 20 years, average parity equivalents for each cohort can be constructed and compared with the reported average parity of women at the time of the census.

The difficulty with applying this general idea is that a cohort represented by a conventional five-year age group at the time of the census would not have been in a conventional five-year age group in each of the earlier years. Thus, the population in age group 30-34 at the time of a census would have been aged 29-33 a year earlier, 28-32 two years earlier and so on. If registered births are tabulated by single year of age of mother, this problem is not serious, because single-year fertility rates can be calculated for each year and then relatively easily summed by cohort. The cross-tabulations and calculations would be lengthy, however, and age-heaping might have a non-trivial effect on the fertility rates. It is therefore convenient to have an approach that can be applied to rates for conventional five-year age groups.

Importantly, the method also finds use as a means of evaluating the completeness of birth registration in a vital registration system.

The method described here circumvents many of the interpolation problems described in its initial formulation in Manual X (UN Population Division 1983: 45-55) by reformulating the approach as another variant of the relational Gompertz model.

Data required

The data required for this method are:

• The number of children ever born, or average parities, by five-year age group of mother, taken from a recent census.
• Registered births by five-year age group of mother for each of 15 or 20 years preceding the census.
• The number of women in each age group from the census, and from one or more earlier censuses, to allow the estimation of the female population by five-year age group for each of the 15 or 20 years preceding the final census.

Assumptions

Most of the assumptions are those associated with the relational Gompertz model, namely:

• The standard fertility schedule chosen for use in the fitting procedure appropriately reflects the shape of the fertility distribution in the population.
• Any inter-survey changes in fertility have been smooth and gradual and have affected all age groups in a broadly similar way.
• Errors in the pre-adjustment fertility rates are proportionately the same for women in the central age groups (20-39), so that the age pattern of fertility described by reported births in the past year is reasonably accurate.

The method also makes the assumption that the populations used as denominators in the derivation of fertility rates have been corrected for under-enumeration, or other errors. This is particularly important because the numerator and denominator of the rates come from different sources.

Preparatory work and preliminary investigations

Before commencing analysis of fertility levels using this method, analysts should investigate the quality of the data at least in the following dimensions:

• age and sex structure of the population;
• reported births in the last year; and
• average parities and whether an el-Badry correction is necessary.

Caveats and warnings

The objective of this method is to estimate the recent levels of fertility and to measure the com­pleteness of birth registration, with a view to adjusting births registered during a recent period to compensate for omission. The effects of other errors, such as changes in the complete­ness of census enumeration through time, should there­fore be allowed for before cumulating age-specific fertil­ity rates for comparison with average parities. Hence, when there is evidence suggesting that there have been changes in the completeness of enumeration, the censuses should, if at all possible, be adjusted before calculating the population denominators. It is not necessary to adjust each census for absolute under-enumeration; it is only necessary to ensure that the relative completeness of enumeration of the earlier censuses is the same as that of the most recent one.

Application of method

The following steps are required for the computa­tional procedure.

Step 1: Calculate the reported average parities

Calculate the average parities at the last census,

${}_5{P}_x$

, of women in each age group [x,x + 5), for x =15, 20 … 45 

if this has not been done already as part of earlier analyses.

Step 2: Estimate the mid-year female population by age group for each year preceding the last census

The exact pro­cedure to be followed in estimating the series of mid­-year female populations by age group depends upon the dates of the census enumerations available. The procedure is therefore discussed here in gen­eral terms.

It is assumed that census enumerations cover, or almost cover, the years for which registered fertility rates are to be cumulated, since it is preferable that the procedure described below is used only for interpolation of the population between census dates, rather than for extrapolation to dates before or after those covered by the censuses.

Define the reference date, t_a, of each census, a, taking into account the fraction of the year up to and including the exact date, or central reference date for the census. Thus, for example, the number representing the date of a census conducted on the night of 9-10 October 2001 would be calculated by summing up the days of the year preceding the census night as

$$\text{2}00\text{1}+\frac{\left(\text{31}+\text{28}+\text{31}+\text{3}0+\text{31}+\text{3}0+\text{31}+\text{31}+\text{3}0+\text{9}\right)}{365}=\text{ 2}00\text{1}+\frac{\text{282}}{365}=\text{ 2}00\text{1}.\text{773}$$

The exponential growth rate, r(i,a), of each age group i between census a and census a+1 is then obtained by divid­ing the difference between the natural logarithms of the female population of age group i at the second and first censuses, N(i,t_a+1) and N(i,t_a) respectively, by the length of the intercensal period in years:

$$r(i,a)=\frac{\ln \left(N(i,t_{a+1})\right)-\ln \left(N(i,t_a)\right)}{t_{a+1}-t_a}=\frac{\ln \left(\frac{N(i,t_{a+1})}{N(i,t_a)}\right)}{t_{a+1}-t_a}$$

Equation 1

If there are three censuses covering the period of investigation, one would derive r(i,1) for the period between the first and second census, and r(i,2) covering the period between the second and third census.

The estimated mid-year population for each year for each age group can then be calculated for each year between t_a and t_a+1 by expanding exponentially the initial population:

$$N(i,t)=N(i,t_a)\exp \left(r(i,a)(t+0.5-t_a)\right)$$

Equation 2

for t_a < t < t_a+1.

If it is necessary to extrapolate to dates before the first census or after the last one, the growth rate in the closest intercensal interval should be used.

As the N(i,t) are estimated mid-year populations in age group i at time t, they represent approximate denominators for the calculation of age-specific fertility rates in that year.

Step 3: Calculate age-specific fertility rates from births registered during the years preceding the census

Age-specific fertility rates are required for a total of 15 calendar years less than the upper age for which parities will be used. For example, if parities up to age 35 (i.e. the 30-34 age group) will be used, a series of fertility rates stretching back 20 years is required). The rate for age group i and calendar year t, f(i,t), is calculated as

$$f(i,t)=\frac{B(i,t)}{N(i,t)}$$

 where B(i,t) is the number of births registered in calen­dar year t as having occurred to women of age group i in calendar year t.

If registered births by age of mother are not available for a few of the 20 calendar years required, the applica­tion of the method will be only slightly affected if rates for the missing year(s) are interpolated from neighbouring rates. For example, if fertility rates are only available for the last 16 of the 20 years required to analyse registration completeness for women under age 35, the rates for the earliest available calendar year can be adopted for the four preceding years without much danger of introducing sizeable errors, since the imputed values used in the analysis would pertain to women aged 15-19 at the time, an age range where fertility is generally relatively low. However, it would be unwise to extrapolate data from the past to impute values for the most recent years because the imputation would affect more age groups, including the years of peak childbearing, in which fertility changes over time are more likely to take place.

Step 4: Cumulate registered fertility for different female birth cohorts to estimate parity equivalents

Birth data are tabulated by calendar year, so age-specific fertility rates will be calculated for calen­dar years. Cumulating rates to the end of each age group will therefore produce fertility cumulants that correspond to the ends of calendar years. The census providing average parities is unlikely to have as its reference date exactly the end of a year, but since average parities for a specified age group change only slowly even when fertility is changing rapidly, the parities from the census can be regarded as referring to the year-end nearest to the census date, and registered rates can be cumulated up to the nearest year-end. Thus, if the census date is on or before 30 June, registered fertility rates would be cumu­lated to the end of the preceding calendar year, whereas if the census date is after 30 June, registered fertility rates would be cumulated to the end of the calendar year during which the census took place. More accurate and refined approaches (for example using exponential extrapolation) are unwarranted given the overall uncertainties and errors in the method.

To estimate the current fertility parity equivalents, a relational Gompertz model is applied to the age-specific fertility rates derived in the previous step. First, the age-specific fertility rates in each year are cumulated to ages 20, 25, 30 and 35. Then, as with the standard relational Gompertz model, the gompits of the ratios of adjacent cumulated fertility estimates are taken. These correspond to the z(x) in the conventional formulation of the model.

Second, estimates of the parameters are derived from the values of z(x) calculated for each year. This is done in a manner analogous to the fitting of a relational Gompertz model only to the F-points in the conventional formulation of the model. We again assume a 20-year run of registration data on ages 15-34. A regression of z(x) - e(x) against g(x), where e(x) and g(x) are derived from the standard fertility schedule, across the three ages 20, 25 and 30, then allows estimates of α(t) and β(t) to be derived. β(t) is the slope of the linear regression based on the three values of z(x) in year t; the intercept is given by α(t)+(β_­(t) - 1)².c/2, where c is a constant derived from the fertility standard used.

Third, the relational Gompertz model is used to produce estimates of fertility by single ages for each calendar year. The values of α(t) and β(t) derived in the previous step are used to determine the shape of the fertility schedule, while the level is derived from the fertility cumulated to each of ages 25, 30 and 35. The estimate of total fertility associated with the fertility cumulated to age x in a given year t is given by

$$TF(x,t)=\frac{F(x,t)}{\exp \left(-\exp \left(-\alpha (t)-\beta (t)Y^s(x)\right)\right)}$$

Equation 3

where F(x,t) is cumulated fertility to age x in year t and Y^s(x) is the gompit at age x of the modified Zaba standard fertility schedule. The resulting three estimates of total fertility are averaged to provide an estimate of the total fertility in each year. Having solved for TF(x,t), it is a simple matter to estimate the fertility between ages x and x+1 at time t, f(x,t):

$$\begin{array}{l}f(x,t)\\ =TF(x,t)\cdot \left(\begin{array}{l}exp\left(-exp\left(-\alpha (t)-\beta (t)Ys(x+1)\right)\right)\\ -exp\left(-exp\left(-\alpha (t)-\beta (t)Ys(x)\right)\right)\end{array}\right).\end{array}$$

Equation 4

Fourth, the parity equivalents are derived by summing the single-age, single-year fertility rates for each five year cohort, and dividing by five (the width of the age interval). If we denote these parity equivalents as E(i,s), where i=1 corresponds to the 15-19 age group, i=2 to the 20-24 age group etc. and s denotes the last year for which estimates will be derived, then

$$E(i)={\displaystyle \sum _{j=0}^{5i+3}{\displaystyle \sum _{m=5i+9}^{5i+13}f(m-j,s-j)}}$$

Equation 5

Step 5: Estimate the completeness of birth registration

The cumulated cohort fertility from registered births, E(i), calculated in the previous step has been con­structed so as to be comparable to reported cohort par­ity, P(i), at the final census. Therefore, the ratio E(i)/P(i) provides a measure of the average complete­ness of registration of the births that occurred to cohort i. If the completeness of registration had remained approximately constant over a period of 15 years or so, the E/P ratios should have more or less the same values for all cohorts, and an average of the ratios for age groups 20-24, 25-29 and 30-34 can be used as an esti­mate of the completeness of birth registration over the period. Its reciprocal can be used as an adjust­ment factor for any or all of the age-specific fertility, schedules calculated in Step 3.

Two forces are in opposition in the interpretation of these E/P ratios. First, in general, if the completeness of birth registration has been improving over time, the E/P ratios for the younger cohorts will be higher than for older cohorts. In such a situation, the most recent fertility schedule (based on the registered births) may be adjusted by P(2)/E(2), the ratio reflecting the most recent level of completeness. (P(1)/E(1) should not be used in general as an adjustment factor because of the intrinsic difficulty in approximating E(1) accurately.) However, when the E/P ratios indicate that completeness has been improving over time, no obvious basis exists for adjusting the fertility schedules referring to earlier years.

The second force arises because the E/P ratios may be lower for younger women than for older women as a result of less reliable and punctual registration of births among younger mothers. If this is the case, choosing the E/P ratio at a younger age will give an inaccurate portrayal of the completeness of the vital registration data. By contrast, the estimates of completeness based on the reports of women aged 25-29 and 30-34 are affected relatively little by excess omission at early ages, so a case could be made for using the average of these two E/P ratios to determine completeness.

Interpretation and diagnostics

An important assumption of this method is that the denom­inators used to calculate the age-specific fertility rates are accurate. Because the numerator and denominator for the estimated age-specific fertility rates do not come from the same source, age-reporting errors that affect birth registration and population enumeration differently will distort the pattern of period age-specific fertility rates. Erratic varia­tion in the age-specific growth rates might suggest problems with age-reporting, but the effects of age misreporting on the final estimates of com­pleteness are very hard to predict. Denominators may also be distorted by changes in the completeness of enumeration from one census to the next and differential completeness of enumeration by age group might affect the results. Changes in enumeration completeness might also affect average parities. For example, if women with children are more likely to be enumerated than women without, average parities will be inflated by omission.

Worked example

The example presented here uses data from three censuses conducted in Chile in April 1970, April 1982 and April 1992 respectively. The original tabulations were taken from the United Nations Statistics Division Demographic Yearbooks, using tabulations on number of women by age group at various censuses; reported births by age of mother and year; mean children ever born at the last census. The method has been implemented in an accompanying Excel workbook.

Step 1: Calculate the reported average parities for ages 15-19 … 30-34 from the final census

The average parities by age from the 1992 census shown in Table 2 are those presented in the Demographic Yearbook:

Table 1 Average parities, Chile, 1992 Census

Age group	15-19	20-24	25-29	30-34
Average parity	0.14	0.69	1.37	2.02

No check is possible on these data as the underlying tabulations are not readily available.

Step 2: Estimate the mid-year female population by age group for each year preceding the last census

Table 2 gives the numbers of women by age group enumerated in each of the three censuses, as downloaded from the Demographic Yearbook.

Table 2 Numbers of women by age group enumerated in the 1970, 1982 and 1992 Chile censuses

		Age group
a	Census date (ta)	15-19	20-24	25-29	30-34
1	22-Apr-70	466,736	398,383	324,130	267,312
2	22-Apr-82	652,552	595,598	479,199	399,344
3	22-Apr-92	600,563	608,933	623,305	576,710

The reference dates for the censuses are 1970.304, 1982.304 and 1992.306. (The reference date for the 1992 census reference date is slightly different from the other two because 1992 is a leap year. The reference date is calculated as 1992 + (31 + 29 + 31 + 21)/366=1992.306, on the assumption that the census date refers to the night of the 21-22 April in each case.

The growth rate in each intercensal period is then derived using Equation 1 as shown in Table 3.

Table 3 Intercensal age-specific growth rates, Chile, 1970, 1982 and 1992 Census

	Age group
Intercensal period	15-19	20-24	25-29	30-34
1970-1982	0.0279	0.0335	0.0326	0.0335
1982-1992	-0.0083	0.0022	0.0263	0.0367

Thus, for example, the growth rate in the 30-34 age group between the 1982 and 1992 Censuses is given by

$$r(3,2)=\frac{\ln \left(\frac{576,710}{399,344}\right)}{1992.306-1982.304}=0.0367$$

The growth rates tend to increase with age group and decline over time. This is suggestive of a declining fertility pattern that may have begun some decades earlier, reinforcing the need for an analytical method that does not assume unchanging fertility.

On the basis of these growth rates, the estimated mid-year population of women by age group can be derived for each calendar year using Equation 2. The resulting values of N(i,t) are shown in Table 4.

Table 4 Estimated mid-year populations by age group, Chile

	Age group
Year	15-19	20-24	25-29	30-34
1972	496,255	428,806	348,169	287,686
1973	510,309	443,420	359,700	297,472
1974	524,761	458,532	371,612	307,591
1975	539,623	474,159	383,919	318,054
1976	554,906	490,318	396,633	328,873
1977	570,621	507,029	409,769	340,060
1978	586,781	524,308	423,340	351,628
1979	603,400	542,177	437,359	363,589
1980	620,488	560,655	451,844	375,957
1981	638,061	579,762	466,808	388,746
1982	651,492	599,521	482,267	401,969
1983	646,106	597,177	494,503	417,283
1984	640,765	598,500	507,674	432,902
1985	635,469	599,827	521,196	449,104
1986	630,215	601,156	535,078	465,913
1987	625,006	602,489	549,331	483,352
1988	619,839	603,824	563,962	501,443
1989	614,715	605,162	578,984	520,211
1990	609,634	606,503	594,405	539,681
1991	604,595	607,847	610,238	559,880

For example the population of 15-19 year old women in mid-1990 is given by

$$N(1,1990)=652,552\exp \left(-0.0083(1990.5-1982.3)\right)=609,634$$

Step 3: Calculate age-specific fertility rates from births registered during the years preceding the census

The number of births reported by age group and year, and downloaded from the Demographic Yearbook is shown in Table 5.

Table 5 Reported births by age group of mother and year, Chile

	Age group
Year	15-19	20-24	25-29	30-34
1972	39,839	80,430	64,624	38,937
1973	40,241	82,108	63,949	38,499
1974	39,884	79,316	63,477	37,880
1975	39,086	75,519	59,365	35,863
1976	37,658	73,889	57,171	34,129
1977	36,104	71,445	53,467	32,190
1978	37,138	73,224	53,725	31,832
1979	36,833	75,905	55,361	32,537
1980	38,562	79,724	59,771	33,769
1981	40,252	86,037	64,849	36,494
1982	39,298	86,061	68,029	38,406
1983	36,077	81,213	65,236	37,506
1984	37,571	83,960	67,266	39,105
1985	34,946	80,735	69,180	39,828
1986	35,925	83,434	72,876	42,605
1987	35,633	84,674	75,416	45,037
1988	37,354	87,484	80,527	48,290
1989	39,095	86,990	82,919	50,875
1990	39,543	85,292	84,336	52,942
1991	38,324	79,406	81,907	53,425

Age-specific fertility rates are derived by dividing the births (Table 5) by the estimated mid-year population for each age group and year (Table 4). The results are shown in Table 6.

Table 6 Age-specific fertility rates by age group and year, Chile

	Age group
Year	15-19	20-24	25-29	30-34
1972	0.0803	0.1876	0.1856	0.1353
1973	0.0789	0.1852	0.1778	0.1294
1974	0.0760	0.1730	0.1708	0.1232
1975	0.0724	0.1593	0.1546	0.1128
1976	0.0679	0.1507	0.1441	0.1038
1977	0.0633	0.1409	0.1305	0.0947
1978	0.0633	0.1397	0.1269	0.0905
1979	0.0610	0.1400	0.1266	0.0895
1980	0.0621	0.1422	0.1323	0.0898
1981	0.0631	0.1484	0.1389	0.0939
1982	0.0603	0.1435	0.1411	0.0955
1983	0.0558	0.1360	0.1319	0.0899
1984	0.0586	0.1403	0.1325	0.0903
1985	0.0550	0.1346	0.1327	0.0887
1986	0.0570	0.1388	0.1362	0.0914
1987	0.0570	0.1405	0.1373	0.0932
1988	0.0603	0.1449	0.1428	0.0963
1989	0.0636	0.1437	0.1432	0.0978
1990	0.0649	0.1406	0.1419	0.0981
1991	0.0634	0.1306	0.1342	0.0954

Step 4: Cumulate registered fertility for different female birth cohorts to estimate parity equivalents

The derivation of the parity equivalents proceeds as follows. The age-specific fertility rates in Table 6 are cumulated to the upper limit of each age group. Thus, for example, the cumulated fertility to age 25 in 1972 would be calculated from 5(0.0803 + 0.1876) = 1.3392. The cumulated fertility to age 30 in 1972 is 5(0.0803 + 0.1876 + 0.1856) = 2.2673. The cumulated rates are shown in Table 7.

Table 7 Cumulated fertility to age x, by year, Chile

	Age x
Year	20	25	30	35
1972	0.4014	1.3392	2.2673	2.9440
1973	0.3943	1.3201	2.2091	2.8562
1974	0.3800	1.2449	2.0990	2.7147
1975	0.3622	1.1585	1.9317	2.4954
1976	0.3393	1.0928	1.8135	2.3324
1977	0.3164	1.0209	1.6733	2.1466
1978	0.3165	1.0147	1.6493	2.1019
1979	0.3052	1.0052	1.6381	2.0856
1980	0.3107	1.0217	1.6831	2.1322
1981	0.3154	1.0574	1.7520	2.2214
1982	0.3016	1.0193	1.7247	2.2024
1983	0.2792	0.9592	1.6188	2.0682
1984	0.2932	0.9946	1.6571	2.1087
1985	0.2750	0.9479	1.6116	2.0550
1986	0.2850	0.9790	1.6600	2.1172
1987	0.2851	0.9878	1.6742	2.1401
1988	0.3013	1.0257	1.7397	2.2212
1989	0.3180	1.0367	1.7528	2.2418
1990	0.3243	1.0275	1.7369	2.2274
1991	0.3169	0.9701	1.6412	2.1183

For each year, the ratio of cumulated fertility in a given age group to that in the next oldest age group is calculated. Thus in the example in the previous paragraph, the ratio of cumulated fertility at age 25 to that at age 30 would be 1.3392/2.2673 = 0.5907. The ratios are shown in the first three columns of Table 8.

Next, a gompit (double negative-log) transform is applied to the ratios to produce a value of z(x) for each of ages 25, 30 and 35 in each year. Using the same example, the value of z(25) in 1972 would be –ln(ln(0.5907)) = 0.6415. The gompits are shown in the last three columns of Table 8.

Table 8 Ratios of cumulated fertility to age x and their gompits, by year, Chile

	Ratios				Gompits z(x)
Year	20	25	30		20	25	30
1972	0.2997	0.5907	0.7701		-0.1864	0.6415	1.3425
1973	0.2987	0.5976	0.7734		-0.1893	0.6639	1.3590
1974	0.3053	0.5931	0.7732		-0.1711	0.6493	1.3577
1975	0.3126	0.5997	0.7741		-0.1508	0.6709	1.3622
1976	0.3105	0.6026	0.7775		-0.1566	0.6802	1.3798
1977	0.3099	0.6101	0.7795		-0.1583	0.7050	1.3900
1978	0.3119	0.6153	0.7847		-0.1529	0.7222	1.4167
1979	0.3036	0.6136	0.7855		-0.1756	0.7167	1.4209
1980	0.3041	0.6070	0.7894		-0.1742	0.6948	1.4417
1981	0.2983	0.6035	0.7887		-0.1904	0.6833	1.4381
1982	0.2959	0.5910	0.7831		-0.1971	0.6427	1.4085
1983	0.2911	0.5925	0.7827		-0.2104	0.6475	1.4065
1984	0.2948	0.6002	0.7858		-0.2001	0.6724	1.4228
1985	0.2901	0.5882	0.7842		-0.2132	0.6336	1.4145
1986	0.2911	0.5898	0.7840		-0.2102	0.6386	1.4135
1987	0.2886	0.5900	0.7823		-0.2173	0.6393	1.4044
1988	0.2938	0.5896	0.7832		-0.2029	0.6381	1.4092
1989	0.3067	0.5915	0.7819		-0.1670	0.6441	1.4022
1990	0.3156	0.5916	0.7798		-0.1425	0.6444	1.3914
1991	0.3267	0.5911	0.7748		-0.1122	0.6429	1.3658

Values of e(x) and g(x) are tabulated without an age-shift since the data on fertility comes from a vital registration system and hence reflects the age of the mother at the birth of the child. The values are derived from Zaba’s modified version of the Booth fertility standard, the only peer-reviewed standard for women currently available. (The standard, and the process of deriving e(x) and g(x), are described in detail in the description of the relational Gompertz method). The values of e(x) and g(x) for the ages required to fit a relational Gompertz model to the observed fertility data are shown in Table 9.

Table 9 Values of e(x) and g(x) from the modified Zaba standard (no age shift), selected ages

Age x	e(x)	g(x)
20	1.3539	-1.3753
25	1.4127	-0.6748
30	1.2750	0.0393

In each year, therefore, it is possible to derive values of z(x) - e(x) for three ages, and to regress these values against the tabulated values of g(x) for the same ages. The value of β in the relational Gompertz model is the slope of the regression equation, while α is estimated from

$$\alpha =\text{intercept}-{(\beta -1)}^2.\frac{c}{2}$$

where c is a constant (=0.95739) derived from the modified Zaba fertility standard. The resulting values of α and β for each year are shown in the first two columns of Table 10.

Table 10 Alpha and Beta parameters of a relational Gompertz model fitted to ages 20, 25 and 30, by year, Chile

				TF based on cumulant to age
Year	Alpha	Beta		25	30	35	Average
1972	0.0049	1.1367		3.5205	3.5525	3.5434	3.5388
1973	0.0242	1.1504		3.4057	3.4173	3.4130	3.4120
1974	0.0177	1.1367		3.2326	3.2701	3.2598	3.2542
1975	0.0270	1.1255		2.9825	3.0074	3.0004	2.9968
1976	0.0430	1.1420		2.7704	2.7897	2.7840	2.7814
1977	0.0585	1.1503		2.5509	2.5508	2.5498	2.5505
1978	0.0828	1.1653		2.4796	2.4782	2.4773	2.4784
1979	0.0844	1.1843		2.4518	2.4466	2.4462	2.4482
1980	0.0912	1.1982		2.4761	2.4973	2.4902	2.4879
1981	0.0850	1.2071		2.5762	2.5995	2.5916	2.5891
1982	0.0516	1.1910		2.5607	2.6054	2.5928	2.5863
1983	0.0520	1.1990		2.4083	2.4396	2.4300	2.4260
1984	0.0712	1.2032		2.4537	2.4750	2.4677	2.4654
1985	0.0527	1.2067		2.3781	2.4227	2.4099	2.4035
1986	0.0537	1.2038		2.4538	2.4963	2.4840	2.4781
1987	0.0479	1.2024		2.4892	2.5248	2.5141	2.5094
1988	0.0506	1.1956		2.5788	2.6257	2.6124	2.6057
1989	0.0472	1.1654		2.6166	2.6728	2.6579	2.6491
1990	0.0388	1.1404		2.6152	2.6780	2.6617	2.6516
1991	0.0179	1.1009		2.5207	2.5856	2.5689	2.5584

The estimates of Total Fertility (TF) associated with cumulated fertility to age x are given by Equation 3. Thus, for example, the estimated Total Fertility implied by the cumulated fertility to age 25 in 1972 is

$$\frac{1.3392}{\exp \left(-\exp \left(-0.0049-1.1367Y^s(x)\right)\right)}=3.5205$$

Averaging the three estimates gives a final estimate of the implied total fertility in each year. Using these estimates in combination with the series of estimates of α and β, one can derive a series of single-year age-specific fertility rates for each calendar year, again using the relational Gompertz model.

Fertility rates by single years of age for each calendar year are derived using Equation 4. As the matrix produced is large (20 years and 25 ages), only an extract from it is shown in Table 11.

Table 11 Age-specific fertility rates by single-years of age and year, Chile

	Age
Year	10	11	12	13	14	15	16	17	18
1982	0.0000	0.0000	0.0000	0.0001	0.0012	0.0081	0.0290	0.0574	0.0883
1983	0.0000	0.0000	0.0000	0.0000	0.0010	0.0072	0.0263	0.0530	0.0822
1984	0.0000	0.0000	0.0000	0.0000	0.0011	0.0078	0.0282	0.0562	0.0864
1985	0.0000	0.0000	0.0000	0.0000	0.0009	0.0067	0.0253	0.0517	0.0809
1986	0.0000	0.0000	0.0000	0.0000	0.0010	0.0071	0.0265	0.0538	0.0839
1987	0.0000	0.0000	0.0000	0.0000	0.0010	0.0071	0.0264	0.0538	0.0841
1988	0.0000	0.0000	0.0000	0.0000	0.0011	0.0079	0.0285	0.0571	0.0884
1989	0.0000	0.0000	0.0000	0.0001	0.0016	0.0099	0.0325	0.0615	0.0919
1990	0.0000	0.0000	0.0000	0.0001	0.0021	0.0113	0.0348	0.0634	0.0927
1991	0.0000	0.0000	0.0000	0.0002	0.0027	0.0129	0.0363	0.0627	0.0890

As an example, the age-specific fertility rate between 16 and 17 in 1990 is given by

$$\begin{array}{l}f(16,1990)\\ =TF(16,1990).\left(\begin{array}{l}\exp \left(-\exp \left(-\alpha (1990)-\beta (1990)Y^s(17)\right)\right)\\ -\exp \left(-\exp \left(-\alpha (1990)-\beta (1990)Y^s(16)\right)\right)\end{array}\right)\\ =2.6516.\left(\begin{array}{l}\exp \left(-\exp \left(-0.0388-1.1404Y^s(17)\right)\right)\\ -\exp \left(-\exp \left(-0.0388-1.1404Y^s(16)\right)\right)\end{array}\right)\\ =0.0348\end{array}$$

Finally, the values of E(i) are derived by applying Equation 5 to the fertility rates in Table 11. E(1), the parity equivalent in the 15-19 age group is therefore calculated from

$$\begin{array}{l}E(1)={\displaystyle \sum _{j=0}^8{\displaystyle \sum _{m=14}^{18}f(m-j,s-j)}}\\ =f(14,1991)+f(15,1991)+\mathrm{...}+f(18,1991)+f(13,1990)+\mathrm{...}f(17,1990)+\mathrm{...}+f(6,1983)+\mathrm{...}f(10,1983)\\ =0.07394\end{array}$$

Although fertility at ages below 10 can safely be assumed to be zero; these ages are included in the formula for ease of presenting the summations.

Step 5: Estimate the completeness of birth registration

For each cohort (defined by age at the end of the final year), the completeness of birth registration is estimated as the ratio of the parity equivalent, E(i), cal­culated from registered births to reported average parity, P(i), as obtained from the final census (shown in Table 1). The results are presented in Table 12.

Table 12 Estimates of completeness of birth registration, Chile 1991

		Age group
		15-19		20-24		25-29	30-34
MCEB from last census (applies to end year)	1991	0.14		0.69		1.37	2.02
Parity equivalents (applies to end year)	1991	0.07		0.57		1.29	1.92
Completeness		0.5281		0.8207		0.9382	0.9515
Average completeness					0.9449

The completeness estimate for the cohort aged 15-19 at the end of 1991 is 0.07/0.14 = 0.5281, or 52 per cent complete. The estimates of completeness of registration of births for women aged 25-29 and 30-34 are both higher and more consistent with each other, suggesting an average level of completeness of around 94 per cent. The estimate for women aged 15-19 is much lower and that for women aged 20-24 suggests that completeness is of the order of 80 per cent. It seems likely that registration of births is substantially less complete among very young mothers while the cumulated com­pleteness of mothers aged 20-24 is also lower because births that they experienced as teenagers are a substantial fraction of their total births.

The estimates of completeness based on the reports of women aged 25-29 and 30-34 are affected relatively little by the excess omission at early ages, so that in this case the best estimate of the complete­ness of birth registration is probably the average of 0.9382 and 0.9515. The final estimate is therefore 0.9449. Hence, an improved estimate of fertility for 1991 could be obtained by inflating the registered age-specific fertility rates for that year by a factor of 1/0.9449, or about 5.83 per cent. It should be noted, however, that the adjusted fertility schedule might not be a good indicator of the age pattern of childbearing because of the apparent relatively higher omission of births by young women.

The results presented in Table 12 do not suggest that birth registration completeness has been improving over time, so the adjustment factor of 1.0583 can also be applied to the observed age-specific fertility rates for years preceding 1991. However, because the analysis has been truncated at age 30-34 the contributions of fertility rates registered before 1982 to cumulated cohort fertility are small. The estimated adjustment factor thus cannot be validly applied to the fertility rates registered before that date. Although not observed in the case of Chile, where there is evidence of a trend towards more complete registration, such as a tendency for the estimates of completeness to decline with age, registered births for particular years should not be adjusted using a scaling factor derived from an average over many years.

However, none of these considerations explains ade­quately the low estimates of completeness obtained for the cohorts aged 15-19 and, to a lesser extent, those aged 20-24. Of course, the procedure used to split the period fertility rates is not perfect and it is most likely to be inaccurate at 15-19, but possible methodological inaccu­racy cannot explain the large differential observed. Furthermore, since average parities were calculated without making any adjustment for non-response, they are more likely to be too small than too large. Hence, on the basis of this evidence alone, it would appear that birth registration in Chile is less complete for young mothers than for older women.

Further variants

An extension of the method, using parities from two surveys five or 10 years apart, combined with vital registration data covering births over a 20-year period, is described on pages 55ff of Manual X (UN Population Division 1983). 

The main assump­tions made in the version described above to adjust period fertility on the basis of cumulated cohort fertility from the beginning of childbearing – namely, that regis­tration completeness be constant both by age of mother and by period – are no longer required. However, parity increments are very sensitive to changes in the completeness of reporting of children ever born, as a result of which the estimates of registration complete­ness obtained by the method described on pages 55ff of Manual X are also quite sensitive to such changes, which are generally most marked for older women.

The main differences between the procedure described on pages 55f of Manual X and that described above are that all calculations in the variant approach are limited to the inter-survey period, and all cohorts of reproductive age at the time of the second survey are considered. In doing so, the method compares the increments in fertility according to the registration system over the inter-survey period, rather than completely cumulated fertility for cohorts of women from the start of their childbearing. There is a similar difference in respect of inter-survey parity increments for cohorts. The measure of completeness is thus based on the ratio of inter-survey cumulated fertility, and inter- survey parity increments.

Further reading and references

The method was originally set out on pages 45-55 of Manual X (UN Population Division 1983). Given its dependence on a long series of vital registration data, the method has not been applied widely. However, as more developing countries seek to implement or improve systems of vital registration, the method might become more important in future as a check on the quality of those data.

The only substantial change made in the implementation of the method as described here is that, whereas the original formulation relied on tabulated coefficients based on the Brass fertility schedule to apportion fertility to non-standard age groups – a long, tedious and error-prone process – the approach adopted here is simpler, making use of the relational Gompertz model to do the same thing. In effect, what is now done is that a separate relational Gompertz model is fitted to the fertility rates observed in the vital registration data in each year. The values of the parameters alpha and beta so derived can then produce estimates of the implied total fertility based on the observed fertility to each of ages 25, 30 and 35. The average of these three estimates of total fertility is then taken as the estimate of total fertility in each year, thereby permitting the calculation of age-specific fertility rates by single years of age in each calendar year. Using these, the calculation of cumulated cohort fertility is simple, as the relevant fertility rates in each year simply need to be summed.

 

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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