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 intersurvey 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 20year period. However, if fertility schedules are available from other sources for regular fiveyear 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 x15 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 3034 at some census, then ten years before the census they were aged 2024, and 20 years before the census they were aged 1014. 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 fiveyear age group at the time of the census would not have been in a conventional fiveyear age group in each of the earlier years. Thus, the population in age group 3034 at the time of a census would have been aged 2933 a year earlier, 2832 two years earlier and so on. If registered births are tabulated by single year of age of mother, this problem is not serious, because singleyear fertility rates can be calculated for each year and then relatively easily summed by cohort. The crosstabulations and calculations would be lengthy, however, and ageheaping might have a nontrivial effect on the fertility rates. It is therefore convenient to have an approach that can be applied to rates for conventional fiveyear 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: 4555) 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 fiveyear age group of mother, taken from a recent census.
 Registered births by fiveyear 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 fiveyear 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 intersurvey changes in fertility have been smooth and gradual and have affected all age groups in a broadly similar way.
 Errors in the preadjustment fertility rates are proportionately the same for women in the central age groups (2039), 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 underenumeration, 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 elBadry 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 agespecific 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 underenumeration; 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,
${\text{\hspace{0.17em}}}_{5}{P}_{x}\text{\hspace{0.17em}}$
, 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 midyear female population by age group for each year preceding the last census
The exact procedure to be followed in estimating the series of midyear 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 910 October 2001 would be calculated by summing up the days of the year preceding the census night as
$$\text{2}00\text{1}+\frac{\left(\text{31}+\text{28}+\text{31}+\text{3}0+\text{31}+\text{3}0+\text{31}+\text{31}+\text{3}0+\text{9}\right)}{365}=\text{2}00\text{1}+\frac{\text{282}}{365}\text{}=\text{2}00\text{1}.\text{773}$$
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:
$$r(i,a)=\frac{\mathrm{ln}\left(N(i,{t}_{a+1})\right)\mathrm{ln}\left(N(i,{t}_{a})\right)}{{t}_{a+1}{t}_{a}}=\frac{\mathrm{ln}\left(\frac{N(i,{t}_{a+1})}{N(i,{t}_{a})}\right)}{{t}_{a+1}{t}_{a}}$$
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 midyear 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:
$$\text{\hspace{0.17em}}N(i,t)=N(i,{t}_{a})\mathrm{exp}\left(r(i,a)(t+0.5{t}_{a})\right)\text{\hspace{0.17em}}$$
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 midyear populations in age group i at time t, they represent approximate denominators for the calculation of agespecific fertility rates in that year.
Step 3: Calculate agespecific fertility rates from births registered during the years preceding the census
Agespecific 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 3034 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
$$\text{\hspace{0.17em}}f(i,t)=\frac{B(i,t)}{N(i,t)}\text{\hspace{0.17em}}$$
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 1519 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 agespecific 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 yearend nearest to the census date, and registered rates can be cumulated up to the nearest yearend. 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 agespecific fertility rates derived in the previous step. First, the agespecific 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 Fpoints in the conventional formulation of the model. We again assume a 20year run of registration data on ages 1534. 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
$$TF(x,t)=\frac{F(x,t)}{\mathrm{exp}\left(\mathrm{exp}\left(\alpha (t)\beta (t){Y}^{s}(x)\right)\right)}$$
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):
$$\begin{array}{l}f(x,t)\\ \text{\hspace{0.17em}}=TF(x,t)\cdot \left(\begin{array}{l}exp\left(exp\left(\alpha (t)\beta (t)Ys(x+1)\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}exp\left(exp\left(\alpha (t)\beta (t)Ys(x)\right)\right)\end{array}\right)\text{\hspace{0.17em}}.\end{array}$$
Fourth, the parity equivalents are derived by summing the singleage, singleyear 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 1519 age group, i=2 to the 2024 age group etc. and s denotes the last year for which estimates will be derived, then
$$E(i)={\displaystyle \sum _{j=0}^{5i+3}{\displaystyle \sum _{m=5i+9}^{5i+13}f(mj,sj)}}$$
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 2024, 2529 and 3034 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 agespecific 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 2529 and 3034 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 agespecific fertility rates are accurate. Because the numerator and denominator for the estimated agespecific fertility rates do not come from the same source, agereporting errors that affect birth registration and population enumeration differently will distort the pattern of period agespecific fertility rates. Erratic variation in the agespecific growth rates might suggest problems with agereporting, 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 1519 … 3034 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 
1519 
2024 
2529 
3034 
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 midyear 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 (t_{a}) 
1519 
2024 
2529 
3034 
1 
22Apr70 
466,736 
398,383 
324,130 
267,312 
2 
22Apr82 
652,552 
595,598 
479,199 
399,344 
3 
22Apr92 
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 2122 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 agespecific growth rates, Chile, 1970, 1982 and 1992 Census

Age group 

Intercensal period 
1519 
2024 
2529 
3034 
19701982 
0.0279 
0.0335 
0.0326 
0.0335 
19821992 
0.0083 
0.0022 
0.0263 
0.0367 
Thus, for example, the growth rate in the 3034 age group between the 1982 and 1992 Censuses is given by
$$r(3,2)=\frac{\mathrm{ln}\left(\frac{576,710}{399,344}\right)}{1992.3061982.304}=0.0367$$
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 midyear 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 midyear populations by age group, Chile

Age group 

Year 
1519 
2024 
2529 
3034 
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 1519 year old women in mid1990 is given by
$$N(1,1990)=652,552\mathrm{exp}\left(0.0083(1990.51982.3)\right)=609,634$$
Step 3: Calculate agespecific 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 
1519 
2024 
2529 
3034 
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 
Agespecific fertility rates are derived by dividing the births (Table 5) by the estimated midyear population for each age group and year (Table 4). The results are shown in Table 6.
Table 6 Agespecific fertility rates by age group and year, Chile

Age group 

Year 
1519 
2024 
2529 
3034 
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 agespecific 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 negativelog) 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 ageshift 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 peerreviewed 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
$$\frac{1.3392}{\mathrm{exp}\left(\mathrm{exp}\left(0.00491.1367{Y}^{s}(x)\right)\right)}=3.5205$$
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 singleyear agespecific 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 Agespecific fertility rates by singleyears 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 agespecific fertility rate between 16 and 17 in 1990 is given by
$$\begin{array}{l}f(16,1990)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}TF(16,1990)\text{\hspace{0.17em}}.\left(\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{exp}\left(\mathrm{exp}\left(\alpha (1990)\beta (1990){Y}^{s}(17)\right)\right)\\ \mathrm{exp}\left(\mathrm{exp}\left(\alpha (1990)\beta (1990){Y}^{s}(16)\right)\right)\end{array}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}2.6516\text{\hspace{0.17em}}.\left(\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{exp}\left(\mathrm{exp}\left(0.03881.1404{Y}^{s}(17)\right)\right)\\ \mathrm{exp}\left(\mathrm{exp}\left(0.03881.1404{Y}^{s}(16)\right)\right)\end{array}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}0.0348\end{array}$$
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 1519 age group is therefore calculated from
$$\begin{array}{l}E(1)={\displaystyle \sum _{j=0}^{8}{\displaystyle \sum _{m=14}^{18}f(mj,sj)}}\\ =f(14,1991)+f(15,1991)+\mathrm{...}+f(18,1991)+f(13,1990)+\mathrm{...}f(17,1990)+\mathrm{...}+f(6,1983)+\mathrm{...}f(10,1983)\\ =0.07394\end{array}$$
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 


1519 
2024 
2529 
3034 

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 1519 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 2529 and 3034 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 1519 is much lower and that for women aged 2024 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 2024 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 2529 and 3034 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 agespecific 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 agespecific fertility rates for years preceding 1991. However, because the analysis has been truncated at age 3034 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 1519 and, to a lesser extent, those aged 2024. Of course, the procedure used to split the period fertility rates is not perfect and it is most likely to be inaccurate at 1519, but possible methodological inaccuracy cannot explain the large differential observed. Furthermore, since average parities were calculated without making any adjustment for nonresponse, 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 20year 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 intersurvey 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 intersurvey period, rather than completely cumulated fertility for cohorts of women from the start of their childbearing. There is a similar difference in respect of intersurvey parity increments for cohorts. The measure of completeness is thus based on the ratio of intersurvey cumulated fertility, and inter survey parity increments.
Further reading and references
The method was originally set out on pages 4555 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 nonstandard age groups – a long, tedious and errorprone 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 agespecific 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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