# 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:

## 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,

${\text{\hspace{0.17em}}}_{5}{P}_{x}\text{\hspace{0.17em}}$

, 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 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, ta, 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

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,ta+1) and N(i,ta) respectively, by the length of the intercensal period in years:

$r(i,a)= ln( N(i, t a+1 ) )−ln( N(i, t a ) ) t a+1 − t a = ln( N(i, t a+1 ) N(i, t a ) ) t a+1 − t a$
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 ta and ta+1 by expanding exponentially the initial population:

$N(i,t)=N(i, t a )exp( r(i,a)(t+0.5− t a ) )$
Equation 2

for ta < t < ta+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

$f(i,t)= B(i,t) N(i,t)$

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

$TF(x,t)= F(x,t) exp( −exp( −α(t)−β(t) Y s (x) ) )$
Equation 3

where F(x,t) is cumulated fertility to age x in year t and Ys(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):

$f(x,t) =TF(x,t)⋅( exp( −exp( −α(t)−β(t)Ys(x+1) ) ) −exp( −exp( −α(t)−β(t)Ys(x) ) ) ) .$
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

$E(i)= ∑ j=0 5i+3 ∑ m=5i+9 5i+13 f(m−j,s−j)$
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

$r(3,2)= ln( 576,710 399,344 ) 1992.306−1982.304 =0.0367$

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

$N(1,1990)=652,552exp( −0.0083(1990.5−1982.3) )=609,634$

#### 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

$α=intercept− (β−1) 2 . c 2$

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

$1.3392 exp( −exp( −0.0049−1.1367 Y s (x) ) ) =3.5205$

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

$f(16,1990) = TF(16,1990) .( exp( −exp( −α(1990)−β(1990) Y s (17) ) ) −exp( −exp( −α(1990)−β(1990) Y s (16) ) ) ) = 2.6516 .( exp( −exp( −0.0388−1.1404 Y s (17) ) ) −exp( −exp( −0.0388−1.1404 Y s (16) ) ) ) = 0.0348$

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

$E(1)= ∑ j=0 8 ∑ m=14 18 f(m−j,s−j) =f(14,1991)+f(15,1991)+...+f(18,1991)+f(13,1990)+...f(17,1990)+...+f(6,1983)+...f(10,1983) =0.07394$

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.