Demographic Analysis and Evaluation Workshop on Demographic Analysis and Evaluation 1

Fertility: Methods Based on Special Questions

Methods Based on Questions about Children Ever Born by Age of Woman Many census and survey questionnaires contain a fertility component asking women about the number of children they have ever had and whether they had a birth in the year preceding the inquiry. Several techniques have been developed to derive estimates of age‑specific fertility rates based on these data. Fertility rates are related to the average number of children ever born per woman. For example, if fertility were to remain constant, the average number of children ever born per woman in the reproductive ages would be similar to the cumulative fertility rates from the beginning to the end of the reproductive ages.

Methods Based on Questions about Children Ever Born by Age of Woman Based on this observation, Mortara (1949) used census data on children ever born per woman to estimate age-specific fertility rates. Working under the assumption that fertility was constant during the past, he graphically converted the data on children ever born from 5‑year age groups to single ages of women by drawing a smooth line passing through the midpoint of each age interval of the data on children ever born. Such a graph represented not only the average number of children ever born by single ages of women, but also the cumulative fertility up to each age (because of the assumption of constant fertility during the past).

Methods Based on Questions about Children Ever Born by Age of Woman ASFR in a year

Methods Based on Questions about Children Ever Born by Age of Woman Hence, he derived age‑specific fertility rates for single ages by taking the differences between the average number of children ever born at successive ages. Mortara applied this process in a country where fertility was almost constant during the past, and where census information was of rather good quality.

Methods Based on Questions about Children Ever Born by Age of Woman However, if fertility is changing and data are of poor quality, the estimates could be biased. In some populations, it is observed that women over age 35 or 40 may underreport the average number of children ever born, for various reasons. For example, there may be a reluctance to report dead children or those who already are living in other households. Age of mother may also be misreported, giving a distorted pattern of fertility based on information on children ever born by age of mother.

Brass’ P/F Ratio Technique In the 1960s, William Brass (Brass et al. 1968) developed a procedure for adjusting reported age-specific fertility for commonly observed data errors by combining parity data with an age-specific fertility pattern from a census or survey under the assumption of constant fertility. Brass’ P/F ratio technique produces a factor for adjusting reported age-specific fertility rates (based on vital registration or births in the 12 months prior to a census or survey) to the "actual" level of fertility. The adjustment factor is determined by comparing data on children ever born by age of women (Px), with a set of cumulated age- specific fertility rates (Fx).

Brass’ P/F Ratio Technique In the chart shown on the next slide, Age-specific fertility is shown in blue. Five-year average values have been scaled up to indicate number of births per woman over a 5-year period. Parity is shown in red. The difference between successive point estimates indicates implied childbearing for women moving from one age group to another within their reproductive years under the assumption of constant fertility, as was true with Mortara’s method. Parities at older ages represent births that occurred during years further in the past. Parities at older ages, and at ages 45-49 in particular, may be under-reported.

Brass’ P/F Ratio Technique

Brass’ P/F Ratio Technique

Brass’ P/F Ratio Technique Data required: Mean number of children ever born alive, per woman, by 5-year age groups (15 to 19 years to 45 to 49 years), as collected in a census or survey. Age-specific fertility rates for the same 5-year age groups of women. These rates may be based on reported births in the 12 months prior to the census or on independent registration for a time period close to the date of the census or survey.

Brass’ P/F Ratio Technique Data required: Information to determine whether the age of women reporting births in the last 12 months was reported as of the date of childbirth or as of the date of the census or survey.

Brass’ P/F Ratio Technique Assumptions: The reporting of the average number of children ever born is complete (at least for younger women, under 30 years or 35 years of age), and represents the level of cumulative fertility up to these ages.  The completeness of reporting of births used to estimate the age-specific fertility rates is the same for all age groups of women. The pattern and level of fertility have not changed in the recent past (15 to 20 years prior to the census or survey).

Brass’ P/F Ratio Technique The steps: Mean children ever born, or parity (P), is tabulated by age of women in 5-year age groups, 15-19, 20-24, …45-49 with midpoints 17.5, 22.5, …47.5. ASFRs (fx) representing the pattern of fertility are cumulated up to ages 20, 25,..., 50. Cumulated fx are converted to parity equivalents (Fx) either using Brass’ original multipliers or, more commonly, using the Trussell variant adjustment factors. Cumulating asfrs involves * 5

Brass’ P/F Ratio Technique The steps: The ratios Px/Fx are calculated for each 5-year age group of women. Ratios for women ages 20-24 and 25-29 are commonly used to adjust the entire set of ASFRs (fx) upward.

Brass’ P/F Ratio Technique The Trussell variant (United Nations 1983: chapter II) of Brass’ method is commonly used, rather than the original Brass formulae. The input panel of the PASEX spreadsheet PFRATIO.xls, which uses the Trussell variant, is shown here. Data entry fields are shown in blue. F(i) = timing Spreadsheet: PFRATIO.xls

Brass’ P/F Ratio Technique After required data are entered into the spreadsheet by the analyst, the spreadsheet will automatically apply the calculations described above, to generate a series of P/F ratios. Spreadsheet: PFRATIO.xls 18

Brass’ P/F Ratio Technique The analysis of the P/F ratio series is important. For instance, if the data are accurate and fertility has been decreasing in recent years, the P/F ratios may show a trend that rises with the age groups of women. This is because most of the births to older women occurred in the past when fertility was higher, while births to younger women occurred more recently when fertility had already declined. This situation occurs especially when a rise in the age at marriage leads to a reduction in fertility among the youngest age groups of women. Under these conditions, the results of the technique do not represent the current pattern of cumulated fertility. F(i) timing

Brass’ P/F Ratio Technique The P/F ratios (derived from data for women in middle age groups), will then be used by the spreadsheet to generate four alternate sets of adjusted ASFRs. Spreadsheet: PFRATIO.xls 20

Brass’ P/F Ratio Technique The first three ASFR sets are based on the following P/F ratios that were applied by the spreadsheet in adjusting the input ASFRs: P2/F2 (based on women ages 20-24), P3/F3 (women 25-29), or P4/F4 (women 30-34). The fourth is an average of P3/F3 and P4/F4. Spreadsheet: PFRATIO.xls

Brass’ P/F Ratio Technique Resulting adjusted ASFRs calculated using alternative P/F ratios.

Brass’ P/F Ratio Technique The task of the analyst then is to select what is likely to be the most accurate ASFR set from among the pre-defined choices, or by manually defining a composite P/F ratio in the last column (blue text).

Brass’ P/F Ratio Technique Advantages: The method is not affected by the failure of some women to report the number of children ever born if the women who did not report have the same number of children as those who reported.  The method uses a relatively small amount of information, although it requires special questions in the survey or census.  

Brass’ P/F Ratio Technique Limitations: If fertility has not been constant, the results of the technique may be biased upward. Errors in the data on children ever born will affect the results:  Age misreporting of women providing these data will have an unpredictable effect.  Underreporting of children ever born will cause a downward bias in the adjusted estimates; over- reporting, an upward bias.

Brass’ P/F Ratio Technique Limitations: Underreporting of children ever born will cause a downward bias in the adjusted estimates; over- reporting, an upward bias. Children who died in infancy (especially in very early infancy), as well as those living away from home, are the births most likely to be omitted, especially by older women. Over-reporting of children can sometimes occur when stillbirths, late fetal deaths, or adopted children are mistakenly included.  

Brass’ P/F Ratio Technique Limitations: Errors in the age-specific fertility rates will affect the results: Age misreporting of women providing these data will, again, have an unpredictable effect.  If the pattern of fertility taken as the "actual" pattern contains errors, the estimated age-specific fertility rates will be incorrect. This may also affect the level of the total fertility rate.

Arriaga’s Method Arriaga (U.S. Bureau of the Census, 1983) developed a technique which does not require constant fertility. Based on a simulation model, he observed that under conditions of declining fertility, the number of children ever born by age of mother changes almost linearly for mothers' ages under 35 years. Based on this observation and on the fact that the reported number of children ever born by mothers under age 35 is usually acceptable, linear interpolation of the data on children ever born per woman by age of mother from two or more censuses or surveys can provide an estimate of the children ever born for 1 year prior to the date of the interview. And this together with CEB at interview, generates an asfr.

Arriaga’s Method Data required: The average number of children ever born per woman, by 5-year age groups, for two dates. An age pattern of fertility, that is, a distribution of age- specific fertility rates by 5-year age groups. Although vital statistics can provide a pattern of fertility, frequently the pattern is obtained from information on the number of births occurring during the 12 months prior to a census or survey, classified by age of mother. Although the estimates can be obtained even if a pattern of fertility is not available, better results are achieved if such information is used.

Arriaga’s Method Assumptions: The completeness of reporting of births used to estimate the age-specific fertility rates is the same for all age groups of women. Reporting of the average number of children ever born per woman is complete (at least for women under 30 years or 35 years of age). Changes in fertility produce a linear change in the average number of children ever born per woman at each particular age of woman (mainly at ages 15 to 35 years) between the two dates.  

Assumptions (continued): Arriaga’s Method Assumptions (continued): Fertility occurs only between exact ages 15 and 50 years.  

Arriaga’s Method Procedure: Estimated single-age cumulative fertility (children ever born, CBx ) for census or survey dates is obtained by an interpolation process using a ninth-degree polynomial. The average number of children ever born for the period between the censuses or surveys is calculated by linearly interpolating between age-specific CBx for one year after the earlier and one year before the later census or survey.

Arriaga’s Method

Arriaga’s Method Procedure (continued): Age-specific fertility rates are calculated as the differences between the average number of children ever born per woman by age of mother for 2 census or survey years, for each single year of age of the female population (next chart, BC and EF). These differences represent the age-specific fertility rates by single years of age. Single age-specific fertility rates within each conventional 5-year age group are averaged and taken as the 5-year age-specific fertility rates representing the level of fertility up to age 35.

Arriaga’s Method

Arriaga’s Method Procedure (continued): Cumulated fertility rates (CF) are calculated for the rates obtained. Cumulated rates (CP) are also calculated for the reported age-specific fertility rates (PF) comprising the pattern of fertility. Adjustment factors are calculated by dividing the cumulated fertility rates by the corresponding cumulated fertility rates pertaining to the pattern of fertility. Zi = (CF/CP)x

Arriaga’s Method Procedure: Finally, adjustment factors Zi are used to adjust the pattern of age-specific fertility rates. Fx = Zi * PFx Adjusted rates are dated half a year after the first census or survey and half a year prior to the second census or survey. It is recommended that the adjustment factor that corresponds to the age group whose mean is closest to the mean age of the fertility pattern be used.

Arriaga’s Method The input panel for Arriaga’s method, using two censuses or surveys, ARFE2.xls. Notice the exact, decimalized dating used and the specification of data type (code 0). Spreadsheets: ARFE-2.xls ARFE-3.xls

Arriaga’s Method After required data are entered into the input panel by the analyst, the spreadsheet will automatically apply the calculations described above to generate two key output panels... Spreadsheets: ARFE-2.xls ARFE-3.xls 39

Arriaga’s Method The first output panel of ARFE2.xls shows implied total fertility rates based on various adjusting factors (Z scores), by the four age groups associated with the data from which they derived. Also shown are the specific Z scores, in their constituent 5-year age groups. } } The first output panel of ARFE2.xls shows implied total fertility rates based on choice of adjusting factors (Z scores) that are associated with input data for women of specific age groups, along with those Z scores, in their 5-year age groups. The 5-year age groups represent the age of the women who’s input data were used to derive them. Age sex, again! Spreadsheets: ARFE-2.xls ARFE-3.xls

Arriaga’s Method The second output panel repeats adjusting factors (Z scores) in their 5-year age groups…. } …and shows the ASFRs derived from them, within the four alternate age groups. Spreadsheets: ARFE-2.xls ARFE-3.xls

Arriaga’s Method The task of the analyst then is to identify the appropriate ASFR set from among the four choices. The appropriate choice is the ASFR set associated with the age group that contains a midpoint most closely approximating the mean age of childbearing. Spreadsheets: ARFE-2.xls ARFE-3.xls

Arriaga’s Method The mean age of childbearing was also automatically calculated by the spreadsheet, and is displayed in the second output panel. Spreadsheets: ARFE-2.xls ARFE-3.xls

Arriaga’s Method The mean age at childbearing is 27.45; therefore, the adjusting factor based on women ages 25-29 would be an appropriate choice. The ASFRs corresponding to this age group imply an adjusted TFR of 3.454.

Arriaga’s Method Advantages: Since the technique does not assume that fertility is constant, it can provide an estimate of fertility when it has been changing. Fertility estimates are obtained for the year of the censuses or surveys.  An analysis of the adjustment factors allows for an evaluation of the data used. The adjustment factors are not affected by changing fertility but are affected by the quality and comparability of the data.

Arriaga’s Method Limitations: Errors in the data on children ever born will affect the results: Age misreporting of women providing these data will have an unpredictable effect. Underreporting or over-reporting of children ever born by women under age 35 years will affect the estimates.

Arriaga’s Method Limitations: Underreporting or overreporting of children ever born by women under age 35 years will affect the estimates. If the reporting of children ever born is more (or less) complete in one census than in the other, the estimates of fertility will be affected, and the trend in fertility may be incorrect. This can occur when data from censuses and surveys are combined in the estimation process because survey data are frequently of better quality than census data.

Errors in the ASFR pattern will affect the results: Arriaga’s Method Limitations: If the technique is used when a pattern of fertility is not available, the results should be interpreted with caution. Errors in the ASFR pattern will affect the results: Age misreporting of women providing these data will have an unpredictable effect. If the pattern of fertility taken as the "actual" pattern contains errors, the estimated ASFRs will be incorrect. And estimated TFR may be affected.  

Arriaga’s Method: Additional Notes The Arriaga technique also can be used when information on the average number of children ever born by age of mother and pattern of fertility are available for only one date. However, if data are available for only one date, then the assumptions, advantages, and limitations are basically the same as those for the P/F ratio technique. Otherwise, the Arriaga method’s estimated TFR will differ from Brass P/F ratio method (and Mortara method) TFR estimates. In the chart shown previously, reproduced on the following page, Arriaga’s estimate of TFR is EF at t+5.5; the Brass and Mortara estimates, EG.

Arriaga’s Method Under conditions of declining fertility: EF: TFR from Arriaga’s method EG: TFR from Brass’ P/F Ratio method

References Arriaga, Eduardo E. 1983. “Estimating Fertility from Data on Children Ever Born, by Age of Mother.” Bureau of the Census, International Research Document No. 11. _____ and Associates. 1994. Population Analysis with Microcomputers, Volume 1. Washington, D.C.: U.S. Bureau of the Census, U.S. Agency for International Development, and United Nations Population Fund, chapter IV. Brass, William and others. 1968. The Demography of Tropical Africa. Princeton, NJ: Princeton University Press.

References _____. 1981. “the Use of the Gompertz Relational Model to Estimate Fertility,” International Population Conference, Manila 1981. Liege: IUSSP, pp. 345-361. Mortara. Giorgio, 1949. "Determinação da fecundidade feminina segundo a idade, conforme o censo de 1940, e aplicação ao cáculo de taxa de natalidade, tábua de fecundidade e do coeficiente de reprodução, para a população de Brasil," Estadística Demográfica no. 5, pp. 7-36. Instituto Brasileiro de Geografia e Estadística. Rio de Janeiro.

References Popoff, Carole and D. H. Judson, 2004. “Some Methods of Estimation for Statistically Underdeveloped Areas,” in Jacob S. Siegel and David A. Swanson, eds. ,The Methods and Materials of Demography, second edition. New York: Elsevier, pp. 603-641. United Nations. 1967. Manual IV. Methods of Estimating Basic Demographic Measures from Incomplete Data. By Ansely J. Coale and Paul Demeny. Department of International Economic and Social Affairs, Population Studies No. 42. New York.