Fertility Control, Indirect Measurement of

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Knowledge of the extent and effectiveness of intentional control of fertility is important in understanding population trends and in theorizing about fertility transitions. Evidence regarding fertility control has played an important role in a long-running debate about contemporary family planning programs.

On one side of the debate, the "continuity" school stresses that intentional fertility control has been known and practiced much earlier than the onset of the demographic transition and certainly well before the initiation of any family planning program. The "discontinuity" school argues that even in the nineteenth century, many populations existed in a pre-rational mode with respect to fertility behavior.

According to the "continuity" school the fertility transition was dominantly triggered by a change in mortality rates and external socio-economic conditions, which lowered the number of births couples wanted. According to the "discontinuity" school, the fertility transition was initiated by the diffusion of the idea that fertility behavior should be brought into the realm of conscious choice.

Members of the "continuity" school were skeptical about the effectiveness of family planning programs in the absence of changes in mortality rates and socio-economic circumstances. Members of the "discontinuity" school were skeptical about any approach that did not directly address people's mindsets. The debate between the two schools was quickly translated into questions about the extent of fertility control and whether or not it was mainly characterized by stopping or spacing.

Information about fertility control practices in contemporary populations is routinely collected by surveys that ask individuals to report on their current and past personal experience. Where it is infeasible to conduct such surveys, it is nevertheless possible to employ indirect methods to obtain some information regarding the extent and nature of deliberate fertility regulation. These are statistical techniques that yield population-level measurements concerning "control," essentially by comparing the pattern of fertility observed in the population of interest with the pattern of a "natural-fertility" population, i.e. one in which fertility control was not practiced. (The population of interest is referred to as the "target" population; the natural-fertility population is termed the "model" population.) This article describes the principal indirect methods of estimating the extent and character of fertility limitation.

Two polar types of fertility limitation are termed perfect stopping and pure spacing. In the case of perfect stopping, all couples have no more children after they initiate fertility control. Pure spacing behavior occurs when all couples who are ever going to limit their fertility begin control before the birth of their first child. In real populations, neither of these extremes ever occurs. Perfect stopping does not happen because of contraceptive failure. Pure spacing is not observed because some couples desire a first birth as soon as possible after marriage.

Early Approaches

David V. Glass and Eugene Grebenik, in a British census report, developed a "statistical model for the study of desired and achieved family size" (Glass and Grebenik, p. 270), a by-product of which was estimates of the extent of use of fertility control. Their estimator derived from the comparison of two tabulations of women by the number of children they had borne (called parity distributions). Each pair of distributions compared women who had married in a specified age range and were observed (still married) at or beyond the end of the childbearing age-span. Glass and Grebnick's target populations were the married women observed by the 1946 family census of Great Britain, and their chosen (non-controlling) models were the corresponding current-and marriage-age groups recorded by the 1911 census of Ireland as residing in rural areas–specifically, outside the highly urbanized county boroughs.

Glass and Grebenik's approach assumed that couples practice stopping behavior–that is, they begin to control their fertility after reaching a certain number of children ever born, or parity, presumed to be their desired family size. Before that point, couples proceeded from parity to parity according to the parity progression rates of the model population. When they initiate control, their parity progression rates become the rates of the model population multiplied by the fraction (1-p), where is the probability of having an additional birth at each parity once "family limitation" has begun. Their procedure produced the first indirect measures of the extent of fertility control as a step in the estimation of "desired and achieved family size."

Coale's M and m and Extensions

Ansley J. Coale, like Glass and Grebenik, apparently came to his indirect measure on his way to another goal. In a 1971 article entitled "Age Patterns of Marriage," Coale devoted little more than a page to his key equation:

m (a) = M·n(a)·em·v(a)

where m(a) is age-specific marital fertility at age a, n(a) is the age-specific marital fertility of Hutterites during the period from 1921 to 1930, and M, m, and v(a) are parameters. (The Hutterites, a small religious community located in border regions of midwestern Canada and the United States, are frequently taken by demographers to illustrate a natural fertility population.) The parameter M is the ratio of marital fertility at ages 20 to 24 to Hutterite marital fertility at that age, and m measures "the extent to which control of fertility causes a systematic deviation from the age pattern of natural fertility" (Coale, p. 207).

When m = 1, the marital fertility schedule diverges from the Hutterite pattern by roughly the average proportional deviation observed in the 43 age-specific marital fertility schedules reported by the United Nations in its Demographic Yearbook for 1965. This implicitly defines the age-specific parameter, v(a). Coale and T. James Trussell made statistical refinements in the method of obtaining these parameters in 1974 and 1978. Coale's approach yields a measure that may be interpreted as reflection of the effects of fertility control, but it provides no estimate of the extent to which the target population engaged in family limitation.

In 1979 Warren C. Sanderson used both parity distributions and the Coale specification for the marital fertility schedule to investigate the evolution of fertility control in the native born white population of the United States from the beginning of the nineteenth century onwards. The parity distributions are those for all women who had ever been married, and were observed at or after the end of the childbearing age-span.

Sanderson denoted F(q) to be the mean fertility of the fraction q of the population with the highest number of children ever born, with F(1) being the mean fertility of the entire cohort. Sanderson then defined a particular value of q, q*, implicitly fromthe equation:

F(q*) = Bn/Φ,

where Bn is the mean fertility of the population in the absence of fertility limitation, and Φ is the proportion of couples who are physiologically capable of bearing a child. Bn depends on Coale's M and m parameters and on the age distribution of marriage. Under plausible conditions, (1 -q*) is a measure of the extent of fertility control.

Cohort Parity Analysis

Cohort parity analysis (CPA) is the name given to an analytic approach to measuring the extent and character of fertility regulation from marriage-age and marriage-duration specific parity distribution data. Paul A. David, Sanderson, and their coauthors developed the methodology, describing it in a series of articles in the 1980s.

Like the Glass and Grebenik procedure, CPA is based on a comparison of parity distributions and treats couples who have ever initiated family limiting behavior as "controllers." CPA also allows for the possibility that couples could initiate fertility reducing behaviors prior to the parity at which they ultimately stop. Unlike Glass and Grebenik, CPA does not assume that people only begin practicing fertility control when they attain their desired family size.

In the CPA framework, couples who do not initiate control proceed upward through the parity distribution according to the model (natural-fertility) parity progression rates. Once controlling behavior has started, all that is known is that parity progression rates must be below those of the model population. CPA does not assume that control is maintained continuously after it is initiated; for example, it allows for the possibility that a couple may use contraception after marriage for three years, again for two years following the birth of their first child, and then continuously after the birth of their second child.

A lower bound on the proportion of cohorts who are practicing birth control is calculated from the target and model populations, and is compared with the level that would result in the population from perfect stopping behavior. A corresponding upper bound on the extent of fertility limitation, similarly calculated, is that which would result were those who controlled to have engaged in pure spacing behaviors. The upper and lower bound estimates are efficient in the sense that, within the CPA framework, there can be no lower upper bound and no higher lower bound.

Because CPA can be applied to cohorts who have not yet completed their childbearing, it is especially useful for studying family limiting behavior among younger couples. David and Sanderson made use of this advantageous property of CPA in a study of fertility control among the married women residing in Ireland's urban areas in 1911. This revealed that not only that there had been a significant amount of family limitation during the preceding decades, but that the extent of control among cohorts of younger women at lower durations of marriage exceeded that among the older women who had reached the end of their childbearing span.

Assessment of the Main Approaches

The two main approaches to estimating the extent of birth control are Coale's M and m method and cohort parity analysis. Each has its critics and its defenders.

A drawback in Coale's methodology is that it rests solely on the shape of the target age-specific fertility schedule, so that decreases in the level of age-specific marital fertility at young ages (resulting from "spacing") must show up as reductions in the level of M, not increases in the index of the intensity of control, m. Yet there is ample evidence of the occurrence of such decreases in M in populations undergoing the fertility transition, implying that measured changes in m alone understate gains in the extent of control.

Barbara S. Okun, who appraised both methods in a 1994 article, reported that Coale's m remained close to zero even in populations where 40 percent of married couples were exercising effective fertility control. Further, m was insensitive to changes in the extent of control when such control was low, and large changes in control often registered only as small changes in m.

Okun's main criticism of CPA is that the bounds found on the extent of control are highly sensitive to the choice of model distribution. Hence, errors in estimated proportions of controllers would arise when an inappropriate model distribution was used. An inappropriate model distribution is one that is derived from a population that differs from the target population in dimensions that are not considered to be fertility control, but which nevertheless affect fertility. The duration of breastfeeding is often thought to be one such factor. Similarly, in a 1996 article, Okun, Trussell, and Barbara Vaughan criticized CPA on the grounds that the method over-states the gain in the extent of control when the population on which the model parity distribution is based (improperly) contains some controllers. Sanderson subsequently addressed these criticisms by providing a test that helps avoid the use of inappropriate model parity distributions. He also demonstrated formally that CPA lower bounds remain lower bounds, even when the model distribution (inappropriately) includes some controllers.

See also: Estimation Methods, Demographic; Fertility, Age Patterns of; Natural Fertility.


Coale, Ansley J. 1971. "Age Patterns of Marriage." Population Studies 25: 193–214.

Coale, Ansley J., and T. James Trussell. 1974. "Model Fertility Schedules: Variations in the Age Structure of Childbearing in Human Populations." Population Index 40: 185–258 (Erratum, Population Index 41: 572).

David, Paul A., et al. 1988. "Cohort Parity Analysis: Statistical Estimates of the Extent of Fertility Control." Demography 25: 163–188.

David, Paul A., and Warren C. Sanderson. 1987. "The Emergence of a Two-Child Norm among American Birth Controllers." Population and Development Review 13: 1–41.

——. 1988. "Measuring Marital Fertility Control with CPA." Population Index 54: 691–713.

——. 1990. "Cohort Parity Analysis and Fertility Transition Dynamics: Reconstructing Historical Trends in Fertility Control from a Single Census." Population Studies 44: 421–455.

Glass, David V., and Eugene Grebenik. 1954. "A Statistical Model for the Study of Desired and Achieved Family Size." In The Trend and Pattern of Fertility in Great Britain: A Report on the Family Census of 1946. London: H. M. Stationery Office.

Okun, Barbara S. 1994. "Evaluating Methods for Detecting Fertility Control: Coale and Trussell's Model and Cohort Parity Analysis." Population Studies 48: 193–222.

Okun, Barbara S., James Trussell, and Barbara Vaughan. 1996. "Using Fertility Surveys to Evaluate an Indirect Method for Detecting Fertility Control: A Study of Cohort Parity Analysis." Population Studies 50: 161–171.

Sanderson, Warren C. 1979. "Quantitative Aspects of Marriage, Fertility, and Family Limitation in Nineteenth Century America: Another Application of the Coale Specifications." Demography 16: 339–358.

internet resource.

Sanderson, Warren C. 2000. "A Users' Guide to the Joys and Pitfalls of Cohort Parity Analysis," Presented to the Stanford Institute for Economic Policy Research Conference, History Matters: Technology, Population and Institutions, June 3–5, at Stanford University. <http://www.siepr.stanford.edu>.

Paul A. David

Warren C. Sanderson