It is widely accepted among scholars that stopping has played the major role in the European historical fertility transition. This article contributes to the reopened debate about the role of spacing before and during the fertility transition (see Okun, 1995 ; Hionidou, 1998 ; Friedlander, Okun and Segal, 1999 ; Fisher, 2000 ; Clegg, 2001). First, it will show how historical research on the fertility transition has been methodologically incapable of detecting intentional spacing behaviour in a convincing manner. Secondly, it makes suggestions about the kind of methods to be used in future historical demographic research in order to assess the role of spacing.

In the first place, John Knodel (1987) points out that most historical demographic studies have focused more on deliberate efforts to stop child-bearing than on efforts to space births. The main reason, according to him, is that deliberate stopping is easier to detect than deliberate spacing. Secondly, it is well known that indices of fertility control generally used are designed to detect stopping only ; they are not meant to find spacing behaviour. This is true in the first place of the widely used m parameter in the Coale-Trussell model (1974, 1978):

where r(a) is the age-specific marital fertility rate of a population at age a, n(a) is a standard age-specific natural fertility rate, v(a) is a vector of age-specific fertility control effects, and M is the level of natural fertility in that population. The m parameter is interpreted as an “index of fertility control”, but it is designed to detect stopping behaviour only (Wilson, Oeppen and Pardoe, 1988).

Another index of fertility control that has often been used is the mother’s age at last childbearing. Although a declining age has generally been interpreted as reflecting stopping behaviour, increased spacing has been shown to reduce age at last birth as well. There has been some controversy regarding the magnitude of that effect (see Knodel, 1987 ; Anderton, 1989 ; and the response by McDonald and Knodel, 1989), but simulations have shown that reductions in the mean age at last birth cannot be interpreted uncritically as a sign of stopping behaviour only (Okun, 1995). The same holds for an increase in the length of the last closed birth interval, which has been interpreted as evidence of failed attempts to stop childbearing. However, increased spacing affects all birth intervals, including the last. Since both forms of fertility regulation inflate the ultimate closed interval, research cannot differentiate between stopping and spacing by examining changes in the length of this interval (Okun, 1995).

The main point to be made in the following discussion is that the lack of conclusive evidence in favour of or against spacing behaviour is also a consequence of the fact that many historical demographic analyses are carried out at a highly aggregated level. The next paragraphs investigate the consequences of such aggregations on different measures of spacing and stopping.

An important point that has often been neglected is that this standard parity progression scheme changes in case of stopping but not necessarily in case of spacing. Suppose that a sub-group of the population exhibits perfect stopping behaviour. Other things remaining equal, this will result in a reduction of the average interval between confinements within final parity groups. This is a compositional effect. The total group of couples achieving a particular final parity will be made up of two types: on the one hand those who did not control their fertility but reached that parity under conditions of natural fertility, and on the other hand those who would have reached a higher parity but chose to stop childbearing earlier. The latter group of fertility controllers has on average shorter birth intervals than the former (perhaps because of higher fecundability, and provided they do not exhibit spacing behaviour), and hence, the mean birth interval within that final parity group is shortened (Knodel, 1987). In case of imperfect stopping (because of inefficient contraception), the effect of failed truncation will be that the last and penultimate intervals are even longer than expected under natural fertility conditions.

Alternatively, suppose that a subgroup within the population starts to limit family size solely by longer spacing. This is possible without changing the average birth interval within a group of couples with the same final number of confinements. By lengthening their birth interval, the fertility controllers would shift from a higher final parity group to a lower one, deliberately taking the same average interval length as the non-controllers already had under conditions of natural fertility. In this case, the birth intervals for the fertility-controlling group cannot be distinguished from those of the natural fertility group. The parity progression schedule of couples achieving a particular final parity remains unchanged. The only way to detect spacing behaviour is to look at compositional shifts: if more people come to fall under the schedule of lower family sizes, this is evidence of spacing (Bean, Mineau and Anderton, 1990). Of course, the overall mean birth interval (aggregated across all final parities) should rise in case of pure spacing.

When family limitation within a population is achieved by a mixture of stopping and spacing, only stopping behaviour will affect the parity progression patterns. Indeed, if there is both spacing and stopping, the shorter birth intervals within parity groups resulting from stopping will not be counterbalanced by any effect of spacing because parity progression schedules within final parity groups will remain unaffected by spacing. Hence, although shortening birth intervals within final parity groups may be an indication of stopping behaviour, they do not rule out the possibility that some sections of the population practice deliberate spacing as well.

Things get even more difficult to interpret when the underlying level of fecundity is rising (as may have been the case during the 19th-century fertility transition). Higher fecundity leads to shorter birth intervals, and this process may obscure efforts made to postpone a next birth (Knodel, 1987). In fact, it may even trigger these efforts. If couples, as a result of higher fecundity, are confronted with more babies arriving in fewer years, this may well be a stimulus to slow down the tempo (for example by abstinence or by prolonged breastfeeding).

The mean number of children ever born (CEB) is a function of:

The purpose of the model is to find out what proportions of an observed change in CEB are due to starting, spacing and stopping respectively. starting is represented by M and F spacing by I, and stopping by L. Note that L-M-F is the number of years elapsed between the first and the last birth.

It has been noted that the interpretation of these indices in not unambiguous. As previously stated, L is not a pure measure of stopping behaviour because it will be influenced to some extent by spacing as well. F may be inflated due to spacing practiced from the beginning of marriage. I may be inflated by stopping behaviour because failed attempts at stopping increase the ultimate and/or penultimate intervals (Okun, 1995). The latter process would lead to an overestimation of the role of spacing. Of course, these problems with interpreting interval lengths have been discussed long ago (Dupâquier and Lachiver, 1969, 1981 ; Henry, 1970 ; Knodel, 1981). However, there is a counteracting process that has often been overlooked.

Although Knodel and Okun agree that effective stopping behaviour would reduce the mean interbirth interval within final parity groups, Okun (1995, footnote 2) states that “it does not follow that the average interval over all final parities would decrease with stopping behaviour”. I would argue, on the contrary, that effective stopping will reduce the overall average interval as well. As stated above, it is well known that birth intervals at higher parities and at higher ages are longer than intervals at lower parities and ages. Hence, if people stop reproducing at lower ages and parities, fewer relatively long intervals contribute to the mean length while the shorter intervals at lower ages and parities will weigh more. In this way, the overall mean interval (I in model [2]) will be reduced. The more effective the stopping behaviour, the higher the potential reduction of the mean interval length.

To clarify things, let us divide a population into pure spacers and pure stoppers. In the simple simulation reported in Table 1, the spacers differ from the natural fertility population in that they increase all age-specific birth intervals by one year. As a result, four children are born instead of five, and their mean interbirth interval I is 3.0 years compared to 2.8 years in the earlier generation that experienced natural fertility. However, their behaviour also reduces the mean age at last birth from 42 to 40. So pure spacing is affecting the indicator L of stopping.

Table 1
Simulation of mean birth interval and mean age at last birth in two hypothetical populations

Alternatively, take the stoppers. Their attempts to truncate childbearing before the onset of sterility will indeed reduce the mean age at last birth. However, it will also reduce the mean length of birth intervals, since intervals at higher parities are longer than intervals at lower parities. If a subsection of the population does not advance to higher parities any more, fewer people will contribute the longer birth intervals typical of higher parities. Therefore, increased stopping behaviour will reduce the mean birth interval length, and this may obscure the spacing behaviour of other subsections of the population. In the hypothetical example of Table 1, the reduction of the mean interbirth intervals among stoppers (from 2.8 to 2.3) completely offsets the increase (from 2.8 to 3.0) in the intervals contributed by the spacers. The net result in the mixed population of 100 women is a decrease of mean interval length from 2.8 to 2.6 years, even though 50% of the population are spacing births.

To conclude, some effect of spacing will erroneously be interpreted as stopping (reduction of L), while some effects of stopping will to some extent counteract the effects of spacing (increased I). Although Okun (1995) concluded from a simulation study that McDonald’s technique can really differentiate between spacing and stopping, I would argue that this is only the case when the population studied practices either spacing or stopping, but not both. Okun’s simulation study indeed included only spacing or stopping conditions and no mixed condition.

What methods, then, are available to historical research to tell stopping from spacing in a population that hypothetically practices both? I distinguish between two kinds of questions and data requirements. The first question is asked at the macro level: to what extent is the observed fertility decline in a given population due to spacing and to what extent to stopping? The second question addresses the micro level: what elements can help to explain observed differences or increases, if any, in birth spacing, and what elements can explain differences in stopping behaviour? The second question, on the determinants of spacing and stopping, clearly calls for individual-level data and analysis. The macro question can be tackled with aggregated data as well.

Noting this severe bias, Ewbank (1989) developed a way to estimate the effect of childlessness and single-child fertility on M. Individual-level data are not necessary. What is needed to calculate the new indices are age-specific marital fertility rates (just like in the Coale-Trussell model) and the first two parity progression ratios, P₁ being the proportion of married women who have a first child, and P₂ the proportion of married mothers of one child who go on to have a second child. These ratios can be calculated from a frequency distribution of final parities. In order to estimate his model, Ewbank introduces the simplifying assumption that all parity progression ratios after the second are equal to the observed value of P₂. Of course, this is unrealistic. A slightly more realistic assumption would be to equate all progression ratios after the second to P₃, but Ewbank states that the advantages of this simplification outweigh the loss in accuracy. Anyway, the model does remove some of the bias in M in the Coale-Trussell model.

The first step is to estimate the extent to which marital fertility is reduced by the prevalence of childlessness and one-child families, indexed by I_p. (Exactly how I_p is estimated is not important for the present purpose ; for details, see Ewbank, 1989). Dividing M by this index I_p yields an estimate of how much larger M would be if all married women eventually had at least two children. This estimate is called M" = (M/I_p). Two main factors that determine M are removed, namely P₁ and P₂. The result is a more accurate index of the average interval between low-parity births.

The Coale-Trussell model of total marital fertility (TMF) can now be rewritten as follows:

In this equation:

5∑n(a)	equals 8.995, five times the sum of the Coale and Trussell natural fertility schedule ;
M″	represents the extent to which the difference between the observed TMF and 8.995 can be attributed to fertility regulation at lower parities ; the lower the estimated value of M″, the larger early birth intervals will be ;
Ip	indicates the degree to which childlessness and single-child fertility lower TMF ;
emν(a)	indicates the extent to which fertility reduction is a result of lower than natural fertility at higher ages and parities.

I would argue that this model, as well as the original model by Coale and Trussell, is biased in favor of detecting stopping behaviour, because spacing at high parities will be subsumed under the “stopping” term ( e^mv(a)) ; it will only affect m and not M. The major contribution of the model is to yield a version of M that controls for disturbances from childlessness and single-child fertility. It yields a less ambiguous index that can be used in a meaningful way in the spacing-stopping debate. If some subsections of a population started to space births more widely at low parities, this would be detected in Ewbank’s model through decreases in the M″ parameter.

The details of the calculation are simple and fully explained in David et al. (1988). Important here are the three basic assumptions underlying CPA.

If these assumptions are true, CPA yields efficient and unbiased estimates of the percentage of controllers at each parity below the cut-off parity — that is to say: upper and lower bounds for this percentage (David and Sanderson, 1988, 1990).

Using Monte Carlo simulation, Okun (1994) investigated how well CPA performs under divergent fertility control situations. If the assumptions are not violated, the model performs quite well. She also investigated how sensitive the CPA-model is to violations of the assumptions.

In conclusion, CPA can be used in the spacing-stopping debate under the assumption that a correct natural fertility model cohort can be found with which the target cohort can be compared.

At each attained parity, women can be divided into two groups: those who proceed to the next higher parity and those who stop at that parity. It is possible to model the probability that no higher parity will be attained, including the attained parity as one of the covariates, together with other relevant covariates including natural fertility determinants. If a significant proportion of the population exhibits effective parity-dependent stopping behaviour, the probability that there is no next birth, P, should significantly depend on parity. For instance in a logistic regression model:

Van Bavel (2004b) uses this approach to look at the diffusion of parity-dependent fertility control in a provincial town of Belgium.

Secondly, a duration model could be developed for all closed birth intervals, representing the determinants of birth spacing and again including age and attained parity as two of the covariates. If spacing is not parity-dependent, the relevant parameter should not differ significantly from zero, after controlling for other relevant parameters such as age and duration of marriage. This could, for instance, be implemented in a Cox regression model (Cox, 1972):

Rodriguez and Cleland (1988), Trussell et al. (1985, 1992), Yamaguchi (1989), Yamaguchi and Ferguson (1995), and Van Bavel (2004a) used this type of model. Two further comments on modelling spacing and stopping are relevant here.

Recently, event history models in which unobserved heterogeneity is explicitly accounted for are becoming more familiar in demography (Manton et al., 1992 ; Lewis and Raftery, 1999). These kinds of models are particularly important for historical demography, because very often we know that some important factors are influencing the hazard rate while we cannot control for them because empirical indicators are lacking.

Secondly, to establish parity-dependence of both the probability of stopping and the length of birth intervals, it is not sufficient to include the cumulative number of births in the regression. Parity-dependency should be looked at from the perspective of reproduction, distinguishing net parity from crude parity. The former is the number of children still alive at the beginning of the current birth interval, while the latter is just the cumulative number of births including children alive as well as deceased. If net parity has a statistically significant effect on fertility, even after controlling for crude parity (or, equivalently, for the number of deceased children), this strongly suggests that fertility is being controlled with a desired number of offspring in mind.

The inclusion of both net and crude parity is essential for the analysis in order to control for two opposing mechanisms behind the association between the number of children already born and subsequent fertility. On the one hand, there is generally a positive association between crude parity and parity progression because parity, at a given age and marriage duration, is positively associated with fecundability. The higher the fecundability, the shorter the birth intervals, and the higher the crude parity attained. On the other hand, every birth entails some risk of secondary sterility or sub-fertility, implying zero or lower subsequent fertility (Van Bavel, 2003).

The effect of net parity on fertility, after controlling for crude parity, is the opposite of the effect of the number of children lost. To some extent, then, net parity would be capturing the effect of infant mortality on fertility, which is known to be positive, even in the absence of fertility control (Preston, 1978). This would invalidate the regression analyses because net parity is meant to detect parity-dependent fertility control while in fact it is capturing the natural effect of infant mortality as well. Therefore, it is essential that the latter effect be also controlled for. This can be done by including a dummy variable indicating the survival of the previous child (Van Bavel, 2003 ; 2004a).