1The distinction between renewable and nonrenewable events is a central component of demographic analysis. When studying renewable events, agespecific rates (events per person) are summed to obtain the mean number of events occurring over the life course. For fertility, this corresponds to total fertility, including births of all orders. If an order is attributed to each birth, successive births can be analysed as a succession of nonrenewable events. It then becomes possible to calculate the probability of transition from one parity to the next and to describe more precisely the occurrence of successive events. In this article, Daniel Devolder establishes a relationship between these two methods. Using historical Spanish fertility data, he provides a graphic representation of total fertility and of parity progression ratios on a comparable scale, along with a decomposition of fertility trends into their components by birth order. This new approach offers original solutions to the practical problems encountered in fertility analysis.
2Analyses of fertility trends over time become much more meaningful if birth order is taken into account. The introduction of birth orderspecific indicators, and that of parity progression ratios (PPR) in particular, was thus a considerable methodological advance. This indicator was popularized by Henry (1953), although Mortara (1935), Whelpton (1946), Ryder (1951), and INSEE (1953) had contributed notably before him. The PPR is a conditional measure of fertility intensity: the proportion of women with a specific number of children (women of the same parity) who have an additional child, and thus can be interpreted as a probability of having one more child. In contrast, traditional aggregate or overall measures of fertility – such as the total fertility rate (TFR) for period data, or completed fertility rate (CFR) for birth cohorts – are unconditional: a mean number of children per woman, regardless of parity. These two classes of indicators are thus not directly comparable: the first refers only to a portion of individuals, while the second refers to all. This article presents relationships between these two classes of indicators that make them more comparable, either through transformations in scale or by taking into account the functional relationship between their respective variations over time, which yields a decomposition of overall fertility into probabilities by birth order. [1]
3These relations between PPRs and general fertility indices provide a new, probabilistic interpretation of unconditional indicators. They also lead to a unified scale that can be used to graphically compare the levels and trends of these two types of indicators. In this context, we introduce a mean PPR that can be used to obtain the exact relationships between aggregate indices and probabilities by birth order. This mean PPR also presents practical advantages when using survey data to estimate fertility levels, as it can be used to obtain a true probability when grouping together births beginning at a given order, and thereby to decrease the volatility of indicators for higher birth orders.
4PPRs, like aggregate indices, can be calculated both for cohorts and crosssectionally (for time periods). In the first case, they are generally the last step in the calculation, after the mean number of children for each birth order has been obtained. When fertility is calculated over an annual period, however, it is sometimes preferable to take the opposite approach, first estimating the PPRs and then deducing the values of the overall fertility indices from them. This contrast is explained by the fact that, as for all demographic phenomena, the measurement of fertility for periods is complicated by variations in timing. For example, variation in age at childbearing over time distorts the rates that are used to calculate the indicators. These distorting effects do not occur for cohort indicators, but can be substantial for period indicators. As shown by Karmel (1950) and Yamaguchi and Beppu (2004), period fertility levels can be estimated more accurately if PPRs are obtained beforehand, since the bias introduced by changes in fertility timing are much greater when fertility is measured using the traditional method of summing agespecific rates than when the total fertility rate (TFR) is derived from PPRs obtained using methods based on life tables, described notably by Rallu and Toulemon (1993). This explains why it is advantageous to present the procedures for calculating the mean PPR, for different types of data, for the cohorts and periods, as we do in Appendix A.1.
5The relationships set out here are then applied to the fertility of the cohorts born between 1898 and 1970 in Spain, which are reconstructed using data from the four latest population censuses. [2]
I – Relations between parity progression ratios and overall fertility indices
1 – Henry’s relation
6Henry (1953) presented key findings for the calculation of PPRs. In particular, he showed that if these ratios have already been determined, the mean number of children can be obtained, for all birth orders combined or for a given birth order. Noting the overall fertility indice as F (mean lifetime number of children per woman, which can be the completed fertility rate (CFR) for a cohort, or the total fertility rate (TFR), if period PPRs are used), Henry established the following relationship:
8where a_{n} is the PPR of order n+1 for women of parity n. F_{n}, the mean number of children of order n per woman, can also be obtained from the identity:
10As F_{1} = a_{0}, we thus have F_{2} = a_{0}a_{1}, F_{3} = a_{0}a_{1}a_{2}…,
11This leads to a recursive formula, beginning with n = 1:
13The latter can be extended to the case where n = 0 if we observe that the quantities F_{n} are also proportions of women having reached parity n, and in this case F_{0} = 1 is the initial number of childless women at the start of their reproductive life. The indices F_{n} are unconditional, insofar as they measure the number of children of order n for all women, whereas parity progression ratios a_{n} are conditional: they measure the probability of having an additional child (thus of order n+1) among all women “at risk of having a child of order n+1” – that is, women who already have n children.
14The values of the unconditional indicators of fertility intensity can thus be calculated, for each birth order separately or for the total, on the basis of the PPRs. This is useful above all for period indicators, when fertility intensity is not precisely known due to the biases introduced by variations in timing. If, instead, we already know the unbiased mean number of children of each order for all women, which is always the case when we use cohort data, we can deduce the corresponding values for the PPRs:
16This equation is also valid beginning at n = 0, if we pose F_{0} = 1.
17A drawback of relation (1) is that it is defined as an infinite series, so in practice we obtain an approximation. The error is generally negligible if the highest order for which we can calculate these probabilities is, for example, 5 for populations with low fertility, or higher than 9 for situations of uncontrolled fertility. Henry showed that a better approximation can be obtained by assuming constant PPRs above a certain order, which converts this formula into a geometric progression. For example, if we make the simplifying assumption that a_{2} = a_{3} = a_{4} = …, and since in practice these probabilities are strictly lower than 1, we get, through reasoning at the limit:
2 – The mean parity progression ratio [3]
19We can obtain an exact formula without this simplifying assumption and without reasoning at the limit, by transforming (4) while assuming that the constant final PPR is an unknown. Using again the case of progression from order 2 to 3:
21which yields
23(Since
25and a_{0}a_{1} = F_{2}, we have:
27x is equal to the ratio of two cumulative sums F_{n+} = F_{n} + F_{n}_{+1} + …, which are numbers of children of order n or higher, for n > 0.
28This ratio x will be denoted ā_{n+}, since as we shall see, it is a mean ratio adapted to the problem of the “closure” of equation (1): [4]
30We can also calculate the sums F_{n+} beginning with n = 0 if we interpret the terms F_{n} as numbers of women having reached parity n, taking again F_{0} = 1. It can then be deduced that: F_{0+} = F_{0} + F_{1+} = 1 + F
31This provides a particularly simple way of obtaining the first mean PPR in terms of the level of total fertility, with an equation that we will generalize below:
33It is important to observe that ā_{n+} is a weighted mean of PPRs beginning at n, where the weights are the numbers of women having reached parities from n onward, divided by their sum:
35(As
37and F_{n}_{+1} = a_{n}F_{n}, we have:
39The mean PPR ā_{n+} can be interpreted as the likelihood of having an additional child, of order equal to or greater than n+1, for women who have had at least n children. This differs from the classic PPR a_{n}, which is the likelihood of having an additional child for women with exactly n children.
40Equation (5) can be directly used for calculations on cohort data. With crosssectional data, a different approach is needed: ā_{n+} must be calculated first, for example using the procedures described in Appendix A.1, based on life tables.
3 – Relations between mean PPR and aggregate fertility indices
41To establish the general relationship between the mean PPR and aggregate indices, it is helpful to introduce a conditional aggregate fertility indice: the mean number of children for persons who have at least n children. The bestknown case, and also the one of greatest interest, is the mean fertility of mothers (that is, women with at least one child):
43This yields a decomposition of overall fertility: F = a_{0}F_{1*}, which has often been used to study variations in the indice of total fertility F on the basis, first, of the proportion of women with at least one child, and second, of mothers’ fertility.
44There is a recursive formula that allows to obtain this indice for higher birth orders – for example, to calculate the mean number of children of mothers with two or more children, three or more children, etc. [5]
46Repeated application of this recursive formula yields a simpler expression, close to Henry’s equation, which will allow us to obtain the desired relation between the mean PPR and the aggregate indices: F_{n*} = n + a_{n} + a_{n}a_{n}_{+1} + …
47This can also be written more concisely as:
49where M_{n+} = a_{n} + a_{n}a_{n+1} + … is marginal fertility, i.e. the mean number of children of order n + 1 and above born to women with at least n children.
50We observe that marginal fertility is also equal to:
52We can then deduce its relationship to the mean PPR, through a comparison with the definition of the latter (5):
54We can also write the inverse transformation:
56The most interesting case, which can be used to derive the relation (6) once again, corresponds to the overall fertility indice, that is, for n = 0:
58This result gives rise to a new interpretation of completed fertility rate (or of the TFR), which now takes the form of odds, for which the underlying probability is the mean PPR beginning at parity 0. [6] The TFR is also the ratio of the probability of having an additional child to the inverse probability of not doing so, for all persons of reproductive age. This corresponds to the odds for a bet on whether an individual of reproductive age will have an additional child in the future. Hence, for a fertility level of two children, the mean odds of having an additional child for all women of reproductive age, on a given date, will be 2/3.
59The above equations establish the existence of simple transformations that allow us to derive aggregate indices from mean PPRs, and vice versa, which we will use to construct graphs with a common scale in the next section. With this in mind, we can also use Henry’s equation to define an index of potential fertility, based on a similar transformation of scale, allowing comparison of classic PPRs and aggregate indices. To do so, we use Henry’s own reasoning: we assume that the level of PPRs is constant beginning at a given birth order. If we apply this logic beginning at order 1, we then have a fertility level that expresses the potential of the first PPR:
61Note that this relation between the PPR and its associated fertility potential has the same form as equations (6) and (12) but was obtained differently. Here it is the equation for the geometric progression that allows us to establish that the odds of the PPR for childless women is the potential number of children that these women would have if all posterior probabilities were equal to the first. This potential can be compared to the overall fertility level F, which we indicate here by adding a tilde. Beginning with parity 1, the values of the potential for each PPR are compared with those of the marginal fertilities M_{n+}, which we indicate again here with a tilde, and we thus write:
63Note also that the level of the total fertility indice F is directly comparable with that of the marginal fertilities M_{n}_{+}, as the two are the odds corresponding to an underlying probability. This is useful for the graphic presentation. However, F is not directly comparable with conditional completed fertility rate F_{n}_{+} when n > 0, as the latter then is not an odds, as shown by equations (9) and (11).
4 – Graphical applications: scales
64The equations above can be helpful in constructing graphs that represent and compare aggregate fertility indices, whether conditional or unconditional, with parity progression ratios, whether simple or mean ones. The problem that arises in this case is the incompatibility between the scales of these different types of indicators. Those who wish to represent them together thus often use two separate scales, as in panel A of Figure 1, where the lefthand scale is for completed fertility rate (CFR) and marginal fertility (M_{1+}), and the righthand scale is for PPRs. The use of two scales on the same axis is generally a bad idea (Isenberg et al., 2011), and in this specific case, the solution recommended notably by Haemer (1948) of using a transformation into index numbers is not applicable, as these scales are incommensurable: the scale for probabilities has a maximum value of 1, whereas the number of children has no particular upper limit. The problem thus cannot be solved using a linear transformation, and their superposition is necessarily arbitrary.
Total and birth orderspecific fertility, comparison of different scales and transformations, Spain, cohorts born between 1898 and 1970
Total and birth orderspecific fertility, comparison of different scales and transformations, Spain, cohorts born between 1898 and 1970
Note: The graph in panel A uses two distinct scales, number of children on the lefthand side and PPRs on the righthand side. Panels B and C use a single scale, after transformation of one of the two classes of indicators, but there is an asymmetry in variations. Panel D shows the correct presentation. To reestablish symmetry it uses two scales, which are unified using a logarithmic transformation for the number of children and a logistic transformation for the ratios.CFR: completed fertility rate; M_{1+}: marginal completed fertility beginning at parity 1; a_{0}, a_{1}: PPRs for first and second births; ā_{2+}: mean PPR for third and subsequent births. Panel A: arbitrary juxtaposition of scales for total fertility (left) and PPR (right). Panel B: transformation of indicators of total fertility into mean PPRs. Panel C: transformation of PPRs into their equivalent in terms of completed fertility. Panel D: unification of scales using a logit transformation of PPRs and a logarithmic transformation of fertilities.
65Using the equations above we can choose one of these two scales. Either we can transform completed or marginal fertility into mean PPR form using equation (6) and (10), which yields panel B, or we can transform the PPRs into odds, thereby converting them into quantities equivalent to an aggregate fertility level, using equations (11) to (14), which yields the figure seen in panel C. These two transformations improve on the figure in panel A. For example, in panel A the levels of completed and marginal fertility are very close to, and sometimes lower than, those of the PPR ā_{2+}, for the oldest cohorts. The hierarchy of levels in panels B and C, based on the conversion equations presented here, show that this is incorrect, resulting from a necessarily arbitrary choice of intervals of variation in panel A. The relative evolution of the values from one cohort to the next is also different: the visual impression given by panel A is that the relative variations in completed fertility are greater than those in the two first PPRs, whereas in panels B and C the opposite can be observed. The transformation of the PPRs into odds in the figure in panel C also allows us to obtain the potential for each PPR: for example, for the 1970 cohort, the PPR for the first child (a_{0}) is 0.773 and its odds is 3.49 children per woman. This would be the value of completed fertility if the PPRs for all birth orders were equal to that of birth order one, which can be compared with a level of 1.48 for the completed fertility of this cohort. Similarly, the marginal fertility beginning at birth order two that would be observed if all PPRs were equal to the value of a_{1} for the 1970 cohort would be equal to its odds: 2.46 children per woman, whereas its real level is 0.9.
66But the ability to choose between these two different transformations is also a drawback, as it leaves us with more options than we can use. Moreover, these scales are imperfect because they are asymmetrical: for panel B, when the value of the ratios approaches 1, an increase, in a sense, “costs” more than a decrease of comparable size, while the reverse is true for values close to zero. The same problem arises for panel C, but here it is only for high values: a given variation in PPRs leads to greater upward than downward variation in odds, when the value of these PPRs is close to 1.
67The representation in panel D offers a solution to these problems consisting in a transformation of the two scales. This yields, first, a representation in terms of either a fertility level or a probability, as in the two previous cases, but also a real “unification” of the two scales; and second, a symmetry in variations. The scales in the figures in panels B and C are fused by using a logistic scale for the PPRs and a logarithmic scale for the aggregate fertility indices. These two transformations of scale convert the two previous figures into a single comparable figure, the one represented in panel D. A legend in terms of either numbers of children or values of a ratio can be chosen, related as per equations (6) and (10) to (14). The figure in panel D also offers the most satisfactory representation, since the logistical scale is the “natural” scale for PPRs, as shown by Toulemon (1995) (see also Leridon and Toulemon, 1997). The surprise here is that, by extension, the logarithmic scale is also the natural scale for aggregate fertility indices, which in fact are odds of mean PPRs.
68Once the correct scale has been determined (panel D), the problem of comparing the levels and evolution of fertility and infertility indicators can be addressed. Panel A of figure 2 presents trends in completed fertility and primary infertility (or final childlessness, i.e. the proportion of women without children at age 50). In visual terms, the main problem here is that the trends run in opposite directions: increasing childlessness across cohorts leads to decreasing fertility, and vice versa. A solution for obtaining trends that run in the same direction is to use the proportion of women with at least one child a_{0}, the complement to the indicator of childlessness (panel B), whose level varies directly with that of completed fertility. But another more attractive solution is to express completed fertility in terms of the associated mean PPR ā_{0+}. In this way we can calculate a mean level of infertility, equal to 1 – ā_{0+}, which is directly comparable to primary infertility (panel C). The drawback here is that only the PPR scale can be used, as the odds for the proportion of childless women evidently cannot be interpreted in terms of fertility levels.
Total fertility and infertility: choice of indicators, Spain, cohorts born between 1898 and 1970
Total fertility and infertility: choice of indicators, Spain, cohorts born between 1898 and 1970
Note: a_{0}: PPR for first birth; Mean infertility: calculated as 1 – ā_{0+} ; Primary infertility (childlessness): calculated as 1 – a_{0}. Panel A: comparison of CFR (lefthand scale) and primary infertility (righthand scale) using the unified log/logit scale. Panel B: comparison of CFR with the PPR for the first birth a_{0} with unified log/logit scale. Panel C: comparison of mean infertility and primary infertility with logit scale.69More generally, we observe that levels of primary infertility (or childlessness) are high in Spain in the youngest cohorts, with more than 20% of women having no children, but that they were even higher for the cohorts born in the early twentieth century. These high levels in the past and their decrease in the cohorts born in the 1930s and 1940s are comparable to patterns observed in many western European countries (France, the Netherlands, Scandinavia, the United Kingdom…) or settled by Europeans (Australia, United States), as reported by Rowland (2007). However, the causes of these two episodes of childlessness are probably different: higher proportions of unpartnered individuals, partly linked to the conflicts of the first half of the twentieth century, effects of the Great Depression of the 1930s, birth control among certain groups of women in the cohorts born around 1900 (Morgan, 1991), and postponement of first union and first birth in the youngest cohorts.
II – Decomposition of variations in overall fertility in terms of variation in parity progression ratios
70The graphical comparison of trends in overall fertility and parity progression ratios can be complemented and fleshed out with a decomposition of absolute variations in the former in terms of those in the latter. Here we will apply Henry’s formula in the exact form obtained above (Appendix A.2.):
72The simplest way to obtain this decomposition is to use the equation for the total differential of a function with several variables (Pullum et al., 1989):
74In this case this yields:
76Variation in the TFR or the CFR are equal to the sum of the effects of variation in each PPR at each parity, weighted by the value of an aggregate indice that only takes account of fertility at higher orders.
77This decomposition is exact in continuous time, but to use it we must perform a discrete time approximation, obtained here with:
79where variation in fertility over a period is decomposed as a sum of terms describing variations in each PPR, the latter being multiplied by the mean value of the partial derivative of F in terms of each PPR in the period (this is the approximation recommended by Pullum and Tan, 1997), rather than the value of the partial derivative obtained on the basis of the mean value of the other PPRs. There is also a residual term, which must be reduced to the lowest possible value.
80In general terms, decomposition methods are used in demography to solve this problem of the discrete approximation of the decomposition of variations in an indicator based on various factors via an equation that is not simply additive. For example, to compare crude rates using standardization methods, Kitagawa (1955) and Das Gupta (1993) showed how exact solutions can be obtained by distributing the residual, understood as an interaction effect, among the different factors. These authors sought to explain variations in a crude rate first in terms of variations in specific rates, and second, in terms of variations in the population distribution by age, marital status or any other factor. For example, in the case of two factors (a crude rate that is a function of agespecific rates and of the age distribution), the decomposition of variation in the crude rate is a sum of variations of products of the type y = ab whose variation can be written:
82The term ΔaΔb is the interaction, which is thus distributed and allocated equally across the two factors:
84where āΔb is the effect of variation of factor b (its variation multiplied by the mean value of a) and is the effect of variation of factor a.
85When the number of factors increases, the number of the interaction terms grows combinatorially: for 3 factors, there are 4 interaction terms; for 4 factors, 10 terms; etc. The resulting decomposition equation is no longer as compact as in the case of two factors.
86For demographic indicators other than crude rates or similar indicators, where relations are more complex – as in the case of relations between life expectancy and agespecific probabilities of death, for example – the most widely used decomposition methods eliminate interaction effects using permutations of the values of the factors, alternating the initial and final values of each one (Andreev et al., 2002 ; Das Gupta 1999 ; Pollard 1988). In the case of the previous example, there are at least two possible permutations:
88We then take the mean of all permutations, which, in this simple example, yields the result of (18). Increasing the number of factors, and particularly including a dimension such as age, can greatly increase the number of permutations, making this method impractical.
89In contrast, in the method used here, there are no interaction effects, so there is no need to distribute them across factors. Instead of these effects, we have the single value of a residual or an approximation error, as can be seen by comparing equations (16) and (17). Here we minimize this residual in two ways: first, using the mean value of the partial derivatives, and second, and most importantly, by calculating variations in fertility over a year rather than over longer periods of increase or decrease, as is usually done in this type of decomposition. As a result, the value of the residual for each of the variations presented in figure 3 is less than 0.1% of the sum of the absolute values of the three factors. Here we base our method loosely on that of Horiuchi, Wilmoth and Pletcher (2008), who use a stepwise technique for decreasing the value of the residual in the discrete approximation of the equation for the total differential, which is not needed when intermediate annual data are available, as is the case here. This approach seems to be a simpler and a more general solution than traditional decomposition methods.
90Figure 3 presents the decomposition of annual variation in the completed fertility rate (CFR), comparing the values for one cohort with those of the immediately preceding cohort, in terms of the effect of variation in the three first PPRs. We have aggregated the positive and negative values of these effects separately so as to be able to visually relate the contribution of variation in each ratio to their sum in absolute terms, and not only to variation in completed fertility. This approach could also be applied to calculating the proportion of total variation in CFR represented by each factor. In general, then, we could choose between two methods. We could compare the effect of each factor on variation in CFR, that is, calculate the proportion of variation in CFR due to variation in the PPR a_{n} with
92or divide them by the sum of their absolute values,
Decomposition of annual variations in completed fertility in terms of variations in parity progression ratios, Spain, cohorts born between 1898 and 1970
Decomposition of annual variations in completed fertility in terms of variations in parity progression ratios, Spain, cohorts born between 1898 and 1970
ΔCFR: annual variation in CFR between two consecutive cohorts; Δa_{0} effect: portion of annual variation in CFR between cohorts explained by variation in PPR for first birth. Δa_{1} effect: ditto for second birth; Δā_{2+} effect: ditto for third and subsequent births. The graphical representation is based on separate summations of the positive and negative contributions of variations in PPRs.Note: In the 1970 cohort CFR was lower by 0.024 children per woman than that of the 1969 cohort. The decrease in the PPR to first births explains 58% of this change, the change to second births explains 27%, and the change to third and subsequent births explains 15%. The sum of these effects shows that for the 1925 cohort, whose CFR was equal to that of the 1924 cohort, and thus ΔCFR = 0, the effect of each of these PPRs was respectively 35%, 15%, and 50% (the graphical representation relates the variation at each order to the total of the absolute values of these effects).
94The first approach is the most common, but it is problematic, particularly when there are compensation effects between factors, because they vary in opposite directions, which can bring ΔCFR close to zero. The second solution is the more logical of the two (Ellenberg, 2014, Chap. 5), and it is the one that we adopted for Figure 3.
95This representation in terms of annual variation is complementary to that of panel D in Figure 1, which compares the absolute levels of the same indicators. But here, the interpretation of variations in CFR in terms of variations in the PPRs is more exact. For example, a 10% decrease in the first PPR will not decrease CFR by 10%, as we might be tempted to conclude based on Figure 1. This is so because the level of the other PPRs must be taken into account, as seen in Figure 3.
96We can thus see that CFR decreased in the cohorts born before 1925, as annual variations were negative, increased for women born between 1925 and 1937, and then decreased again up to the most recent cohort. The decomposition allows a precise analysis of the factors underlying the changes in CFR:
 The decrease in CFR over the entire period is principally a function of decreasing fertility at parity 3 and above, as this has been continuous across all cohorts, and has only recently begun to slow.
 For many years, increasing fertility at parity 1 slowed this negative effect on CFR, up to the cohorts born in the late 1940s. However, in the youngest cohorts, increased childlessness has progressively become the predominant factor behind the decline in the overall fertility indice.
 The baby boom, which essentially concerned the cohorts born in the 1920s and 1930s, was explained mainly by changes at birth order 2: the increasing probability of a transition to second birth among members of these cohorts was the principal reason for increased total fertility.
 The decrease in CFR began to accelerate with the cohorts born in the 1940s: from this point on, the PPRs for all parities decreased, and there was no longer any compensation effect between parities.
Conclusion
97The main contributions of article bear on the comparability of classical aggregate fertility indices, such as TFR and CFR, and birth orderspecific parity progression ratios. We showed, in particular, that the levels of these two classes of indicators can be directly compared if the PPRs are converted to odds, yielding their fertility potential. This led to the definition of a unified scale that allows these two classes of indicators to be represented in a single figure. This comparison led us to define a mean parity progression ratio (PPR) associated to the aggregate indices. We then showed that TFR and CFR are also the value of the odds of such a mean PPR, which gives scope for probabilistic interpretations. This mean PPR is also useful for calculating fertility on the basis of survey data with relatively small samples, as it allows births from a certain parity to be grouped together while using calculation methods based on fertility tables, yielding better estimates of fertility levels for annual periods. Finally, we showed that graphical comparison of the absolute values of the different fertility indicators is considerably enriched by looking at their variations over time, particularly when a decomposition approach can be used. In this context, it may also be important to use a graphical approach which allows for full use of the available data, rather than a table summarizing the contribution of the factors over a multiyear period.
Acknowledgements
My thanks to the three anonymous reviewers, as well as Laurent Toulemon, whose comments allowed me to considerably improve the text. This study is among the research projects financed by the Spanish Ministries of Science and Innovation (CSO201129136) and of Economics, Industry and Competitiveness (CSO201460113R).Appendix
Document A.1. Calculating the mean parity progression ratio
98The method used to calculate the mean parity progression ratio depends on the type of data used and on whether the analysis is by cohort or period.
Cohort data
99To calculate cohort parity progression ratios, data are usually drawn from population censuses or fertility surveys to determine the distribution of women beyond reproductive age by number of live births (data P_{I} in the table below). Fertility at each parity can then be obtained from the rightward sum of these data (Table A.1). Fertility at birth order 2 is thus F_{2} = P_{2} + P_{3+}, and at birth order 1 it is F_{1} = P_{1} + P_{2} + P_{3+}.
100To calculate fertility at order 3 and beyond, however, we need to know the total number of births, and thus completed fertility, which is unfortunately not always the case when using census summary tables. For higher birth orders we must begin with completed fertility rate CFR, obtaining F_{3+} = CFR – F_{1} – F_{2}. The mean PPR at order 3 and above is thus obtained as follows:
Calculation of the mean PPR for the 1970 cohort in Spain
Calculation of the mean PPR for the 1970 cohort in Spain
Period data
102The final or mean PPR for an annual period can be calculated on the basis of survey data or of vital statistics. The preparation phase and the meaning of data differ between these two types of sources, and we thus present them separately. Here we discuss the method based on life tables which include the duration since the previous birth or the mother’s own birth (for birth order 1). The same reasoning can be applied to life tables that instead include age (Whelpton, 1946) or both age and duration (Ng, 1992; Rallu and Toulemon, 1993). The general principle is the same, and consists in grouping women of parities n and above in the denominator, and the corresponding births in the numerator (at the same age, or at the same age and for the same duration) of birth order (n+1) and above.
Using survey data
103Retrospective fertility surveys can be used to reconstruct women’s (or men’s) reproductive history, as the fictional example in Table A.2 shows for three women who in the past had two, three and one live births respectively, along with the years of birth of each woman’s children.
Women’s reproductive history, fictitious example
Women’s reproductive history, fictitious example
104To calculate ā_{n+} for a given year – the probability that women who had had n or more children in previous years had an additional child in that year – we must convert each record for women who had had at least n children into N – (n – 1) records of births of order n or above, where N is the number of live births to each woman. For example, if n = 2, to calculate ā_{2+}, we obtain Table A.3 from Table A.2, indicating the identifier of each woman who had at least two children, and for each birth of order 2+ we indicate the year when it occurred as well as the year of the next birth, if applicable.
Conversion of reproductive history for calculation of the final parity progression ratio for birth order 2+
Conversion of reproductive history for calculation of the final parity progression ratio for birth order 2+
105Drawing on tables A.2 and A.3, the next step is to construct a table that matches years of births of order 2 and above against the years of the subsequent births of order 3 and above, similar to Table A.4.
Calculation of ā_{2+} for the year 2006 on the basis of births of order 3+ consecutive to births of order 2+, by year, survey data for Catalonia
Calculation of ā_{2+} for the year 2006 on the basis of births of order 3+ consecutive to births of order 2+, by year, survey data for Catalonia
106On the basis of this table we can construct a fertility table by duration to study the progression from parity 2+ to the following parity, precisely as we can do for progression to parity 1 (matching data on women’s year of birth and the year of birth of their first child) or beginning with parity 2 (year of birth of siblings of birth order 1 matched against year of birth for siblings of order 2). Thus, the fertility table for the year 2006 can be constructed on the basis of all the probabilities for each parity cohort, analogous to a probability of death in a life table. For example, for the cohort of women who had 163 births of order 2 and above in the year 1985, the ratio for the year 2006 is 1 / (1 + 120), because in that year there was one birth of order 3+ in the cohort, while 121 women had not yet had a further birth on 1 January 2006. The probabilities for the following years are calculated in the same way, up to 2006. For example, for the 2005 parity cohort, the probability is 13.3 per 1,000 (or 2 / (2 + 148)). For the 2006 cohort, for which births of order 2+ and subsequent births of order 3+ occur in the same year, we cannot directly calculate a probability, as on 1 January 2006 there had not yet been a birth of order 2 or above. A solution is to calculate a rate, here 1 / 72.5 (where 72.5 is the mean number of women who had a birth of order 2+ in 2006 but no additional child in that same year) and to convert it then to a probability, which yields the indicated value of 13.7 per 1,000.
107We can then calculate the survivors in the table as follows:
109where q_{t}(t – d,d) is the probability for time interval d, year t, and parity cohort t–d, and n is the maximum time interval.
110This fertility table is a table of survival to the risk of an additional child, analogous to a life table. Henry (1953) called this the classical or direct method of calculating parity progression ratios. A useful presentation of this method can also be found in Hinde (1998). We then obtain the final or mean parity progression ratio as: ā_{2+} = 1 – S_{n}(t).
Using vital statistics data
111If vital statistics offer a long series of births by biological birth order for the mother, the mean parity progression ratio beginning at a given parity can also be calculated. The ideal situation is to have, for each birth, the length of the interval since the previous birth, or else the year of the previous birth (for births of order 1, the previous event used is the mother’s date of birth). The classical method based on a life table can then be applied. This is illustrated in the following table with data for Spain, and again for the calculation of the final ratio ā_{2+}. We begin by aggregating births of order 2 and above for each year (first and second columns). We then include births of order 3+ by calendar year and the year of the corresponding previous birth (from the third column). The calculation is then similar to the previous case based on survey data. For example, the probability for the year 2014 of the 1999 parity cohort of births of order 2+ is obtained as follows:
113where the probability is multiplied by a factor that corrects for the weight of births of order 3+ for which the year of the previous birth is unknown.
Calculation of ā_{2+} for the year 2014 on the basis of births of order 2+ by year, and births of order 3+ by year and time since the previous birth, Spain
Calculation of ā_{2+} for the year 2014 on the basis of births of order 2+ by year, and births of order 3+ by year and time since the previous birth, Spain
114The main difference here with respect to survey data is that the initial set of women with a birth of order 2+ in a given year can vary later, due to mortality and especially migration. Some correction may be attempted if annual estimates of the population by sex and age are available. A simple method is to use mean cohort size (Calot, 1984) by parity:
116where ā_{2+} (t) and are, respectively, the mean parity progression ratio at parity 2+ calculated using the life table method, and corrected for the effects of migration and mortality; G_{i}(t) is the mean cohort size for order in year t;
d_{i – >i+}_{1}(t) is the mean interval between births of order i and i + 1, beginning in the year t in which the births of order i + 1 under consideration took place.
117Figure A.1 presents the period PPR for births of order 2+ to births of order 3 + (ā_{2+}) in comparison with the PPR for women with a single child (a_{1}) accompanied by values corrected for mean cohort size (using Spanish data). This shows highly stable fertility at order 2 and above, as well as a large effect of immigration, which, around 2005, increased the values of the two PPRs by 10%. Note that the mean interval between births remained virtually constant over this period. If that had not been the case, we could have used Brass’s (1991) correction equation for indicators calculated from life tables, in order to correct for timing bias and further improve these results:
119where a_{1}^{*} is the period PPR for parity i to parity i + 1, corrected for the effect of variation in the mean interval between these two parities; Δd is the mean variation in this interval measured on the basis of period life tables.
Evolution of a_{1} and ā_{2+}, with or without correction for the effect of migration, Spain, 19902014
Evolution of a_{1} and ā_{2+}, with or without correction for the effect of migration, Spain, 19902014
Notes: a_{1}: PPR for second births for the year considered; ā_{2+} PPR for third and subsequent births; et : PPRs corrected for change in size of mean cohorts (see text). Interval of 4.5 years between birth orders 1 and 2, and of 5.5 years between birth orders 2+ and 3+. Logistic scale.Document A.2. Main indicators
F  is the overall fertility indice, which measures the number of children per woman. For periods it represents the total fertility rate (TFR), and for cohorts it corresponds to the completed fertility rate (CFR). 
F_{n}  is the mean number of children of birth order n per woman, with F = F_{1} + F_{2} + … It is also the proportion of women having reached parity n. 
F_{n+}  is the cumulative mean number of children per parity beginning at parity n: F_{n+} = F_{n} + F_{n+}_{1} … 
a_{n}  is the parity progression ratio for women at parity n: that is, the proportion of women having reached this parity who have an additional child. It can be calculated as a_{n} = F_{n+}_{1} / F_{n}. 
ā_{n+}  is a mean of parity progression ratios beginning at parity n. Its value represents the proportion of women at parity n+ who have an additional child. It can also be used to “close” a fertility table at order n+1 and above; this is referred to as the final PPR. It can be calculated as ā_{n+} = F_{(n}_{+1)+} / F_{n+} or as ā_{n+} = (a_{n}F_{n} + a_{n}_{+1}F_{n}_{+1} + …) / F_{n+} 
M_{n+}  is the mean marginal number of children per woman beginning at parity n. 
F_{n*}  is the mean number of children for women who have reached parity n. It can be calculated as F_{n*} = n + M_{n+}. 
120At parity 0 we have: F = F_{0*} = F_{1+} = M_{0+}, but these quantities all differ beginning at parity 1.
Principal relationships established
1211. Closure of the fundamental relationship between an aggregate fertility indice and parity progression ratios (Henry’s equation):
1232. The principal relationship used for converting scales and for the construction of Figures 1 and 2 is that between aggregate indices and mean parity progression ratio:
125And in particular for the overall fertility indice for all women:
127As well as the reverse relationship: conversion of the scale for aggregate indicators (number of children per woman) to a probability scale, for example at parity 0:
1293. The indicator of potential fertility associated to a PPR corresponds to the total number of children that would be born if all parity progression ratios were equal beginning at parity n, and in particular for the initial situation of women without children, at parity 0 (the tilde indicates that we are comparing this potential to F):
Notes

[1]
Indicators by birth order are rarely calculated for men, due to lack of data. Moreover, the birth order cited here is biological birth order, not within the current marriage, as has most often been analysed in the past. We also do not discuss below the problem of taking into account multiplicity in the calculation and interpretation of indicators by birth order.

[2]
Available on the site of the Instituto Nacional de Estadísticas (www.ine.es).

[3]
The most important equations obtained hereafter are repeated and summarized in Appendix A.2 which presents the principal indicators and equations.

[4]
Feeney (1986) was the first to calculate this mean ratio. See also Devolder (1994).

[5]
To differentiate this conditional completed fertility rate F_{n*} from the unconditional completed fertility rate F_{n+}, which is a sum of birthorderspecific fertility rates, we use the symbol * after the initial birth order indicator. This is appropriate as the recursive calculation is multiplicative, whereas for unconditional fertility indices by order, it is additive.

[6]
Hereafter the odds of a probability p is defined in the standard way as its transform p/(1p).