Department Meeting, July 3rd, 2001.

by M. Nakazawa

Chandola T, Coleman DA, Hiorns RW (1999) Recent European fertility patterns: Fitting curves to ‘distorted’ distributions. Population Studies , 53: 317-329.

Background◦ To compare fertility, a lot of indices can be used: TFR, mean age of childbearing, distributional measures based on
birth order or parity, and detailed distribution of the age-specific fertility curves, which can be obtained
as model parameters. Recent trends in European fertility may converge into high mode, median and mean (see,Figure 1). However, there are considerable variation, in both of timing and volume (see, Figure 2). UK and Irish
showed marked hump in early ferility. This study aims to model these ‘distorted’ patterns.

◦ Several models of age-specific fertility are known to give excellent fits to data from populations with natural fertility,with transitional fertility, and with controlled fertility: Pearson Type 1 (equivalent to B -distribution) and Type3 (equivalent to Γ-distribution), polynomial functionss, Gompertz relational models (Brass, 1977), and Coale andTrussell (1974) model. Hadwiger distribution (Hadwiger, 1940) is a relatively less-known alternative. B -distributionand Γ-distribution cannot fit well especially for younger ages. Modified Γ-distribution (Hoem et al., 1981) fitted
well to Danish data, but the parameters have no demographic meaning, so that is not appropriate for comparison.
Gompertz relational model does not fit well.

Coale and Trussell
F (x) = G(x)Fm (x),Fm (x) = M n(x) exp(mν(x))

where G(a) is ever-married proportion at age x, Fm (x) is age-specific marital fertility rate for age x, n(x) is agespecific natural fertility rate for age x, ν(x) is age-specific downward discrepancy from natural fertility for age x (n(x)
and ν(x) are given as schedules in Coale and Trussell’s paper), M is overall scale factor and m is degree of control of
fertility relative to the standard fertility schedule. It gives good fit to overall age-specific fertility rates and parameters
have a priori demographic meanings, but it has originally treated age-specific marital fertility rate with intentional
control and its computation is daunting.

HadwigerF (x) =ncoab  c 3/2xexp −b2+ −2c xxc

where F (x) is age-specific fertility rate for age x, a, b, c are parameters without any a priori demographic meaning(However, the authors suggested demographic meaning in this paper). Hoem et al. (1981) showed a good fit of Hadwiger
function to the fertility curves of modern populations. Simple Hadwiger model fit well to the fertility curves of Austria,
Denmark, France and Sweden in 1994 (Figure 3) but not to those of UK and Irish Republic in 1994 because of excessesof fertility at younger ages (Figure 4).

A mixture of Hadwiger
The possibility is that some populations including UK and Irish Republic are composed of two somewhat different
populations. Let the mixture parameter m,
b1  c1 3/2
c1xb2  c2 3/2
c2x
22F (x) = m
exp −b1+−2+ (1 − m)exp −b2+−2
c1 xx
c1
c2 xx
c2
The data were obtained from Eurostat (not downloadable nor free, but cf. http://europa.eu.int/comm/eurostat/). The
mixture model fitted to the fertility curves of UK and Irish Republic in 1994 well (Figure 4). The mixture parameterm determines the relative size of the two component distributions and the mixture 1 and mixture 2 almost correnpond
to non-marital and marital fertility curve, respectively (Figure 5). It suggests that the hump in UK and Irish Republic
could be caused by a large proportion of teenage fertility, most of non-marital. Mixture 1 underestimates non-marital
portion in later ages, which is absorbed by the relatively larger levels of mixture 2 than real marital fertility curves.
Since UK and Irish Republic have by no means the highest levels of non-marital fertility (illegitimacyratio) in Western Europe, the relationship was tested for other countries (Figure 6), which was relatively linear.
However, most of the other higher illegitimacy ratio populations obeyed the simple (homogeneous) Hadwiger model(no need for mixture), because there marital and non-marital fertility curves were relatively identical and there was noexceptionally high teenage fertility (Figure 7). In Australia, non-marital fertility is lower but not notably hererogeneous
because timing was almost same as marital fertility.
Norwegian marital and non-marital fertility curves in 1994 were distinct both in timing and volume. Combinationof these may generate the hump in younger age, which may fit with the Hadwiger mixture model. Figure 8 shows theemergence of ‘distorted’ fertility in UK (evident since 1980s).
Figure 9 shows the relationship between illegitimacy ratio and the mixture parameter m in UK and Irish Republic(linear for both countries, increasing over time). The dip in m in 1990s for UK suggests the relationship being notstraitforward.
Other parameters of the simple Hadwiger model have demographic interpretations. Figure 10 shows the significant
linear relationship between a × cb and age-specific marital fertility rate (b affects the maximum age-specific fertility rate).
Figure 11 shows the strong correaltions between a and TFR (a = 0.56 × TFR) and between c and the mean age ofmotherhood.
Extending these interpretations, c1 and c2 in the Hadwiger mixture model may reflect mean ages of motherhood fornon-marital and marital groups, respectively. In UK and Irish Republic, c2 continously increased and c1 decreased in
1990s (Figure 12). Mean age of non-marital motherhood in UK and Irish Republic has increased over this period, sothat the Hadwiger mixture model does not exactly correspond to the mixture of marital and non-marital fertility curves.

Discussion◦ The simple Hadwiger model successfully fits to fertility curves in some of modern European countries. Averaging
these curves may give the standard European fertility curve. The Hadwiger mixture model fits well to the ‘humped’fertility curves in some other European countries. These fits make it possible that is comparison of fertility and its
components among European countries and over time.

◦ There was strong correspondence between a, b, c and TFR, maximum age-specific fertility rate, mean age of motherhood, respectively. It is new finding because Hoem et al. complained no demographic meaning of Hadwiger
parameters. Moreover, it suggests that the Hadwiger models may be re-interpreted by more traditional demographicterms (not given in this paper).

◦ The Hadwiger mixture model may be related to marital and non-marital fertility. The mixture term m partly determines the relative size of two distributions, which related to illegitimacy ratio both cross-sectionally and longitudinally(This cannot be found by traditional measures!). c2 may represent the mean age of marital motherhood, but considerable attention must be paid to interpret this, because c1 does not correspond to the mean age of non-maritalmotherhood.

◦ Good fits of the simple Hadwiger model to some heterogenous fertility are attributable to the relatively small proportion of non-marital fertility as in Austria or to a similarity of timing and volume of marital and non-marital fertilities
as in Denmark or Sweden. Societies with fitting the simple Hadwiger model to their fertility due to the latter reason
may suffer from ‘Second Demographic Transition’ (indicating the disappearance of different reproductive behavior
by marital status).

Further application of the Hadwiger function (by M. Nakazawa)
The method presented in this paper can easily be applicable to the fertility of postwar Japan. In Japan, non-marital
fertility is virtually ignorable, so that I applied the simple Hadwiger model to the Japanese data during 1950 to 1995
compiled from the record of the vital statistics and the census report. Fitting were excellent for any year (RMSE≤ 0.012)
and the parameter a was closely correlated with TFR (R2 = 0.999) and c was closely correlated with the mean age ofmotherhood (R2 = 0.96) also in Japan. Thus the method is useful to analyze the fertility decline in postwar Japan.
The trends of three parameters were well approximated by the cubic functions (R2 ≥ 0.9). Projected TFR in 2000
and 2005 were 0.63 and −0.34, respectively. Is it clearly wrong, probably due to a being less than zero in cubic trend
approximation (R2 = 0.91). Then, how is the real trend? If only the data during 1960-1995 were used for cubic trend
approximation (R2 = 0.95), a keeps positive after 2005, and projected TFRs are 1.37, 1.44 and 1.61 in 2000, 2005, and2010, respectively. The selection of data points is more or less arbitrary and unreliable, but it may have some implication
for fertility projection.