Population Trends 109
Autumn 2002
The effect of changes in
timing of childbearing on
measuring fertility in
England and Wales
Steve Smallwood
Population and Demography
Division
Office for National Statistics
Changes in the ages at which
women give birth to their
children mean that fertility
measured at a particular point
in time (period) may not be a
good representation of the
ultimate fertility of those
women. The common measure
of period fertility is the total
fertility rate, which in 2001 has
fallen to the lowest level since
records began in England and
Wales. This article presents
various methods that have been
proposed to adjust period
fertility data to take account of
changes in the timing of
childbearing, applied to England
and Wales data. The article
concludes that while these
adjustment methods provide
useful insights, for example, that
the total fertility rate has
underestimated period
quantum fertility since the
1970s, the measures produced
are difficult to interpret. This is
in part because the concept
they are trying to measure,
period quantum is itself
imprecise. The adjustments do
not necessarily provide a
reliable indicator of underlying
cohort fertility.
National Statistics
INTRODUCTION
The recent publicity surrounding the publication of the (provisional)
lowest ever total fertility rate (TFR) of 1.64 in 2001 for England and
Wales highlights the importance of the TFR as a measure of fertility.1
The TFR, which gives the average number of children per woman if a
group of women experienced the age-specific fertility rates of a
particular year, is common currency among demographers and the
public for measuring fertility. The problem with the measure is that it is
a period measure, that is, it is based on the births and female population
in one particular year. When women are delaying childbearing, the
measure is likely to underestimate the overall number of children
women will eventually have. Similarly if women are advancing
childbearing the TFR is likely to overestimate the overall number of
children women will have. It is commonly known that women are
choosing to start childbearing later in life,2 the evidence for this can be
seen in Figure 1, with fertility falling at ages under thirty but rising at
older ages for the last two decades.
If fertility rates at older ages continue to rise the TFR will understate
the overall level of fertility women aged under 30 will achieve. The
question demographers really want to answer is what can be inferred
about ultimate levels of fertility for cohorts that have not yet completed
their childbearing from period measures, given the change in timing of
births.
These phenomena of falling total fertility rates and delayed childbearing are not confined to England and Wales, see Table 1, and therefore
the deficiencies of the TFR as a measure of fertility have been brought
to the fore in the demographic community. Various methods have
recently been proposed to adjust period fertility measures for the
changes in timing of births (tempo), to leave a measure that gives a
better reading of the ‘true’ level of fertility (quantum).
36
Population Trends 109
This article presents the results of applying some of these adjustment
methods to England and Wales data and provides a short discussion on
the efficacy of such adjustments. Before the adjustment methods and
results are presented it is necessary to consider the concepts of quantum
and tempo in the context of period fertility, that is the fertility in a
particular year, and cohort fertility, the fertility of a group of women
born in a particular year.
Figure 1
Components of TFR below and above age 30,
1940–2001
England and Wales
2.5
Period v cohort fertility
Children per woman
2.0
Measures of period and cohort fertility are sourced from the same
information viewed from different perspectives. Table 2 illustrates the
construction of the total fertility rate and completed family size from
age specific fertility rates. It should be noted that historical birth data
for England and Wales are available by age of mother at childbirth but
not for her own year of birth. Thus the number of births in each year at
each age come from two different cohorts. For example, births to 14
year olds in 1969 could have been born to mothers who were born in
either 1954 or 1955. Births are related to the population by using the
age specific mid-year population estimates as the denominators. The
age specific population will have been born between two mid year
periods. So the births in our previous example are related to the midyear population age 14 in 1969 who would have been born between
July 1954 and June 1955. For presentation purposes the cohort fertility
is referred to as the later of the two cohort years. So in our example the
resulting fertility rate would be presented as part of the 1955 cohort.
Table 1
Autumn 2002
TFR age under 30
1.5
1.0
TFR age 30 and over
0.5
0.0
1940
1950
1960
1970
Year
1980
1990
2000
Total fertility rate and age standardised mean age at motherhood, 1970–2000
Countries of the European Union
Total fertility rate (children per woman)
Mean age at childbearing (years)
Country
1970
1980
1990
2000
1970
EU-15
2.38
1.82
1.57
1.53 *
:
Belgium
Denmark
Germany
Greece
Spain
France
Ireland
Italy
Luxembourg
Netherlands
Austria
Portugal
Finland
Sweden
United Kingdom
2.25
1.95
2.03
2.39
2.90
2.47
3.93
2.42
1.98
2.57
2.29
2.83
1.82
1.92
2.43
1.68
1.55
1.56
2.21
2.20
1.95
3.23
1.64
1.49
1.60
1.65
2.18
1.63
1.68
1.90
1.62
1.67
1.45
1.39
1.36
1.78
2.11
1.33
1.61
1.62
1.45
1.57
1.78
2.13
1.83
1.65 *
1.76 *
1.34 p
1.30 *
1.22 *
1.89 p
1.89
1.25 *
1.78
1.72 p
1.32 *
1.54 *
1.73
1.54
1.64
27.2
26.7
26.6
:
29.6
27.2
:
28.3
27.2
28.2
26.7
29.0
27.1
27.0
26.3
1980
1990
1998 1
27.0 *
28.2 *
29.1 *
26.6
26.8
26.4
26.1
28.2
26.8
29.7
27.4
27.5
27.7
26.3
27.2
27.7
27.6
26.9
27.9
28.5
27.6
27.2
28.9
28.3
29.9
28.9
28.4
29.3
27.2
27.3
28.9
28.6
27.7
:
29.5
28.6
28.7
30.6
29.3
30.4
:
29.2
30.3
28.0
28.5
29.5
29.7
28.3
1 Latest year for which most countries have data available in Eurostat table
: Data not available
*
Estimate in that year made by Eurostat
p
Provisional data
Source: European social statistics, Demography 2001, Theme 3, Eurostat.Tables E-4 & E-5.
37
National Statistics
Table 2
Age-specific fertility rates per 1,000 women, illustrating period and cohort calculation, 1969-–2000
England and Wales
1970
1971
1972 1973
1974
1975
1976
1977
1978
1979
1980 1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998 1999
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
0.8
3.8
16.8
45.3
75.4
101.3
123.7
146.7
157.9
175.1
181.2
179.8
172.2
158.5
143.9
126.9
111.5
94.4
82.9
70.4
62.4
53.2
44.3
36.1
30.5
24.3
17.8
12.5
8.9
5.5
3.3
3.3
0.8
3.6
17.1
45.5
76.8
102.5
124.1
144.3
161.1
164.9
178.8
177.7
167.6
153.6
139.3
122.6
107.7
90.1
77.5
65.6
57.2
49.9
41.7
34.2
27.7
21.3
16.7
11.2
7.9
4.8
2.8
3.0
0.8
3.8
18.3
46.9
78.4
104.4
124.8
142.3
154.8
167.2
168.7
178.2
168.1
154.5
139.3
120.7
106.9
88.1
74.8
63.5
53.6
46.7
38.9
32.4
25.6
20.2
15.7
10.9
7.2
4.5
2.5
2.7
0.8
3.9
19.0
47.0
72.9
97.2
115.4
128.8
140.7
152.1
160.1
159.2
158.6
144.9
127.6
112.2
96.3
79.7
67.3
56.0
47.4
40.7
33.9
28.2
22.3
18.4
13.4
9.6
6.5
4.1
2.3
2.4
0.8
3.6
15.9
41.1
62.4
83.4
99.5
114.8
126.2
133.4
140.8
143.9
141.7
129.5
118.7
101.4
84.8
68.1
56.7
46.1
38.1
31.1
25.2
20.8
16.9
13.0
10.4
7.0
5.1
3.1
1.8
1.9
0.7
3.4
13.2
35.2
57.2
76.4
92.0
105.2
117.5
124.9
130.0
137.0
134.3
126.9
111.2
100.1
83.4
66.4
54.4
43.4
36.3
29.2
23.9
18.9
15.5
11.6
9.1
6.4
4.5
2.9
1.6
2.0
0.7
3.1
11.9
29.8
50.1
69.8
85.8
100.8
111.5
120.8
127.6
131.3
131.3
124.9
114.2
97.1
84.9
66.1
53.4
43.4
34.3
28.3
22.9
17.6
14.1
10.8
8.2
5.8
4.1
2.3
1.5
1.8
0.6
2.8
11.5
27.5
45.5
63.1
81.3
95.0
106.9
113.5
122.3
128.9
125.9
122.6
112.6
100.3
82.7
68.5
55.6
43.1
34.8
27.5
21.9
17.4
13.5
10.7
7.3
5.2
3.7
2.4
1.5
1.7
0.6
2.9
11.3
27.3
45.1
63.5
81.0
98.3
110.3
119.4
127.3
132.7
132.0
126.2
117.9
105.6
91.6
72.7
59.5
46.9
37.0
30.2
23.2
18.5
14.2
10.2
7.7
5.2
3.7
2.5
1.4
2.0
0.6
2.8
10.9
28.0
46.6
65.7
83.5
100.1
115.3
126.5
134.3
138.7
141.4
135.5
126.6
114.4
99.5
81.7
64.4
53.7
41.8
32.5
25.4
19.5
15.1
11.4
8.3
5.5
3.7
2.3
1.5
2.0
0.6
2.6
10.6
27.3
47.1
66.0
84.9
101.6
116.0
128.1
136.1
143.9
142.1
138.2
128.6
115.1
103.2
84.6
68.9
53.3
44.5
34.2
26.6
20.7
15.7
11.6
8.3
5.6
3.7
2.4
1.5
2.3
0.6
2.4
9.9
24.3
40.6
58.2
74.3
90.2
104.6
115.8
126.9
132.8
132.8
129.2
122.5
114.0
99.5
82.8
68.4
55.1
45.0
34.4
28.0
21.0
15.3
11.3
8.2
5.6
3.8
2.3
1.3
2.2
0.5
2.8
10.2
24.0
39.0
55.2
70.6
85.9
101.7
114.4
124.3
131.9
133.5
129.3
123.2
112.8
102.1
85.6
70.5
56.5
45.7
36.0
27.2
21.7
15.9
11.4
8.0
5.8
4.0
2.3
1.3
2.3
0.6
2.9
10.7
25.3
40.0
55.3
68.2
82.7
98.1
110.0
122.6
128.6
132.6
129.6
124.0
113.8
103.0
88.5
72.3
58.4
47.3
37.1
28.9
21.7
16.7
11.8
8.4
5.6
3.9
2.4
1.5
2.0
0.7
3.2
11.6
27.2
43.1
57.8
69.1
83.2
95.6
108.3
118.9
129.3
132.3
131.0
126.3
117.2
105.7
89.9
75.9
62.1
49.0
39.1
30.9
22.5
16.2
12.8
8.6
5.7
3.7
2.4
1.5
2.2
0.6
3.1
11.7
27.9
45.2
59.1
70.0
80.8
93.0
105.1
115.0
124.1
127.7
127.9
123.3
115.5
105.9
91.3
77.4
64.1
50.9
42.0
31.5
23.6
17.6
12.3
9.2
5.8
3.5
2.1
1.3
1.9
0.6
3.2
11.8
28.5
45.7
61.4
71.4
82.2
93.3
104.1
114.5
123.2
129.0
129.9
125.0
118.2
108.6
94.1
80.5
67.0
54.3
43.9
34.0
25.6
18.6
13.2
9.0
6.2
3.8
2.1
1.3
1.7
0.7
3.2
12.8
30.0
47.6
63.0
74.1
84.6
94.1
103.9
114.0
122.4
126.2
127.4
124.6
119.2
108.4
94.5
81.6
67.9
56.9
45.2
35.2
26.5
20.0
14.0
9.5
5.8
4.0
2.3
1.2
1.6
0.7
3.6
12.9
29.3
45.6
61.2
71.8
83.1
91.5
101.1
109.0
117.1
122.3
123.9
121.9
116.8
108.1
95.7
81.8
68.4
58.0
46.7
36.9
28.0
20.9
14.9
10.2
6.3
3.8
2.4
1.2
1.5
0.7
3.7
13.5
31.2
46.9
62.0
71.6
82.0
91.3
100.4
109.1
118.6
123.0
126.6
125.0
120.0
112.9
99.4
85.8
72.2
60.6
49.2
39.0
29.5
22.0
16.2
11.0
6.7
4.1
2.3
1.4
1.5
0.9
4.1
13.8
30.5
46.6
61.4
71.1
79.9
88.6
98.5
106.6
115.0
119.9
123.0
122.3
116.6
110.8
98.4
86.4
73.5
61.5
49.7
40.2
30.3
22.9
16.8
11.2
7.1
4.4
2.5
1.3
1.2
0.8
4.0
13.2
29.6
45.3
59.1
68.4
75.6
86.2
94.3
104.4
111.5
118.4
119.6
120.0
116.8
109.5
98.5
87.8
74.2
62.6
51.6
41.2
31.9
23.7
17.6
12.0
7.8
4.8
2.6
1.3
1.3
0.9
4.1
14.2
29.6
44.3
57.6
66.3
73.0
81.4
90.8
99.1
107.0
113.7
116.7
117.8
114.8
108.7
98.4
86.4
74.9
63.1
51.9
42.1
32.7
24.2
17.6
12.5
8.3
4.9
2.6
1.4
1.5
0.9
3.5
13.0
28.5
43.5
55.2
64.7
70.0
77.9
85.6
95.1
103.3
111.0
115.2
115.9
114.5
109.4
100.4
88.8
75.6
65.9
54.2
43.6
33.8
25.4
19.1
12.4
8.4
5.0
3.0
1.6
1.4
0.9
3.8
13.3
28.3
43.7
56.5
62.8
70.0
75.7
81.9
89.5
99.5
106.3
111.6
112.3
112.5
107.7
98.0
88.2
75.8
65.1
54.9
43.7
34.5
25.9
19.6
13.6
8.8
5.3
3.0
1.7
1.6
0.9
4.2
14.2
30.7
45.3
58.9
67.0
70.5
77.1
81.6
88.4
97.1
104.5
108.9
112.3
111.1
107.6
100.3
89.7
78.0
66.8
55.4
46.2
35.1
26.8
20.0
14.1
9.4
5.7
3.2
1.7
1.8
0.9
4.2
14.2
30.8
45.8
58.1
66.9
71.1
74.7
80.4
87.1
93.3
101.4
106.7
110.1
111.4
106.9
99.3
90.4
79.7
68.2
58.0
47.0
37.7
28.2
20.8
14.5
9.8
6.2
3.4
1.9
1.7
0.9
4.0
13.8
31.5
45.7
58.2
65.4
70.3
75.7
78.5
85.7
90.8
98.5
103.7
108.0
108.7
108.5
100.2
92.0
80.2
69.9
59.3
47.8
37.9
29.2
21.9
14.9
10.1
6.1
3.6
1.9
1.7
0.9
3.9
17.9
42.5
67.6
89.3
107.0
121.3
130.5
141.6
148.1
153.3
145.2
138.6
121.2
104.7
89.4
72.1
60.1
49.9
41.2
35.0
29.1
23.5
18.9
15.3
11.5
8.2
5.6
3.6
1.8
2.2
Data used in calculating CFS for 1955, 1965 and 1975 cohorts.
Data used in calculating 2000 TFR.
0.6
2.4
10.2
24.2
42.3
61.2
77.6
95.8
106.9
118.6
131.2
137.9
135.3
133.3
124.6
114.0
99.8
85.3
68.6
56.2
41.1
33.8
25.6
20.5
14.7
12.0
8.0
6.9
3.8
2.3
1.4
2.5
2000
0.9
0.9
3.9
3.7
13.2 12.3
30.4 28.6
45.8 44.3
58.6 56.8
65.5 62.5
69.4 66.5
74.1 70.6
77.4 74.7
81.6 79.5
88.3 84.5
95.1 90.8
100.4 95.1
103.8 100.7
106.6 103.3
105.2 102.9
100.0 97.4
90.9 90.9
80.9 80.0
70.6 70.4
59.3 60.1
48.2 49.7
37.7 39.0
28.9 29.5
22.3 22.4
15.5 15.8
10.0 10.6
6.2
6.5
3.8
3.8
1.9
2.0
1.9
1.9
Autumn 2002
38
1969
Population Trends 109
National Statistics
Age
Population Trends 109
Autumn 2002
The total fertility rate for any year is the sum of the age specific fertility
rates in that year (divided by 1,000), thus the TFR for 2000 is 1.66. The
TFR is a synthetic measure in that it refers to a hypothetical group of
women experiencing the fertility of a particular year. The completed
family size (CFS) for the 1955 cohort is the sum of the diagonal of the
fertility rates, starting at age 14 in 1969 (divided by 1,000) and equals
2.02. Younger cohorts have yet to complete their childbearing, so their
CFS cannot be calculated without making assumptions about the future
fertility rate, by which they will complete their childbearing. The CFS
is closer to being a real rather than synthetic measure, in that it refers to
a real cohort of women. It does, however, assume that the women
survive through their child-bearing years, a reasonable assumption in
developed societies.
and mortality) all there is to tell about how our future pensions will be
taken care of”. For good reasons we use more detailed measures such
as the TFR to look at period fertility. The TFR is not sensitive to
changes in the age composition of the female population of childbearing age so is useful in making comparisons across time and
geographies. However, it is sensitive to changes in the timing of births,
being ‘depressed’ if women are delaying childbearing or ‘exaggerated’
if women are advancing childbearing. It is clear that the ‘depression’ or
‘exaggeration’ could be thought of in relation to the ultimate average
number of children women will have, that is cohort quantum, but, as
will be shown later, the relationship to cohort quantum can only be
shown when all the cohorts in the population have completed their
childbearing.
Quantum and tempo
There has been much recent debate on proposed measures that claim to
adjust period fertility to take account of changes in tempo. One of the
key points in the debate has been whether the resulting adjusted
measure is trying to approximate cohort quantum, that is whether the
proposed adjustments are trying to translate from the period perspective
to the cohort perspective.6 If the intention is to adjust the period data to
produce the underlying cohort fertility the various proposed methods of
adjustment can be tested empirically. If the intention is not this, but as
Bongaarts and Feeney7 say of their method, “We are not attempting to
predict cohort fertility, only to get an improved reading of period
fertility.” some thought needs to be given to what the Bongaarts and
Feeney (BF) and other similar adjusted measures are actually giving. In
other words, what is meant by the ‘period’ quantum, which their tempo
adjusted measure represents? This is discussed later in the article after
the methods have been applied to England and Wales data.
The two perspectives to the way that fertility can be looked at, the
period perspective and the cohort perspective, have different concepts
of quantum and tempo. For cohort fertility the quantum is a measure of
the number of children produced over the life course (for example,
CFS), while tempo simply refers to the timing of births within the life
course. Period tempo is the change in timing of births in the population
over time, the population being made up of individual cohorts who each
may each be postponing or advancing births.
From a period perspective, quantum is a more elusive concept. It might
be described as the level of fertility after adjusting for the changes in
the population’s timing of births or as Van Imhoff3 says: “the TFR that
would have been observed in year t if the age pattern of fertility (for
each birth order) had been the same as in year t-1 under the assumption
that the shape of the order-specific age pattern of the AFSRs is equal in
both years.” He argues that the ultimate indicator of the period quantum
of fertility is the annual number of births relative to the total population
size (the crude birth rate) as, “It tells us (almost: we need migration
Figure 2
BACKGROUND TO THE ADJUSTMENT METHODS
Total fertility rate 1940–2000, completed
family size and tempo adjusted completed
family size for cohorts born 1924–55
The proposed adjusted measures have their roots in the work of Ryder.8
Both the TFR and CFS are calculated from the same set of fertility data,
but from different perspectives. If the age-profile i.e. the pattern of age
specific fertility rates, is invariant and the changes in timing develop
smoothly then Ryder’s demographic translation technique can be used
to translate the cohort indicator CFS into the period indicator TFR.
Ryder’s simple translation formula is:
England and Wales
3.0
CFS adjusted to TFR
Children per woman
Some of the structure of the article and many of the points made are
derived from a paper presented on the subject by Evert Van Imhoff at
the Euresco conference “The second demographic Transition in
Europe” and subsequently published in the on-line demographic journal
Demographic Research.3
TFR=CFS x (1-∆MAC)
2.5
JJ
J JJJJJ
JJJ
J
JJ
JJ
JJ
J
J
where ∆MAC is the change in cohort mean age at childbearing,
This holds under the conditions of constant cohort quantum and agespecific proportions in total fertility changing linearly over successive
cohorts.9
We can approximate ∆MAC as follows:
JJ J
J
J J JCFS
JJJ
2.0
JJ
∆MAC ≈ (MACt+1-MACt-1)/2, where t is the year of birth of the cohort.
TFR
1.5
1940
1950
1960
1970
1980
1990
2000
Year (TFR), Birth cohort year+mean age at childbearing of cohort (CFS)
Note: The trough in the CFS around the 1946 to 1948 cohorts is a consequence of
rapidly changing fertility just after World War Two, which saw a peak in births in the
second half of 1946 and the first half of 1947.4 As the birth data are only available
by age of mother at birth, some of the births to women born in 1946/47 will have
been attributed in the age specific fertility rates to the adjacent cohorts, inflating
their fertility at the expense of the mid-1946 to mid-1947 cohort.5
We can see from the adjusted CFS line in Figure 2 that empirically for
England and Wales, the relationship, to a certain extent, holds. The
differences occur as the age-profile is not invariant and the level of
cohort fertility is changing. Using mean age alone therefore does not
completely adjust for the cohort tempo effect to convert to the period
fertility measure. Further, the change in mean age is calculated by
assuming a linear variation between the two adjacent years and is
thereby an approximation.
39
National Statistics
Autumn 2002
Note that the approximation is made to fit better the pattern of the TFR
by plotting the Ryder adjusted CFS at the cohort’s year of birth plus the
mean age at which the cohort bore their children when compared with
period data. In most publications10 the cohort data is presented at a
standard difference from the year of the mothers birth, say 28 years.
This approximates to the mean age at birth of the cohort. In the case of
England and Wales this stretches out the rise in cohort fertility in the
1960s. As can be seen in Figure 3, the CFS data points are bunched
more closely together horizontally in the 1960s. This is caused by the
decrease in age at birth for more recent cohorts. For example, the mean
age at birth for the 1936 cohort was 27 years, for the 1941 cohort the
mean age had fallen to 26 years.
Being able to translate back from the known cohort fertility to produce
(an approximation of) period total fertility does not tell us a great deal.
We already know what the period fertility is. The question demographers really want to answer is what can be inferred from historical and
current levels of fertility about the number of children women will
have, given the change in timing of births.
Figure 3
38
36
34
Fifth and higher births
Forth births
32
Third births
30
Second births
28
All births
26
24
APPLYING TEMPO ADJUSTED FERTILITY MEASURES TO
ENGLAND AND WALES DATA
Mean age of mother by birth order,
standardised for population age distribution,
1940–2000
England and Wales
Mean age
Population Trends 109
First births
22
1940
1950
1960
1970
Year
1980
1990
As we have seen with the Ryder formula there is clearly a relationship
between the CFS and the TFR. But what can be inferred from the TFR?
The adjustment methods are based on changes in the period mean age
of childbearing. Figure 3 shows the period mean age at childbearing by
true birth order,2 that is, for first live-births, second live-births and so
on, calculated from the parity – specific fertility rates. The fall in
fertility rates for younger cohorts and the rise for older ones naturally
affects the period mean age at birth, for example the mean age at first
birth has risen from 23.7 in 1970 to 26.5 in 2000.
The ‘period adjustment’ approach
In this section the most recently proposed period fertility adjustment
measures are applied to England and Wales data. The measures are
based on fertility decomposed by birth order as the mean age of all
births will not accurately capture tempo effects. For example , when
cohort fertility is declining, the reduction occurs primarily at higher
birth orders and therefore at older ages. Consequently the mean age at
birth declines even when there is no change of tempo.
The first method is that proposed by Bongaarts and Feeney7 (hereafter
referred to as BF). Their inspiration is the translation formula of Ryder,
but instead of translating cohort data to period data using the change in
the mean age at childbearing for cohorts, the adjustment applies parity
specific11 changes in period mean age to adjust the observed TFR. See
Box 1 for the calculation of the TFR using birth order data and the BF
adjustment.
Box one
CALCULATING THE TFR USING BIRTHS BY BIRTH
ORDER
The TFR is calculated using births by births order as
follows:
i=birth order
t=year
a=age
B=births
P=female population
i=n
TFR t =
∑ TFR i,t
i=l
a=45
Where TFR i,t =
B
∑ P a,i,t
a=15
a,t
The BF adjusted TFR (TFR′) is created by:
It is important to note that although the BF calculation may look like a
reformulation of Ryder’s formula (with the complication of birth order)
the resulting adjusted TFR is not the same as the CFS, it is still a period
measure. It attempts to state, assuming no changes in the shape of the
age-profile at each birth order, what the fertility level would have been
if the age shift of births in that year had not taken place.
Figure 4 shows the application of the BF adjustment to England and
Wales data. It is clear that the shape of the adjusted TFR is not closely
related to the shape of CFS, so empirically this is not an adjustment that
converts period data into cohort form. Over the 1960s and 1970s the BF
adjusted TFR series shows very similar characteristics to the actual
TFR, with the rapid rise to a peak in the mid-1960s followed by a rapid
fall. The trough in the TFR in the late 1970s is less severe, suggesting
National Statistics
40
(a) adjusting TFR of parity i (TFR i,t ) by the reciprocal of
one minus the change in mean age at parity i (r i,t )
TFR′ i,t = TFR i,t / (1- r i,t )
and then,
(b) summing the resulting adjusted parity TFRs.
TFR′ t = ∑ TFR′ i,t
The change in the mean age r i,t being approximated as
(PMA i,t+1 -PMA i,t-1 )/2
where PMA is period mean age.
2000
Population Trends 109
that the underlying ‘period quantum’ of fertility was higher than the, at
the time, record low TFR suggested. Since 1970 the adjusted BF
measure is consistently above the observed TFR, suggesting that the
‘period quantum’ of fertility in each year since the 1970s has been
underestimated by the observed TFR. But cohort fertility cannot be
inferred from this adjusted level of fertility, unless the synthetic cohorts
implied by a TFR are known to approximate to actual cohorts. At times
of very volatile fertility, for example the period during and following
World War II, the adjusted measures of fertility can be even more
volatile than the total fertility rate. The problem is that using the mean
age alone is insufficient to capture the changes taking place in the shape
of the age-profile, exacerbated by the crude method for calculating the
change in mean age. For example, in 1942 the contribution of first
births to the BF adjusted TFR is increased from the recorded 0.86 births
per woman to an impossible 1.10 births per woman as the mean age
rose by over 0.2 of a year [1.10=0.86/(1–0.22)]. The rise in mean age in
that year is not due to a shift of the age-profile rightwards but rather a
general increase in fertility rates at all ages but concentrated above age
25.It is also possible for an unadjusted contribution of first births to a
TFR to accumulate to being more than one child per woman. This is a
weakness of the TFR calculated from age specific data, and therefore of
any adjust measure based on it.
complex and for a full explanation of the calculation the reader is
referred to their article. The principle of the adjustment is as follows. If
the variance of the age-profile is changing over time its shape is also
changing. This change in shape will affect the mean age and potentially
also the change in mean age. KP adjust the parity specific fertility rates
for the change in variance so that the mean age and change in mean age
can be calculated between two fertility profiles with the same shape.
Note that here the KP adjustment has been computed without smoothing the change in variance and change in mean age.
As can be seen from Figure 4, the KP adjusted TFR is very similar to
the BF adjusted TFR for England and Wales. This more complex
measure does suggest that adjusting for the changing shape of the ageprofile caused the ‘period quantum’ to be underestimated in the 1980s
by the BF method, and also suggests that for very recent years the BF
method overestimates the ‘period quantum’.
Some of the volatility for these measures can be removed by smoothing
the change in mean age. Kohler and Ortega13 provide a BF measure
calculated using smoothed changes in mean age. This reduces the
volatility of the BF measure, as can be seen in Figure 4. In particular it
smoothes out the disturbance in the measure in the mid 1990s. Trends
in the unsmoothed adjusted measures of fertility are consistently around
0.2 above the recorded TFR in the 1990s except in the mid-1990s,
particularly 1995. On 18 October 1995 the Committee on Safety of
Medicines issued a warning that certain ‘new generation’ contraceptive
pills carried a relatively higher risk of thrombosis. There was concern
that the scare may have led to unplanned births in 1996. Wood, Botting
and Dunnell14 found that the conception rate following the pill scare
Kohler and Philipov12 (KP) have taken the rationale of the BF adjustment and extended it to include the effect of changing variance in the
age-profile. They noted that its shape varied over time for a number of
European countries and that this was a ‘potentially relevant’ violation
of the underlying assumptions of the BF formula. The adjustment they
propose makes the computation of the adjusted measure much more
Figure 4
Autumn 2002
Total fertility rate, completed family size, Bongaarts Feeney adjusted TFR, Kohler Philipov adjusted TFR, Kohler
Ortega Fertility Index and adjusted Fertility Index, and Timing Index
England and Wales
3.0
2.8
Children per woman
2.6
2.4
J J J
2.2
J J
J J
J
J J J
J J J J
J
J J
J
J
J J J
J
2.0
See Footnote
in Fig 2 re CFS
J
J J
J J J J J
1.8
1.6
1.4
1940
1950
1960
1970
1980
Year (all period measures), Birth cohort year+mean age at child
TFR
KP adjusted TFR
BF adjusted TFR
Smoothed BF adjusted TFR
Timing index
J J J J
1990
Period Fertility index
KO adjusted Period
fertility Index
CFS
41
National Statistics
2000
Population Trends 109
Autumn 2002
had increased, although it could not be concluded that the pill scare was
the cause. They also found that the rate for conceptions leading to
maternity in the three quarters following the pill scare were higher than
rates in the preceding year. Whether or not it was the effect of the pill
scare, the mean age at birth in 1996 changed very little for first and
second births compared to 1995. There were rises in birth rates at both
younger and older ages, but falls at some ages for women in their 20s.
Where the change in mean age was not smoothed but approximated
from the mean ages in 1994 and 1996 the upward trend in mean age
was effectively halved for 1995 leading to a much smaller upward
adjustment in that year.
Kohler and Ortega (KO) have recently published a further method of
adjusting fertility for changes in birth timing.15 This uses the KP
principle of adjusting for mean age and variance but applies it to
occurrence-exposure rates rather than to parity-specific fertility rates.
Occurrence-exposure rates use as denominators the number of women
at risk of having birth of a given order and are less sensitive to tempo
distortions. For example if, because of postponement, childless women
aged 25 have fewer first births in year y there will be more childless
women aged 26 in year y+1 compared to the cohort of women aged 26
in year y. The fertility rate at age 26 in year y underestimates next
year’s fertility rate at age 26 as there are more women at risk of having
a first birth aged 26 in year y+1. Using O-E rates removes this distortion. From the O-E rates KO calculate an actual and adjusted ‘Period
Fertility Index’ similar in construction to the total period parity fertility
rate shown in the previous edition of Population Trends.2 Their method
goes beyond simply presenting an adjusted historical fertility measure,
they also use it to produce scenarios of completed fertility for cohorts
just beginning or in the midst of childbearing (see later section). The
computer programs required for the calculations are helpfully made
available on the Internet.13 The effect of the KO adjustment on the
Period Fertility Index is similar in quantity and direction to the effect of
the BF and KP adjustments on the TFR. However in recent years the
amount of adjustment has been lower suggesting less postponement of
births.
Box two
A TRANSLATION FROM PERIOD TO COHORT – THE
TIMING INDEX
Butz and Ward, 19 and also Ryder, 20 have suggested a true
measure of timing, called the timing index (TI) 21 The TI
looks at the proportion of a cohort’s fertility that
occurs in particular year. Summing the proportions at
each age gives an index which indicates, if less than one,
a year of postponement, and if greater than one a year
with a concentration of births. Dividing the recorded
TFR in the year by the TI gives a measure of timing
adjusted period fertility. The problem with such a
measure is that the entire cohort fertility needs to be
known for each ‘active’ cohort in the period year in
question. Thus although in Figure 4 it can be seen that
measure provides a good translation from period to
cohort, the measure can only be calculated for a small
number of historical years. Unlike the other proposed
measures, the TI does provide a reasonable translation
from period to cohort fertility when applied to England
and Wales data for that period.
National Statistics
42
The efficacy of tempo adjusted fertility measures
In previous work I have described tempo adjusted fertility measures
such as BF and KP as ‘taking synthetic measures and making them even
more synthetic’.16 The TFR, as a synthetic measure, does not reflect the
childbearing of a particular cohort of women. The measures take the
age specific fertility rates of all cohorts in their child-bearing ages at a
particular point in time and treats the data as if they applied to a single
synthetic cohort of women.
The period change in the mean age is therefore an amalgam of the
behaviours of different cohorts of women at a particular point in time. It
could therefore be possible that some cohorts of women in a period are
experiencing postponement, while other age groups are not. Kim and
Schoen17 have pointed out that the assumption in the BF method of a
linear shift affecting every cohort of reproductive age is severely
constraining and demonstrated that where timing changes are sinusoidal, the measure could be quite volatile even where cohort and period
fertility were constant. On the basis of analysis of US and Taiwan data,
Yi and Land18 have concluded that that the refinements of allowing for
change in variance in the KP method are normally not necessary. The
analysis of England and Wales data certainly suggests that little is
gained from the refinement in this case. The England and Wales
analysis also makes it clear that these tempo-adjusted measures are not
producing a measure of underlying cohort fertility. They clearly
demonstrate what can be intuitively grasped, that where women are
having later births it is likely that that current fertility measured by the
TFR ‘undercounts’ the total number of children that women will have
on average. Only under very particular circumstances will such adjusted
measures give a measure of underlying cohort fertility. The utility of the
level they do give is questionable, as what it is measuring is conceptually difficult to interpret. We may be able to interpret these adjusted
figures when looking at past data but they give no clue to cohort
fertility, unless we make the assumption that the synthetic cohort is
close to real cohort fertility. In reality, we cannot know what the
outcome of women’s fertility decisions at younger ages today will be
for their fertility in ten to fifteen years time. Where we do know a
cohort’s completed fertility it is possible to translate from period
fertility to cohort fertility, see Box 2, however this is of little use in
interpreting current data without making assumptions for cohorts that
have yet to complete their fertility.
The KO approach has the advantage of using the more stable O-E rates,
although the adjusted period fertility index is still an adjusted period
measure and tells us, for England and Wales little more that the BF and
KP methods.
The period extrapolation to complete censored
cohorts approach
Kohler and Ortega also use their method to make statements about real
cohorts. The method uses each cohort’s fertility history until the latest
period available, and then uses the latest period O-E rates to complete
each cohort’s fertility. Effectively they are making projections on
explicit alternative assumptions. KO propose three scenarios to do this:
1. Holding the current O-E rates constant (Observed);
2. Using O-E rates adjusted by the KP method and then kept constant
(Postponement stops);
3. Taking the O-E rates adjusted by the KP method and projecting them
forward assuming continuation of the changes in period mean age
and variance. (Postponement continues).
For first births scenario 3 will eventually yield the same number of first
births as scenario 2, however, the change in the timing of first births
will affect the numbers at risk of having higher order births. Figure 5
Population Trends 109
cent. If cohorts experience the levels of fertility in 2000 they would
complete at an average of 1.63 children per woman. Using the adjusted
O-E rates the figure is 1.70, similar to the assumed completed family
size of 1.74 in the 2000-based national population projections.22 The
postponement continues scenario produces a level of fertility of around
1.65 for the 1986 cohort (Figure 5d). Kohler and Ortega term the
difference between the postponement stops and the postponement
shows the results of the above three KO scenarios for England and
Wales.
Simply taking the current O-E rates produces levels of childlessness of
just over 25 per cent of women (Figure 5a and 5b). Adjusting the
occurrence-exposure rates for the current period changes in mean age
and variance produces levels of childlessness of a little under 23 per
Figure 5
Autumn 2002
Projection of fertility behaviour for cohorts who have not finished childbearing in 2000, based on level of fertility
and postponement pattern observed in 2000
(b) Proportion remaining ultimately childless
(a) Proportion still childless at year
1.0
0.27
0.25
0.8
Proportion
Proportion
0.23
0.6
0.21
1985 Cohort
0.4
0.19
0.2
1975 Cohort
0.17
0.0
0.15
2000
2010
2020
Year
2030
1955
1965
1975
1985
Cohort
(c) Cumulative cohort fertility
(d) Completed fertility
2.1
2.0
1.8
1975 Cohort
2.0
1985 Cohort
1.4
1.2
1.0
Children per woman
Children per woman
1.6
1.9
1.8
0.8
fertility ageing
effect
0.6
1.7
0.4
0.2
1.6
0.0
2000
2010
2020
Year
2030
1955
1965
1975
1985
Cohort
stops
observed
continues
Note: In graph (b) the postponement stops and the postponement continues scenarios will both imply the same levels of childlessness, so only postponement stops is shown.
43
National Statistics
Population Trends 109
Autumn 2002
continues the ‘fertility ageing effect’. The effect occurs in the KO
model because the higher birth order occurrence exposure rates are
shifting to older ages at a slower pace than lower birth order rates. Thus
the movement of first and second order births to older ages will tend to
move the population at risk of higher order births to a point further
along the higher order O-E rate curve and consequently a lower fertility
rate. As Van Imhoff3 points out, there undoubtedly is a real biological
fertility ageing effect in that the probabilities of conceiving fall with
age. This is could also lead to an artefactual ageing effect as women are
not deliberately postponing childbearing, rather simply taking longer to
conceive at older ages. What is noticeable with the postponement
continues scenario is that cumulative fertility for the 1985 cohort shown
in Figure 5(c) is below that implied by applying the current observed
fertility until about 2030. This implies continuing low levels of period
fertility. Cooper23 has previously demonstrated with some simple
numerical examples that period and cohort fertility may diverge for
long periods of time, with the cause being an increase in the age of
mothers. Although as she points out “. . an older age distribution is also
compatible with a fall in completed family size.”
It is extreme to assume that postponement continues for the childbearing length of the 1985 cohort. It would imply that differential rates of
postponement would persist at different parities for many years.
KO, although an improvement on the BF and KP approaches, suffers
from a number of drawbacks. The O-E rates are each treated independently at each parity and the assumption that there is no relation between
a woman’s timing in having a birth and the timing of her next birth is
clearly not realistic. The method also suffers from the use of the current
period O-E rate schedules. As with the BF methods this only works if
the period data is a good representation of the shape of cohort data.
Further the method assumes that somehow the recent changes in the
period shape provide all the information required when clearly the past
history of each cohort will mean that period effects will impact
different cohorts in different ways.
Figure 6
England and Wales
Thinner lines indicate
projected data from
the 2000-based
national population
projections
See footnote
in Fig 2
METHODOLOGY OF CURRENT POPULATION PROJECTIONS
Projections of future fertility were carried out in the latest set of
national population projections by examining the latest achieved
completed family size of cohorts and making a plausible assumption of
the future trends in completed family size. Trends in individual age and
parity specific O-E rates were projected a few years forward to assist in
this process and to help produce plausible and age specific rates. Figure
6 shows by age and cohort the actual and projected cumulative fertility.
The process exhibits some slight further postponement in fertility at
younger ages with some recuperation of fertility at older ages. Coincidentally the resulting projected complete family size is close to the
level produced by the KO ‘postponement stops’ scenario. This is likely
to be because both methods assume that the recent shape of period
fertility is broadly applicable to cohort fertility and the trends in the age
specific O-E rates are broadly in line with the shifting of the age profile
adjusted for by the KP method. However, this should not be taken as a
validation of the KO method. It could have been that examination of
individual age and parity trends led to a different conclusion; rather it
demonstrates the relative stability of England and Wales fertility
patterns and recent changes to those patterns.
Key findings
• When the level of fertility of cohorts and timing of
births is changing, the total fertility rate may a
misleading indicator of the ultimate number of
children women will have.
• The various methods of adjustment presented
suggest that the TFR has been underestimating the
level of period ‘quantum’ fertility since the early
1970s.
Actual and projected cumulative cohort
fertility by age, 1925 to 1986 cohorts
2,500
Van Imhoff3 concluded that empirical analysis of cohort fertility was
the only way to see whether adjusted period measures provide a good
indication of cohort quantum. Such analysis is carried out for England
and Wales when making the national population projections.
• For past data, the adjustment methods presented
would not have provided reliable indicators of
cohort fertility from period fertility for England and
Wales. They also would not have provided a good
indication of fertility trends.
Births per thousand women
2,000
Age
45
40
35
1,500
• The Kohler Ortega model demonstrates that with
continued fertility postponement TFRs could
theoretically remain below the completed family
size for many years.
CONCLUSIONS
30
1,000
25
500
20
0
1925
1935
1945
1955
Cohort
National Statistics
1965
44
1975
1985
The concept of period quantum is conceptually imprecise and so its
estimation is unclear. The relationship between current period fertility
and cohort quantum cannot be known until all current child bearing
cohorts have completed their childbearing. Period adjustment measures
such as those proposed by Bongaarts and Feeney, Kohler and Philipov,
and Kohler and Ortega take a synthetic measure and attempt to adjust it
based only on changes in the current period. These adjustments do not
provide a measure of underlying cohort quantum except in very
particular circumstances. Interpreting the results they provide is,
therefore, difficult both in terms of level and trend. Statistics produced
by these adjustment methods are certainly not candidates for producing
Population Trends 109
as regular statistics alongside traditional fertility measures. Rather they
are research tools that may be helpful to those analysing trends in
fertility data. The KO approach, as it uses O-E rates, is potentially the
best method but still suffers from the fact that it is a period adjustment.
The conclusion that Ryder came to in 1964 still holds “No cohort
parameter can be computed accurately until that cohort has been
studied.”
For England and Wales, all of the methods of adjustment to period
fertility rates produced results in the same direction and mostly of a
similar level. None of the adjustments comes close to providing an
approximation of the underlying cohort quantum as measured by the
CFS over the entire period for which comparable figures are available.
In particular they would only have been slightly better at estimating the
level of cohort quantum fertility in the 1960s than the TFR and would
have been as bad, if not worse, at determining the trend. The KO model
demonstrates that with continued postponement England and Wales
TFRs could theoretically remain below the completed family size for
many years.
ACKNOWLEDGEMENT
The author wishes to thank Mike Murphy of the London School of
Economics for assistance with the software required to calculate the KO
measures.
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