1

University of Copenhagen

Institute of Public Health

Autumn 2003

Modelling regional variation of

first-time births in Denmark

1980-1994

Individuelt studieforløb

Afleveret 1. oktober 2003

Lau Caspar Thygesen

Vejleder: Professor Niels Keiding

2

1. Introduction

The fertility pattern in Denmark has changed dramatically through the last part of

the 20th

century. The total fertility rate (TFR) was relatively high after the Second

World War (TFR = 2.5*) while it decreased dramatically at the end of the 1960s

and less dramatically through the 1970s until 1983 where the absolute minimum

was reached (TFR = 1.377). From that point on the TFR increased to 1.8 in the

start of the 1990s where after it has remained stable1.

This development conceals different trends for specific age groups through the

last 20 years. The age specific fertility rates for the younger age groups (15-19

and 20-24 years) have been decreasing since 1982, the fertility rate for the 25-29

years have increased until 1994 after which it has been decreasing and the fertility

rates for the oldest age groups (30-34, 35-39 and 40-44 years) have been

increasing since 19821;2

. These patterns could indicate that the women give birth

to almost same number of children but that they are older when giving birth.

Denmark is divided into 14 counties and 2 municipalities that are considered

counties (the municipalities of Copenhagen and Frederiksberg). The differences

of the total fertility rates between these 16 counties are rather large in the sense

that the fertility rates of Copenhagen and Frederiksberg municipalities are small

compared with the counties in Jutland (particularly Viborg County, Ringkøbing

County, South Jutland County and Ribe County). These differences conceal large

differences between the age specific fertility rates, whereby Copenhagen and

Frederiksberg municipalities have low fertility rates for the younger age groups

(20-29 years) compared with the other counties and higher for the older age

groups (30-39 years)3. This pattern also applies to a lesser account for Aarhus

County. For some of the Jutlandic counties (e.g. Viborg, Ringkøbing, South

Jutland and Ribe counties) the fertility rate is high for the younger (20-29 years)

and low for the older age groups (30-39 years) compared with the average

measure for Denmark. At the same time all counties in the period 1982-2002 have

experienced a decrease in the age groups younger than 24 years of age, stagnation

for the women 25-29 years of age and an increase for the older age groups.

This pattern indicates that the development where the women give birth to

children at still older ages first has become pronounced in Copenhagen,

* The total fertility rate could be interpreted as the average number of children a woman would

bear if she survived through the reproductive age span and experienced at each age the particular

set of age-specific fertility rates for that period (Preston, page 95).

3

Frederiksberg and to a lesser account Aarhus, whereas some of the counties in

Jutland may come to experience the same development.

On this background the objective of this investigation is to analyse the fertility of

childless women both for Denmark in general but also by analysing the regional

variation of first-time births. By analysing nulliparous women it is possible to

describe one dimension of the phenomenon where women are still older when

giving birth.

The standard method of analysing fertility rates by demographers is to tabulate

key summary measures (e.g. total fertility rate) by age, period and/or cohort

according to the whole population or by some well-defined subgroup (e.g. by

parity or marital status). This decomposition of the fertility into subgroups is then

used to show how subgroups of the female population influence the fertility

pattern and then aggregate these figures into some measure of fertility4. These

solutions will not be of primary interest in this paper where the aim is to show an

alternative way of analysing fertility.

The strategy in this paper is to analyse the fertility of childless women by a

statistical model called the age-period-cohort model. This model simultaneous

analyse the effect of age, calendar year and birth cohort. When several time-

related factors in complex combinations are in play, it becomes difficult to discern

clear patterns in the temporal variation in fertility rates. This could be solved by

using statistical modelling techniques to separate the effect of age, period (secular

influences) and cohort (generational factors). By analysing a complex

phenomenon like fertility with the use of statistical models it becomes possible to

give a more concise description of the phenomenon by pointing out the effects

that have the most pronounced effect on the fertility rates.

But some drawbacks to model fitting has to be considered where the parametric

specification of the model is often of great importance. In the models presented

below the parametric specification will be of central interest because the selection

of parameters in the age-period-cohort model often has an arbitrary element

included.

On this background the aim of this study is to analyse the fertility pattern of

childless women in respect to three time scales (age, period and birth cohort) and

to analyse the regional variation of fertility between the 16 counties of Denmark.

Both investigations will apply the age-period-cohort model to simultaneous model

the three time-related factors.

4

2. The model

2.1. Effect of age, period and cohort

In this study the fertility rate of nulliparous women will be analysed according to

three time-scales: The age of the woman giving birth, the period of giving birth

and the birth cohort of the woman giving birth. These effects will be termed age,

period and cohort, respectively. The effect of the 16 counties will be introduced in

the end of this section.

Age refers to the age of the woman giving birth in full years. I have data for the

age groups from 13-49 years (A=37) indexed a (a=13,…,49). Period refers to the

year the women are giving birth to the first child. I have data for the period 1980-

1994; both years included (P=15) and indexed p (p=1980,…,1994). Cohort refers

to the year the women are born. A cohort is defined as the aggregate of all units

that experience a particular demographic event during a specific time interval in

this investigation being born at the same time4. In this paper a cohort is defined by

C (C=A+P-1=51) and the relation is that cohort = period – age. The oldest cohort

is women who are 49 years of age in 1980 who therefore are from the 1931 birth

cohort. The youngest cohort is women who are 13 years in 1994 (the 1981-

cohort). Of course I cannot know whether a woman giving birth to her first child

in 1985 at the age of 25 is from the 1960 or 1959 cohort, because I have no

information about if she has had birthday in 1985 when giving birth. This problem

could not be solved in the present material and I therefore used the above

equation.

In the paper I will analyse the fertility rate of nulliparous women. The fertility rate

is estimated as the ratio of a count (number of births) to the ‘risk’ time (the

number of women under risk of giving birth to her first child). One class of

models are appropriate for analysing this kind of data: Generalised linear models.

2.2. Generalized linear models

With this broad class of models it is possible to analyse different types of

response variables. The class includes ordinary regression and analysis of

variance for continuous response variables as well as different models for

categorical response variables. All generalized linear models (subsequently called

GLM) are defined by three components: The random component identifies the

response variable, Y, and attaches a probability distribution for it. The systematic

component specifies the explanatory variables used as predictors in the model.

5

And finally the link describes the functional relationship between the systematic

component and the expected value of the random component. The GLM relates a

function of that value to the systematic component through a prediction equation

having a linear form. I find it necessary to briefly introduce the three components

to develop a satisfactory statistical model for the fertility data in this investigation.

2.2.1. The random component

For a sample size N, denote the observations on the response variable by Y1, …,

YN. At this point an assumption is stated to treat Y1, …, YN as independent

observations. The random component of a GLM consists of identifying the

response variable and selecting a probability distribution for Y1, …, YN.

In the scenario of this investigation the response variable is a count (number of

first births in a given year and at a given age), which is a non-negative value.

In the analysis of non-negative count observations a Poisson distribution for the

random component could be assumed. See appendix I for an argument for using

the Poisson distribution in the analysis of survival data, where we model the

intensity of first birth†. A random variable Y is said to have a Poisson distribution

with parameter µ if it takes integer y = 0, 1, 2, … with probability

!)(

y

eyYP

yµµ ⋅==

−

for µ > 0. The mean and variance of the Poisson distribution can be shown to be

E(Y) = var(Y) = µ

Since the mean is equal to the variance it follows that any factor that affects one

will also affect the other. If the actual variance of the observed number of births is

larger than expected under the Poisson assumption, the model is said to exhibit

over-dispersion. This is not uncommon when counts are large, because the

heterogeneity of the population will be pronounced when the data quantity is

large. The consequence of over-dispersion is that the parameters of the model

estimated will have standard errors that are too small. Therefore significance tests

will have a tendency to be significant. Several methods have been developed to

take account of this problem5, but when analysing counts that are as large as in

this investigation (counts from a whole population) the standard errors will no

matter what be so small that tests of significance will almost always be significant

6

no matter if these methods for adjustment of the standard errors are used or not.

Therefore the simple significance tests of the single parameter estimates will not

be used in this investigation but instead two other methods checking the adequacy

of the model will be used. These will be presented below in part 2.3.

2.2.2. The systematic component

The systematic component of a GLM specifies the explanatory variables. These

enter linearly as predictors on the right side of the model equation. The systematic

component specifies the variables that play the roles of xi in the formula

α + β1x1 + … + βkxk,

where i = 1, …, k and α and βi are the regression coefficients of the systematic

component. This linear combination of the explanatory variables is called the

linear predictor. Some of xi may be included in the linear predictor in other ways,

e.g. x3 = x1x2 to allow for interaction between x1 and x2.

2.2.3. The link

The link between the random and systematic component of the GLM specifies

how µ relates to the explanatory variables in the linear predictor. The model

formula states

g(µ) = α + β1x1 + … + βkxk,

where g(.) is called the link function. I could suppose a simple linear model but

this has the disadvantage that a linear predictor on the right side can assume any

real value, whereas the Poisson mean on the left side, which represents a count,

has to be non-negative. A straightforward solution to this problem is to model

instead the logarithm of the mean using a linear model. Thus, I consider a

generalized linear model with a link log: g(µ) = log(µ). A GLM using the log link

is called a log-linear model with the form

log(µ) = α + β1x1 + … + βkxk. (2.1)

2.3. Generalized linear models for rate data: Poisson regression

As has been indicated above the Poisson assumption could be used in the analysis

of count data. The data investigated in this paper are not count data but rather rate

† The analysis in this investigation could be interpreted as a survival analysis because we are

interested in analysing the time to giving birth for nulliparous women. Time could be expressed as

a function of age, period or cohort.

7

data, because when the events investigated occur over time it is reasonable to take

into account the time under risk. The formula for the rate r is

r = µ / t,

where µ is the count and t is the time under investigation called person-years

under risk (subsequently p-yrs). A log-linear model for rate data can be modelled

log(µ / t) = α + β1x1 + … + βkxk, (2.2)

which has equivalent representation

log(µ) – log(t) = α + β1x1 + … + βkxk. (2.3)

The adjustment term, - log(t), to the log link of the mean is called an offset.

Standard GLM software (e.g. SAS proc genmod) can fit models having offsets.

For model (2.3) the expected number of outcomes (e.g. births) satisfies

µ = t exp(α + β1x1 + … + βkxk). (2.4)

This means that the mean of µ is proportional to the index t, with proportionality

depending (constantly) on the value of the explanatory variables. If the values of

x1, …, xk is fixed, doubling the population size (doubling the p-yrs) also doubles

the expected number of outcomes. This element is also called the assumption of

piecewise constant intensity, which means that when x1, …, xk is fixed the count is

also constant. This assumption is fundamental for Poisson regression and should

be carefully considered in a concrete analysis. Normally when the intervals of the

variables included in the model are small it is assumed that the assumption is

valid. As will be obvious below I have data for age in whole years and for each

year in the period 1980-1994. Intervals of this size are normally assumed to be

small. Though the assumption cannot be tested in this investigation it seems

plausible.

The model (2.4) can be rewritten in the following way

µ = t ⋅ eα ⋅ eβ1X1 ⋅ ... ⋅ eβkXk, (2.5)

which is the reason why this model is also called a multiplicative model. In this

case e.g. eβkXk represents the relative risk of disease for exposure at level k relative

to a baseline at level 1 (eβ1X1 = 1). It should be emphasized that this specification

is not necessary the correct one and should be evaluated carefully in the course of

model selection.

8

When fitting data of enormous sample size like this study of the fertility pattern

for all fertile women in Denmark for 15 years almost any significance tests of

specific parameters or of whole models will be highly significant, because the

large sample provides small standard errors. But a statistically significantly effect

need not be important in a practical sense. With huge samples, it is crucial to

focus on estimation rather than hypothesis testing, because simpler models are

easier to summarize. Two methods will in this investigation be used to assess the

adequacy of the models estimated. The primary method will be to compare the

estimated fertility rates from the model with the observed fertility rates from the

dataset. This method will give an impression of the models predictive abilities.

The second and less important method will be to compare the deviance of the

model with the degrees of freedom. If these two numbers approach each other the

fit of the model is assumed to be satisfactory6. The deviance is also used to

compare two different but nested models, because the difference between the two

models’ deviances is an approximately chi-squared statistic with degrees of

freedom equal to the number of additional non-redundant parameters that are in

the largest model but not in the smallest and nested model.

2.4. Poisson regression with effects of age, period and cohort

The multiplicative model just described could include many types of explanatory

variable. In this investigation four explanatory variables will be included: The

effect of age, period, cohort and county.

In the first model without the inclusion of the county effect the expected count (µ)

for a given age (a), period (p) and cohort (c) are obtained by the following

formula

µ = t ⋅ exp(α + βaxa + βpxp + βcxc), (2.6)

where βa, βp and βc are the regression coefficients of the age group, period and

cohort, respectively and α is the intercept (when βa=βp=βc=0). This model is

equal to

µ / t = eα ⋅ eβaXa ⋅ eβpXp ⋅ eβcXc. (2.7)

In this model the antilogs of the effects βa, βp and βc are to be interpreted as the

adjusted relative rate ratios with respect to the reference categories for a, p and c.

If the constant parameter eα is multiplied by e

βaXa the age specific rate is

calculated for the reference period and the reference cohort.

9

There is a simple linear relationship between a, p and c

p = c + a, (2.8)

which induces a linear constraint between the three factors. This is a fundamental

problem in interpreting parameter estimates from the full age-period-cohort

model, because when there is this constraint there is no unique solution7;8

. In its

most general form the model is overparameterized since the mathematical relation

between age, period and cohort allows the same model to be written in infinitely

many ways, which leads to a problem of identification. The problem is that the

model has more parameters than may be estimated from the data.

One solution is to find a parameterization which has eβaXa representing fitted age

specific rates by choosing a reference period and reference cohort setting βp=1

and βc=1. This would leave as unknown A age parameters, (P-1) period

parameters and (C-1) cohort parameters. The total number of parameters would

then be A+P+C-2. Unfortunately this does not solve the problem of identification

because of the problem of drift7;8

. Drift is a variation of the rates, which does not

distinguish between period and cohort influences. The term could be understood

as a continuous parameter with the same change in the log-rate on the whole

scale.

In an age log-linear drift-model the number of parameters to estimate are A + 1

(one drift parameter). If an age-period model (P-2) extra parameters expressing

irregular period effects are added and for an age-cohort model (C-2) parameters

are added to the regular age-drift model.

Therefore the whole age-period-cohort model includes three components in

addition to the age parameters: The drift component, a non-drift period

component and a non-drift cohort component. That is 1+(P-2)+(C-2) parameters

in addition to the age parameters (A) (sums up to A+P+C-3). When I estimate by

model (2.7) I try to estimate A+(P-1)+(C-1) estimates which leaves us with

A+P+C-2 parameters. In other words the model includes one parameter for the

period drift and one parameter for the cohort drift. But that is not possible because

it is not possible from the data to distinguish between these two effects7;8

.

One obvious solution to solve this problem is to infer an extra constraint. Because

age is an important determining factor for the fertility of nulliparous women I

infer one constraint on the period or cohort variable and two on the other. This

means that the model could be estimated, but the problem with this solution is that

the two constraints on the period or cohort variable are arbitrarily selected.

10

Therefore another solution to the identification problem has been suggested,

where the ratios of two adjacent relative risks are contrasted8. The method derives

from a consideration of what defines non-drift effects. If this method for a period

effect is used the relative risk of period 3 versus period 2 could be contrasted with

the relative risk of period 2 versus period 1:

12

23

/

/ββ

ββ

ee

ee.

This estimate can be interpreted as a measure of acceleration of the period trend

during the time around period 2. If the ratio is higher than 1 it tells that the period

effect will be higher from period 2 to period 3 compared with the change from

period 1 to period 2. On the logarithmic scale the following relation holds:

β3 - β2 – (β2 - β1) = β3 - 2β2 + β1

This is a measure of curvature, where a negative value indicates a concave

relationship, a positive value indicates a convex relationship and a value of zero is

a straight line. Therefore the method can illustrate the development in small

intervals.

In this investigation I will use both solutions to get the best description of the

data.

The effect of the 16 counties will be modelled within the same framework. One

obvious solution is to introduce the county effect into (2.7):

µ / t = eα ⋅ eβaXa ⋅ eβpXp ⋅ eβcXc ⋅ eβcountyXcounty. (2.9)

This model has the same problem of identification as described above, but the

interpretation of the county estimates could be as the relative risk (antilog of the

estimates) for the reference categories of the age, period and cohort effects. The

analysis will show the interpretation of the model.

After this presentation of the APC-model, the data for this investigation will be

introduced and described.

11

3. Materials

The materials in this investigation originate from The Fertility Database (the

formal name is the Statistical Register of Fertility Research). The register contains

information on women and men of the fertile age resident in Denmark on the 1st

of January of the calendar year in question. It also gives information on the

children of these women and men. A range of information, which is collected

annually, covers the education and employment situation, family and housing

conditions, etc. of all adults. The content of the register therefore gives

opportunity for analysing men and women’s fertility and to analyse parental and

familial relationships (mother-child, father-child, parents of the same child)9.

In this investigation I will analyse the fertility rate of women’s first child with the

respect to age, year and county. I therefore have information of age and year in 1-

year groups and county in 16 groups (see table 3.1). With this information it is

possible to approximate the cohorts (cohort = year-age).

Table 3.1: Counties and municipalities of Denmark

1. Copenhagen Municipality 9. Funen County

2. Frederiksberg Municipality 10. South Jutland County

3. Copenhagen County 11. Ribe County

4. Frederiksborg County 12. Vejle County

5. Roskilde County 13. Ringkøbing County

6. Western Zealand County 14. Aarhus County

7. Storstrøm County 15. Viborg County

8. Bornholm County 16. North Jutland County

I have data about the 15 years from 1980 until 1994 and for the 36 age groups

from 13 until 48 years of age. The Fertility Database covers the whole fertile age

range (13-49 years of age), but because no nulliparous women in this period gave

birth at the age of 49 I excluded this age group. The absolute numbers of childless

women giving birth for all counties are reproduced in appendix II.

12

Figure 3.1. The data in a Lexis diagram

In figure 3.1 the data is illustrated in a Lexis diagram4. The figure clearly

illustrates that the oldest cohorts in this investigation are the oldest women early

in the period and that the youngest cohorts are the youngest at the end of the

period. The figure also shows that the youngest and oldest cohorts only consist of

a few observations that will result in more uncertain statistical inference about

these cohorts.

To calculate the fertility rate for the first child in the 15 years for the 36 age

groups it is also necessary to know how many women in the particular age and

period groups that were under risk. That is the number of women who have not

given birth to any children, because a woman can only have her first child once.

This number is approximated by the number of childless women 1st of January the

same year added to the number of childless women the 1st of January the

following year divided by 2. In the last year (1994) the number at risk is

approximated by the number of childless women 1st of January in 1994. The

number of childless women for each calendar year and age is reproduced in

appendix III, while the age and period-specific fertility rates could be found in

appendix IV.

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4. Descriptive presentation of data

In this part I will describe the fertility rates for different age groups, periods,

cohorts and counties. This part will only be a description of the data, while the

models already presented in part 2 will be applied in part 5 and 6.

The association between age and the fertility rate of getting the first child is

presented in Figure 4.1.

Figure 4.1. Observed rates of age

0,00

0,02

0,04

0,06

0,08

0,10

0,12

13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47

Figure 4.1. shows a strong association between age and the rate of getting the first

child. The rate increases sharply from age 21 and until 27 where after it decreases

sharply.

Figure 4.2. Observed rates by period

0,03

0,04

0,04

0,04

0,04

0,04

0,05

0,05

1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994

14

Figure 4.2. shows that the rate of first child fell until 1983 where after it again

began a modest increase through the period.

Figure 4.3. Observed age-specific rates

0,00

0,02

0,04

0,06

0,08

0,10

0,12

0,14

1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994

20 år 25 år 30 år 35 år 40 år

Figure 4.3. demonstrates that the pattern shown in figure 4.1 and figure 4.2 hides

a very heterogeneous development for the different age groups. The fertility rate

of first child of the younger age groups shows an uniform decrease through the

period, while the older age groups show an increase through the same period.

This pattern could also be found in figure 4.4 where the effect of age on the

fertility rate of first child is shown for the 15 periods.

15

Figure 4.4. Observed age-specific fertility rates for each

year 1980-1994

0,00

0,02

0,04

0,06

0,08

0,10

0,12

0,14

13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47

1980 1981

1982 1983

1984 1985

1986 1987

1988 1989

1990 1991

1992 1993

1994

In figure 4.4. it is possible to see the change in age distribution for each year

through the period. The decrease of fertility rate for the younger age groups is

most pronounced early in the period while the increase for the older age groups is

most pronounced later in the period. This pattern could be interpreted to say that

the women who did not gave birth to the first child early in the period waited until

some later point.

Figure 4.5. Observed rates of counties

0,035

0,040

0,045

0,050

Cop

enha

gen

M.

Frede

riksb

erg

M.

Cop

enha

gen

C.

Frede

riksb

org

C.

Ros

kilde

C.

C. W

este

rn Z

ealand

C. B

ornh

olm

Funen

C.

C. S

outh

ern

Jutla

nd

Ribe

C.

Vejle C

.

Vibor

g C.

C. N

orth

ern

Jutla

nd

16

Figure 4.5 illustrates the fertility rate of first child for women for the 16 counties.

The figure shows that the fertility rate is lowest for Copenhagen, Frederiksborg

and Roskilde counties and highest in Ribe County and Copenhagen Municipality.

This result indicates that the different age distributions in the counties may

influence the results because it was expected that the fertility rate would be lowest

in the more urban areas of Denmark and higher in the rural areas of Zealand and

Jutland. This dimension will be further analysed in part 6.

17

5. Modelling

After this preliminary presentation of the data I will now construct a model for the

effect of age, period and cohort. The inclusion of the regional variation will be

included in part 6.

As has been described earlier in this paper the fertility rate has changed

dramatically through the last decades. Both the age specific fertility rates and the

period-specific rates have been changing. But what is still fundamental is that age

is the main explanatory effect of fertility. Therefore it is obvious to introduce it as

the first factor. See model 1.

log(µ) = log(t) + α + βagexage (model 1)

This model consists of the offset value (log(t)), an effect for the reference age

group (�) and the effect of age (�age). This model therefore implies absence of

temporal change in the age specific fertility rates. This model is not of primary

interest and will only be used as the reference model for the models introduced

below.

Next, the period effect is added to model 1 as a continuous variable to investigate

if the different age specific curves show a common constant linear slope or drift

over time. See model 2.

log(µ) = log(t) + α + βagexage + δp(p-p0) (model 2)

This model states that there has been a linear change of the logarithmically

transformed fertility rate when taking account of the effect of age. As seen in the

last part this model is probably not a good description of the data because the

effect of period is different for the specific age groups. See figure 5.1.

18

Figure 5.1. Age-drift model

0,00

0,02

0,04

0,06

0,08

0,10

0,12

0,14

1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994

25 y-model 25 y-obs 30 y-model

30 y-obs 35 y-model 35 y-obs

Figure 5.1 illustrates that the heterogeneous development for the different age

groups cannot be captured by the age-drift model. For the 25 years old women the

model is a satisfactory description but for the older age groups with an increasing

fertility rate throughout this period the model is not a good description.

As a note it should be emphasized that the drift parameter in the age-(period)drift

model is equal to the drift parameter in the age-(cohort)drift model7. The age-

(cohort)drift model would give the same linear relationship as model 2 – a

negative parameter estimate.

If the age groups express different linear relationships it may imply that there is a

non-linear effect of period. One simple way to express this relationship would be

an age period model where the period parameter would imply a non-linear

relationship. The age specific log-fertility curves plotted against period should be

parallel (but not linear), if the AP-model would fit well. The model can be

formulated the following way. See model 3a.

log(µ) = log(t) + α + βage xage + βperiod xperiod (model 3a)

As has already been illuminated in Figure 4.3 the age specific curves were not

parallel but rather showed a decreasing trend for the younger women and an

increasing trend for women above 30 years of age. This different development

should not be well fitted with a AP-model, which is confirmed in figure 5.2 and

figure 5.3.

19

Figure 5.2. AP-model

0,00

0,02

0,04

0,06

0,08

0,10

0,12

0,14

1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994

20 years - model

20 years - obs

25 years - model

25 years - obs

Figure 5.3. AP - model

0,00

0,02

0,04

0,06

0,08

0,10

0,12

0,14

1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994

30 years - model

30 years - obs

35 years - model

35 years - obs

As can be seen the AP-model predicts that the development for every age group is

a decrease until 1983 and thereafter a stagnation of the log-fertility rates. This

prediction does not fit well with the observed data for all age groups. For the

youngest age groups the model does not capture the decreasing trend through the

period and for the older age groups the model gives a completely erroneous

description of the data.

Instead of modelling the age-period model an alternative model could be the age-

cohort model that would model the age and non-linear cohort effects. See model

3b.

log(µ) = log(t) + α + βage xage + βcohort xcohort (model 3b)

The fit of this model is shown in figure 5.4.

20

Figure 5.4. AC-model

0,00

0,02

0,04

0,06

0,08

0,10

0,12

0,14

1980 1982 1984 1986 1988 1990 1992 1994

25 y - model

25 y - obs

30 y - model

30 y - obs

35 y - model

35 y - obs

At first the cohort effect may seem to be relatively less important than the age and

period effects, because one would expect certain period and age effect, while it

may be more difficult to posit cohort effects, particularly differences between

adjacent or nearly adjacent cohorts. On the contrary figure 5.4 shows that model

3b captures the development for the oldest age group (35 years of age) accurately

and more or less captures the trends for the two other age groups. Although the

overall trend is fine there is some divergences between the observed rates and the

model prediction.

This underscores that the simple age-period and age-cohort models are not able to

describe the heterogeneous fertility development in Denmark in the period 1980-

1994. A further extension of the model is necessary to capture this development.

The method used in this investigation is the introduction of both the period and

cohort variables at the same time.

What the cohort effect could introduce to the age-period model is different age

specific period fertility rates. With this introduction the oldest cohorts will have a

higher fertility rate for younger women, while the youngest cohorts will have a

higher fertility rate for older women. This interaction between age and period

could possibly be described by the cohort effect.

Therefore I estimate the full age-period-cohort model. See model 4a.

log(µ) = log(t) + α + βage xage + βperiod xperiod + βcohort xcohort (model 4a)

The age-period-cohort model allows for non-parallel age specific mortality curves

as a function of cohort or period. But the model has the problem of identification

as described in part 2.

21

This has been solved in two ways in this investigation. The first solution is to

make an additional constraint on the cohort parameter, so it has been set to zero

for cohort 1945 and 1967. By introducing one constraint on period and two on the

cohort effect the problem of identification is eliminated. The observed and

expected plots for this model can be found in Figure 5.5 and Figure 5.6.

Figure 5.5. Age-period-cohort model

0,00

0,02

0,04

0,06

0,08

0,10

0,12

0,14

1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994

20 years - model

20 years - obs

25 years - model

25 years - obs

Figure 5.6. Age-period-cohort model

0

0,02

0,04

0,06

0,08

0,1

0,12

0,14

1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994

30 years - model

30 years - obs

35 years - model

35 years - obs

This model seems to fit the data much better‡. The model predicts that the slope of

the different age groups is not equal so that it is decreasing for the younger groups

‡ When estimating model 4a using the SAS-procedure proc genmod it should be noted that the

convergence is questionable, but that this problem was solved by excluding the four oldest cohort

in 1980 (equivalent with the three oldest in 1981, the two oldest in 1982 and the oldest in 1983)

and the youngest cohort in 1994. This convergence problem was caused by no observations in

these 11 cells. Despite the problem of convergence the estimates in the model with all

observations were no different than the dataset with the five cohorts excluded.

22

and increasing for the older. The minor deviations between the model and the

observed rates for the four age groups presented in the two figures could not

easily be explained. The cohort effect is illustrated in Figure 5.7 as the relative

risk compared with the two reference cohorts: 1945 and 1967.

Figure 5.7. Cohort effect in age-period-cohort model

0,0

0,5

1,0

1,5

2,0

2,5

1932 1936 1940 1944 1948 1952 1956 1960 1964 1968 1972 1976 1980

Re

lati

ve

ris

k (

ref:

19

45

an

d 1

96

7)

For the four oldest cohorts and for the youngest cohort the women have

experienced no birth and therefore the standard errors of these parameters are

relatively large. Therefore the confidence intervals have not been included in the

figure.

The concrete estimates of the cohort effect in Figure 5.7 are not identifiable, while

the shape of the figure could be identified. The shape indicates that the cohorts

between the two reference groups have a higher ‘risk’ of fertility, while the oldest

and youngest cohorts have a lower ‘risk’. The lower risk might be explained by

few first-time births for the oldest cohorts, which could represent that these

cohorts do not give birth to the first child in the age-groups included in the data.

The confidence intervals also indicate that the estimates are very uncertain. For

the youngest cohorts the decreased risk could represent that these cohorts

postpone their first birth. The cohorts after 1967 are younger than 27 years of age

at the time of this investigation and therefore could be influenced by the still

lower fertility for the younger age groups later in the period of observation. This

lowered risk may have changed if I had data of the whole fertile age-span for

these cohorts.

The higher fertility rate for the cohort between 1945 and 1967 expresses that these

cohorts have a higher fertility rate compared with the reference cohorts when we

have taken account of the age and period effects.

23

Figure 5.8. Period effekt in APC-model

0,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

1,6

1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994

Figure 5.8 illustrates the parameters for the period effect expressed as the relative

risk of fertility compared with the fertility rate of 1987. The figure clearly

underscores that the fertility rate fell until 1983 where after it was stable until

1985 and increased for the rest of the period. This is the estimate for the reference

groups for the age and cohort effects which means that it is the fertility rate of the

25-year-old women from one of the two reference cohorts.

To compare the models introduced above a deviance analysis have been carried

through. See Table 5.1.

Table 5.1. Deviance analysis of model 1-4

Nr. Model (1)

Devian.

(D)

(2)

df

(3)

comp.

model

(4)

∆D

(5)

∆df

(6)

p

(7) mod.

v obs dat

(1)/(2)

0 Null 430895.3 8630 49.9299

1 A 30768.5 8595 0 400126.8 35 <0.001 3.5798

2 A-drift 30257.4 8594 1 83.1 1 <0.001 3.5208

3a AP 29183.0 8581 2 1502.4 13 <0.001 3.4009

3b AC 26386.3 8546 2 4299.1 48 <0.001 3.0876

4a APC #11 23481.5 8533 3a 5701.5 48 <0.001 2.7518

4b APC #22 23481.5 8533 3b 2904.8 13 <0.001 2.7518

1Year is constrained for year = 1987 and cohort is constrained twice for cohort = 1945 and 1967 2Year is constrained twice for year = 1982 and 1992 and cohort is constrained for cohort = 1967

The table shows that the full age-period-cohort models (model 4a and 4b) give the

best fit of the data (column (7)), which was also obvious above in the figures. The

age effect have an immense explanatory effect compared with the null-model and

each inclusion of a variable in the models gives a better fit of the model (column

(4)-(6)).

24

The age-period-cohort model gives the best fit of the above models, but still has

the disadvantage that no unique parameterisation is possible. In fact inevitable

many solutions exit. This was solved by introducing an additional constraint on

the period and cohort effects. This solution is satisfactory if you find constraints

that are justified, but it should be emphasized that the constraint in many

investigations in general and this investigation in particular are chosen arbitrarily.

If you want to avoid this problem another solution has been suggested, where you

contrast the ratio of two adjacent relative risks8.

The great strength of this estimate is that it is unaffected by the parameterisation

of the age, period and cohort effect. See Figure 5.9 and Figure 5.10.

Figure 5.9. Second differences cohort effect

(without cohorts 1932-42 and 1978-81)

0,8

0,9

1,0

1,1

1,2

1942 1945 1948 1951 1954 1957 1960 1963 1966 1969 1972 1975

Figure 5.10. Second differences period effect

0,8

0,9

1,0

1,1

1,2

1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993

Figure 5.9 shows minor deviations from 1 in the cohorts between 1942 and 1978.

The deviations that can be shown have no overall trend and the differences

between single years show no consistent pattern.

25

Figure 5.10 shows that the second order differences for the periods are very small.

The trend is that there is a little acceleration in the fertility rates in the start of the

period and a more heterogeneous development in the end of the period. The

acceleration around 1981 indicates a slowing off of the general decrease in the

start of the 1980’s, while the acceleration around 1983 illustrates the change from

a decreasing trend until 1983 and a modest increase from 1983 to 1984. The

second differences are very close to 1 and should be interpreted cautiously.

To get a more overall interpretation of the second order differences it may be

necessary to smooth out the estimates to get an impression of an overall trend.

This has not been done in this investigation, because it is outside the scope of the

statistical inference applied in this investigation, but it could give a more coherent

and interpretable impression of the secular and generational effects on the fertility

rates.

After this statistical analysis of the general fertility in Denmark I will now analyse

the regional variation.

26

6. Regional variation

Denmark is divided into 14 counties and the municipalities of Copenhagen and

Frederiksberg. The variation between the counties is large when analysing the

total and age specific fertility rates3, but to my knowledge it has not previously

been analysed for nulliparous women. The purpose of this investigations will be

to set up a statistical model that may give estimates for the counties and describe

the differences between the counties in a concise way.

The basis for the analyses will be the model 4a described in the last section with

an additional parameter for the counties. See model 5.

log(µ) = log(t)+α+βagexage+βperiodxperiod+βcohortxcohort+βcounty x county (model 5)

Model 5 has the same problem of identification as described above. I therefore

infer an additional constraint on the cohort-parameter (for the cohorts 1945 and

1967). The parameter estimates for the counties in model 5 are reproduced in

textbox 6.1.

The exponential transformation could be interpreted as the relative risk of birth

for nulliparous women in the counties in relation to Aarhus County (reference

category) when taking account of age, period and cohort. The estimates state that

Aarhus County has the lowest fertility rate of nulliparous women, while

Bornholm County has the highest. The municipalities of Copenhagen and

Frederiksberg have surprisingly high estimates while most of the rural counties

have rather low fertility rates. Overall the pattern of the estimates is rather

surprising.

Textbox 6.1. Parameter estimates for the APC-county-model Standard

Parameter Estimate Error exp(estimate) Bornholm County 0.3257 0.0189 1.38 Frederiksberg Municipality 0.1874 0.0120 1.21 Frederiksborg County 0.1359 0.0080 1.15 Funen County 0.1335 0.0071 1.14 Copenhagen Municipality 0.1903 0.0064 1.21 Copenhagen County 0.0715 0.0067 1.01 North Jutland County 0.1979 0.0070 1.01 Ribe County 0.2433 0.0090 1.28 Ringkøbing County 0.2093 0.0085 1.01 Roskilde County 0.1586 0.0093 1.17 Storstrøm County 0.2022 0.0090 1.01 South Jutland County 0.2437 0.0088 1.01 Vejle County 0.2148 0.0079 1.24 West Zealand County 0.2303 0.0084 1.01 Viborg County 0.2512 0.0092 1.29 Aarhus County 0.0000 0.0000 1.00 (ref)

27

If we compare the fertility rates estimated by the model with the observed fertility

rates it becomes obvious that the model does not fit well. Figure 6.1, 6.2 and 6.3

give an impression of the disparity between the model and the observed rates.

Figure 6.1. Ringkøbing and Funen Counties - APC-

county-model

0,00

0,02

0,04

0,06

0,08

0,10

0,12

0,14

0,16

0,18

1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994

25 y,Ringk-model

25 y,Ringk-obs

25 y,Fyn-model

25 y,Fyn-obs

Figure 6.2. Ribe County - APC-county-model

0,00

0,02

0,04

0,06

0,08

0,10

0,12

0,14

0,16

1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994

Ribe-25 y-model

Ribe-25 y-obs

Ribe-30 y-model

Ribe-30 y-obs

28

Figure 6.3. Copenhagen municipality - APC-county-

model

0,00

0,02

0,04

0,06

0,08

0,10

1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994

25 y-model 25 y-obs 30 y-model

30 y-obs 35 y-model 35 y-obs

As can be seen from the figures the fit of the model is not striking. Although the

deviance has increased markedly compared with model 4a (see row 2 in Table

6.1) and for some of the counties (e.g. Ringkøbing and Funen counties for 25

years old women) the fit is more or less satisfactory, the overall fit of the model is

not good. The estimates of model for Ribe County shows the correct trend, but

have some differences compared with the observed fertility rates and for the

municipality of Copenhagen the model estimates the fertility rate too high for the

young age group (25 year) and generally too low for the old age group (35 year).

This pattern may suggest that the trend in Copenhagen Municipality differs from

the general trend in Denmark. The same pattern is also seen for the municipality

of Frederiksberg (data not shown). This finding could suggest that the analysis

perhaps should differentiate between the municipalities of Copenhagen and

Frederiksberg on the one side and the rest of counties on the other side.

This suggestion has been followed by analysing the fertility rate in two different

dataset. Model 6a, which has the same parameters as model 5, includes all

counties except the municipalities of Copenhagen and Frederiksberg, while model

6b only includes these two municipalities. It is not possible to compare this model

with the previous estimated models because the included data are not

homogeneous, but the model vs. observed data of model 6a and 6b has decreased

markedly (see Table 6.1).

29

Table 6.1. Deviance analysis of model 4-6

Nr. Model (1)

Devian.

(D)

(2)

df

(3)

comp.

model

(4)

∆D

(5)

∆df

(6)

p

(7) mod.

v obs dat

(1)/(2)

4a APC #11 23481.5 8533 3a 5701.5 48 <0.001 2.7518

5 APCC 1 14729.5 8299 4a 8752.0 234 <0.001 1.7749

6a APCC2 9256.3 7249 - - - - 1.2769

6b APCC3 1147.3 958 - - - - 1.1976

1Year is constrained for year = 1987 and cohort is constrained twice for cohort = 1945 and 1967 2Year is constrained for year = 1987 and cohort is constrained twice for cohort = 1945 and 1967.

The dataset is reduced by excluding the municipalities of Copenhagen and Frederiksberg 3Year is constrained for year = 1987 and cohort is constrained twice for cohort = 1945 and 1967.

The dataset is reduced by only including the municipalities of Copenhagen and Frederiksberg

This solution offers a satisfactory description of the data as seen in figure 6.4-6.6.

Figure 6.4. Ringkøbing and Funen Counties - APC-

county-model - reduced dataset

0,00

0,02

0,04

0,06

0,08

0,10

0,12

0,14

0,16

0,18

1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994

25 y,Ringk-model

25 y,Ringk-obs

25 y,Fyn-model

25 y,Fyn-obs

Figure 6.5. Ribe County - APC-county-model -

reduced dataset

0,00

0,02

0,04

0,06

0,08

0,10

0,12

0,14

0,16

1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994

Ribe-25 y-model

Ribe-25 y-obs

Ribe-30 y-model

Ribe-30 y-obs

30

Figure 6.6. Copenhagen municipality - APC-county

model with reduced dataset

0,00

0,02

0,04

0,06

0,08

0,10

1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994

25 y - model 25 y - obs 30 y - model30 y - obs 35 y - model 35 y - obs

The problems seen for model 5 above are no longer as obvious. The fertility rates

for both the municipality of Copenhagen and the other counties are more or less

satisfactorily captured. For Ringkøbing, Funen and Ribe counties the model gives

the right description of the observed rates with some disparities between the

model and the observed rates. For the municipality of Copenhagen model 6b

gives a very satisfactory description with only minor differences between

observed and model rates.

The disadvantage of this solution is that two separate models have to been

estimated. Therefore it is not as simple to report as parameter estimates from a

single model.

As the previous analyses have indicated it seems that the age specific fertility

rates of the different counties diverge with the greatest difference between the

municipalities of Copenhagen and Frederiksberg compared with the rest of the

country. It seems like the fertility rates of the younger age-groups is lower in the

municipalities of Copenhagen and Frederiksberg compared to the rest of the

country, while the rates of the older age-groups is higher compared to the other

counties of Denmark.

This pattern could indicate that an interaction between the specific counties and

age could solve the problems indicated above. See model 7.

log(µ)=log(t)+α+βagexage+βperxper+βcohxcoh+βcounxcoun+βage,counxcounxage (model 7)

Model 7 indicates that the age specific fertility rates will not be the same for

different counties. This might potentially give a good fit. See table 6.2.

31

Table 6.2. Deviance analysis of model 5-7

Nr. Model (1)

Devian.

(D)

(2)

df

(3)

comp.

model

(4)

∆D

(5)

∆df

(6)

p

(7) mod.

v obs dat

(1)/(2)

5 APCC 1 14729.5 8299 - - - - 1.7749

6a APCC2 9256.3 7249 - - - - 1.2769

6b APCC3 1147.3 958 - - - - 1.1976

7 APCC-

int.ac4

7276.6 6609 - - - - 1.1010

1Year is constrained for year = 1987 and cohort is constrained twice for cohort = 1945 and 1967 2Year is constrained for year = 1987 and cohort is constrained twice for cohort = 1945 and 1967.

The dataset is reduced by excluding the municipalities of Copenhagen and Frederiksberg 3Year is constrained for year = 1987 and cohort is constrained twice for cohort = 1945 and 1967.

The dataset is reduced by including only the municipalities of Copenhagen and Frederiksberg 4Year is constrained for year = 1987 and cohort is constrained twice for cohort = 1945 and 1967.

The model includes an interaction between age and county. For the model to converge it was

necessary to exclude observations for women younger than 15 years and older than 44 years

because of few observations in these age groups. This exclusion made it impossible to make

comparison with the other models.

Table 6.2. clearly shows that the model gives a nice description of the data. Figure

6.7 and 6.8 shows the fitted rates and the observed rates for the municipality of

Copenhagen and Ribe county.

Figure 6.7. Copenhagen municipality - APC-

county model with interaction between age and

county

0,00

0,01

0,02

0,03

0,04

0,05

0,06

0,07

0,08

0,09

0,10

1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994

25 y - model 25 y - obs 30 y - model

30 y - obs 35 y - model 35 y - obs

32

Figure 6.8. Ribe County - APC-county model with

age-county interaction

0,00

0,02

0,04

0,06

0,08

0,10

0,12

0,14

0,16

0,18

1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994

25 y - model 25 y - obs 30 y - model

30 y - obs 35 y - model 35 y - obs

It is obvious that the model fits the fertility rates of Copenhagen much better than

model 5 and that the model fits the rates of Ribe County satisfactory. This model

seems to be able to include both model 6a and 6b.

This advanced model seems to capture the rather diverging fertility rates of the

counties strikingly well. The different age-specific fertility rates of the counties

with lower rates for younger and higher rates for older in the municipalities of

Copenhagen and Frederiksberg is captured with the model while the rates of the

more rural counties are captured satisfactory.

The model specifies different age-structures for the individual counties. Figure 6.9

and 6.10 illustrate the age-specific fertility rates for the reference period (1987)

and reference cohorts (1945 and 1967) for Storstrøm County and Copenhagen

Municipality (Figure 6.9) and North Jutland County and Frederiksberg

Municipality (Figure 6.10).

33

Figure 6.9. Age-specific fertility rates - Storstrøm

County and Copenhagen Municipality

0,00

0,02

0,04

0,06

0,08

0,10

0,12

0,14

15 17 19 21 23 25 27 29 31 33 35 37 39 41 43

Storstrøm C -

model

Storstrøm C -

obs

Copenhagen M

- model

Copenhagen M

- obs

Figure 6.10. Age-specific fertility rates - North

Jutland County and Frederiksberg Municipality

0,00

0,02

0,04

0,06

0,08

0,10

0,12

0,14

15 17 19 21 23 25 27 29 31 33 35 37 39 41 43

North Jutlan C -

model

North Jutland C

- obs

Frederiksberg

M - model

Frederiksberg

M - obs

The figures clearly show that the APC-county model with interaction between age

and county gives a very nice description of the observed fertility rates both for

counties with high age-specific rates and for the municipalities of Copenhagen

and Frederiksberg where the fertility rates is lower for younger women and then

approaches the level of the other more rural counties for older women.

This pattern is also illustrated in Figure 6.11 and 6.12 where the age-specific rates

estimated by model 7 are illustrated for the 14 counties and 2 municipalities.

34

Figure 6.11. Age-specific modelrates - Eastern

Denmark

0,00

0,02

0,04

0,06

0,08

0,10

0,12

0,14

0,16

15 17 19 21 23 25 27 29 31 33 35 37 39 41 43

Copenhagen M

Frederiksberg M

Copenhagen C

Frederiksborg C

Roskilde C

West Zealand C

Storstrøm C

Bornholm C

Figure 6.12. Age-specific modelrates - Western

Denmark

0,00

0,02

0,04

0,06

0,08

0,10

0,12

0,14

0,16

15 17 19 21 23 25 27 29 31 33 35 37 39 41 43

Funen C

South Jutland C

Ribe C

Vejle C

Ringkøbing C

Aarhus C

Viborg C

North Jutland C

The figures clearly show that the fertility rates of Copenhagen and Frederiksberg

are lower for the younger age groups, but also that the counties of Aarhus,

Copenhagen and to some extent Funen are lower than the remainder. These three

counties and the two municipalities is illustrated in Figure 6.13.

35

Figure 6.13. Age-specific modelrates - urban

regions of Denmark

0,00

0,02

0,04

0,06

0,08

0,10

0,12

0,14

0,16

15 17 19 21 23 25 27 29 31 33 35 37 39 41 43

Frederiksberg C

Funen C

Copenhagen M

Copenhagen C

Aarhus C

This figure of the age-specific fertility rates of the most urban areas of Denmark

shows that Copenhagen and Frederiksberg municipalities have the lowest fertility

rates but also that the fertility rate of the youngest age-groups in Copenhagen

differ from Frederiksberg. The childless women about the age of 20 have a higher

fertility rate than the other counties but after this rather high fertility rate the rates

is consistently the lowest until the age of 35. The three counties have almost

consistently higher fertility rates than the two municipalities and have the highest

fertility rate at an earlier age than the municipalities. As also seen in Figure 6.12

the fertility rates of Funen County approaches the rates of other more rural

counties. This pattern may illustrate that Funen County is both composed of the

urban area around Odense and the smaller more rural cities on the island.

This examination of the APC-county model with an interaction between age and

county clearly shows that the model gives satisfactory description of the regional

variation in Denmark in the period from 1980 until 1994. The interaction between

age and county offers a rather simple formulation of the complex demographic

phenomenon of different age-structures of fertility in the regions of Denmark.

36

7. Conclusion

This investigation has analysed the fertility of childless women in the period

1980-1994. The investigation has been two-fased by both analysing the fertility

rate for the whole country and by analysing the regional differences between the

14 counties and 2 municipalities in Denmark. The purpose of the investigation

was to illustrate how the age-period-cohort model could describe these

phenomena and to illustrate if this multiple regression technique could give

knowledge on how the three connected time-scales have influenced the fertility

rates.

The analyses have shown that a simple description only introducing age and

period effects gives an imperfect description of the data. The full age-period-

cohort model has the ability to describe the heterogeneous development of

fertility rates for different ages with a decreasing trend through the period for the

youngest age-groups and increasing trend for the older age-groups. The cohort

effect captures that the birth cohorts from the 1950’s and the start of the 1960’s

have a higher fertility rate than younger and older cohorts.

When analysing the regional variation by an APC-model introducing a categorial

effect of the different regions, it becomes obvious that this model gives a

erroneous description of the data. For the municipalities of Copenhagen and

Frederiksberg the model overestimates the effect of younger women and

underestimates the effect of older women. For the other counties the model is

more or less satisfactory. This finding opened an analysis where it was

hypothesized that the different effect of age for the specific counties could be

modelled by introducing an interaction between age and county. This model gave

a very satisfactory description of the data.

Through the investigation I have been using the terms year, period and cohort

effect. It should be emphasized that it is not possible to attribute any causal

interpretation of age, year or cohort. These effects are proxy variables of not

directly observed factors that may be part of the physical and social environment.

The factors could range from very basic and obvious factors such as availability

of contraceptives to more subtle factors such as the connotations of giving birth

for younger (or older) women. The interpretation of the slower changing cohort

effect is not as straightforward in interpretation as the effect of age and period.

The necessary condition for a cohort effect is that the impact of some period

effect has a permanent effect on particular cohorts. It is not adequate to think of

cohort effect as fast occurring as e.g. the period effect of legalisation of

37

contraceptives, but rather as slower changes over a longer time span. Because of

these difficulties of interpretation of the three effects, the use of APC-models

could be used as an analysis in the start of a scientific process used to make an

advanced description of a phenomenon occurring over a time span.

This study has extended our knowledge of first-time fertility in Denmark by

simultaneous analyzing the effect of age, period and cohort. The study has shown

that the APC-model could adequately describe the combined effect of these three

time-scales on the demographic phenomenon fertility and could describe the

regional variation between counties in Denmark.

38

References

1. Statistics Denmark. statistikbanken.dk. www.statistikbanken.dk . 27-7-2003.

2. Statistics Denmark. Statistiske efterretninger 1997:3. Statistics Denmark,

1997.

3. Statistics Denmark. Befolkningens bevægelser 1999. Statistics Denmark,

2000.

4. Preston SH, Heuveline P, Guillot M. Demography: Measuring and

Modelling Population Processes. Blackwell Publishers, 2000.

5. Agresti A. An Introduction to Categorial Analysis. John Wiley & Sons, Inc,

1996.

6. Arbyn M, Van Oyen H, Sartor F, Tibaldi F, Molenberghs G. Description of

the influence of age, period and cohort effects on cervical cancer mortality

by loglinear Poisson models (Belgium, 1955-94). Archives of Public Health

2002;60:73-100.

7. Clayton D,.Schifflers E. Models for temporal variation in cancer rates. I:

Age-period and age-cohort models. Statistics in Medicine 1987;6:449-67.

8. Clayton D,.Schifflers E. Models for temporal variation in cancer rates. II:

Age-period-cohort models. Statistics in Medicine 1987;6:469-81.

9. Statistics Denmark. Fertilitetsdatabasen - Vejledning i udtræk fra

Fertilitetsdatabasen. Statistics Denmark, 1996.

39

Appendix I – the relationship between survival analysis and Poisson regression

Consider the age at first birth as a survival time. The actual survival time of a woman

‘under risk’ of giving birth to her first child, t, can be regarded as the value of a variable

T, which can take any non-negative value. The values of T have a probability distribution

and can be regarded as a random variable associated with the survival time. If the random

variable T has a probability distribution with the underlying probability density function

f(t), the distribution function of T is given by

F(t) = P(T ≤ t) = �t

duuf0

)( , (I.1)

which represents the probability that the survival time is less than t.

The survival function, S(t), is defined as the probability that the survival time is greater

than or equal to t:

S(t) = P(T > t) = 1 – F(t) = �∞

t

duuf )( . (I.2)

Another central function is the hazard function, h(t), which is the probability that the

woman gives birth at time t, conditional on she was not given birth to that time. The

hazard function represents the instantaneous birth rate for a childless woman at time t.

The formal definition of the hazard function is:

h(t) = ���

����

� ≥+<≤

→ t

tTttTtP

t δ

δ

δ

)(lim

0. (I.3)

This implies that the probability that the random variable associated with a woman’s

survival time, T, lies between t and t + δt, conditioned on T being greater than or equal to

t. The hazard function is then the limiting value of this probability divided by the time

interval, δt, as δt tends to 0.

The integrated hazard function, H(t), is defined as follows

H(t) = �t

duuh0

)( . (I.4)

From these definitions it is possible to show that

h(t) = )(

)(

tS

tf. (I.5)

It can also be shown that

S(t) = exp(-H(t)) = exp(- �t

duuh0

)( ). (I.6)

The likelihood, L, of a sample data is the joint probability of the observed data, regarded

as a function of the unknown parameters in the assumed model. The survival time is

defined as X.

If the time of birth is X = ti then:

L = f(ti) = h(ti) S(ti). (I.7)

If the time of birth is X > ti, which happens with right censoring, then

L = S(ti). (I.8)

The combined likelihood could then be expressed the following way:

40

L = h(ti)Di S(ti), (I.9)

where Di = 1 if the woman i has given birth and Di = 0 if the woman i is censored.

L = h(ti)Di exp(- �

ti

duuh0

)( ) = h(ti)Di exp(- �

τ

0

)()( dusYuh i ),

where Yi(s) = I(ti ≥ x), which indicates if the woman i gives birth to a child in the whole

time span from time 0 to time τ. Yi(s) will assume the value 1 if the woman remains

childless through the time span and assume the value 0 if she gives birth to the first child

through the time span.

Consider now the likelihood for the i’th woman, Li, where i is 1, …,n independent

observations. The likelihood for all women, L, is then:

L = ∏=

n

i

iL1

= ∏=

n

i 1

h(ti)Di exp(-E) , (I.10)

where E = � �τ

0

)()(i

i duuYsh , which is the same as expected number of births: h(s) is the

probability of giving birth at the time s and �i

i uY )( expresses the number of women

remaining childless through the time of observation 0 - τ.

If I at this point assume that the hazard function is constant (h(t) = h), the following

likelihood function emerges:

L = hD exp(-h �

τ

0

)( duuY , (I.11)

where D = �=

n

i

iD1

and Y(u) = �=

n

i

i uY1

)( .

L = hD exp(-h T) , (I.12)

where T = �τ

0

)( duuY , which is the same as person-years at risk: Y(s) expresses the

number of childless women at the time s which is integrated over the time span 0 - τ.

log L = D log h - h T , (I.13)

where h T is the expected number of births.

This result is equal to the Poisson regression model. If I assume that T is fixed and D is

Poisson distributed with the expected value h T, the likelihood is

)exp(!

)(hT

D

hTD

− = constant ⋅ hD exp(-hT). (I.14)

This means that the Poisson regression model can be used for survival analysis, but that

the interpretation of the person-years of risk (T) should be cautiously interpreted, because

T is not a fixed but random variable.