Transcript Lecture 2: More on simple population models, population

Basic Demographic Analysis (1):
Population Change, Population
Accounts, Age and Time
Phil Rees
QMSS2 Summer School
Projection Methods for Ethnicity and Immigration Status
2-9 July 2009
School of Geography, University of Leeds, UK
Outline
•
•
•
•
•
•
•
Components of change
Basic measures of change
Concepts of growth
Simple growth models
A population model using rates
Population accounts and models
Age and time
Definitions
(sources: Chambers Dictionary, Rowland and others)
• Population = a group of people, objects of items,
considered statistically
• Analysis = action of analysing = breaking something into
its component parts in order to discover the general
principles governing it
• Demography = the study of human populations (demos =
people in Greek, graphein = to write)
• Population dynamics = the motions/changes in a
population = how a population changes over time (and
space)
• See Rees, Demography, article to be published in the
International Encyclopaedia of Human Geography.
Components of population change (1)
• To study demographic processes we need to
measure the components of population change
• Population change = natural increase + net
migration
– Natural increase = births – deaths
– Net migration = migration inflows – migration outflows
– Another way of saying this is:
• Net migration = arrivals - departures
Examples of change
components
Components of change for
the USA (Rowland 2003,
Figure 1.4)
Components of change for
the UK, Population Trends
122, p8
Components of population change (2)
• We need to adopt a simple mathematical notation to
explore the components further
– P = population count or estimate or projection
• Pt = population at time t
• Pt+n = population at time t+n
• n = length of time interval in years
–
–
–
–
–
–
–
–
B = number of births
D = number of deaths
NI = natural increase
Min = in-migration (from the rest of the country to a region)
Mout = out-migration (from a region to the rest of the country)
I = immigration (from other countries to a country)
E = emigration (from a country to others)
NM = net migration = balance of migration inflows and migration
outflows
– Δ = change
Components of population change (3)
• Population change = natural increase + net migration
Pt+n – Pt = NI + NM
• Population accounting equation
Pt+n = Pt + NI + NM
• Breaking down the components
Pt+n = Pt + (B – D) + (Min – Mout) + (I – E)
– Note the two sets of migration terms
– The internal migration flows cancel when we are
dealing with a national population but are needed
when we are working with sub-national populations
Components of population change (4)
• Sometimes we don’t know what the net
migration flow has been, so we use the
population accounting equation to help
make the estimate
NM = (Pt+n – Pt) – (B – D)
• This is the residual method for estimating
net migration
• Accurate census population counts are
needed at time t and t+n
Basic measures of change (1)
• We need to convert the counts into “demographic intensity
measures”
• These can be of several different kinds (after Rowland
2003, Table 1.2)
– Rate = the ratio of number of demographic events to the population
at risk of experiencing the event
• Sometimes referred to as “occurrence-exposure rates”
– Proportion = a ratio of the numerator to the denominator
– Probability = the ratio of the number of demographic events or
transitions to the initial population at risk of experiencing them
(probabilities sum to 1 over all cases/groups)
– Re-scaled rates or probabilities = proportions and rates are
rescaled depending on purpose and intensity level
• e.g. percentages=proportion  100
• e.g. rates per 1,000 = proportion  1000
– Intensities = collective noun that refers to either rates or
probabilities
– Ratio = the relative size of one number to another
Basic Measures of Change (2):
Populations at Risk
•
•
•
•
We compute intensities of fertility, mortality and
migration by relating the events of birth, death and
migration to the population at risk of the event
The population at risk: ideally this should be all
persons who are in a country, region or local area
weighted by the time they spend there (person-time
measure)
The population at risk should focus on the population
really involved (e.g. women aged 15 to 50 rather than
all people are at risk of giving birth)
Where this is not the case, our basic measures are
called “Crude Rates” (crude = not refined)
Basic Measures of Change (3):
Alternatives for Populations at Risk
• There are several alternative populations
at risk:
– Mid-year population in the time interval, n=1
• Pt+½
– Average population in the time interval, n=1
• ½ (Pt + Pt+1)
– The first is used in the UK, the second in most
European countries
– Few countries attempt to measure person-time
exposure but such measures are important in
longitudinal micro-data studies
Basic Measures of Change (4): Crude Rates
• Crude birth rate
CBR = (B/Pt+½)  1000
• Crude death rate
CDR = (D/Pt+ ½)  1000
• Natural increase rate
NIR = ((B-D)/Pt+ ½ )  1000 = CBR – CDR
• Net migration rate
NMR = (NM/ Pt+ ½ )  1000
• Population change rate
PCR = (Pt+ ½ – Pt)/Pt+ ½  1000
• Net migration rate (as a residual, needs good population estimates)
NMR = PCR – NIR
• The 1000 multiplier is used to scale the numbers between 5 and 60
(the range we observe) rather than .005 and .060
Basic Measures of Change (5): Examples
(Source: Population Trends 124, Table 1.1)
Rate
Russia
2001
EU-25
2002
China
2004
UK
2004
USA
2003
India
1998
CBR
9.0
10.3
7.2
12.1
14.1
26.2
CDR
15.6
9.8
5.3
9.7
8.4
9.0
NIR
-6.6
0.5
1.9
2.4
5.7
17.2
Each country set of numbers represents the outcome of an interesting recent
population history.
However, we need more than just one year to confirm what demographic
regime a country displays. We need a set of years.
By demographic regime, we mean the combination of fertility, mortality and
migration levels that characterize the population.
Concepts of growth (1) (Rowland 2003, Figure 2.1, p47)
Concepts of growth (2): the four kinds
• Arithmetic growth = constant addition of an
absolute amount to the population each year
• Geometric = constant percentage of start
population in interval added to the population
each year
• Exponential growth = constant growth that
compounds continuously over the year
• Logistic growth = the growth rate changes in
response to population growth
Example of a logistic curve (from Wikipedia)
Concepts of growth (3): a caveat
• See Rowland (Chapter 2) for a discussion of the
differences between these four types of growth, which is
useful but in some respects a little misleading
• He seems to be saying that you get higher growth over a
time interval as you move from geometric to exponential
growth. This is only true if you use the same numbers for
the growth rate for the interval. If you calibrate the two
models properly, the exponential growth rate that
achieves the same population change in a time interval
will be lower than the geometric growth rate
• Note that these models apply to any demographic series
(e.g. changing mortality or fertility or migration not just
population).
• But first, we need to define the “models of change” more
precisely.
Simple growth models (1)
• Arithmetic (cf. simple interest)
Growth = G = PC = Pt+1 – Pt = B – D + NM
Pt+n = Pt + nG = Pt + n(B – D + NM)
• Geometric (cf. compound interest)
Growth rate g = G/Pt = (B – D + NM)/Pt
Pt+n = Pt  (1 + g)n
• Exponential (cf. continuous interest)
Growth rate g = ln(Pt+1/Pt)
Pt+n = Pt  egn
where e = base of natural logarithms = that number which when
differentiated gives itself (if f(x) = e, then f’(x) = e) = 2.7182818…
• Logistic (growth limit, growth rate reduces with increasing density)
Pt = K  P0egt /[K + P0 (egt −1)]
where Pt = population at time t, P0 = population at time 0, K =
population limit, e = exponential function, g = exponential growth
rate
Simple growth models (2): derivation of
annual increments/rates
• Arithmetic
G = (Pt+n – Pt)/n
• Geometric
g = (Pt+n/Pt)(1/n) – 1
• Exponential
g = ln(Pt+n/Pt)/n
• Logistic
More complicated derivation (see Rowlands for
references)
Simple Growth Models (3): World Population Example
Year
Pop in millions
Arithmetic
Geometric
Exponential
2000
6085.572
6085.572
6085.572
2005
6464.750
6464.750
6464.750
75.836
0.012162
0.012088
2005
6464.750
6464.750
6464.750
2010
6843.928
6867.554
6867.554
2020
7602.284
7705.018
7705.018
2050
9877.352
11137.936
11137.936
Annual
change/growth rate
Simple growth models (4)
• It is important to understand the derivation of the
intensities (rates, probabilities) that enter simple
population models. The intensities follow from
the model form.
• Derivation of annual increments/rates for the
Arithmetic model
Pt+n = Pt + nG
– Subtract Pt from both sides
Pt+n – Pt = nG
– Divide both sides by n and swap LH and RH sides
G = (Pt+n – Pt)/n
Simple growth models (5):
Derivation of annual increments/rates for the
geometric (linear) model
• Geometric (linear) model
Pt+n = Pt (1 + g)n
– Divide both sides by Pt
Pt+n/Pt = (1 + g)n
– Apply the nth root to both sides (i.e. raise to the power
1/n)
(Pt+n/Pt)(1/n) = (1 + g)
– Subtract 1 from both sides and swap LH and RH
sides
g = (Pt+n/Pt)(1/n) – 1
Simple growth models (6):
Derivation of annual increments/rates for the
exponential model
• The exponential growth model
Pt+n = Pt  egn
Divide both sides by Pt
Pt+n/Pt = egn
Swap LH & RH sides
egn = Pt+n/Pt
Apply natural log function to both sides etc
ln(egn) = ln(Pt+n/Pt)
Powers can be taken outside the ln function
gn  ln(e) = ln(Pt+n/Pt)
ln(e) = 1 by definition, so this simplifies to
gn = ln(Pt+n/Pt)
Divide both sides by n
g = ln(Pt+n/Pt)/n
Simple growth models (6):
some principles for computing and using rates in
simple population models
• (1) You must compute the input rates using the
same equations as used in the population growth model
• (2) The numerical values of those rates/intensities will
differ depending on the model adopted
• (3) Using the same input stocks and flows but different
models you get the same growth forecast as long as you
follow point (1)
• (4) It is incorrect to claim that if you use the exponential
model you get faster growth than with the geometric,
because you should not use the same numerical values
but compute the values appropriate to the model used.
A population model using crude rates (1)
• We start with the components of change
equation:
Pt+1 = Pt + B – D + NM
• We then substitute for B, D and NM the
corresponding rates multiplied by the population
at risk
Pt+1 = Pt + b½(Pt+1 + Pt) – d ½(Pt+1 + Pt) + n ½(Pt+1 + Pt)
This simplifies to
Pt+1 = Pt + (b – d + n) ½(Pt+1 + Pt)
• This model poses a problem. We have the
outcome variable, population at the end of the
interval, Pt+1, on both sides of the equation.
A population model using crude rates (2)
• We can, however, manipulate this equation and bring all Pt+1
terms to the LH side
Multiply through the RH side terms:
Pt+1 = Pt + ½ (b – d + n) Pt+1 + ½ (b – d + n) Pt
Subtract ½ (b – d + n) Pt+1 from both sides
Pt+1 - ½ (b – d + n) Pt+1 = Pt + ½ (b – d + n) Pt
Extract the common population factors from both sides:
Pt+1 (1 - ½ (b – d + n)) = Pt (1 + ½ (b – d + n))
Divide both sides by (1 - ½ (b – d + n))
Pt+1 = Pt [(1 + ½ (b – d + n))/(1 - ½ (b – d + n))]
Now we have only Pt on the LH side and Pt on the RH side
The term in square brackets can be called the growth
multiplier
A population model using rates (3)
•
Let us use some example values for the USA for 2000-05 (World Population
Prospects 2004, UN)
b = 14.0/1000 = 0.0140
d = 8.4/1000 = 0.0084
n = 4.0/1000 = 0.0040
P2000 = 284.154 millions
P2005 = 298.213 millions
So
[(1 + ½ (b – d + n))/(1 - ½ (b – d + n))]=
[(1 + 0.5(0.0140 – 0.0084 + 0.0040))/(1 - 0.5(0.0140 – 0.0084 + 0.0040))]
= [1.0048/0.9952] = 1.009646302
•
We can apply this growth multiplier to compute (as a check) the population
in 2005
P2005 = 284.154 (1.009646302)5 = 298.126
– Note that this is slightly different from the 2005 figure above. We used crude
rates rounded to 1 decimal place and this caused the discrepancy.
– If we look again at the earlier table where we were working with the birth, death
and net migration counts, this discrepancy does not occur.
A population model using rates (4): example
This table shows an alternative simple model of population growth. The input
rates are defined with the (linear) average population in the interval as the
population at risk. This is the population at risk that is used in to compute
crude rates.
A population model using rates (5)
• A generalization of this model
– This growth model shows up later when we
add age to the projection and when we add
many regions:
– pt+1 = [(I + ½ M) (I - ½ M)-1] pt
– where the variables are now matrices (I, M) or
vectors, p, indicated by underlining or bolding
– I is the identity matrix, M is a matrix of
mortality, fertility and migration rates
Population accounts and models (1)
Richard Stone, Professor of Economics at Cambridge
developed the system of national economic accounts,
upon which all macro-economic work is founded.
Late in his career (1960s, 1970s) he turned his attention to
demographic accounts and developed systems of
population accounting which were used in educational
planning. He also developed a complete specification of
what a comprehensive system of social and demographic
statistics should look like.
Around the same time Professor Andrei Rogers,
working in Berkeley, Chicago, Vienna developed a
methodology for multiregional, later multistate
population modelling that could be applied to life
table analysis and to population projection. This
incorporated gross migration flows into population
models and captured the impact of migration from
one region to another.
Population accounts and models (2)
Take two great ideas and
put them together to
make a third:
multiregional (later
multistate) population
accounts and models.
Building on these ideas of Stone and Rogers, Professor Alan Wilson
and I developed a family of demographics accounts and models.
These are described in the book Spatial Population Analysis (Edward
Arnold, 1976).
Stone used the accounting principle that all population accounts
should balance (cf. components of growth equation). Rogers
introduced flows of migration between regions in a country. We took
these two ideas and melded them into a combined specification of
multiregional accounts.
Population accounts and models (3):
moves or transitions?
• The accounts described above require
moves/migrations (events of migration)
• Some migration data are different: they
are called transitions (usually derived from
a question in a census or survey about
where did you live one, five or ten years
ago?)
Time-space
diagram
showing one
transition
and one
move
Region
1
Region
2
t
t+n
Region 1
Time space
diagram
showing three
moves and one
transition
Region 2
t
t+n
Population accounts using movements (2)
Some origin to destination migration flows
To
My region
(of interest) #1
My country
(rest of) #2
My world
(rest of) #3
My region
(of interest)
#1
M11 or S1 or R1
M12
E1
Migration within region
1 or Stayers in region
2 or Residual
Out-migration from
region 1 to region 2
My country
(rest of)
#2
M21
M22 or S2 or R2
Out-migration from
region 2 to region 1
Migration within region
2 or Stayers in region 2
or Residual
My world
(rest of)
#3
I1
I2
Immigration to region 1
Immigration to region 2
From
Emigration from
region 1
E2
Emigration from
region 2
M33
Migration within
region 3 or
Stayers in 3 or
Residual
Births and deaths added to the migration flow table
To
My region
(of interest) #1
My country
(rest of) #2
My world
(rest of) #3
Deaths
My region
(of interest)
#1
M11 or S1 or R1
Migration within
region 1 or Stayers
in region 2 or
Residual
M12
Out-migration from
region 1 to region 2
E1
Emigration from
region 1
D1
Deaths in
region 1
My country (rest of)
#2
M21
Out-migration from
region 2 to region 1
M22 or S2 or R2
Migration within
region 2 or Stayers
in region 2 or
Residual
E2
Emigration from
region 2
D2
Deaths in
region 2
My world (rest of)
#3
I1
Immigration to
region 2
I2
Immigration to
region 2
NA
NA
Births
B1
Births in region 1
B2
Births in region 2
NA
NA
From
A full population account (movement type)
To
My region
#1
My country
#2
My world
#3
Deaths
Totals
My region
#1
R1
M12
E1
D1
P1*
My country
#2
M21
R2
E2
D2
P2*
My world
#3
I1
I2
0
0
I*
Births
B1
B1
0
0
B*
Totals
P*1
P*2
E*
D*
T**
Example for Greater London 1976-81
To
Greater
London
Greater
London
Rest of UK
Outside UK Deaths
Totals
5,307,263
1,053,475
259,291
408,271
7,028,300
Rest of UK
770,541
44,395,991
781,210
2,954,158
48,901,900
Outside UK
287,104
603,356
0
0
890,460
Births
443,546
3,136,291
0
0
3,579,837
Totals
6,808,454
49,189,113
1,040,501
3,362,429
60,400,497
Population accounts and models (3):
links to components of change
• What are the residuals?
– They are accounting terms
– R1 = P1 – D1 – E1 – M12
– R2 = P2 – D2 – E2 – M21
• The end of interval population is given by:
– P1 = R1 + M21 + I1 + B1
– P2 = R2 + M12 + I2 + B2
• Substitute the expression for R into the end of population equation
– P1 = P1 – D1 – E1 – M12 + M21 + I1 + B1
– P2 = P2 – D2 – E2 – M21 + M12 + I2 + B2
• Collect together the terms in a different order
– P1 = P1 + (B1 – D1) + (I1– E1) + (M21 – M12)
– P2 = P2 + (B2 – D2) + (I2– E2) + (M12 – M21)
• This is the components of change equation presented earlier
Variables in a transition population accounts tables
End state
Start state
Survive in
region 1
Survive in
region 2
Survive in
rest of
world
Die in
region 1
Die in
region 2
Die in rest
of world
Totals
Exist in
region 1
Ke1s1
Ke1s2
Ke1sr
Ke1d1
Ke1d2
Ke1dr
Ke1**
Exist in
region 2
Ke2s1
Ke2s2
Ke2sr
Ke2d1
Ke2d2
Ke2dr
Ke2**
Exist in
rest of
world
Kers1
Kers2
0
Kerd1
Kerd2
0
Ker**
Born in
region 1
Kb1s1
Kb1s2
Kb1sr
Kb1d1
Kb1d2
Kb1dr
Kb1**
Born in
region 2
Kb2s1
Kb2s2
Kb2sr
Kb2d1
Kb2d2
Kb2dr
Kb2**
Born in
rest of
world
Kbrs1
Kbrs2
0
Kbrd1
Kbrd2
0
Kbr**
Total
K**s1
K**s2
K**sr
K**d1
K**d2
K**dr
K****
Population accounts and models (4):
movement or transition accounts?
• Movement accounts easier to construct
• Fit in with year by year data on migration
from registers (e.g. NHS Central Register
in UK)
• Transition accounts are feasible if you
have linked censuses (e.g. 1976, 1981,
1986, 1991, 1996 and 2001 in Australia
with 5 year migration question)
Age and time (1) : Shakespeare said it all in 1599
All the world's a stage,
And all the men and women merely players,
They have their exits and entrances,
And one man in his time plays many parts,
His acts being seven ages. At first the infant,
Mewling and puking in the nurse's arms.
Then the whining schoolboy, with his satchel
And shining morning face, creeping like snail
Unwillingly to school. And then the lover,
Sighing like furnace, with a woeful ballad
Made to mistress' eyebrow. Then a soldier.
Full of strange oaths, and bearded like the pard,
Jealous in honour, sudden and quick in quarrel,
Seeking the bubble reputation
Even in the cannon's mouth. And then the justice,
In fair round belly with good capon lined,
With eyes severe and beard of formal cut,
Full of wise saws and modern instances;
And so he plays his part. The sixth age shifts
Into the lean and slippered pantaloon,
With spectacles on nose and pouch on side,
His youthful hose, well saved, a world too wide,
For his shrunk shank; and his big manly voice,
Turning again towards the childish treble, pipes
And whistles in his sound. Last scene of all,
That ends this strange eventful history,
Is second childishness and mere oblivion,
Sans teeth, sans eyes, sans taste, sans everything
William Shakespeare's As You Like It, Act II,
Scene 7 by the melancholy Jacques.
The Seven Ages of Man
Drawing by Henry Martin
© 1987 The New Yorker Magazine, Inc.
All rights reserved
Age and time (2): some definitions
•
•
•
•
Age is time elapsed since birth/creation
Attribute that changes with the march of time
No reversals allowed; no jumps forward
Measured in different ways
– Age at last birthday e.g. I am 64 years old, you are 20
years old
– This is the integer (whole number) part of a real
number age: today I am 64.074 years old
– Age = Int(64.074) = 64
– Your age changes on your birthday
Age and time (3): groupings
• Single years of age: 0,1,2,3, …, 99,100+
• Five year age groups: 0-4, 5-9, …, 95-99,
100+
• Ten year age groups: 0-9, 10-19, …, 9099, 100+
• Other groupings: 0,1-4,5-14,15-24, …, 65+
• Pensionable ages: 65+men + 60+ women
(but will change from 2010 to 65+ persons
and to 66 in mid-2020s, to 67 in mid-2030s
and to 68 in mid-2040s)
Age and time (4): associated concepts
• Cohort = persons born in same time
period, who proceed through life together
• Generation = persons born in same period
but usually longer
• Vintage = birth years for wine (applied to
human cohorts by Cambridge don, Sir
Richard Stone)
• But cohorts  ages
Age and time (4): age-time diagrams - when
were they invented? Who invented them?
• Most often known as “Lexis” diagrams after the
German demographer Wilhelm Lexis who
included a version of the diagram in his 1875
book, Einleitung in die Theorie der
Bevölkerungsstatistik [Introduction to the Theory
of Population Statistics]. Strassburg, Karl
Trübner.
• But the Belgian demographer Christophe
Vandeschrick has shown that other researchers
had invented the diagram before Lexis and that
it was not until the 1950s that the current form
was used (his Demographic Research Paper is
posted on the VLE).
Age and time (5):
papers ancient and modern on age-time diagrams
•
Zeuner G. (1869) Abhandlungen aus der Mathematischen Statistik (Treatise on
Mathematical Statistics). Leipzig: Verlag von Arthur Felix (cited in Vandeschrick
2001).
•
Brasche O. (1870) Beitrag zur Methode der Sterblichkeitberechnung und zur
Mortalitätstatistik Russland’s (Contribution on the Method of Mortality Estimation and
on Russia’s Mortality Statistics). Würzburg: A. Struber’s Buchhandlung (cited in
Vandeschrick 2001).
•
Lexis W. (1875) Einleitung in die Theorie der Bevölkerungsstatistik (Introduction to
the Theory of Population Statistics). Strassburg: Karl J. Trübner (cited in
Vandeschrick 2001).
•
Pressat, R. (1969) L’Analyse Démographique: Concepts, Méthodes, Résultats. Paris:
Presses Universitaires de France. 2nd edition.
•
Pressat, R. (1972) Demographic Analysis: Methods, Results, Applications. Translated
by J. Matras. London: Edward Arnold.
•
Vandeschrick C. (1992) Le diagramme de Lexis revisité (The Lexis diagram
revisited). Population, 47(5): 1241-1267.
•
Vandeschrick C. (2001) The Lexis diagram, a misnomer. Demographic Research,
4(3): 97-124. http://www.demographic-research.org/volumes/
•
Bell M. and Rees P. (2006) Comparing migration in Britain and Australia:
harmonisation through use of age-time plans. Environment and Planning A, 38, 959988.
Age and time (6): the “original” age-time diagram
Source: Vandeschrick (2001). Note that “Brache” should be “Brasche”. The original is from Brasche (1870)
Age and time (7): from ancient to modern
 rotate 90o 
Brasche diagram
Pressat diagram
Age
0
1859
1860
1861
1
2

Age
2
1
0
Time

1961
1962
1963
Time
Age and time (8): age-time diagrams showing
age group, period and cohort concepts
Age group
Period
Cohort
The dark lines mark out a
birth cohort. The yellow
squares show the periodage spaces can be used to
approximate cohort
measures.
Age and time (9): the four age-time spaces used to record
demographic events and transitions
Period-age space
Period-cohort space
Age-cohort space
Age-period-cohort spaces
`
`
Concluding remarks
• Population dynamics is linked to the age-sex structure of
the population
– The current age-sex plot reveals the history of population
change for circa 70 years
– The current age-sex distribution has important influences on the
future population
– There are important ageing trends in train which very little will
change
• Each component of population change has a different
relationship to age
– While these relationships are changing in detail in important
ways, their main shape determines the shape of a population’s
age-sex structure
• Although demographic trends and structures are very
important, other variables (economic, social) must be
included along with demographic for a proper analysis