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

• 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

Components of change for the USA (Rowland 2003, Figure 1.4)

Components of change for the UK, Population Trends 122, p8

Examples of change components

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 equationPt+n = Pt + NI + NM

• Breaking down the componentsPt+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 estimateNM = (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 rateCDR = (D/Pt+ ½) 1000

• Natural increase rateNIR = ((B-D)/Pt+ ½ ) 1000 = CBR – CDR

• Net migration rateNMR = (NM/ Pt+ ½ ) 1000

• Population change ratePCR = (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 + NMPt+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

• ArithmeticG = (Pt+n – Pt)/n

• Geometricg = (Pt+n/Pt)(1/n) – 1

• Exponentialg = ln(Pt+n/Pt)/n

• LogisticMore complicated derivation (see Rowlands for

references)

Year

Pop in millions

Arithmetic Geometric Exponential

2000 6085.572 6085.572 6085.572

2005 6464.750 6464.750 6464.750

Annual change/growth rate

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

Simple Growth Models (3): World Population Example

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) modelPt+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 modelPt+n = Pt egn

Divide both sides by Pt

Pt+n/Pt = egn

Swap LH & RH sidesegn = Pt+n/Pt

Apply natural log function to both sides etcln(egn) = ln(Pt+n/Pt)

Powers can be taken outside the ln functiongn ln(e) = ln(Pt+n/Pt)

ln(e) = 1 by definition, so this simplifies to gn = ln(Pt+n/Pt)

Divide both sides by ng = 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 riskPt+1 = Pt + b½(Pt+1 + Pt) – d ½(Pt+1 + Pt) + n ½(Pt+1 + Pt)This simplifies toPt+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 sideMultiply 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 sideThe 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 = 0.0140d = 8.4/1000 = 0.0084n = 4.0/1000 = 0.0040

P2000 = 284.154 millions

P2005 = 298.213 millionsSo[(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.

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 (4): example

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.

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.

Take two great ideas and put them together to make a third: multiregional (later multistate) population accounts and models.

Population accounts and models (2)

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?)

t t+n

Region 1

Region 2

Time-space diagram showing one transition and one move

Time space diagram showing three moves and one transition

Region 1

Region 2

t t+n

Population accounts using movements (2)

To

From

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

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

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

My world (rest of)

#3

I1

Immigration to region 1

I2

Immigration to region 2

M33

Migration within region 3 or Stayers in 3 or Residual

Some origin to destination migration flows

Births and deaths added to the migration flow table

To

From

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

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**

A full population account (movement type)

To Greater London

Rest of UK Outside UK Deaths Totals

Greater London

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

Example for Greater London 1976-81

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

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****

Variables in a transition population accounts tables

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)

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

The Seven Ages of ManDrawing by Henry Martin© 1987 The New Yorker Magazine, Inc. All rights reservedWilliam Shakespeare's As You Like It, Act II,

Scene 7 by the melancholy Jacques.

Age and time (1) : Shakespeare said it all in 1599

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, …, 90-99, 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, 959-988.

Source: Vandeschrick (2001). Note that “Brache” should be “Brasche”. The original is from Brasche (1870)

Age and time (6): the “original” age-time diagram

Age and time (7): from ancient to modern

Brasche diagram Pressat diagram

Age

Time

Time

Age1859

1860

1861

0 1 2

1961 1962 1963

0

1

2

rotate 90o

Age group Period Cohort

Age and time (8): age-time diagrams showing age group, period and cohort concepts

The dark lines mark out a birth cohort. The yellow squares show the period-age spaces can be used to approximate cohort measures.

Period-age space Period-cohort space

Age-cohort space Age-period-cohort spaces

` `

Age and time (9): the four age-time spaces used to record demographic events and transitions

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