Quantifying lifespan disparities: Which measure to use?

Alyson van RaalteBSPS Conference, Manchester

12 September 2008

Outline

Why measure lifespan inequality

Objectives

Considerations in choosing measures

Methods Description of measures examined

Data

Decomposition technique used

Results Lifespan inequality over time, across countries

Statistics of disagreement, testing for Lorenz dominance

Decomposition example, Japan in 1990s

Why measure lifespan inequality

0 20 40 60 80 100

0.0

00

.01

0.0

20

.03

Death density, Japan and USA, male life expectancy 75.2 years

Age

dens

ity

Japan 1986USA 2004

Objectives

How different are the examined inequality measures?

In which parts of the age distribution are the different measures more sensitive?

What are the advantages and drawbacks to using the different measures?

Considerations in choosing a measure

Criteria:

1. Lorenz Dominance

2. Pigou-Dalton Principle of Transfers

3. Scale and translation invariance

4. Population size independence

Considerations: Aversion to inequality

Age spectrum examined

Pooled-sex data or separate male/female data

Sensitivity to data errors or period fluctuations

Compositional change in the population

Lorenz curve

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Lorenz curves of lifespan inequality

Proportion of the population

Pro

port

ion

of t

he t

otal

per

son-

year

s liv

ed

line of perfect equalityCanada 1960

Lorenz dominance

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Lorenz curves of lifespan inequality

Proportion of the population

Pro

port

ion

of t

he t

otal

per

son-

year

s liv

ed

line of perfect equalityCanada 1960Canada 2003

Measures under examination

Comparing individuals to central value

Standard deviation / Coefficient of Variation

Interquartile range / IQRM

Comparing each individual to each other individual

Absolute inter-individual difference / Gini

Entropy of survival curve

Years of life lost due to death (e†) / Keyfitz’ Η

Data

Countries used: Canada, Denmark, Japan, Russia, USA

All data from Human Mortality Database, 1960-2006 (2004 for USA and Canada)

Life table male death distributions

Full age range examined

Methods

Statistics of disagreement Over time: differences in the direction of inequality change

Across countries: differences in ranking

Testing for Lorenz dominance

Age decompositions (stepwise replacement) to determine why measures disagreed

Direction of inequality change unclear (Japan in 1990s)

Results: Relative Measures

1960 1970 1980 1990 2000

0.14

0.18

0.22

0.26

Keyfitz's H, males

year

entr

opy

1960 1970 1980 1990 2000

0.20

0.25

0.30

0.35

Coefficient of variation, males

year

coef

ficie

nt o

f va

riatio

n

1960 1970 1980 1990 2000

0.10

0.12

0.14

0.16

0.18

Gini coeffient, males

year

gini

coe

ffic

ient

1960 1970 1980 1990 2000

0.20

0.25

0.30

0.35

0.40

IQR divided by the median, males

year

IQR

div

ided

by

med

ian

Results: Absolute Measures

1960 1970 1980 1990 2000

0.70

0.80

0.90

1.00

Canada, males

year

prop

ortio

nal c

hang

e

1960 1970 1980 1990 20000.

700.

800.

901.

00

Denmark, males

year

prop

ortio

nal c

hang

e

1960 1970 1980 1990 2000

0.70

0.80

0.90

1.00

Japan, males

year

prop

ortio

nal c

hang

e

1960 1970 1980 1990 2000

0.70

0.80

0.90

1.00

Russia, males

year

prop

ortio

nal c

hang

e

1960 1970 1980 1990 2000

0.70

0.80

0.90

1.00

USA, males

year

prop

ortio

nal c

hang

e

e†Absolute inter-individual differenceStandard deviationInterquartile range

e†Absolute inter-individual differenceStandard deviationInterquartile range

e†Absolute inter-individual differenceStandard deviationInterquartile range

Statistics of disagreement: Country Rankings

Absolute inequality: Country rankings differed 25/45 years

SD alone ranked countries differently 9 times

IQR alone ranked countries differently 8 times

Relative inequality: Country rankings differed 18/45 years

CV alone ranked countries differently 8 times

IQRM alone ranked countries differently 6 times

Lorenz dominance criterion broken: 4 times by standard deviation

twice by interquartile range

never by relative measures

Direction of inequality change

Absolute measures 77/225 cases where absolute measures disagreed

AID disagreed with all other measures zero times

e† disagreed with all other measures six times

SD disagreed with all other measures seventeen times

IQR disagreed with all other measures thirty-seven times

Relative measures 52/225 cases where absolute measures disagreed

Gini coefficient disagreed with all other measures zero times

Keyfitz’ H disagreed with all other measures four times

CV disagreed with all other measures seven times

IQRM disagreed with all other measures thirty times

Example: Japan in the 1990s

Absolute inequality: increased according to e†, AID and IQR

decreased according to SD

Relative inequality: increased according to IQRM

decreased according to H, G, and CV

Decomposing life expectancy increases

0

5-9

15-19

25-29

35-39

45-49

55-59

65-69

75-79

85-89

95-99

1960-1990

proportional contribution of age interval

-0.0

5

0.00

0.05

0.10

0.15

0.20

total increase: 10.62 years

0

5-9

15-19

25-29

35-39

45-49

55-59

65-69

75-79

85-89

95-99

1990-2000

proportional contribution of age interval

-0.0

5

0.00

0.05

0.10

0.15

0.20

total increase: 1.77 years

Age decompositions: Absolute measures

0

5-9

15-19

25-29

35-39

45-49

55-59

65-69

75-79

85-89

95-99

IQR, total increase: 3.55 percent

proportional contribution of age interval

-0.5 0.0

0.5

1.0

0

5-9

15-19

25-29

35-39

45-49

55-59

65-69

75-79

85-89

95-99

e+, total increase: 0.94 percent

proportional contribution of age interval

-0.5 0.0

0.5

1.0

1.5

2.0

0

5-9

15-19

25-29

35-39

45-49

55-59

65-69

75-79

85-89

95-99

AID, total increase: 0.32 percent

proportional contribution of age interval

-2 0 2 4

0

5-9

15-19

25-29

35-39

45-49

55-59

65-69

75-79

85-89

95-99

SD, total increase: -1.39 percent

proportional contribution of age interval

-1.0

-0.5 0.0

0.5

Age decompositions: Relative measures

0

5-9

15-19

25-29

35-39

45-49

55-59

65-69

75-79

85-89

95-99

IQRM, total increase: 1.52 percent

proportional contribution of age interval

-1 0 1 2 3

0

5-9

15-19

25-29

35-39

45-49

55-59

65-69

75-79

85-89

95-99

H, total increase: -1.35 percent

proportional contribution of age interval

-0.5 0.0

0.5

1.0

0

5-9

15-19

25-29

35-39

45-49

55-59

65-69

75-79

85-89

95-99

Gini, total increase: -1.95 percent

proportional contribution of age interval

-0.6

-0.4

-0.2 0.0

0.2

0.4

0.6

0

5-9

15-19

25-29

35-39

45-49

55-59

65-69

75-79

85-89

95-99

CV, total increase: -3.63 percent

proportional contribution of age interval

-0.5

-0.4

-0.3

-0.2

-0.1 0.0

0.1

0.2

Summary of results

Differences in aversion to inequality:

SD/CV very sensitive to changes in infant mortality

Ages 50-85 most impacting IQR/IQRM (modern distributions)

e†/H and AID/G both sensitive to transfers around mean, but e†/H more sensitive to upper ages

Most cases of different rankings owed to different age profiles of mortality

Standard deviation and Interquartile Range both found to violate Lorenz dominance

IQR/IQRM and SD/CV disagreed most often with other measures in ranking distributions

Conclusion

1. The choice of inequality measure matters

2. AID and e† are safe absolute inequality measures (of those studied)

3. Gini and H are safe relative inequality measures

Comments or Questions?

Step-wise replacement decomposition

In theory any aggregate demographic measure can be decomposed

For differences between lifespan inequality measures, need only to replace mx values

  Canada Japan

Age mx mx

0 0.00570 0.00352

1 0.00032 0.00055

2 0.00018 0.00033

3 0.00022 0.00027

… …  

110+ 0.7211 0.7008

SD 15.31 14.86

Step-wise decomposition example: SD

  1st replacement 2nd replacement Final replacement

  Canada Japan Contr. Canada Japan Contr. Canada Japan Contr.

Age mx mx   mx mx   mx mx  

0 0.00570 0.00570 0.42 0.00570 0.00570 0.42 0.00570 0.00352 0.42

1 0.00032 0.00055 … 0.00032 0.00032 -0.04 0.00032 0.00032 -0.04

2 0.00018 0.00033 … 0.00018 0.00033 … 0.00018 0.00018 0.03

3 0.00022 0.00027 … 0.00022 0.00027 … 0.00022 0.00022 0.01

… …     …     …    

110+ 0.7211 0.7008 … 0.7211 0.7008 … 0.7211 0.7211 0

SD 15.31 15.28   15.31 15.24   15.31 15.31 0.45

  Original

mx mx0 0.00570 0.003521 0.00032 0.00055… …  

110+ 0.7211 0.7008SD 15.31 14.86