Which measure to use?
Alyson van Raalte
BSPS 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
Death density, Japan and USA, male life expectancy 75.2 years
Japan 1986
USA 2004
0 20 40
Age
60 80 100

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 curves of lifespan inequality line of perfect equality
Canada 1960
0.0
0.2
0.4
0.6
Proportion of the population
0.8
1.0

Lorenz curves of lifespan inequality line of perfect equality
Canada 1960
Canada 2003
0.0
0.2
0.4
0.6
Proportion of the population
0.8
1.0

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
Keyfitz's H, males Coefficient of variation, males
1960 1970 1980 1990 year
2000
Gini coeffient, males
1960 1970 1980 year
1990 2000
IQR divided by the median, males
1960 1970 1980 1990 year
2000 1960 1970 1980 year
1990 2000

Canada, males
Results: Absolute Measures
Denmark, males USA, males
1960 1970 1980 1990 year
2000
Japan, males
1960 1970 1980 1990 2000 year
Russia, males
1960 1970 1980 year
1990 2000
1960 1970 1980 1990 2000 year
1960 1970 1980 1990 2000 year e† e†
Absolute inter-individual difference
Standard deviation
Interquartile 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
95-99
85-89
75-79
65-69
55-59
45-49
35-39
25-29
15-19
5-9
0
1960-1990 total increase: 10.62 years
95-99
85-89
75-79
65-69
55-59
45-49
35-39
25-29
15-19
5-9
0 proportional contribution of age interval
1990-2000 total increase: 1.77 years proportional contribution of age interval

Age decompositions: Absolute measures
IQR, total increase: 3.55 percent e+, total increase: 0.94 percent
95-99
85-89
75-79
65-69
55-59
45-49
35-39
25-29
15-19
5-9
0
95-99
85-89
75-79
65-69
55-59
45-49
35-39
25-29
15-19
5-9
0 proportional contribution of age interval
AID, total increase: 0.32 percent proportional contribution of age interval
SD, total increase: -1.39 percent
95-99
85-89
75-79
65-69
55-59
45-49
35-39
25-29
15-19
5-9
0
95-99
85-89
75-79
65-69
55-59
45-49
35-39
25-29
15-19
5-9
0 proportional contribution of age interval proportional contribution of age interval

Age decompositions: Relative measures
IQRM, total increase: 1.52 percent H, total increase: -1.35 percent
95-99
85-89
75-79
65-69
55-59
45-49
35-39
25-29
15-19
5-9
0
95-99
85-89
75-79
65-69
55-59
45-49
35-39
25-29
15-19
5-9
0
95-99
85-89
75-79
65-69
55-59
45-49
35-39
25-29
15-19
5-9
0 proportional contribution of age interval
Gini, total increase: -1.95 percent
95-99
85-89
75-79
65-69
55-59
45-49
35-39
25-29
15-19
5-9
0 proportional contribution of age interval
CV, total increase: -3.63 percent proportional contribution of age interval proportional contribution of age interval

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

Step-wise replacement decomposition
 In theory any aggregate demographic measure can be decomposed
 For differences between lifespan inequality measures, need only to replace m x values
Age
Canada Japan 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
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
…
2 0.00018
0.00033
…
3 0.00022
0.00027
…
… …
110+ 0.7211
0.7008
…
0.00032 0.00032
-0.04
0.00032 0.00032
-0.04
0.00018
0.00033
…
0.00018 0.00018
0.03
0.00022
0.00027
…
0.00022 0.00022
0.01
… …
0.7211
0.7008
… 0.7211
0.7211
0
15.31
15.28
15.31
15.24
15.31
15.31
0.45
0
1
…
110+
SD
Original mx mx
0.00570
0.00352
0.00032
0.00055
…
0.7211
0.7008
15.31
14.86