Indicator Models
Meltem ¨ OZT ¨ URK
CNRS / Universit´e Paris IX Dauphine
Presentation
• What is an index?
• Examples of Indicators
• Human Development Index (HDI)
• Air pollution index (ATMO)
• Decathlon
• Measurement theory
• Conclusion
What is an index?
Examples: I.Q, consumer price index, poverty index, air quality index, GNP, unemployment index, safety index, ...
What is an index?
Examples: I.Q, consumer price index, poverty index, air quality index, GNP, unemployment index, safety index, ...
• What do they represent?
=⇒ reality? : One reality? More realities? No reality?
What is an index?
Examples: I.Q, consumer price index, poverty index, air quality index, GNP, unemployment index, safety index, ...
• What do they represent?
=⇒ reality? : One reality? More realities? No reality?
• Who will use them?
What is an index?
Examples: I.Q, consumer price index, poverty index, air quality index, GNP, unemployment index, safety index, ...
• What do they represent?
=⇒ reality? : One reality? More realities? No reality?
• Who will use them?
• For what do we use them?
What is an index?
Examples: I.Q, consumer price index, poverty index, air quality index, GNP, unemployment index, safety index, ...
• What do they represent?
=⇒ reality? : One reality? More realities? No reality?
• Who will use them?
• For what do we use them?
• Evaluate a current situation
What is an index?
Examples: I.Q, consumer price index, poverty index, air quality index, GNP, unemployment index, safety index, ...
• What do they represent?
=⇒ reality? : One reality? More realities? No reality?
• Who will use them?
• For what do we use them?
• Evaluate a current situation
• Evaluate a progress
What is an index?
Examples: I.Q, consumer price index, poverty index, air quality index, GNP, unemployment index, safety index, ...
• What do they represent?
=⇒ reality? : One reality? More realities? No reality?
• Who will use them?
• For what do we use them?
• Evaluate a current situation
• Evaluate a progress
• Make a decision
What is an index?
Examples: I.Q, consumer price index, poverty index, air quality index, GNP, unemployment index, safety index, ...
• What do they represent?
=⇒ reality? : One reality? More realities? No reality?
• Who will use them?
• For what do we use them?
• Evaluate a current situation
• Evaluate a progress
• Make a decision
• Use them as a policy
What is an index?
Examples: I.Q, consumer price index, poverty index, air quality index, GNP, unemployment index, safety index, ...
• What do they represent?
=⇒ reality? : One reality? More realities? No reality?
• Who will use them?
• For what do we use them?
• Evaluate a current situation
• Evaluate a progress
• Make a decision
• Use them as a policy
• Put a pressure, justify, ...
What is an index?
Examples: I.Q, consumer price index, poverty index, air quality index, GNP, unemployment index, safety index, ...
• What do they represent?
=⇒ reality? : One reality? More realities? No reality?
• Who will use them?
• For what do we use them?
• Evaluate a current situation
• Evaluate a progress
• Make a decision
• Use them as a policy
• Put a pressure, justify, ...
• How to construct them?
Presentation
• What is an index?
• Examples of Indicators
• Human Development Index (HDI)
• Air pollution index (ATMO)
• Decathlon
• Measurement theory
• Conclusion
Human Development Index
HDI = LEI + EAI + GDPI 3
where
Human Development Index
HDI = LEI + EAI + GDPI 3
where
LEI: Life expectancy index
LEI = life expectancy at birth − 25 85 − 25
Human Development Index
HDI = LEI + EAI + GDPI 3
where
LEI: Life expectancy index
LEI = life expectancy at birth − 25 85 − 25
EAI: Educational Attainment index
EAI = 2ALI + ERI 3
Human Development Index
HDI = LEI + EAI + GDPI 3
where
LEI: Life expectancy index
LEI = life expectancy at birth − 25 85 − 25
EAI: Educational Attainment index
EAI = 2ALI + ERI 3
GDPI: Adjusted real GDP per capita
Human Development Index
GDPI = transformed income − W(100) W(40 000) − W(100)
where W(x) represents the conversion of the GDP in standard monetary equivalents (USD) following Atkinson’s formula.
Scale normalization
life expectancy EAI GDPI
South Korea 71.5 .93 .97
Costa Rica 76.6 .86 .95
Scale normalization
life expectancy EAI GDPI
South Korea 71.5 .93 .97
Costa Rica 76.6 .86 .95
If the scale is [85,25] then HDI(SK)>HDI(CR)
Scale normalization
life expectancy EAI GDPI
South Korea 71.5 .93 .97
Costa Rica 76.6 .86 .95
If the scale is [85,25] then HDI(SK)>HDI(CR) If the scale is [80,25] then HDI(CR)>HDI(SK)
Scale normalization
life expectancy EAI GDPI
South Korea 71.5 .93 .97
Costa Rica 76.6 .86 .95
If the scale is [85,25] then HDI(SK)>HDI(CR) If the scale is [80,25] then HDI(CR)>HDI(SK)
• Normalization: weight
• normalization not really [0,1]
Compensation
life expectancy ALI ERI real GDP HDI
Gabon 54.1 .63 .60 3 641 .56
Solomon Islands 70.8 .62 .47 2 118 .58
A year of life is equivalent to 100.9 USD (equivalent).
If we transform this equivalent in real USD then poor’s people life is less worth than rich people life!
exemple:
GDP(x)= 1307, a year of life is equivalent to 21084 USD
Dimension Independence
life expectancy ALI ERI real GDP HDI
x 30 .80 .65 500 .30
y 30 .35 .40 3500 .34
w 70 .80 .65 500 .52
z 70 .35 .40 3 500 .56
Dimension Independence
life expectancy ALI ERI real GDP HDI
x 30 .80 .65 500 .30
y 30 .35 .40 3 500 .34
w 70 .80 .65 500 .52
z 70 .35 .40 3500 .56
Dimension Independence
life expectancy ALI ERI real GDP HDI
x 30 .80 .65 500 .30
y 30 .35 .40 3 500 .34
w 70 .80 .65 500 .52
z 70 .35 .40 3 500 .56
Dimension Independence
life expectancy ALI ERI real GDP HDI
x 30 .80 .65 500 .30
y 30 .35 .40 3500 .34
w 70 .80 .65 500 .52
z 70 .35 .40 3500 .56
Presentation
• What is an index?
• Examples of Indicators
• Human Development Index (HDI)
• Air pollution index (ATMO)
• Decathlon
• Measurement theory
• Conclusion
ATMO Index
pollutant CO
2SO
2O
3dust
sub-index x y z w
ATMO = max(x, y, z, w)
ATMO Index: monotonicity
pollutant CO
2SO
2O
3dust
t
13 3 2 8
t
23 3 8 8
ATMO remains unchanged from t
1to t
2Non compensation
pollutant CO2 SO2 O3 dust
x 1 1 6 1
y 5 4 5 5
y does better than x. It is better to have an average bad quality, than good quality on single pollutants.
Non compensation
pollutant CO2 SO2 O3 dust
x 1 1 6 1
y 5 4 5 5
quality of SO2(y) better than quality of O3(y)
Non compensation
pollutant CO2 SO2 O3 dust
x 1 1 6 1
y 5 4 5 5
dust(x) not five time better than dust (y)!
Presentation
• What is an index?
• Examples of Indicators
• Human Development Index (HDI)
• Air pollution index (ATMO)
• Decathlon
• Measurement theory
• Conclusion
Decathlon score
• Before 1908 it was a some of ranks (Borda!)
Decathlon score
• Before 1908 it was a some of ranks (Borda!)
• Don’t show the real performance
• Don’t show the progress
Decathlon score
• Before 1908 it was a some of ranks (Borda!)
• Don’t show the real performance
• Don’t show the progress
• Convex 1908-1962: much more points when it is closed to the world record!
Decathlon score
• Before 1908 it was a some of ranks (Borda!)
• Don’t show the real performance
• Don’t show the progress
• Convex 1908-1962: much more points when it is closed to the world record!
• athlete specialized in some fields!
Decathlon score
• Before 1908 it was a some of ranks (Borda!)
• Don’t show the real performance
• Don’t show the progress
• Convex 1908-1962: much more points when it is closed to the world record!
• athlete specialized in some fields!
• Concave 1962-today:
Decathlon score
• Before 1908 it was a some of ranks (Borda!)
• Don’t show the real performance
• Don’t show the progress
• Convex 1908-1962: much more points when it is closed to the world record!
• athlete specialized in some fields!
• Concave 1962-today:
• be excellent in all disciplines!
Decathlon score meaning
• Identify the winner
• Asses the performance of an athlete
• Assess the trend of an athlete
• Identify training strategies for athletes
Presentation
• What is an index?
• Examples of Indicators
• Human Development Index (HDI)
• Air pollution index (ATMO)
• Decathlon
• Measurement theory
• Conclusion
Measurement Theory/Measurement Scales
• A phenomenon
• Observations
• Possibly an order among observations
• An order preserving function
• Information equivalent numerical scales
• Admissible transformations of numerical scales (create information equivalent representations)
• A class of admissible transformations univocally determines a scale type.
Measurement Scales
• Ordinal Scales
(strictly increasing transformations)
• Interval Scales
(positive affine transformations: ϕ(x) = αx + β)
• Ratio Scales
(positive homothetic transformations: ϕ(x) = αx)
• Absolute Scales
(identity transformations)
Meaningfulness
• If x is 10kg and y is 20kg can we say that y is twice more heavy than x?
• If x is 10 degrees Co and y is 20 degrees Co can we say that y is twice more hot than x?
• If x is large (10) and y is small (1) can we say that x is 10 times bigger than y?
Measurement and Decision
• Unfortunately measurement theory is only necessary, but not sufficient in order to find the appropriate way to observe “reality”.
• WHO decides the use and the structure of a measure?
Presentation
• What is an index?
• Examples of Indicators
• Human Development Index (HDI)
• Air pollution index (ATMO)
• Decathlon
• Measurement theory
• Conclusion
Conclusions
• Indexes are both a representation of some realities and tools for pursuing policies.
• Indexes have to be carefully studied in order to be meaningful and useful.
• Several tools are available in order to construct meaningful and useful indexes.