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(1)

Indicator Models

Meltem ¨ OZT ¨ URK

CNRS / Universit´e Paris IX Dauphine

(2)

Presentation

• What is an index?

• Examples of Indicators

• Human Development Index (HDI)

• Air pollution index (ATMO)

• Decathlon

• Measurement theory

• Conclusion

(3)

What is an index?

Examples: I.Q, consumer price index, poverty index, air quality index, GNP, unemployment index, safety index, ...

(4)

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?

(5)

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?

(6)

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?

(7)

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

(8)

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

(9)

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

(10)

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

(11)

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, ...

(12)

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?

(13)

Presentation

• What is an index?

Examples of Indicators

Human Development Index (HDI)

• Air pollution index (ATMO)

• Decathlon

• Measurement theory

• Conclusion

(14)

Human Development Index

HDI = LEI + EAI + GDPI 3

where

(15)

Human Development Index

HDI = LEI + EAI + GDPI 3

where

LEI: Life expectancy index

LEI = life expectancy at birth − 25 85 − 25

(16)

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

(17)

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

(18)

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.

(19)

Scale normalization

life expectancy EAI GDPI

South Korea 71.5 .93 .97

Costa Rica 76.6 .86 .95

(20)

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)

(21)

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)

(22)

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]

(23)

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

(24)

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

(25)

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

(26)

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

(27)

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

(28)

Presentation

• What is an index?

Examples of Indicators

• Human Development Index (HDI)

Air pollution index (ATMO)

• Decathlon

• Measurement theory

• Conclusion

(29)

ATMO Index

pollutant CO

2

SO

2

O

3

dust

sub-index x y z w

ATMO = max(x, y, z, w)

(30)

ATMO Index: monotonicity

pollutant CO

2

SO

2

O

3

dust

t

1

3 3 2 8

t

2

3 3 8 8

ATMO remains unchanged from t

1

to t

2

(31)

Non 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.

(32)

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)

(33)

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)!

(34)

Presentation

• What is an index?

Examples of Indicators

• Human Development Index (HDI)

• Air pollution index (ATMO)

Decathlon

• Measurement theory

• Conclusion

(35)

Decathlon score

• Before 1908 it was a some of ranks (Borda!)

(36)

Decathlon score

• Before 1908 it was a some of ranks (Borda!)

• Don’t show the real performance

• Don’t show the progress

(37)

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!

(38)

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!

(39)

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:

(40)

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!

(41)

Decathlon score meaning

• Identify the winner

• Asses the performance of an athlete

• Assess the trend of an athlete

• Identify training strategies for athletes

(42)

Presentation

• What is an index?

• Examples of Indicators

• Human Development Index (HDI)

• Air pollution index (ATMO)

• Decathlon

Measurement theory

• Conclusion

(43)

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.

(44)

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)

(45)

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?

(46)

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?

(47)

Presentation

• What is an index?

• Examples of Indicators

• Human Development Index (HDI)

• Air pollution index (ATMO)

• Decathlon

• Measurement theory

Conclusion

(48)

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.

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