FIRST
GLOBAL FORUM ON HUMAN DEVELOPMENT
29-31
July 1999 . United Nations Headquarters . New York
NEW METHODOLOGIES FOR CALCULATING THE
HDI
Teresa
G. del Valle Irala
Carmen
Puerta Gil 1
University of Bilbao,
Spain
1. INTRODUCTION
The United Nations Development Program (UNDP)
published the first "Human Development Report" in 1990. A new concept of
development and an index to measure it were presented on it. The index
is elaborated combining three sets of variables: Health, Education and
Income. The UNDP calls these three sets "essential components " and they
are given the same importance in the concept of development and therefore
in the index.
This multidimensional concept elaborated by
the UNDP has led us to propose Multiple Factorial Analysis (MFA) as a method
of studying and measuring Human Development, and in particular, Weighted
Principal Components Analysis (WPCA) as an adequate factorial technique
to proportion us with an index which will allow us to discriminate among
different levels of development. The index, which we obtain by applying
this analysis, gives us an order very similar to the one obtained by the
Human Development Index (HDI).
The methodology we propose is complementary
to the one of the UNDP, it stems from the same concept of development and
obtains a very similar index and cluster. Using this as a new tool has
the following advantages:
1. -It is flexible and general, in the sense
that it can be used with any number of sets and of variables inside a set.
2. - It allows us to evaluate the type of
development, more centered in income or in education.
3. - It proportions graphs in which associations,
oppositions and proximities among countries can be visualized. It also
proportions graphs in which the center of gravity of each country is represented
together with the positions that these countries take in each of the sets.
4. - It allows the information that the variables
give to be visualized, helping in the process of selection of these.
5. - It allows temporary comparison.
6. - It allows an index of human development
in time to be obtained easily.
7. - It possesses a strong statistical base.
The paper has the following structure:
2. - Data base of 1994
With the data base used by the UNDP to elaborate
the HDI of 1994, the following analysis are presented:
2.a- Weighted principal components analysis
of the data base
2.b – Principal Components Analysis of the
database and why this technique can not be used to solve the problem posed
by the UNDP
2.c- Weighted Principal Components analysis
of the base but changing the variable of the income set.
2.d- Weighted Principal Components Analysis
of the database in which more variables have been incorporated to the health
and to the education set.
3. - Data base of 1997
With the data base used by the UNDP to elaborate
the HDI of 1997, the following analysis are presented:
3.a – Weighted Principal Components Analysis
of the database and clusters of three and five classes.
3.b – Multiple Factorial Analysis of the
database.
4. - Data base of 1994 and
of 1997
4.a – Comparative study of human development
in 1994 and in 1997 using Multiple Factorial Analysis.
4.b – A proposal of a Human Development index
in time.
5. - Conclusion
6. - Bibliography
2. DATA BASE OF 1994
2.a- Weighted Principal Components Analysis
of the 1994 table2
The data of the table analyzed are quantitative,
the files-individuals are countries in this case and the columns are the
selected variables that in this case are structured in sets, essential
components in UNDP terminology. The table of data X has therefore I files
and K columns structured in J sets. The I files are elements of RK,
the distance between two individuals i, l is defined in the following formula:
The
coefficients mk weight the influence of each column k in calculating
the distance between individuals. These weights are put in a diagonal matrix
M, which determines the metric in RK. The weighting chosen,
is the one proposed by Multiple Factorial Analysis.3
The weight given by MFA to the variables in a set, is the inverse of the
first eigen value of the Principal Components Analysis of that set. This
weight balances the influence of each set in the first global axes. As
the weight is the same for all the variables in a set, its internal structure
is maintained.
The first factor of WPCA is a linear combination
of the original variables with maximum variance. It has been constructed
balancing the sets of variables; this factor is proposed as an index of
development.
In this section we present the results of
the WPCA of the database selected by UNDP to elaborate HDI in 1994. This
table crosses 175 individuals with 4 variables structured in 3 sets:
-
Health set: Life expectancy at birth (LE4)
-
Education set: Literacy rate in Adults (LRA4)
and Mean Years of Schooling (MYS4)
-
Income set: Gross domestic product per capita
in parity purchasing power adjusted by Atkinson formula (GDA4).
We weight the variables with the inverse of the
first eigen value to balance the sets. As the units of measurement are
very different we introduce the inverse of the variance in the weights.
The metric we use to apply WPCA has the following associated matrix M:
Where si2 with i=1,2,3,4
is the variance of each variable, ?j1 with j=1,2,3
represents the first eigen value of the PCA of each set. Table 1 presents
the eigen values and the distribution of the inertia between the axes.
TABLE 1-Eigen values and
the distribution of the inertia between the axes.
|
Eigen value
|
Percentage
|
Percentage cum
|
|
2.6554
|
86.20
|
86.20
|
|
0.2264
|
7.35
|
93.55
|
|
0.1195
|
3.88
|
97.43
|
|
0.0792
|
2.57
|
100.00
|
The first eigen value accounts for 86.2% of
the variability of the data. The first factor provides a good synthesis
of the original data. The equation4 is:
This factor is a "height factor". The correlations
between the variables and F11,94 are high and are
all of the same sign. We call it the "Development factor" and propose it
as a new index to differentiate different levels of development. Countries
with very high development as well as those with very low development are
represented quite accurately by this axe. The equation of the 2nd
factor is:
The second factor has a higher correlation
with GDA4 and LRA4 (table 2). This factor places the income variable in
opposition to the others. It completes the interpretation of those countries
with middle development; most of these countries have unbalanced development.
Countries with high income and low levels of education for their degree
of development are placed in the negative part of this factor. Countries
with high education levels and low income are placed in the positive part.
We named it the "type of development factor"
TABLE 2 - Correlations variables-factors.
|
Active Variables
|
Correlations
|
|
Labels
|
F11,94
|
F21,94
|
|
LE4
|
-0. 96
|
0. 06
|
|
LRA4
|
-0. 89
|
0. 38
|
|
MYS4
|
-0. 90
|
0. 19
|
|
GDA4
|
-0. 93
|
-0.36
|
|
Illustrative Variable
|
|
|
|
HDI4
|
-1.00
|
-0.06
|
In graph 1 the countries are projected onto
the first factorial plane. The first two factors account for 93.55% of
the total inertia. In this way the distances remain similar to those in
the cloud. The countries with balanced development are accurately explained
by the first factor (development factor). Those with unequal development
(high income-low education or vice-versa) are explained in a better way
by the 2nd factor.
The first factor of this analysis proportions
an order very similar to the one proposed by the UNDP for the same year
(1994, the correlation between F1 and HDI4 is –1). We expected
this result, since the variables selected are the same and the sets are
weighted equally. WPCA further clarifies the final result: it proportions
graphs in which associations, oppositions and proximities among countries
can be visualized and the second factor allows us to evaluate the type
of development
GRAPH 1
2.b –Principal Components Analysis of
the data base
In this section we present the results of
Principal Components Analysis (PCA) of the table analyzed in the previous
section with WPCA. The difference is that now, it is not taking into account
that the variables are structured in sets, and therefore, they are not
weighted by the inverse of the first eigen value of its set.
The first eigen value of this analysis is
3,412 and accounts for 85.31% of the variability of the data. The first
two eigen values account for more than 92% of the inertia. The four variables
have high negative correlation with the first factor (table 3).
Table 3 – Correlations variables
–factor
|
Variables
|
F12,94
|
|
LE4
|
-0.95
|
|
LRA4
|
-0.91
|
|
MYS4
|
-0.92
|
|
GDA4
|
-0.90
|
The equation of this factor is:
As we can notice the four variables have nearly
the same coefficient. This causes the educational set to have more importance
whereas our objective is to have balanced sets. Variables structured in
sets and analyzed by WPCA do not display this imbalance. Therefore PCA
is not suitable for this type of analysis.
2.c – Third Analysis
The objective of this analysis is to show
how the factorial planes are a useful tool in selecting variables. In their
reports, the UNDP debates the suitability of the Atkinson formula in weighting
GDP. Without entering into any debate on the suitability of the Atkinson
formula, we want to show by comparing the current analysis and that developed
in section 2.a, that the information provided by factorial planes helps
in the process of selecting the variables.
This analysis was done with the same sets
and the same number of variables as in section 2.a. the only exception
is that we have selected the variable GDP (Gross Domestic Product in parity
purchasing power but without adjusting it by the Atkinson formula) for
the income set.
Matrix M is the same as in section 2.a with
the exception that now s24 is the variance of GDP.
The first eigen value accounts for 81.30% of the variability of the data.
The two first factors explain 94.58% of the whole inertia. The equations
of the two first factors are:
Table 4 shows the correlations of the variables
with the two first factors.
Table 4 – Correlations variables-factors
|
VARIABLES
|
F13,94
|
F23,94
|
|
LE4
|
-0.94
|
0.24
|
|
LRA4
|
-0.85
|
0.43
|
|
MYS4
|
-0.94
|
0.00
|
|
GDP4
|
-0.87
|
-0.49
|
In many respects, this analysis is similar
to that made in section 2.a. The first factor is again the development
factor and the second factor is the type of development. The ranking given
by the first factor presents more differences with the HDI than that obtained
in section 2.a.5
In graph 1 we can see how the countries placed
on the extremes of the first factor (highly developed countries and less
developed countries) are near the origin of the second factor. Because
of this, we can say they have balanced development in respect to the three
sets.
When we do WPCA substituting GDA4 with GDP4
we can see in the first factorial plane (graph 2) that the countries plotted
on the extremes of factor 1, are placed on the negative extreme of factor
2. This indicates that the most developed countries have high values in
all the variables of the 3 sets but they have especially high values in
the income set. The lowest developed countries have low values in the three
sets but their values in the income set are better than the others.
The most important difference between both
analysis is that whilst in the first analysis countries with high or with
low development present balanced development, in this analysis this kind
of balanced development with respect to the three sets does not appear.
They appear clearly on the side of the graph, which indicates a higher
level of income compared to the other variables. A comparison of the plots
shows the difference between using GDA4 or GDP4.
Also it is possible to see that when the countries
pass from a low to a middle development, in general, the set which most
increases is the Education set.
GRAPH 2
2.d – Fourth Analysis
In this analysis the number of the variables
in the table has been increased, keeping the same sets. Two variables have
been incorporated to the Health set: Infant mortality rate (IMR4) and Maternal
mortality rate (MMR4) and in the Education set the variable Mean Years
of schooling has been substituted by Male mean years of schooling (MMS4)
and Female mean years of schooling (FMS4).
Matrix M is of the same type as in section
2.a but now its order is 7. The first eigen value represents 84.48% of
the variability of the data. The first two eigen values explain 91.71%
of the total inertia. Their equations are:
Table 5 shows the correlations between the
variables and the two first factors.
Table 5 – Correlations variables-factors
|
VARIABLES
|
F14,94
|
F24,94
|
|
LE4
|
-0.95
|
0.04
|
|
MIR4
|
0.94
|
0.00
|
|
MMR4
|
0.94
|
-0.04
|
|
LRA4
|
-0.87
|
0.29
|
|
FMS4
|
-0.88
|
0.35
|
|
MMS4
|
-0.89
|
0.32
|
|
GDA4
|
-0.94
|
-0.33
|
Now factor 1 is not a "height factor", but
it goes on being a "Development factor". The second factor, as in the analysis
above, places the Education set opposite to the Income set.
Our objective, in this section, is to show
the flexibility of WPCA to work with any number of variables inside the
sets. For this and also to keep the comparability between the different
analysis of the 2nd section, 173 countries have been used. Some
of these did not have values in the new variables introduced in this analysis;
the mean of the correspondent variable has been assigned to these countries.
This alters in a certain manner the ranking of the countries and their
positions in the graph of the first two factors and due to this, a more
detailed analysis has not been done.
3. – DATA BASE OF 19976
3.a – Weighted Principal Components Analysis
In this section we present the results of
the WPCA of the database selected by UNDP to elaborate HDI in 1997. This
table crosses 175 individuals with 4 variables structured in 3 sets:
-
Health set: Life expectancy at birth (LE7)
-
Education set: Literacy rate in adults (LRA7)
and Combined first, second and third level gross enrolment ratio (GER7).
-
Income set: Gross domestic product per capita
in parity purchasing power adjusted by Atkinson formula (GDA7).
Matrix M is of the same type as in section 1.a).
Table 6 presents the eigen values and the distribution of the inertia between
the axes.
TABLE 6 - Eigen values and
the distribution of the inertia between the axes.
|
Eigen value
|
Percentage
|
Percentage cum
|
|
2.5775
|
83. 07
|
83. 07
|
|
0.2697
|
8. 69
|
91.76
|
|
0. 1646
|
5. 30
|
97. 07
|
|
0.0911
|
2. 93
|
100.00
|
The first eigen value accounts for 83.07%
of the variability of the data. The first factor provides a good synthesis
of the original data. The equation is:
This factor is a "height factor". The correlations
between the variables and F11 are high and are all
of the same sign. We call it the "Development factor" and propose it as
a new index to assess different levels of development. Countries with very
high development as well as those with very low development are represented
quite accurately by this axe. The equation of the 2nd factor
is:
The second factor has a higher correlation
with GDA7 and LRA7 (table 7). This factor places the income variable in
opposition to the others. It completes the interpretation of those countries
with middle development; Most of these countries have unbalanced development.
Countries with high income and low levels of education for their degree
of development are placed in the negative part of this factor. Countries
with high education levels and low income are placed in the positive part.
We named it the "type of development factor". The results of this analysis
are very similar to those of section 2.a.
TABLE 7 - Correlations variables-factors.
| Active
Variables |
Correlations
|
| Labels |
F11,97
|
F21,97
|
| LE7 |
-0.95
|
0. 02
|
| LRA7 |
-0.85
|
0. 44
|
| GER7 |
-0.90
|
0. 22
|
| GDA7 |
-0.92
|
-0.37
|
| Illustrative
Variable |
|
|
| HDI7 |
-1.00
|
-0.06
|
In graph 3, the countries are projected onto
the first factorial plane. The first two factors account for 91,76% of
the total inertia. The countries with balanced development are accurately
explained by the first factor (development factor). Those with unequal
development (high income-low education or vice-versa) are better explained
by the 2nd factor. Our ranking of the countries for 1997 in
the first factor is practically the same as that of the UNDP for the same
year.
GRAPH 3
3.b - CLUSTER ANALYSIS
In this section the results of a cluster
analysis of the data base for 1997 is presented. Ward´s criterion
was used with the two first factorial co-ordinates. Five classes were selected.
The relation inertia inter/total inertia is 0.9022. The classes are the
following:
Class 1: Canada, Switzerland, Japan,
Sweden, Norway, France, Austria, USA, Netherlands, United Kingdom, Australia,
Belgium, Iceland, Denmark, Finland, Luxembourg, New Zealand, Israel, Barbados,
Ireland, Italy, Spain, Hong Kong, Greece, Cyprus, Czech Republic, Korea
Rep., Uruguay, Trinidad and Tobago, Bahamas, Argentina, Chile, Costa Rica,
Malta, Portugal, Singapore, Panama, Poland, Antigua and Barbuda, Bahrain,
Fiji, United Arab Emirates, Dominica, Grenada, Slovakia and Slovenia.
Class 2: Hungary, Venezuela, Colombia,
Kuwait, Mexico, Thailand, Qatar, Malaysia, Mauritius, Brazil, Saudi Arabia
Turkey, Saint Vincent, Saint Kitts and Nevis, Rep. Arab Syria, Santa Lucia,
Libyan Arab J., Tunisia, Seychelles, Suriname, Iran, Belize, Oman, Lebanon
and Algeria.
Class 3: Lithuania, Estonia, Lithuania,
Federation Russia, Belarus, Ukraine, Bulgaria, Armenia, Kazakhstan, Jamaica,
Georgia, Azerbaijan, Rumania, Ecuador, Republic de Moldova, Albania, Turkmenistan,
Kyrgyzstan, Paraguay, Cuba, Sri Lanka, Uzbekistan, South Africa, China,
Peru, R. Dominican, Tajikistan, Jordan, Philippines, Korea Dem. Rep, Samoa,
Indonesia, Guyana, Maldives, Croatia and Macedonia.
Class 4: Botswana, Iraq, Mongolia,
Nicaragua, Guatemala, Egypt, Morocco, El Salvador, Bolivia, Gabon, Honduras,
Vietnam, Swaziland, Vanuatu, Lesotho, Zimbabwe, Cape Verde, Congo, Cameroon,
Kenya, Solomon Isles, Namibia, Sao Tomé and Principe, Papua, Myanmar,
Pakistan, Laos, Ghana, India and Equatorial Guinea.
Class 5: Madagascar, Côte d’Ivoire,
Haiti, Zambia, Nigeria, Zaire, Comoros, Yemen, Senegal, Togo, Bangladesh,
Cambodia, Tanzania, Nepal, Sudan, Burundi, Rwanda, Uganda, Angola, Benin,
Malawi, Mauritania, Mozambique, R. Central African Rep., Ethiopia Bhutan,
Djibouti, Guinea-Bissau, Gambia, Mali, Chad, Nigeria, Sierra Leona, Burkina
Faso, Guinea and Eritrea.
Table 8
|
Classes
|
Effective
|
Co-ordinates
|
| |
|
F1
|
F2
|
|
Class 1
|
48
|
-1.71
|
-0.13
|
|
Class 2
|
25
|
-0. 91
|
-0. 55
|
|
Class 3
|
36
|
-0. 39
|
0. 62
|
|
Class 4
|
30
|
0. 88
|
0. 12
|
|
Class 5
|
36
|
2. 57
|
-0. 17
|
Table 8 shows the co-ordinates of the classes
and their effectives. It can be observed that classes 2 and 3 have close
co-ordinates in factor 1 but opposite in factor 2. Countries belonging
to class 2 have high income however countries belonging to class 3 have
high education. Table 9 ratifies this, the mean of LRA7 and GER7 of class
3 are higher than those of class 2 while the mean of GDA7 is higher in
class 2 than in class 3. The centers of gravity of the 5 classes selected
can be seen in graph 3. The cluster with 5 classes allows us to discriminate
the type of development.
UNDP has published a cluster of 3 classes
based only on the values of HDI. Using the first two factorial co-ordinates
and Ward’s criterion a cluster of three classes would be obtained and would
give practically the same result. The election made by UNDP is therefore
ratified by Ward's criterion.
Table 9
| Classes |
HDI
|
LE7
|
LRA7
|
GER7
|
GDA7
|
|
Mean
|
s.d.
|
Mean
|
s.d.
|
Mean
|
s.d.
|
Mean
|
s.d.
|
Mean
|
s.d.
|
| Class
1 |
0.91
|
0.03
|
75.52
|
2.48
|
96.01
|
4.49
|
79.83
|
8.58
|
5975.88
|
204.61
|
| Class
2 |
0.81
|
0.04
|
69.96
|
2.35
|
79.27
|
13.75
|
67.56
|
8.01
|
5697.48
|
364.33
|
| Class
3 |
0.70
|
0.06
|
68.77
|
2.83
|
93.42
|
5.77
|
70.28
|
7.01
|
3248.22
|
993.38
|
| Class
4 |
0.54
|
0.06
|
59.98
|
6.31
|
66.27
|
13.32
|
56.33
|
10.31
|
2475.30
|
1002.88
|
| Class
5 |
0.31
|
0.06
|
48.23
|
6.67
|
43.37
|
14.26
|
34.56
|
11.82
|
1014.92
|
365.33
|
3.c – Multiple Factorial Analysis
In this section, the results of a Multiple
Factorial Analysis of the database of 1997 are presented. Multiple Factorial
analysis is a methodology adapted to the treatment of tables in which a
set of individuals is described by several sets of variables and in which
the influence of the different sets is balanced. It allows, not only the
study of similarities among individuals and the relations among variables,
as the classical factorial methods do, but also the comparison of sets
of variables. It was proposed by B. Escoffier and J. Pagés (1983).
As said before, the weight given by MFA to
the variables in a set, is the inverse of the first eigen value of PCA
of that set. This weight balances the influence of each set in the first
global axes. As the weight is the same for all the variables in a set,
the internal structure is maintained
Table 6 shows the eigen values and the distribution
of the inertia in global analysis. Due to the weighting, the maximum value
of the first eigen value can be the number of sets, in this case 3. The
first eigen value is 2.5773 and points out a direction of the inertia which
is very important for the three sets.
Table 10 shows the co-ordinates, the contributions
and the square cosines of the active sets.
Table 10
| |
|
|
CO-ORDINATES
|
CONTRIBUTIONS
|
SQ COSINES
|
|
SET
|
RE WEIGH
|
DISTANCE
|
1
|
2
|
1
|
2
|
1
|
2
|
|
SET 1
|
0.33
|
1.00
|
0.89
|
0.00
|
34.7
|
0.2
|
0.80
|
0.00
|
|
SET 2
|
0.33
|
1.01
|
0.84
|
0.12
|
32.7
|
49.9
|
0.70
|
0.02
|
|
SET 3
|
0.33
|
1.00
|
0.84
|
0.13
|
32.6
|
49.9
|
0.71
|
0.02
|
|
ENSEMBLE
|
100.0
|
100.0
|
0.74
|
0.01
|
The co-ordinates are always between 0 and
1. They are interpreted as the inertia projected by the variables of set
j on the axes. The co-ordinates of the three sets on the first factor are
very high, this ratifies that the first axe in the global analysis is a
very important direction of the inertia of each set. The square cosines
measure the quality of the representation of a set as a unique point, in
the space of the sets, on a factor. The quality of the representation on
the first factor is very good.
Table 11 shows the correlations between the
factors of the global analysis and the sets
Table 11
|
FAC.
|
1
|
2
|
|
SET 1
|
0.95
|
0.02
|
|
SET 2
|
0.92
|
0.38
|
|
SET 3
|
0.92
|
0.37
|
The high correlations between the first factor
and the sets permit us to confirm that this is a common factor of inertia.
When this common direction exists, it is interesting to measure and study
the importance of the different sets. The importance of a factor in a set
is measured by the accumulated inertia of the variables of that set on
that factor and is shown in table 10. As it can be seen in table 10, the
three sets have practically the same importance. Factor 2 is a common factor
of inertia only of sets 2 and 3.
Table 12 presents the relation Inertia inter/
total inertia
Table 12
It measures the global interest of the simultaneous
representation of the clouds. The representation of these sets on factor
1 is totally justified.
Graph 4 shows the representation of the 3
sets of variables on the first factorial plane.
GRAPH 4
The strong co-ordinate of the three active
sets on the first factor (our proposal as a new human development index)
ratifies that this factor is a very important common direction of inertia
of each of the sub-clouds of variables. The weak co-ordinates on factor
2 are due to the fact that the first factor is extremely strong.
The graph of the mean individuals (individuals
as a center of gravity of the individuals in each of the three sets) is
not presented as it is the same as graph 3. The graph of partial clouds
permits us to see the mean individual –center of gravity of the three sets
– and the individual from the point of view of each of the sets (graph
5 and 5.1).
Spain has an equilibrated development, the
three sets present similar abscissas. Cuba and Oman present unbalanced
and opposite developments. Cuba is on the side of education. The Health
set has the most negative abscissa (that is the highest development), then
comes the Education set and finally, Income. Oman is on the positive part
of axe 2, on the side of income in this graph. The Income set presents
the highest negative abscissa (highest development), the second set is
Health and the last one Education. Benin presents an equilibrated development
but not as equilibrated as Spain.
MFA allows the simultaneous representation
of the three sets and therefore gives a more detailed analysis of the development
of each country. It completes the WPCA study (section 3).
GRAPH 5
GRAPH 5.1
4. - DATA BASES OF 1994 AND 19977
4.a – Comparative Study of Human Development
in 1994 and 1997 using Multiple Factorial Analysis
In this section, the results of a Multiple
Factorial Analysis applied to two sets are presented. Th first set is formed
by the database used by the UNDP to elaborate the HDI of 1994 and the second
one by the data used to elaborate the HDI of 1997. Both tables cross 169
countries and 4 variables structured in three sets. The table of data for
1994 is the one used in section 2.a and for 1997 the one of section 3.a.
There were three important problems to solve
in order to make the comparison:
a. - Different variables had been used in
both years.
b. - The closest set in time had to have
the same importance as the most distant one.
c. - Inside each year, the influence of each
of the sets had to be equilibrated.
MFA balances the sets of the variables in
the analysis, the computer program in which it is implemented solve points
a and b.
But the computer AFM program does not recognize
sets inside the sets. To solve point c) variables have been introduced
in each of the two sets weighted by the inverse of the first eigen value
of the set and by the variance of each variable. For example the variables
belonging to the Education set for 1994, LRA4 and MYS4, have been introduced
already divided by the square root of the first eigen value of the PCA
of both variables, and also divided each by their the standard deviation.
In this way, on one hand the two sets 94 and 97 have been balanced and
on the other the three sets inside 94 and the three inside 97 have also
been balanced.
In tables 1 and 6 the first eigen values of
each set were presented.8 The matrix of correlations
between partial factors is presented in table 13, the correlation between
the two first factors is 0.97 and between the two second factors 0.72,
this points out that the structure of the inertia is kept in time.
Table 13- Matrix of correlations
between partial factors
| |
101
|
102
|
201
|
202
|
|
101
|
1.00
|
|
|
|
|
102
|
0.00
|
1.00
|
|
|
|
201
|
0.97
|
0.03
|
1.00
|
|
|
202
|
-0.07
|
0.72
|
0.00
|
1.00
|
| |
101
|
102
|
201
|
202
|
Table 14 presents the first two eigen values
of the global analysis.
Table 14- Eigen values and
distribution of the inertia in the global analysis
|
Eigen values
|
Percentage
|
Percentage cum
|
|
1.9680
|
82.81
|
82.81
|
|
0.1726
|
7.26
|
90.07
|
The first factor, due to the weighting, can
value at most the number of sets, 2 in this case. The first value is 1.9680
and points out that it is a very important direction of the inertia in
both sets.
Tables 15 and 16 show the matrices L and RV,
which give information about the relations between sets. The elements of
matrix L are indexes of relation between sets in the sense of the MFA.
Table 15- Matrix L
| |
YEAR 94
|
YEAR 97
|
|
YEAR 94
|
1.01
|
|
|
YEAR 97
|
0.94
|
1.02
|
The diagonal elements of this matrix point
out the multidimensionality of the set. Both sets are practically unidimensional,
this ratifies the results obtained in sections 2.a and 3.a and the indexes
there proposed.
Matrix RV is obtained by dividing each term
of matrix L by the product of the norm of the corresponding sets. The elements
belonging to the principal diagonal are therefore 1. The other terms of
the matrix have values between 0 and 1, and they will be closer to one,
when the sets are more related, in MFA sense. When it takes value 1, it
means that both clouds are homothetic.
Table 16 - Matriz RV
| |
YEAR 94
|
YEAR 97
|
|
YEAR 94
|
1.00
|
|
|
YEAR 97
|
0.93
|
1.00
|
RV(1,2)=0.93 pointing out that both clouds
are practically homothetic. This result is important to get the equation
of the human development index in time.
Table 17 shows the co-ordinates, the contributions
and the square cosines of the active sets.
Table 17
| |
CO-ORDINATES
|
CONTRIBUTIONS
|
SQ COSINES
|
|
SET
|
RWEIGH
|
DISTANCE
|
1
|
2
|
1
|
2
|
1
|
2
|
|
SET 1
|
0.50
|
1.01
|
0.98
|
0.08
|
50.0
|
43.8
|
0.96
|
0.01
|
|
SET 2
|
0.50
|
1.02
|
0.98
|
0.10
|
50.0
|
56.2
|
0.95
|
0.01
|
|
CONJUNTO
|
100.0
|
100.0
|
0.95
|
0.01
|
The co-ordinates are practically 1, this ratifies
that the first factor is a very important direction of the inertia of each
of the sets. The square cosines measure the quality of the representation
of each set as a unique point in the space of thE sets on a factor. The
quality of the representation of the two sets in the first factor is excellent.
Table 18 shows the correlations between the
factors of the global analysis and the sets.
Table 18
| |
CORRELATIONS
|
|
FAC
|
1
|
2
|
|
SET 1
|
0.99
|
0.87
|
|
SET 2
|
0.99
|
0.91
|
The high correlations observed between the
first factor and the sets allow us to confirm this as a common factor of
inertia. When this common direction exists, it is interesting to measure
and to study the importance of the different sets. The importance of a
factor in a set is measured by the accumulated inertia of all the variables
of that set and it can be seen in table 17, both sets are equally important
. Factor 2 is also a common factor of inertia.
Table 19 - Relation Inertia
Inter/ Total Inertia
| |
RELATION
|
|
FAC
|
1
|
2
|
|
0.98
|
0.79
|
Table 19 presents the relation between inertia
inter / total inertia, this measures the global interest of the simultaneous
representation of both clouds. In this case, it is high and the simultaneous
representation is totally justified.
The analysis of the active variables (graph
6) allow us to interpret the first axe as a general level of human development,
all the variables have high co-ordinates in this factor. The second axe
describes the type of development more centered in Education (positive
part of the 2nd axe in this graph) or in Income (negative part
of the 2nd axe).
GRAPH 6
Graph 7 shows the representation of the two
sets of variables in the first factorial plane
GRAPH 7
The strong co-ordinate of the two active
sets on the first factor, which can be interpreted as general level of
development, ratifies that this factor is a very important direction of
inertia of the two clouds of variables. The graph of mean individuals on
the plane of the two first axes (graph 8 ) is very similar to the one obtained
by WPCA of the years 94 and 97 separately, this ratifies the homothecy
of both clouds.
GRAPH 8
The graph of partial clouds (graph 9) allow
us to see an individual from the total point of view and from each of the
sets. Spain, Philippines and Oman have improved from 94 to 97, their abscissas
are more negative in 97 than in 94. Rwanda has worsened.
GRAPH 9
4.b - An index of human development in
time
In the study of human development in 1994,
the following factor was obtained by using WPCA
The
equation obtained for F1 with the data of 1997 was:
We have seen in the study in section 4.a that
both clouds are homothetic, this allows us to propose the following equation
as an index of human development in time:
In equation (1) d indicates the
variable divided by the standard deviation. Typified variables are not
used here because to subtract the mean, causes the size effect to disappear.
In general, the variables used in equation (1) tend to increase and the
human development, although slowly, to improve. Variables are divided by
the standard deviation so that the units of measurement do not affect the
index. The correlation between HDIt and the HDI proposed by
the UNDP for 1994 is 0.997 and for 1997 is 0.996.
Table 20 shows the 22 countries that worsened
from 1994 to 1997
Table 20
|
Rwanda
|
-0.91
|
|
Latvia
|
-0.76
|
|
Iraq
|
-0.51
|
|
Lithuania
|
-0.47
|
|
Armenia
|
-0.37
|
|
Estonia
|
-0.36
|
|
Ukraine
|
-0.35
|
|
Moldova R
|
-0.29
|
|
Sierra Leone
|
-0.27
|
|
Georgia
|
-0.25
|
|
Russian F.
|
-0.23
|
|
Albania
|
-0.22
|
|
Madagascar
|
-0.14
|
|
Burundi
|
-0.13
|
|
Azerbaijan
|
-0.12
|
|
Luxembourg
|
-0.12
|
|
Haiti
|
-0.08
|
|
Bulgaria
|
-0.03
|
|
Tajikistan
|
-0.03
|
|
Hungary
|
-0.01
|
|
Ethiopia
|
-0.01
|
|
Niger
|
-0.01
|
Rwanda had a value in HDIt for
1994 of 3.6; her relative loss was 25% while Luxembourg had a value of
8.139 its relative loss was 0.014%.
The variable GER is not a good variable for
Luxembourg; this country sends the students of university level to study
in France and in Belgium.
Rwanda, Burundi and S.Leone are countries
with armed conflicts. The ex USSR republics are in a deep crisis.
5. – CONCLUSIONS
1. - MFA enables us to study development through
three balanced sets: health, education and income.
2. -The first factor of MFA is a linear combination
of the original variables with maximum variance. It was constructed balancing
the sets of variables. We have proposed this factor as an index.
3. - The second factor gives information about
the type of development.
4. - Factorial planes permit us to evaluate
the information given by the variables and by the sets and help in the
process of selecting them.
5. - The cluster analysis of 5 classes classifies
the countries by type and level of development.
6. - MFA allows the simultaneous study of
human development in different moments in time, with any kind of variables
and sets. It balances the sets so that the closer ones will weigh the same
as the more distant ones.
7. - This study shows how the structure of
the inertia in both years 1994 and 1997 is practically the same, that both
clouds are practically homothetic and that the first factor of global analysis
is very strong and shows the direction of inertia that is shared by both
sets. This allows us to propose equation 1 as an index of human development
in time.
6. – BIBLIOGRAPHY
[1] Balzategui Andreiñua C. , G. del
Valle Irala T., Martínez Arnaíz A. y Puerta Gil C.(1999)
Estudio
comparado del desarrollo Humano mediante el Análisis Factorial Múltiple
y propuesta de un índice de desarrollo en el tiempo. Documentos
de trabajo. Economía Aplicada, Serie Economía Matemática
D.T.4,1999
[2] Escofier B. and Pagès J (1983)
Méthode
pour l’analyse de plusieurs groupes de variables. Application à
ls caractéristation des vins rouges du Val de Loire. Revue Statist.
Appl. 31, p 43-59
[3] Escofier B. and Pagès J (1988-1990-1993)
Analyses
Factorelles Simples et Multiples. Objectifs, Méthodes et Interprétation.
Dunod.
[4]G.del Valle Irala T. y Puerta Gil C. (1998-a)
Weighted
Principal Components Analysis applied to the study of Human Development.
Congés International Dánalyses Multidimensionnelles des Données
, editors K. Fernández Aguirre and A Morineau.
[5]G. del Valle Irala T. y Puerta Gil C. (1998-b)
Human
Development: Weighted Principal Components Analysis applied to the data
of 1997 report. Congreso: ISI (International Statistical Institute)
on "Statistics for economic and social development", september 1998
[6]G. del Valle Irala T. y Puerta Gil C (1999)
Análisis
Factorial Múltiple aplicado al estudio de la tabla de datos sobre
desarrollo humano del Informe de 1997. Documentos de trabajo. Economía
Aplicada, Serie Economía Matemática D.T.5,1999
[7] Lebart L., Morineau A. et Piron M. (1995)
Statisque
exploratoire multidimensionnelle. Dunod
[8] Puerta Gil C.(1996) Aplicación
del Análisis de Componentes Principales y del Análisis Factorial
Múltiple para el estudio del Desarrollo.UPV/EHU
[9] United Nations Development Programme(1994)
HumanDevelopment Report 1994. New York: Oxford University Press
[10] United Nations Development Programme(1997)
Informe
sobre el Desarrollo Humano 1997 Ediciones Mundi-Prensa.
[11] World Development Report (1993) Investing
in Health New York: Oxford University Press
1 G.
del Valle Irala T, Statistics Professor , Escuela Universitaria de Estudios
Empresariales. Applied Economy Department, Basque Country University ,
Bilbao, Spain eupgairt@lg.ehu.es
Puerta Gil Carmen, Mathematics
Reader, Facultad de Ciencias Económicas y Empresariales,Applied
Economy Department , Basque Country University,Bilbao, Spain eupugic@bs.ehu.es
2 G.
del Valle Irala T. and Puerta Gil C.(1998-a)
3 B.
Escoffier and J. Pagès (1983, 1988, 1990, 1993)
4 The
equations of the factors of the different analysis are written in standard
variables
5 Puerta
Gil (1996)
6 G.
del Valle Irala T. and Puerta Gil C (1998-b)
7 This
study has been done in colaboration with C. Balzategui Andreiñua
, Mathematics Reader, Escuela Universitaria de Estudios Empresariales.
Applied Economy Department, Basque Country University , Bilbao, Spain eupbaanm@lg.ehu.es
A. Martínez Arnaiz ,
Statistics Professor , Escuela Universitaria de Estudios Empresariales.
Applied Economy Department, Basque Country University , Bilbao, Spain eupmaara@lg.ehu.es
8 Table
1 presents the eigen values for 1994 anf for 173 countries, only 169 are
used, and something similar happens with the data of 1997.
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