2023年12月11日发(作者：小学生必备的数学试卷分析)
Team #2750 Page 1 of 30
Can We Assess a Health Care
System\'s Performance?
参赛队员：董希望（自动化学院），
刘琳燕（城环学院）
刘福亮（软件学院）
指导教师：肖 剑
参赛单位：重庆大学
参赛时间：2008年2月15∼18日 Team #2750 Page 2 of 30
Can We Assess a Health Care System\'s Performance?
1. Background
Health systems consist of all the people and actions whose primary purpose is to
improve health. They may be integrated and centrally directed, but often they are not.
After centuries as small-scale, largely private or charitable, mostly ineffectual entities,
they have grown explosively in this century as knowledge has been gained and
applied. They have contributed enormously to better health, but their contribution
could be greater still, especially for the poor. Failure to achieve that potential is due
more to systemic failings than to technical limitations. It is therefore urgent to assess
current performance and to judge how health systems can reach their potential.
The World Health Organization (WHO) is a specialized agency of the United
Nations (UN) that acts as a coordinating authority on international public health.
Established on 7 April 1948, and headquartered in Geneva, Switzerland, the agency
inherited the mandate and resources of its predecessor, the Health Organization,
which had been an agency of the League of Nations.
The WHO\'s constitution states that its objective \"is the attainment by all peoples
of the highest possible level of health.\" Its major task is to combat disease, especially
key infectious diseases, and to promote the general health of the people of the world.
As well as coordinating international efforts to monitor outbreaks of infectious
diseases, such as SARS, malaria, and AIDS, the WHO also sponsors programs to
prevent and treat such diseases. The WHO supports the development and distribution
of safe and effective vaccines, pharmaceutical diagnostics, and drugs. The WHO also
carries out various health-related campaigns — for example, to boost the consumption
of fruits and vegetables worldwide and to discourage tobacco use.
The annual World Health Report (/whr/en/)
assesses global health factors and World Health Statistics provides health statistics for
the countries in the UN. The production and dissemination of health statistics is a
major function of the WHO. To many people, these data and the associated analyses
are considered unbiased and very valuable to the world community.
2. Basic Assumption and Hypotheses
1. Assume that in a certain interval such as 5years the main components (metrics) of
the health care system stays steady, that is to say the metric won’t change
continually.
2. Assume that all the statistics we get from the database of the WHO is authentic.
3. Assume that the ranking of the world\'s health systems in 2000 made by WHO is
scientific and dependable.
4. During the data processing if a data little than x we can replace it with x.
5. If there existing data missing for some year’s indicator we can value it with the
corresponding value of the near years. Team #2750 Page 3 of 30
3. Symbols
Symbol
Z\'
Z
zju
umlijR1
ciSiQ
X
U
aijA
aibi
Definition and Property
The matrix before standardization
The matrix after standardization
The statistic of the j indicator to each country
The main component to be evaluated
The m th main component of the indicator
The load of the original indicator
The correlation matrix of Z
The contribution rate of the i th main component
The summation of the front i main components’ contribution rate
The integrated score of each country
The project set ( the Member States)
The attribute set which also means main component set
The attribute value of xi in reference to ujThe decision making matrix
The mean value of the line I in the primitive matrix
The standard deviation of row I in the primitive matrix
4. Problem Analysis
To determine several important and viable metrics for assessing the performance
of a health care system and comparing health care systems in different countries. We
have to know what metrics or indicators are there in a health care system, as is shown
in the problem we search the web of the WHO and get the database of the indicators.
There exists statistics for 50 core indicators on mortality, morbidity, risk factors,
service coverage, and health systems, which take on more than one hundred and fifty
terms of raw indicators. We must use some data mining technology or method to
distill the crucial metrics.
Considering the data is promiscuous and inconsistent and not all the countries
have the corresponding data to each indicator from the year 1960 to 2006, we first
need to choose certain year’s data as our study object. Then to the mass actual
statistical data we can’t expect all the indicators are complete so what to do with the
incomplete data to make sure that all the indicators or all the data we used below are
universal or effective is an inevitable problem. There are 159 raw indicators how
could we select the most important ones and combine them scientifically to make
them more useful in measuring quality is another basal problem. Then how could we
accomplish this goal? The main components analysis method which we could use to
devise our first model will help a lot.
Furthermore how could we assess a country’s health care system and make some
comparisons with the combined metrics? This situation much agrees with the multiple
attribute decision problems. So we could solve this problem by ranking all the
countries health care systems using this multiple attribute decision method. Team #2750 Page 4 of 30
5. The Establishment of Model
5.1 The Primary Data and Indicators Processing
According to the above problem analysis part we know that we could obtain
enough raw data for almost 159 indicators from 1960 to 2006. We first choose a year
2004 whose data is much completer than other years as our study object. Then if some
of the indicators of certain country in 2004 have no value and the year close to 2004
such as the year of 2005 or 2003 has the corresponding value we treat this close value
as the valve of the country in that indicator in 2004.
Based on these we select the indicators that 95% of the country has the
corresponding data for them from all the 159 raw indicators. By doing this primary
selection we make sure that all the indicators or all the data we used below are
universal or effective. After the primary selection we get 48 crucial indicators as our
primary outcomes (metrics).
5.2 Model 1 Design
Following the above analysis we utilize the main components analysis method to
devise our first model.
When it comes to main components analysis the biggest effect to it is the
dimension of the data. So in the practical application we first should make
standardization to the data.
Assume that Z\' is the matrix before standardization Z is the matrix after
standardization zj is the statistic of the j indicator to each country; u is the main
component to be evaluated, so the objective function could be:
⎧u1=l11z1+l12z2+\"l1pzp⎪⎪u2=l21z1+l22z2+\"l2pzp (1)
⎨\"\"\"\"⎪⎪u=lz+lz+\"lzmpp⎩mm11m22
Where u1, u2,… um
is called the 1st, 2nd,
… mth main component of the indicator
z1, z2, zp; lij is the load of the original indicator zj (j=1,2,
…,p) in each main
component.
The detailed process of this solution is as follows:
Step1: Evaluate the standardized matrix Z of the matrix
Z\'The standardization of theZ\'is just replace the
zi\' (i=1,2,
…,p) and the zij Team #2750 Page 5 of 30
of the matrixZ\'with zi(i=1,2,
…,p) and with zij
respectively, which is shown in table1
Table1. The process of the standardization of the matrix
number z1z2z3z4
1
2
3
n
zij=z\'ij−z\'jDz\'jzp
z\'ij,i=1,2,\",n;j=1,2,\"p
Step 2: Evaluate the correlation matrix R1 of matrix Z
R1 could be evaluated by the following matrix:
⎡r11⎢r21R1=⎢⎢#⎢⎢⎣rp1r12\"r1p⎤⎥r22\"r2p⎥ (2)
⎥#\"#⎥rp2\"rpp⎥⎦Where rij (i, j =1, 2,…,p) is the original indicator zi and zj’s correlation coefficient
specially rij=rji. rij could be derived by the following formula:
n rij=∑(zk=1nk=1ki−zi)(zkj−zj)n (3)
22(z−z)(z−z)ij∑ki∑kjk=1Step 3: By formula 1 we can compute the characteristic root and characteristic vector
of matrix R1 then rank the characteristic values of R1 as expression 1
λI−R1=0 (4)
\"
λ
1
≥
λ
2
≥
≥
λ
p
≥
0 (5)
Step 4: Evaluate the contribution rate and the accumulative contribution rate
according to formula 1and 1of the main components, determine the proper number of
the main components.
λic=(i=1,2,\",p)ip
∑
λ
k (6)
k=1 Team #2750 Page 6 of 30
Where ci is the contribution rate of the i th main component.
i
λk
∑ (7)
k=1Si=p(i=1,2,\",p)
λk∑
k=1Here
Si is the summation of the front i main components’ contribution rate. If the
value of the accumulative contribution rate reaches to more than 80% we can
approbate the effect of the main components.
Step 5: Compute the load of the main component lij
lij=p(yi,zj)=λieij(i,j=1,2,\",p) (8)
Step 6: Sum up the above five steps get our objective function
⎧u1=l11z1+l12z2+\"+l1pzp⎪⎪u2=l21z1+l22z2+\"+l2pzp
⎨............⎪⎪u=lz+lz+\"+lzmpp⎩mm11m22Step 7: Evaluate the integrated value of each country and make a ranking of them
with the formula 1.
Q=1∑λi=1m(λ1u1+λ2u2+\"+λmum) (9)
iQ is the integrated score of each country.
5.3 Model 2 Design
5.3.1 The Description of the Principle
The method of the multiple attribute decision making based on dispersion
maximization is used to solve the multiple attribute decision making problems with
the uncertain weight attribute. We can use this method to make ranking and
comparison between different projects with multiple attributes.
In more details the smaller the difference between certain attribute for all the
projects is the less affection it has on the decision making and ranking of the projects.
On the contrary the bigger it is the more affection it has on the decision making and
ranking. As a result in the view of ranking the bigger of one attribute’s deviation is the
bigger weight of this attribute should be given. Especially if there is no deviation for
certain attribute to all the projects which means that this attribute will have little
affection on the ranking we can value a zero to its weight.
5.3.2 Model Development
Step ure and Normalize the Decision Making Matrix Team #2750 Page 7 of 30
1.1 Structure the Decision Making Matrix
Assume that:
M={1,2,…,m},N={1,2,…,n} (10)
The projects set which also is the set of the Member States in WHO is X
X={x1,x2,…,xn} (11)
The attribute set which means main component set here is U
U={u1,u2,…,um} (12)
is the attribute value of in reference to so we obtain the decision
making matrix
A=(aij)n×mwhose form is shown as table 2
x1x2#
xn
Table2. The form of the decision making matrix
u2… umu1a11a12… a1ma21a22… a2m
#
#
#
an1an2… anm5.4 Model 3 Design
Model 3 is our predictive model, from model 1 we can get the objective function
with the data of that year.
When it comes to predicating for the convenience of evaluating the main
components we can change the main component which is expressed by the
standardization indicator zi into the form that expressed by nonstandard indicator zi’ to
predicate the main components.
⎧u1=l\'11z\'1+l\'12z\'2+\"+l\'1pz\'p+l10⎪\'\'\'\'\'\'⎪u2=l21z1+l22z2+\"+l2pzp+l20⎨............⎪⎪u=l\'z\'+l\'z\'+\"+l\'z\'+lm11m22mppm0
⎩m(13)
Here
zi=(z\'i−ai)/bi (14)
Substitute
z\'iinto formula 1 can we obtain the formula 2.
aiis the mean value of the line i in the primitive matrix; biis the standard deviation
of row i in the primitive matrix.
Then utilize the formula 9 to compute the synthetic score and get variability of
Team #2750 Page 8 of 30
the system.
Normalize the Decision Making Matrix
There are many types of attributes such as benefit type, cost type, fixation type,
deviate type, interval type, deviate interval type etc. In our model all the attribute
could be sorted to two types the benefit type and the cost type approximately. The
benefit cost requires the value of the attribute as big as possible; the cost type requires
the value of the attribute as small as possible.
To eliminate the impact of the different dimensions to the decision making result
we should normalize the decision making matrix A whose values could be obtained
from the model 1.
Assume that Ii
(i=1, 2) stands for the subscript set of the benefit type and cost
type. If the attribute is benefit type we value i in Ii as 1. If the attribute is cost type
we value i in Ii as 2.
aij−min(aij)i,i∈N,j∈I1 (15)
rij=max(aij)−min(aij)ii
rij=max(aij)−aijimax(aij)−min(aij)ii,i∈N,j∈I2 (16)
After this step we get the normalized matrix
R=(rij)n×m whose form is the same
with the matrix A.
Step 2: Calculus the optimization weight vector
w
wj=∑∑i=1k=1mnnj=1i=1k=1nnrji−rkj,rij−rkjj∈M (17)
∑∑∑Where wj
is the j th main component’s weight.
Step 3: Computer the synthetic attribute zi(w) (i∈N) of project xi.
zi(w)=∑rijwj,i∈N,j∈M (18)
j=1mStep 4: Make ranking and comparison to the projects (countries) using zi(w)(i∈N)
6. Applying the Model1 and Model 2
6.1 Applying the Model 1 to the Statistics of the Year 2004
6.1.1 Data for Model 1 in the Year of 2004
We first select the indicators that 95% of the countries own these indicators from
all 159 indicators getting 28 indicators which could be seen in appendix
Ⅰ. Then we Team #2750 Page 9 of 30
select the countries that have the data for all these 28 indicators from all 193 Member
States getting 163 countries. By doing these we have made good preparation for our
model 1’s solution.
6.1.2 Solution of the Model 1 for the Year of 2004
Based on the above data we solve our model 1 in matlab using the function of
zscore to normalize the data, and then we calculate the characteristic roots and
characteristic vector. The characteristic roots are shown in the table 3.
Table3. Part Valves of Model 1
From the table 3 we can see that the front six red colored components’
accumulative contribution rate reaches to 81.5% which means that most of the main
components are involved, so these six components are just our combined metrics. We
renamed these six combined indicators with A, B, C, D, E, F metrics all of which are
constituted by several raw indicators and could reflect certain performance of a health
care system.
In more detail the metric A is much positively related with life expectancy, per
capita total expenditure on health at international dollar rate etc and much negatively
related with mortality rate, incidence of tuberculosis (per 100 000 population per year)
etc. The visual relationship between the metric and the 28 indicators is shown in
figure 1. The x axis is the order of the 28 indicators which maps to corresponding 28
indicators in appendix
Ⅰ. The y axis is the affection of each of the 28 indicator on
metric A. All the rest five figures follow this instruction so we won’t explain the rest
five figures again. Team #2750 Page 10 of 30
Figure1. The affection of the 28 indicators on metric A
The metric B is much positively related with expenditure on health, disease
detection rate etc and much negatively related with government expenditure on health,
alcohol consumption etc.
Figure2. The affection of the 28 indicators on metric B
The metric C is much positively related with General government expenditure on
health as percentage of total expenditure on health, immunized with disease etc and
much negatively related with private expenditure on health as percentage of total
expenditure on health, population (in thousands) total etc.
Figure3. The affection of the 28 indicators on metric C
The metric D is much positively related with private expenditure on health as
percentage of total expenditure on health, immunized with disease etc and much Team #2750 Page 11 of 30
negatively related with General government expenditure on health as percentage of
total expenditure on health etc.
Figure4. The affection of the 28 indicators on metric D
The metric E is much positively related with external resources for health as
percentage of total expenditure on health etc and much negatively related with
tuberculosis: DOTS case detection rate, probability of dying (per 1 000 population)
between 15 and 60 years etc.
Figure5. The affection of the 28 indicators on metric E
The metric F is much positively related with immunized with disease,
out-of-pocket expenditure as percentage of private expenditure on health etc and
much negatively related with general government expenditure on health as percentage
of total government expenditure, population (in thousands) total etc.
Figure6. The affection of the 28 indicators on metric F Team #2750 Page 12 of 30
The above descriptions show that our metrics is reasonable, moreover all the
28indicarors we selected could be found in 92% of all Member States, which means
that our metrics could be easily collected.
Furthermore we get the ranking for all the 163 countries that own orbicular and
effective data. The front and the back 20 countries in our ranking and their scores
calculated by our model 1 are listed as table 4:
Table4. Part of our Ranking by Our Model1
The whole ranking is shown in appendix
Ⅱ.
After obtaining the six metrics we treat the still missing value’s indicator as zero
then recompute the ranking of the year 2004 with the model 1 and get another ranking
for all the Member States which we list in appendix
Ⅲ.
In conclusion we put forward 28 important indicators from all the 159 indicators
furthermore we combine the 28 important indicators getting 6 main components
which we renamed as metric A, B, C, D, E and F. Then we assess the health care
system with these six metrics and make a ranking of all the 163 countries.
6.1.3 Applying model 1 to the Statistics of the Year 2000
Using model 1 and the six metrics obtained from 4.11 we assess the health care
system of each country in the year of 2000. This time we only utilize the data of this
year, which means that we just substitute the data of 2004 with that of 2000.
By doing this we get the ranking of this year as table 5 which just show out the
front and the back 20 countries too.
Team #2750 Page 13 of 30
Table5. The Part Ranking of the Year 2000 by Model 1
(There are 194 Member States in 2000; the score here is just a relative value
computed by our model; the whole ranking is shown in appendix
Ⅳ)
6.2 Applying the Model 2
The six metrics obtained from model1 is ordered. Although the six metrics keep
the same in the model 2 as what they are in model 1 according to our assumptions,
there is no certain order among them in our model 2. The data processing methods for
the raw data are the same with what we have described and used before.
6.2.1 Applying the Model 2 to the year of 2004
With the help of the software matlab we realize the algorithm of dispersion
maximization computing the weight of the six metrics, and then we calculate the
synthetic score of each of the 163 country that own holonomic statistics after our data
mining process. After comparing the synthetic score of these countries we get the
ranking of their health system as shown in table 6.
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Table6. Part of the ranking for the 163Member States
(The score here is just a relative value computed by our model; the whole ranking
is shown in appendix
Ⅴ)
6.2.2 Applying the Model 2 to the year of 2000
Similar to 4.1.2 we just replace the data of 4.2.1 with the data of the year 2000
then compute the synthetic score of all the 194 Member States. After comparing the
different countries we get the ranking as table7.
Table7. Part of the ranking for 2000 by model 2
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7. Comparisons
7.1 Comparisons between Different Rankings
From the above solution we obtain 4 different rankings. The precise clues of our
models have already showed their validity. Besides we can load a ranking for all the
190 Member states in 2000 from the WHO’s official web which we list in appendix
Ⅷ.By making comparisons between our two rankings with the official ranking of the
year 2000 we can test the reliability and practicability of our model to a certain extent.
The figure7 shows their relationship clearly.
Figure7. The corresponding relationship between our rankings and the WHO’s
We can see that the dots which stand for parts of the countries in the rankings
match quite well with each other in the three polygonal lines. That means the model1
and model 2’s results not only agree with each other but also agree with the official
results quite well. So we can conclude that our two models are practical and
reasonable.
Since the solution for the year of 2000 is dependable, we have reason enough to
predicate that our solution for 2004 is authentic as the only difference between 2004
and 2000 is the substituted statistics and the data of 2004 is more holonomic than that
of 2000.
To make sure that both of our models’ results for the year 2004 are unitive we make
a comparison between their rankings. We select some characteristic countries in both
rankings and compare those countries rankings as shown in figure 8. Team #2750 Page 16 of 30
Figure8. The comparison between the two rankings for the year 2004
From the figure we find that the two rankings match quite well.
In conclusion our models are scientific and our results are authentic.
7.2 Comparisons between US and France
In the 2000’s ranking of WHO France takes the first place, which also could be
seen clearly in the appendix. In the year 2004 there is no official ranking so we assess
these two countries health care system with our model 1 to see which country has the
better health care system then.
The table8 shows their score according to our six metrics:
Table8.
The comparisons between US and France’s health system according to our metrics
The metric A, C, F belongs to benefit type and the rest belong to cost type. Base on
this we can see that the health care system of US in 2004 is better than France in
metric A, B, D. According to our table 1 we know that the synthetic score of US is
better than France.
7.3 Comparisons between US and India
In the ranking of WHO the health system of US is better than India. With the
help of our mode 1 we consider that India has the poor health care system in 2004, so
we make a comparison between them.
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Table9.
The comparisons between US and India’s health system according to our metrics
Similar to 7.2 we can see that the health care system of US is better than India in
metric A, B, C, D, F. Also from the table 9 we know that the ranking of US is much
better than India.
8. Applying the Model 3
Based on model 1 and model 2 with the help of the software matlab we realize
the algorithm in model 3 and get the predictive function of the synthetic scores as
follows:
\'\'\'\'Q=−0.003381z1\'+0.0096692z2+0.010559z3−0.0019468z4−0.00052279z5\'\'\'\'−0.00052406z6−3.06×10−7z7−0.078213z8−0.0042194z9−0.000362z1\'0\'\'\'\'−0.0087351z11+0.0095689z12+0.0098489z13−0.00039226z14+0.0043929z1\'5\'\'\'\'\'+0.005344z16+0.025752z17−0.0037719z18+0.00014346z19+0.00018277z20\'\'\'\'\'+0.00010941z21+0.00013679z22−0.0053441z23+0.05644z24−0.0029234z25\'\'\'−0.00084042z26+0.02965z27+0.012447z28−3.8668(19)
Considering the affection of the weight on the synthetic score we could find that the
bigger the absolute value of weight is the bigger the impact is on the synthetic score
of the country. On the contrary if the absolute value of weight is small then the
variation of the metric won’t produce big changes to the synthetic score. Then we take
some indicators of the all 28 indicators as examples to discuss what affection it will
has on the health care system if the various changes are occurred.
\' is the formula is the total fertility rate (per woman). It has a negative
z8correlation with the synthetic score. What’s more it has a big affection on the score so
this indicator should be as small as possible, which means that the government should
take some measures to control the population within a proper range to improve the
health care system of the nation.
z\'24is the total expenditure on health as percentage of gross domestic product. It
is an indicator that positively related with the synthetic score which means that the
more it spend on the total expenditure on health as percentage of gross domestic
product the better score it has in the system.
z\'17is the general government expenditure on health as percentage of total
government expenditure. It is an indicator that positively related with the synthetic
score which means that the bigger the general government expenditure on health as
percentage of total government expenditure is the better score it has in the system
z\'3is the life expectancy at birth (years) males. It is an indicator that positively Team #2750 Page 18 of 30
related with the synthetic score which means that the longer the life expectancy at
birth (years) males is the better score it has in the system.
z\'11stands for the neonatal mortality rate (per 1 000 live births). It has a negative
correlation with the synthetic score. That’s to say the smaller the neonatal mortality
rate (per 1 000 live births)is the better the health care system will become.
9. The Strength and Weakness
9.1The Strength
We obtain the statistics directly from the raw database of the WHO’s official web
not from the report of the WHO. We use some data mining technology to draw the
available and effective data from thousands terms of data ourselves.
We develop three different models to solve all the six parts of the problem, those
models are built with precise logic, scientific principle which could solve the
problems efficaciously.
We don’t solve the problem part by part but solve them in our models’
development and solution process, which keeps the whole paper’s with a good
continuity.
We compare our result with the practical result, which tests our models’
practicability and validity greatly.
Our models could be easily extended to other fields to solve the
multiple attribute
decision making problems.
Our models are independent to the metric (indicators) to a certain extent as the
algorithm of our models has the universal applications.
9.2 The Weakness
The raw data we get is the data from the real world, which means that there must
be some imperfect data which do have some negative impact on our result.
As there are so many indictors that it is hard to select proper metrics to assess the
health system properly without some kind of error.
Because the limitation of the time and resource it’s inevitable to have some
imperfect aspects in our models, analysis and paper.
10. References
[1] Zeshui Xu, 8/2004, Uncertain Multiple Attribute Decision Making: Methods and
Applications, Tsinghua University Press.
[2] Qiyuan Jiang, Jinxing Xie, 12/2004, Mathematical Model, Higher Education Press
[3] The World Health Report 2000 - Health systems: improving performance.
/whr/2000/en/whr00_
[4] World Health Organization, /research/en/s
[5] Principal Component Analysis,
/jpkc/jldlx/admin/ewebeditor/UploadFile/
[6] /wiki/World_Health_Organisation,\"World Health Organization\" Team #2750 Page 19 of 30
11. Appendix
AppendixⅠ: The list of all the 28 indicators and their sequence number
AppendixⅡ: The Ranking of all 163 Countries in 2004 by model 1
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Appendix
Ⅲ: The Ranking of all 194 Countries in 2004 by model 1
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Appendix
Ⅳ: The Ranking of all 194 Countries in 2000 by model 1
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Appendix
Ⅴ: The Ranking of all 163 Countries in 2004 by model 2
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Appendix
Ⅵ: The Ranking of all 194Countries in 2004 by model 2
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Appendix
Ⅶ: The Ranking of all 194Countries in 2000 by model 2
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Appendix
Ⅷ: The Ranking of all 190Countries in 2000 by the WHO
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