数学中国w, madio,net The Mas! Expensive is Not the Best 155 The Most Expensive is Not the best Hongxing Hao Xiangrong Zer Boliang Sun National University of Defense Technology Changsha, Chin Advisor: Ziyang mao Abstract Motivated to evaluate healthcare systems more accurately, we analyze existing evaluation methods. Most methods mainly focus on outcomes and their metrics often ignore intemal characteristics of the healthcare systems. We devise two methods: animproved World Health Organization(WHO) method and a comprehensive evaluation method Theimproved WHO method uses the same metrics as the WHOmethod which are determined by the outcomes of the healthcare system. Our im- provement is to use a grey comprehensive evaluation and the principle of minimum loss of information to combine the metrics, rather than simply combining them linearly. In our comprehensive evaluation method, we define five new metrics that concern both outcomes and characteristics of the healthcare system itself, including the effect of the government and the basic situation of a country. Then we use the equal-interval method to get a final score. Compared with other methods, this one does a better job in distinguishing countries and in sensitivit After comparing with other four countries that represent the four main modes of healthcare systems, we conclude that the most important reason why the highest cost can't make the U.S. the best is unfairness. Afterward, we use a neural network algorithm to predict what will happen to the us. if some values of the metrics change. We conclude that the U.S. can get the greatest benefit by improving fairness We finally consider a policy change, a"medical insurance voucher"as a method to increase insurance coverage and reduce unfaimess The UMAP Journnl29(2)(2008)155-168. Copyright2008 by COMAP Inc. All rights reserved. Permission to make digital or hard copies of part or all of this work for personal or dassroom use is granted without tee provided that copies are not made or distributed for profit or commercial dvantage and that copies bear this notice. Abstracting with credit is permitted but copyrights for components of this work owned by others than COMAP must be honored. To copy otherwise, to republish, to post on servers, or to redistribute to lists requires prior permission from COMAP
数学中国 www.madio.net
数学中国w, madio,net 156 The UMAP Journal 29.2(2008) Introduction Many countries have recently introduced reforms in the health sector with the explicit aim of improving performance [Mathers et al. 2000: 2001] There is extensive literature on health-sector reform and recent debates have emerged on how best to measure performance so that the impact of reforms can be assessed [Goldstein 1996]. Measurement of performance requires an explicit framework defining the goals of a healthcare system and a suitable method to make a compelling evaluation So our goal is pretty clear Devise metrics to evaluate the effectiveness of a healthcare system Devise a method to combine the metrics Compare several representative countries. Restructure the healthcare system of the U.S. and build predictive models to test the chan Our approach is: Analyze factors that can affect the performance of a healthcare system. Search the literature on existing evaluation methods and find their short- Develop a comprehensive evaluation method that asks only for existing data or data easy to measure and collect Collect experimental data that can be used in our method. Compare current methods and determine their characteristics. Do a sensitivity analysis of variations of our models Restructure the healthcare system of the U.S. and build a model based on neural networks to test changes. Do further discussion based on our work Four Representative healthcare Systems The healthcare system, as an important part of the social security sys- m, is essential to promote the stability of society, and it reflects social justice. Due to the different histories, cultures, and status of human rights protection, healthcare systems vary from country to country. There are four representative healthcare systems[Ding 2005:
数学中国 www.madio.net
数学中国 The Most is Not the Best 15 National insurance. The main countries using this system are the UK, Eastern Europe, and Russia. The government dominates, healthcare is free, with full medical treatment and complete coverage of the popula tion. But it doesnt have high efficiency or make use of the market, and it is a heavy burden to the government. Commercial insurance. The U.S. is the main country using this system, which makes the market the guideline of the healthcare system. Cost is high, and a large number of people fail to pay. Socialinsurance. This system features mandatory coverage and fairness, as in Japan, Germany, and Canada. It has high cost and slow service. Savings insurance. Singapore is the representative country. The main disadvantage is a low service efficiency. Costs rise rapidly, and it cannot achieve full coverage Analysis of the wHo Estimation Method The WHO,'s methuds focus un the outcomes of a healthcare system with out considering any characteristics of the system itself The metrics that the WHO uses to evaluate a healthcare system aim to measure goal attainment, and they include most of the outcomes that a healthcare system should produce Weaknesses The weights placed on each dimension are somewhat arbitrar The approach heavily penalizes countries with epidemic disease unre- Lated to the healthcare system. This approach does not look at how the system is organized and man- The WHO 2000 rankings do not look at access, utilization, quality, or cost-effectiveness In addition, according to Almeida et al. [ 2001, 1693: The measure of health inequalities does not reflect concerns about eqr uity Important methodologicallimitatiuns and controversiesare not acknowl-
数学中国 www.madio.net
数学中国www, madio,net 158 The UMAP Journal 29.2(2008) The multicomponent indices are problematic conceptually and method ologically; they are not useful to guide policy, in part because of the opacity of their component measures. " Primary health care is declared a failure without examining adequate evidence, apparently based on the authors ideological position ."These methodological issues are not only matters of technical and scien- consequences. Improved WHO Method In the WHO methods, the weights in the construction of the composite index are used without considering uncertainty in the values of the different nen Weuse agrey comprahensive evaluation mode? toimprove the WHOmethod to make the evaluation more credible Methodology e Suppose that ck, for are the raw data of the metrics k= l,.m in suntry i, for i=1, ..,n, giving the n x m matrix=(c).We suppose cR=(ci,,,, c), a best possible situation, as a reference and compare the value of metric k in country i to this ideal via s(k)=min, Ic-cal+ pmaxiIct lc+k-eu+pmax Ic where p E(0, 1)is a differentiation coefficient that we generally can take to be.5. Using,(k), we get the evaluation matrix E=5()\s for them mtrppose W=(wi,., Wm)is a weight-distribution vector for the m metrics, with w the weight of metric k and >wk= 1. Based on the dis- cussion above, we get the grey comprehensive evaluation model R=W·E=(n1,…,rn ' S NOTE: This method, not known under this name in the US. was introduced by Deng Julong in Tutorial of Gray Systean Theory [in Chinese](1982). It uses ideas of TL Saaty's analytic hierarchy process and is well well-known in China(googling"grey aystem"gets 64,100 hits, induding The Journal of Grey Systemt, edited by Deng). For a numerical example, see Sun, Yan and Zong Sun, The grey comprehensive evaluation model for safety of construction sites, 2007 International Conference on winless CommunicatIons, Networking and Mobile Camping, 5240-5243
数学中国 www.madio.net
数学中国w, madio,net The Most Expensive is Not the Best 159 where E is the transpose of E and r,=>=, wS ( k)is the relating de- gree. The vector R=(u ., r,.)contains the final scores of the countries healthcare systems. The larger r the better the system How to Determine the Weights Wewe want to determine the weight vector in a credible way. We use the principle of minimum loss I Wang et aL. 2000. Because our metrics ul, evaluate information from different aspects, combining all the metrics in a informatics We should maximize conservation of information. So we choose the most classical method: We calculate variance to represent information; the larger the variance, the more information. In the final scored=w u, we should choose the best weight w to make the variance of d reach the maximum: D(d)=w D(uw where D(d) is the variance matrix of d. when ww=l, D(d)achieves it maximum We use the method of Lagrange multipliers. Suppose that =2D(u)u-2)=0 入 which reduces to D(uw=Aw, 1 So Ais aneigenvalue of D(u with eigenvector w when ww=l, to make the maximum eigenvahtw= A reach the maximum, we should take Aas D() lue of D (u) In the real calculation, we do not know D(u), so we use the variance matrix D(u)=ou)of the sample(cin,., Cn)of u, to represent it, where (x-x)(
数学中国 www.madio.net
数学中国w, madio,net 160 The UMAP 29.2(2008) The variance matrix D(u)is a nonnegative symmetric real matrix, so all its eigenvalues are real. From the properties of Rayleigh s entropy, we get uNDid)u here Ao is the maximum eigenvalue of D(u), and the eigenvector w of D(u) is the weight vector that we seek. A Partial Discussion The improved WHO method does not change the focus on outcomes of the healthcare system. Its improvement is in making the evaluation more credible. This kind of method makes its own sense in that it reall can reflect the goals of the healthcare system, but it can t reflect the inside For example, a country with an epidemic often gets a low score in WHOs evaluation method, but maybe this is not the problem of the healthcare system. So a new method that reflects the inside is needed Comprehensive Evaluation Method We bring up a method to evaluate a healthcare system, mentioned by Ding 2005] that considers both the outcomes and properties of systems themselves Metrics to Evaluate Overall Effectiveness To make an overall comparison between countries' health care systems more objectively, fairly and quantitatively, the metrics must be made well. The World Dank has specified the goals of a healthcare systein [Schieber and Maeda 1997, 2: "Improving apopulation s healthstatus and promoting social well-bein ."Ensuring equity and access to care ."Ensuring microeconomic and macroeconomic efficiency in the use of resources ."Enhancing clinical effectiveness Improving the quality of care and consumer satisfaction ."Assuring the system s long-run financial sustainability Pursuant to this definition, we make five metrics for the overall healthcare stem
数学中国 www.madio.net
数学中国w, madio,net The Most Expensive is Not the Best 161 Efficiency, the proportionality between inputs and outcomes. It can be divided into technical efficiency, economic efficiency, and allocative efficiency. For our purposes, we choose te Fairness, both in medical treatment and in contributing to the costs Responsiveness"refers to the non-health improving dimensions of the interactions of the populace with the health system, and reflects respect of persons and client orientation in the delivery ofhealth services, among other factors"[Tandon et al. 2000, 2-31 The effect of the government The basic situation of a country. This means a composite index of sec- tors, which include economy, education, scientific research, and popula The Model to deal with the Index and Data Accordingly, we make five new indexes, one for each metric above. Choose the Operation Model We use the method of equal intervals to combine the indexes, which is also used in the Human Development Index by the United Nations to compare countries. We also solve the problem of how to determine the eights The Equal Interval Method The Operating Process Divide the subindexes into positive indexes and negative indexes Use different algorithms to make the standardization to the two kinds of indexes According to the subindexes, we can get the five main indexes composite Calculate the fimal score of different countries based on the five metrics values Classification of the Indexes Classification, Positive index: the higher the value, the better the health care system; for example, availability of safe drinking water. Negative ndex: the higher the value, the worse the healthcare system; for example, the proportion of smokers
数学中国 www.madio.net
数学中国w, madio,net 162 The UMAP Journal 29.2(2008) Standardization. The indexes have different units, so we should stan- dardize before calculating the final score. After the classification, we can deal with the two kinds of indexes differently. Npm”+10 Positive index: Fiut Ruus-Ru Ru 100 R t in where i is the one of five metrics, L is the subindex of the metric i j is the one of the Rumin is the minimum value of the subindex of the metric i in the statistical data, and Rulmax is the maximum value of the subindex of the metric i in the statistical data, and Fai is the value of the i subindex of the metric i after standardization Determine the weights. We can get the value of every metric using the function F where n is the number of the subindex in metric i and a is a weight of the tetric i Get the final score of the evaluated country. Based on the discussion above we get the function S k where k is the number of metrics (in our case k= 5) Comparisons between Methods Before the comparison, each component measure was rescaled on a o to 100 scale for healthy life expectancy H Health-20 80-20×100 for health inequality, HI=(1-HealthInequality)x 100
数学中国 www.madio.net
数学中国w, madio,net The Most Expensive is Not the Best 163 for responsiveness leve el, R- Responsiveness x a for responsiveness inequality, RI= 100(1-Responsivenesslnequality ); for faimess in financing, FF- FaimessofFinancing Contribution x 100 The overall composite was, therefore, a number from 0 to 100 Dipartite Degree analysis As we know, a good metric should distinguish. But the WHOs method cant for example, its method gives 36 countries the same value in its metric of responsiveness, We evaluate the degree of distinction via DD=√m+…+n where N- iis the number of countries that can t be distinguished in crite rioni. The smaller DD, the better the degree of distinction. Monte Carlo Simulation To test the dipartite degree(degree of distinction)of every method, we use Monte Carlo simulation to make a small change to every data value since the value must contain some error. The process is as below. First, we use the beta distribution to determine the change in each value Because the beta distribution is restricted to the interval [0, 1], a linear func tion of a beta-distributed random variable can be used to scale the sampling interval appropriatel The beta distribution can be described by the probability density func (a+ Beta(a,B)()=r(r(s) 1(1-x)1,0<x<1; 0 el It has expected value E X]=a/(a+ B) Suppose that Tij, withl s i< 191 and 1 <3< 10, is the unknown true mean of the random variable X: representing the jth metric in country i. x=(xy-1)+2Beta(22)(X) which takes values in ry-1, Iy+l] and has expected value E[(xy-1)+2Beta(2,2)(x) 1+22/(2+2)=z Ne use Monte Carlo simulation to create 1, U numbers randomly in the interval([ze-1, ry+1] and calculate a 95% confidence interval for rs
数学中国 www.madio.net
数学中国w, madio,net 164 The LAP Journal 29.2(2008) Sensitivity Analysis About the values of the metrics In this part, we change the values but keep the weights to see how can this change affect the evaluation result. Then we can arrive at the most important metric, the one that can affect the final score acutely Suppose that Gp and Ga are the final scores of countries p and g. Let Uor be the the value of metric r in country q. Change it to make G, = Ga;then we can get the marginal value U Uo=Ue+ We can do sensitivity analysis to the values of the metrics following the rocess below If Ug is outside of the allowable interval, whatever it changes, it won't change the order of the two countries; so r is a value- insensitive metric. When Uor is close to U, changing the value will change the order of the two countries, so r is a value- sensitive metric. About the Weights In this part, we change the weights but keep the values of the metrics to see how doing so affects the evaluation result. Then we can get the most important weight, the one that can affect the final score acutely To make a simple analysis, when a weight changes, let only one another change at the same time, and keep the others fixed Suppose that the weights values before they change are ,, U,,, G,and after changing they are w;, Ui, Gr. Suppose that the changing weights are r and 8, so that W,+w,=wr+w The changing interval of and w, is 0, w, +ws. When they change, maybe the final score of one country will equal that of another. Let the two countries be p and g. Then we can get the marginal weights =0-0)=(0-0m5 (m+觉) When the two countries have the same score, we can get r and w, as
数学中国 www.madio.net
