EFFICIENT ALLOCATION OF RESOURCES IN ...

EFFICIENT ALLOCATION OF RESOURCES IN HUMAN ORGAN MARKETS: IS IT ACHIEVABLE THROUGH ECONOMIC TOOLS? by ALPER ALTINANAHTAR, B.A., M.A. A DISSERTATION IN ECONOMICS Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY

Approved

Chairperson of the Committee

Accepted

1^*^ wr ,.f

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Dean of the Graduate School May, 2004

ACKNOWLEDGMENTS

I am gratefiil to Dr. Jamie Kmse, the chairperson of my committee, for encouraging me to pursue a dissertation on human organ markets. I am thankfiil for her patience, advice, and excellent guidance throughout this work. I would also like to thank Dr. Thomas L. Steinmeier and Dr. Ke Xu for their contributions and helpful comments as committee members. I also would like to thank Dr. Joseph E. King, the chairman of the department, and the department staff for their support throughout my studies in the Ph.D. program and my work as an instractor at the Department of Economics in Texas Tech University. My appreciation and special thanks goes to my parents Zuhal and Aydin Altinanahtar, and my sister Fulya Sener in Turkey. This study would not be possible without their help, encouragement, and enthusiasm. Special appreciation is extended to my lovely wife, Nilay Altinanahtar, for her support, patience, and dedication throughout this study. My mother-in-law Ayfer Sevinc deserves special thanks. Her presence with my family in the U.S. has given me a lot of courage and strength in my work. Last but not least, I would like to thank our newborn baby girl, Dila Altinanahtar, for bringing excitement and enthusiasm into our lives and providing me with the most important incentive to complete my studies.

11

TABLE OF CONTENTS

ACKNOWLEDGEMENTS

ii

LIST OF TABLES

vi

LIST OF FIGURES

viii

CHAPTER I.

n.

INTRODUCTION

1

1.1 What Is Transplantation?

4

1.2 Transplzmt History

11

DESCRIPTION OF THE CURRENT PROCUREMENT SYSTEM

14

2.1 The Orgaruzation of the Network of Organ Procurement

14

2.2 The Organ Supply Chain

20

2.3 Current Applications Around the World

24

2.4 Organ Procurement and Transplantation in Turkey

27

2.5 The Shortage

29

2.5.1 Supply Stioictiires m.

31

ALLEVIATING ORGAN SHORTAGE: FOUR APPROACHES

36

3.1 Ranking Patients

36

3.2 Exhortation

41

3.3 Presumed Consent

46

3.4 Market-Based Solutions

50

111

rV.

V.

CONTINGENT VALUATION SURVEY

68

4.1 The Sample

70

4.2 Analysis of the Data and Results

71

4.2.1 Valuation Questions

71

4.2.2 Perception Questions

78

4.2.3 Demographics and Other Questions

80

4.3 Ordered Probit Model

82

4.4 Regression Analysis of Valuation Questions

84

4.5 The WTA and WTP Disparity

89

4.5.1 WTA/WTP Disparity for Organs

92

4.5.2 WTA/WTAR Disparity

94

4.6 What is the Market Clearing Price?

95

CONCLUSION

99

REFERENCES

103

APPENDDC A. ORGAN PROCUREMENT ORGANIZATIONS' SERVICE AREAS

111

B. THE PROCUREMENT PROCESS

113

C UNOS ALGORITHM FOR ORGAN TRANSPLANT RECIPIENT SELECTION 115 D. SURVEY INSTRUMENT

118

E. BRIEF STATEMENT DESCRIBING THE PURPOSE OF THE QUESTIONNAIRE

139

F. REQUEST LETTER TO FACULTY AND GRADUATE STUDETS TO PARTICIPATE TO THE SURVEY STUDY

142

IV

G. INTERVAL REGRESSION RESULTS FOR VALUATION QUESTIONS

144

LIST OF TABLES

1.1

Transplants by Donor Type (U.S. Transplants Performed)

8

1.2

Number of People on the Waiting List and Estimated per-transplantation Billed Charges, by Organ (1997-2003)

9

1.3

Reason for Removal from Organ Transplant Waiting List

10

1.4

Cadaveric Donors Recovered in the U.S. by Donor Age

11

2.1

Cold Ischemia Time (Approximate Preservation Times) for Solid Organ Transplantation

21

3.1

The Number of Cadaveric Donors in Europe and the U.S. in 1997 and the Prevailing Donor Legislation

48

4.1

Responses to Willingness to Accept to Opt In (Question 6)

74

4.2

Responses to Willingness to Pay to Opt Out (Question 7)

75

4.3 4.4

Responses to Willingness to Accept for Relative's Organs (Question 8) Responses to Valuation Questions for All (Full Sample), USA (USA Respondents), and TC (Turkey Respondents)

76

4.5

77

Respondent's Attitudes towards Two Altemative Procurement Systems (Yes=Offended, No-Not Offended)

80

4.6

Distribution ofRespondent's Annual Household Incomes

82

4.7

Ordered Probit Model Estimate for Willingness to Accept Question, (Q6) N=332

87

Ordered Probit Model Estimate for Willingness to Pay Question, (Q7) N=320

88

Ordered Probit Model Estimate for Willingness to Accept for Relative, (Q8) N=319

89

t-test Results for Mean Differences between WTA and WTP

93

4.8

4.9

4.10

VI

4.11

The Wilcoxon Rank-sum Test Results for Mean Differences between WTA and WTP

94

4.12

t-test Results for Mean Differences between WTA and WTAR

94

4.13

The Wilcoxon Rank-sum Test Results for Mean Differences between WTA and WTAR

95

4.14

National Figures for the Supply Schedule of Cadaveric Donors

96

4.15

Regression Results

96

C. 1

Transp lant Priority Rating

116

C.2

Blood Compatibility

116

C.3

Time Spent on the Waiting List

117

D.l

Respondents'Fields

120

G. 1

Interval Regression Results for Willingness to Accept Question (N=215)

145

G.2

Interval Regression Results for Willingness to Accept Pay Question (N=320) 146 Interval Regression Results for Willingness to Accept for Relative's Organs Question (N=318) 147

G.3

vu

LIST OF FIGURES

1.1

Stmcture of the Current Procurement System hi the U.S.

17

1.2

Stmcture of the Current Procurement System fri Turkey

29

3.1

Market for Transplant Operations

55

3.2

The Profitability of Restricted Supply of Organs

57

3.3

Five-Year Plan for Stabihzing the Growth of Organ Shortage Usmg Cadaveric Organs

60

3.4

Possible Outcomes of Providing Monetary Incentives to an Organ Market

62

3.5

Equilibrium Price versus Non-Usability Rate

66

4.1

Cadaveric Organ Market Equilibrium for 2002

98

A. 1

Organ Procurement Orgaiuzations Service Areas

112

B.l

The Procurement Process

114

D. 1

Results for Question 1

119

D .2

Results for Question 3

121

D.3

Results for Question 4

122

D.4

Results for Question 5

123

D. 5

Resuhs for Question 6

124

D.6

Results for Question 7

125

D.7

Results for Question 8

126

D.8

Results for Question 9

127

D.9

Results for Question 10

128

D. 10

Resuhs for Question 11

129

viii

D. 11 Results for Question 12

130

D. 12 Results for Question 13

131

D.13

Results for Question 14

132

D. 14

Results for Question 15

133

D. 15

Results for Question 16

134

D. 16 Results for Question 17

135

D. 17

Resuhs for Question 18

136

D. 18 Results for Question 19

137

D.19

138

Results for Question 20

IX

CHAPTER I INTRODUCTION

In the mid 20' century the first successfiil human organ transplant was performed (December 23, 1954).' A kidney was fransplanted from a living donor who was an identical twin of the recipient, hi these early days fransplantation technology excluded any use of cadaveric organs due to the state of knowledge regarding immunosuppressive therapy. Therefore, organ (kidney) transplant candidates had to bring the necessary donor witii them when they checked into the hospital for the fransplant operation. If the patient was not able to find an acceptable living donor, there would not be a transplant operation. Thus, there were no waiting lists and no apparent shortages in organs (Baraett and Kaserman, 2000). After the first successful transplant, advanced medical procedures and improved tissue matchmg made transplantation of cadaveric organs possible. Since fransplantation of organs from cadaveric donors became possible, organ waiting lists began to appear in the late 1970s. Subsequently, in the 1980s transplant candidates began to form lines for cadaveric organs. In the 1980s, waiting lists for cadaveric transplantations were managed by health professionals located at transplant centers. Moreover, improvement in the success rates in cadaveric organ transplantation and expansion of insurance coverage^

The fu:st successful comeafransplantationwas in 1905 and the first heartfransplantwas in Dec. 1967. ^ For instance, Federal coverage for kidneyfransplantswas instituted in 1972 with passage of the End Stage Renal Disease Program, administered by the Social Security Adminisfration. This program covers both kidneyfransplantsand dialysistteatmentsfor ahnost all citizens. In addition, during the late 1980s and early 1990s, private insurance companies increasingly began to provide coverage for other organ

caused the demand for cadaveric organs to grow rapidly. Therefore, the organ procurement policy, which was focused exclusively upon living related donors in the past, now rehes primarily upon cadaveric organs from unrelated donors. This put pressure on scientists to find ways to alleviate the supply of transplantable organs (Essig, 1993). hi that context, many proposals have been offered. (Essig, 1993 and Barney et al, 1989 provide an overview of proposed solutions) Altiiough the supply of useable organs from both live and cadaveric donors has expanded, the current shortage is getting worse due to the fact that the number of people on the waiting lists has grown more rapidly than the number of procured organs. The number of organ transplants performed in the U.S. has risen from 9.1% per year between 1990 and 2001 (from 12,618 to 24,090); the number of centers performing this surgery has grown from 235 to 278; and the number of patients awaiting fransplantation has gone from about 14,000 to some 80,000 persons. Over the same time period organ donation has expanded more slowly (5,906 to 12,580). The fact that almost 6,500 patients die each year is evidence that there is a shortage of cadaveric organs for transplant. Bamett, Saliba, and Walker (2001) argue that shortage has two subparts: Economic shortage is defined as the shortfall between the quantity of organs supplied and the quantity demanded at the current ($0) price. On the other hand, medical shortage (Bamett et al., 2001) exists when there is not a suitable organ for every patient. In this study, we will focus only on the economic shortage.

fransplants (hearts, livers, etc.) as these procedures moved from the experimental stage to become accepted medical freatments.

This is an area of public poUcy that is in extreme need of objective analysis and serious debate. Therefore, in order to understand the problem the reader will need a basic understanding of the current transplant system in the U.S. and in other countries. We are aware of the delicacy of the subject under consideration. That is why we leave it to the reader to decide which attemative is ethically permissible and which is not. Is it possible to increase the supply of organs for transplant using the tools of economics? This dissertation seeks to explore the use of incentives as a means to mitigate the economic shortage. In the first chapter of this dissertation, we provide the reader with the concepts and definitions about steps for transplantation, its procedure, and evolution of property rights for human organs. Note that it is not our intention to introduce every medical (technical) detail of transplantation process or the law to the reader. On the confrary, we intend to provide the reader with a basic understanding of current policies and process about organ fransplantation. Hence, we explain technical details only to the extend that it allows us to point out the critical issues about the subject matter. In the second chapter, we explain how the current systems in the United States and other countries work. We will look at similarities and differences in terms of laws and practices regarding transplantation and procurement of organs. Finally, we will address some possible solutions proposed in the literature. In chapter three, we focus on four possible remedies suggested in the literature, discuss their applications, and compare their advantages and disadvantages with current organ allocation pohcies. Most of our attention in this chapter will be focused on a

possible free market for human organs. Relative to other remedies, this one will be discussed more carefiiUy by giving cross references and using graphical representations. Since no organized human organ market exists we use a contingent valuation survey to explore attitudes toward monetary compensation for pre and posthumous organ donation. In Chapter IV, we describe how the survey was administered, and to whom it was administered. Using the data collected by survey techniques, we examine a hypothetical organ market and compare the outcome with the current system's outcome. Finally, we conclude in Chapter V. 1.1 What Is Transplantation? Transplantation is a treatment when an organ has failed because of illness or injury. Replacing the organ may be the only viable freatment choice for the patient or one choice among several options. The process of organ procurement and transplantation begins when a patient in need of an organ transplant is referred to one (or more) of the 278 organ fransplant programs currently in operation in the United States. If accepted by a transplant program, the patient is placed on that program's waiting list until a donated organ that is determined to be medically appropriate becomes available. Under current practice, individuals are encouraged to indicate their willingness to be a donor by registering; most commonly at the time they obtain or renew a driver's license. Yet, the decision to be a registered donor is and can be reversed at any time. All individuals can indicate their intent to donate (persons less than 18 years of age must have a parent's or guardian's consent). There are no limitations on who can donate. The

deciding factor on whether a person can donate is the person's physical condition, not the person's age. The circumstances surrounding the potential donor's death determine in part whether cadaveric organs are viable for transplant. Acceptable donors must be in a hospital and declared brain dead. Central nervous system trauma is the typical cause of death among organ donors, usually caused by "automobile or other accident" (40-50%); homicides and sfrokes are also common causes of death for organ donors (Prottas and Batten, 1991a). Before a donated organ is transplanted into a patient's body, there are several limitations that must be overcome. One hurdle to successfiil transplantation is the human immune system. The human immune system is the body's main defense against infectious organisms. Although the immune system is essential for life, when a person needs an organ transplant, the immune system suddenly becomes a deadly force, attacking and destroying the "foreign" organ. The level to which the immune system is aroused is a function of the compatibility of the organ. Immunosuppressive dmgs such as cyclosporine* have made human organ transplantation possible by hmiting the body's attack on the fransplanted organ while sparing the immune system's most life-saving benefits. The use of cyclosporine, approved by the FDA m 1985, has dramatically increased the one-year survival rate of transplant patients. The one-year survival rate for kidney patients rose from 61 to 80 percent; for liver patients, from 32 to 70 percent; and for heart patients from 50 to 80 percent. Cyclosporine also made heart/lung transplant

^ Prottas, J. M. (1994). The Most Useful Gift. San Francisco: Jossey-Bass "* It came in to use in 1979.

possible. However, graft survival rates^ are still much shorter when the organ does not match perfectly. A perfect match would lead to the lowest inmiune system response. The poorer the match the sfronger the immune response. For instance, the one-year organ-graft-survival rate for well-matched kidneys is 13% higher than the poorly matched ones (Carlsfrom et al., 1999). Almost 40 years after the first successfiil kidney transplant, new drags, improved tissue matching, and advanced surgical procedures allowed transplantation of cadaveric organs and improved transplant success rates. These new changes expanded the application of fransplant technology to vital organs other than kidneys. Now surgeons are capable of transplanting the following soUd organs; heart, lungs, liver, pancreas, kidney, intestine, and the following tissues; comea, bone, cartilage, skin, heart valves, saphenous vein. Between 1982 and 1986, the system doubled in size, the success rate rose, and the discard rate fell substantially from 20 to 12%. Between 1984 and 1999, the national transplant system has extended the lives of more than 200,000 individuals (Meier, 1999). Currently, the majority of organs supphed in the United States are procured from cadaveric or non-living individuals. Kidney donations alone may be provided by living individuals who are related, or by a living stranger who voluntarily gives up one kidney for transplantation.

^ Dowie, M (1988;). We have a donor. New York: St. Martin's Press, p. 75. ^ Graft Survival Rate: The percent of patient who have fimctioning fransplanted organs (grafts). The rate is usually measured at one, three and five years time periods. ' It should be noted that all organs fransplanted except kidneys and corneas are necessary for life and could not be safely removed from a living individual. Comeas are necessary for sight and are not removed firom living individuals.

New technology allows us to overcome many medical constraints. Therefore, the main consfraining factor on transplantation is the shortage of available organs. Between 10,000 and 12,000 people who are considered medically suitable for organ donation, die each year yet only an estimated 6,000 donate.^ Unfortunately, Bamett et al, (1993) argue that many more organs are buried each year than the number required to meet the demand for transplantation. If we provide adequate incentives and correct other flaws m our procurement system, there is a chance we may be able to harvest enough organs to resolve the shortage problem. In 2003, 21,351 Americans—about 59 people a day— received organ fransplants. 15,722 of the organs were procured from cadavers while rest of the organs came from donated by living donors. An even greater number received tissue transplants, such as comeas and skin. Table 1.1 and Table 1.2, summarize statistics on the number of transplants, the waiting lists for organs, and cost estimates for transplants. Table 1.3 summarizes the reasons for removal from the waiting lists. Finally, Table 1.4 summarizes the age distribution of donors in the U.S. These data reflect the growth in the procurement/fransplantation system.

' Taken from the CORE (center for organ recovery and education) website http://www.core.org/itsafact.html (March 3, 2001)

Table 1.1 Transplants by Donor Type (U.S. Transplants performed: Jan 1, 1997-Aug 31, 2002). 2003

2002

All Organs Kidney Liver Pancreas Kidney/Pane Heart Lung Heart/lung Intestine

21,351 12,594 4,763 427 745 1,774 921 27 100

24,901 14,776 5,329 554 905 2,155 1,042 33 107

All Organs

15,722

18,289

Kidney Liver Pancreas Kidney/Pane Heart Lung Heart/lung Intestine

7,247 4,496 427 743 1,774 911 27 97

8,539 4,969 553 905 2,155 1029 33 106

2001

2000 All Donor Types 24,182 23,192 14,237 13,566 5,188 4,987 472 438 891 914 2,202 2,199 1,054 959 27 48 111 81 Deceased Donor 17,631

17,329

1999

1998

1997

21,978 12,725 4,748 358 943 2,188 892 51 73

21,498 12,431 4,516 245 972 2,348 869 47 70

20,299 11,694 4,187 209 853 2,293 933 62 68

17,004

16,966

16,259

8,228 8,125 8,040 8,024 4,70 4,497 4,424 4,593 471 437 358 245 888 908 935 970 2,202 2,199 2,188 2,348 1034 941 864 840 27 48 51 47 111 78 71 68 Living Donor 4,974 6,612 6,551 5,863 4,532 5,629 All Organs 6,009 5,441 4,685 4,407 6,237 5,347 Kidney 394 92 518 251 360 267 Liver 1 1 0 0 1 0 Pancreas 6 8 2 3 0 2 Kidney/Pane 0 0 0 0 0 0 Heart 20 18 28 29 13 10 Lung 3 2 2 0 1 3 Intestine Procurement Source: Based on information gathered from the website of Organ Transplant Network (OPTN) (Jan 2, 2004) URL: httD://www.ODtn.or2

7,771 4,101 209 847 2,293 910 62 66 4,040 3,923 86 0 6 0 23 2 and

Table 1.2 Number of People on the Waiting List and Estimated per-transplantation Billed Charges, by Organ (1997-2003).

2003

2002

2001

2000

1999

1998

Estimated Perfransplantation 1997 billed charges 1996 ($)^

Kidney

56,634

56,520

53,560

50,426

46,489

42,364

38,236

171,700

Liver

17,379

17,564

19,066

17,132

14,709

12,056

9,637

393,900

Pancreas

1,525

1,365

1243

1032

725

455

361

148,700

Kidney/Panc

2,430

2,579

2,557

2,553

2,225

1,841

1,591

198,200

Heart

3,556

3,933

4,137

4,164

4,121

4,185

3,897

316,600

Lung

3,928

3,868

3,841

3,679

3,491

3,165

2,664

312,200

Heart/lung

190

199

212

209

234

257

236

317,000

Intestine

175

-

-

-

473,900

TOTAL

85,817 86,028 84,616 79,195 71,994 64,323 56,622

Source: Based on information gathered from the website of Organ Procurement and Transplant Network (OPTN) (Jan 2, 2004) URL: http://www.optn.org ^ Source: Haubolt, Richard A., 1996. Note: Billed charges are the only source of costs for many fransplants. Actual costs may vary.

Table 1.3 Reasons for Removal from Organ Transplant Waiting List Reason

1998

1999

2000

2001

2002

All Removal Reasons*

27,866

30,309

31,862

33,231

35,799

Cadaveric Transplant

16,501

16,462

16,736

17,032

17,689

Living Donor Transplant

2,253

2,689

3,454

3,995

4,154

Cadaveric Emergency Tx

4

18

28

28

14

Cadaveric Multi-Organ Tx

3

28

61

55

78

Patient Died during Tx Procedure

0

0

0

0

1

Medically Unsuitable

0

0

1

0

1

175

249

266

265

367

Transferred to another center

1,186

1,272

1,808

1,883

1,694

Died

5,181

6,507

6,136

6,648

6,602

Other

1.425

1,970

1,936

1,873

2,765

Condition Improved

595

671

669

609

1,209

Too Sick to transplant

1,182

1,341

1,443

1,557

1,844

990

1,298

1,406

1,416

1,500

Refused Transplant

Transplants at another center

Source: Based on information gathered from the website of Organ Procurement and Trfln=pi^nt Mf^twnrV TOPTN^ CTan 2. 2004^ URL: http://w-ww.ODtn.or2 * Totals may be less than the sums due to patients included in multiple categories.

10

Table 1.4 Cadaveric Donors Recovered in the U.S. by Donor Age

1998

1999

2000

2001

2002

All Ages

18,617

18,585

18,916

19,250

19,957

lim ^(^) = i z-»+co

lim o(z) = 0 2ind

z^-«

(4.4)

A n u m b e r of statistical m o d e l s available for analyzing data from valuation questions are c o m m o n l y referred to as "ordinal data analysis," "categorical data analysis," or "discrete choice analysis." O n e c o m m o n characteristic of these related statistical m e t h o d s is that they all m o d e l the probability of an event. However, these models are only valid w h e n there are only two outcomes of an event. Standard probit or logit models estimate parameters based on the m o s t preferred choice, but m o r e information can b e incorporated into parameter estimation b y using an ordered probit or ordered logit m o d e l that accounts for respondents' rankings of all alternatives.

83

There are many settings in which the economic outcome we seek to model is a discrete choice among a set of alternatives, rather than a continuous measure of some activity, hi cases such as this one, tiie dependent variable is an indicator of a discrete choice. Such responses are normally coded 0, 1, 2, 3, and so on. hi these questions, respondents check the box next to payment, which for the case of this survey, represents WTA or WTP for an individual's own or relative's organs. There is a clear ranking among the categories, but tiie differences among adjacent categories cannot be treated as the same. In general, conventional regression methods are inappropriate in these cases, (Greene, 2003), because linear regression would treat the differences between a 3 and a 2 the same as that between a 2 and a 1, whereas in fact they are only a ranking. For this reason, we used ordered probit model, which is a simple extension of a probit model, to estimate the relationship between an ordinal dependent variable and a set of independent variables. An ordinal variable is a variable that is categorical and ordered like our payment scale. 4.4 Regression Analysis of Valuation Questions The proportion of 'zero' responses to various valuation questions ranged from about 9.7% in Question 6 (WTA) to 27% in Question 7 (WTP). hi Question 8 (WTAR), 22.9% of the respondents answered 'zero'. This can be interpreted as the proportion of subjects that were willing to donate without receiving any financial benefits. Such responses may reflect a religious belief or moral responsibility that saving another human's life is compensation enough to the donor and/or the donor's next of kin. Alternatively, these responses to the willingness to accept (Q 6) may be what Smith and

84

Desvourges (1987) describe as 'protest bids', i.e. reflecting a basic unwillingness to engage in a hypotiietical valuation exercise related to a topic as sensitive as this one. An individual's answer to a particular valuation question is logically related to various personal characteristics, frituition and previous research suggest that factors such as, age, education, marital status, number of children, hicome and membership in an organized religion may be relevant in explaining the likelihood that an individual will choose to donate his/her organs. Using the dataset described above, we estimate: WTAi" = p'X, + e,

(4.5)

WTPi" = P'Xi + ei

(4.6)

WTARi" = P'X, + s,

(4.7)

where Xi represents independent variables matrix and P is the vector of unknown parameters and Sj is a random disturbance. The noise term is introduced to account for differences in preferences across the population those are not explained by the independent variables. We assume e, has a normal distribution centered at zero with an unknown variance. This parametric distributional assumption allows us to use the ordered probit model to estimate the coefficients of the model. In order to estimate our model, we used the following explanatory variables: 1

Household's annual income from all sources ($),

2

Respondent's age (years),

3

Respondent's gender (= 1 if male),

4

Having Children (=1 if yes),

5

Member of an organized rehgion (-1 if yes).

85

6

Country (=1 if Turkey).

Here i'^ respondent's response to the j ' ^ payment choice is represented by Rj, and Sj represents j * payment choice. Response fimction can be shown as follows: j^^

riifSj>WTAi>Sj.,

" \Oif

WTAi>Sj

i = l,...n j =0

(4.8) 9

In ordered probit, an underlying score is estimated as a linear fiinction of the independent variables and a set of cut points. The probability of observing / corresponds to tiie probability tiiat the estimated linear fiinction, plus random error, is within the range of the cut points estimated for the outcome: ?r(outcome^ = 1) = Pr(/c,., )$5000 c)$1000 d)$500 _e)$100

f)$50 g)$25 h)$10 i) $0-$5

160 T 138

140

128

120 ^ 100 + c

76

0)

=

80

LH

60

W IP

40

+ 5a