How many myocardial infarctions should we rule out?

ORIGINAL CONTRIBUTION cost-effectiveness analysis, myocardial infarction; myocardial infarction, admission strategies

How Many Myocardial Infarctions Should We Rule Out? We used computer simulation to estimate the consequences of four admitting strategies (coronary care unit, intermediate care unit, routine ward care, or outpatient follow-up) on cost, outcome, admission threshold probabilities, and false-positive admission rates for patients with acute myocardial infarction. A t virtually any probability of acute myocardial infarction, replacing more intensive by less intensive strategies saved money but increased m o r t a l i t y and decreased life expectancy. Therefore, choices among strategies m a y be made by using the most effective strategy for progressively lower and lower risk patients until the additional cost per additional life saved reaches a cutoff value; then, a less expensive strategy is selected. With sample cutoff values of $I and $2 million per life saved, the marginal threshold admission probabilities were: $1. Million $2 Million Coronary care unit vs outpatient .08 .04 Intermediate care unit vs outpatient follow-up .07 .04 Routine inpatient vs outpatient follow-up .03 .02

Robert L Wears, MD, FACEP Sergio Li, MD John D Hemandez, MD, FACP Robert C Luten, MD, FACER FAAP David J Vukich, MD, FACEP Jacksonville, Florida

These results imply that the acceptable proportion of false-positive admissions m a y be as high as 70% to 80%; lower rates could indicate excessively restrictive admitting policies. Clinicians m a y be operating closer to the optimal decision point than has previously been asserted. [Wears RL, Li S, Hernandez JD, Luten Re, Vukich DJ: How m a n y myocardial infarctions should we rule out? Ann Emerg Med September 1989;18:953-963.]

Address for reprints: Robert L Wears, MD, FACER Division of Medical Computer Applications, Division of Emergency Medicine, Department of Surgery, University Hospital, 655 West Eighth Street, Jacksonville, Florida 32209.

From the Department of Surgery, University of Florida, Jacksonville. Received for publication March 10, 1988. Revision received March 23, 1989. Accepted for publication May 8, 1989. Presented at the University Association for Emergency Medicine Annual Meeting in Philadelphia, May 1987.

INTRODUCTION To ensure hospitalization of virtually all patients with acute myocardial infarctions (AMIs), physicians hospitalize even more patients, in whom the diagnosis is ultimately ruled out. This proportion of false-positive hospital admissions has attracted considerable attention. More than two thirds of coronary care unit (CCU) admissions are false-positives; 1-5 this rate is generally believed to be too high, although no critical analysis of this belief has been done. The introduction of prospective reimbursement under Medicare has further complicated the issue by inducing some hospitals to make their admitting policies more restrictive to cut costs, while others claim that incentives to increase admissions of lower-risk patients increase reimbursement. 6 This problem has led to work 1-4,7-43 on a variety of methods of quantifying patients' risks of AMI with the aim of reducing admissions of low-risk patients or of shifting low-risk admissions to a less intensive and presumably less expensive level of care. However, no studies have been conducted to determine whether the current false-positive admission rate is too low or too high. Three elements are required for such a determination. First, a method of estimating the probability of myocardial infarction (PMI) in individual patients m u s t be available. P o z e n et al, 1,2 G o l d m a n et al, 4 and others7,11,16,17,2i~,23,24,31,32,36,41,42 have provided such tools, although none has gained general acceptance. Second, the distributors for those probability estimates in the population presenting to the emergency department with chest pain must be known; in particular, we must know the distribu-

18:9September 1989

Annals of Emergency Medicine

953/95

ADMITTING STRATEGIES Wears et al

TABLE 1. Probabilities Probabilities

Mean

SD

Primary ventricular fibrillation

.06375

.014433

49-52

Complete heart block Efficacy, lidocaine CCU INT ROUT OUT

.0292

.018644

53,54

.8 .6

.115 .087

50,52,55

.12 .23 .54 .907

.03186 .06177 .12355 .08361

55

.35 .43 .475 .52

.0361 .0447 .0489 .0495

53,54,58

Mortality, iatrogenic CCU INT ROUT OUT

.0001 .0001 . . . .

.00071 .00071

59 59

Late mortality, uncomplicated myocardial infarction

.0030

.00412

6O

Mortality, general population adjusted for age and sex

.000135

.0011619

. .

Mortality, ventricular fibrillation CCU INT ROUT OUT Mortality, block CCU INT ROUT OUT

. .

. .

. .

. .

. .

. .

. .

Reference(s)

. .

56,57

. .

CCU, coronary care unit; INT, intermediate care unit; ROUT, routine ward care; OUT, outpatient follow-up. If unnoted, reference is Fineberg et al. lo

FIGURE 1. Tree representing the out-

come of an admitting strategy. Probabilities are PVF, ventricular fibrillation; PCHB, complete heart block; PML acute myocardial infarction; PLR, severe lidocaine reaction. Eft, efficacy of ]idocaine preventing ventricular fibrillation. Mortalities are MVF, ventricular fibrillation; MCHB, complete heart block; MU, uncomplicated myocardial infarction (late mortality); MAT, population mortality; MLR, severe lidocaine reaction. Utilities are UVFL, UVFD, ventricular fibrillation survival, death; UUL, UUD, uncomplicated myocardial infarction survival, death; UCHBL, UCHBD, complete heart block survival, death; UNL, UND, normal population survival, death; ULRL, ULRD, lidocaine rea}2tion survival, death.

Live UVFL

Not Prevented l-elf

l-k~/F Die UVFD MVF

V Fib

Live

UUL

Prevented

1-MVF

l-elf

Die UUD

MVF Live -

MI

Complete Heart B l o c k

-

UCHBL

I-MCHB Die

PHI

UCHBD

MCHB -

Live -

UUL

Uncomplicated

No Reaction I-PLR

No MI

Die MU

UUD

Live I-MN

UNL

1 ° PMI -

Fatal Iatrogenic Reaction

Die -

UND

Die ULRD

PER

tions for the subsets of patients with and without AMI. To our knowledge, only Tierney et a134 have reported 96/954

1 Annals of Emergency Medicine

18:9 September 1989

FIGURE 2. Algebraic expression corresponding to outcome value of tree in Figure 1. Each line is the subexpressio~ for the utility of a branch in Figure 1; where branches join together, t h e i r s u b e x p r e s s i o n s are s u m m e d . Thus, + signs represent nodes joining several branches.

Utility = PLR " ((1 - MLR) • ULR L q- MLR • ULRD) + (1 - PLR) ° ((1 -- PMI)" ((1 -- M N ) . UNL + M N . UND ) 4PMI ° ((1 -- PVF -- PCHB)" ((1 -- M U ) . Uuh + M U " UuD) +

F I G U R E 3. Expected cost for four management strategies as a function of PML Note threshold probability where graphs for R O U T and O U T

PCHB ° ((1 -- MCHB) ° UCHBL + MCHB " UCHBD ) + PVF ° (eft • ((1 ÷

MU) • UUL + MU • UVD)

(1 -- eff)-((1 -

cross.

MVF) ° UVFh + MVF ° UVFD))

this i n f o r m a t i o n . Finally, we m u s t have some idea of the level of risk (or threshold 44) at which it is reasonable to select a less i n t e n s i v e m a n a g e m e n t or admission strategy. The first two elements are closely related; Greenes et a145-47 have called a t t e n t i o n to the importance of evalu a t i n g diagnostic tests and clinical policies on their ability to assign patients to one side or another of a decision threshold and n o t s i m p l y on their a b i l i t y to produce an accurate rank ordering of risk. The purpose of our study was to estimate the cost effectiveness of several plausible mana g e m e n t strategies as a f u n c t i o n of PMI and to use that result in conj u n c t i o n w i t h e l e m e n t 2 as listed above to indirectly estimate a reasonable false-positive admission rate.

15000

10000

C?_

~0

/

50o0

ROUT

LP

METHODS r

i

r

Probobility of AMI

3

TABLE 2. Litigation estimates* Probabilities

Mean

SD

Lawsuit, given death CCU INT ROUT OUT

.025 .05 .10 .20

.0349 .069 .134 .253

Loss, given lawsuit CCU INT ROUT OUT ~

.15 .30 .60 .90

.196 .112 .438 .134

Defense costs ($) Settlement if loss ($)

20,000

2,000

190,000

38,000

*Dunn JD: Failure to diagnose myocardial infarction. Malpractice Digest. St Paul Fire & Marine insurance Co, 1986. 18:9 September 1989

Annals of Emergency Medicine

We developed our model after one suggested by Fineberg et a l J ° Four possible a d m i t t i n g strategies were examined: admission to the coronary care u n i t w i t h c o n t i n u o u s m o n i t o r ing, high nurse to patient ratios, and prophylactic lidocaine treatment (CCU); admission to an intermediate care u n i t w i t h c o n t i n u o u s m o n i t o r ing and prophylactic lidocaine treatm e n t (INT); a d m i s s i o n to a regular hospital bed w i t h o u t m o n i t o r i n g or l i d o c a i n e t r e a t m e n t (ROUT); a n d o u t p a t i e n t follow-up i n 24 a n d 48 hours (OUT) with admission to a regular hospital bed if repeat evaluation revealed evidence of AMI. Patients suffering acute c o m p l i c a t i o n s (ventricular fibrillation or complete heart block) were a s s u m e d to be t r a n s ferred to the CCU. Each strategy was e v a l u a t e d by t h r e e o u t c o m e m e a sures: mortality, cost, and life expectancy. No attempt was made to adjust survival or life expectancy for quality. 955/97

ADMITTING STRATEGIES Wears et al

C o s t - E f f e c t i v e n e s s Rotios

C o s t - E f f e c t i v e n e s s Ratios

CCU, INT, ROUT v s OUT

CCU, INT, ROUT v s OUT

2000 "

200000

~oo.

X:~ 150000

_1000

~

•

IOOO00

>~ c~

o ~oo

50000

ROUT

~

ccu

CCU

INT

.'~

o

.~ Probability

INT

.~

.~

.'1

.~

.1,

.4

P r o b a b i l i t y of AMI

of AMI

Risk Rctio (v OuT)

INT v OUT Oe©r~=~dMorl:#1y 9.8.7-

>...6

o 2~

.6-

~.5-

~

/

.4-

,21

o

.,

.~

.~

.;

0

.~

P r o b o b i l i t y o f AMI

P r o b a b i l i t y of AMI

7

F I G U R E 4. A d d i t i o n a l c o s t (in $1,O00s) per additional life saved comparing CCU, INT, and R O U T with OUT. FIGURE 5. Additional cost per additional year of life saved comparing CCU, INT, and ROUT with OUT. FIGURE 6. Probability of obtaining an increase in cost of $1,000 or more, an increase in life expectancy of 22 days or more, and a decrease in mortality rate of 0.0025 or more as a result of switching from OUT to INT. FIGURE 7. Relative ~isk of death of OUT strategy compared with each inpatient strategy as a function of PMI. It was assumed that patients who manifested significant arrhythmias, 98/956

acute congestive heart failure, persistent ischemic pain, or cardiogenic shock on their initial presentation would be admitted to the CCU regardless of any assessment of the probability of myocardial infarction; thus, that group of patients was excluded from the analysis. Similarly, patients presenting with acute ischemic changes such as new ST elevation do not present a disposition problem in the ED and were likewise excluded from consideration. We elaborated on Fineberg et al's model in several ways. First, we modelled the outcomes for those patients who ruled in as well as those who ruled out for all four strategies. We also included in our model the risk of severe lidocaine toxicity, the small " l o s s of l i f e " i n c u r r e d by spending several days in the hospital unnecessarily for those who ruled Annals of Emergency Medicine

out, and the small but nonzero probability that some equally serious but noncardiac problem might be detected during hospital admission that might otherwise be missed. Because the ED discharge of a patient suffering an AMI or the mistriage of such a patient to a lower level of care certainly represents a potential liability risk, those costs were also included in the model. To some extent, liability costs are already included as a component of hospital costs and should not be c o u n t e d twice; however, a major change from current admitting practices would likely be associated with new increases or decreases in liability costs, which should be used to adjust the overall cost estimates. Therefore, we adjusted those costs by a function of PMI; this countered the effect of the liability cost term in sit18:9 September 1989

TABLE 3. Costs

Costs Costs for rule-outs ($) CCU INT R©UT OUT Costs for rule-ins ($) CCU INT ROUT OUT

SD

Mean*

5,203 4,561 1,505 772

Reference(s)

520 456 150 77

10,237 8,848 3,171 . . .

61,8

14,348 12,579 4,149 .

.

.

*All costs adjusted for inflation to 1987 dollars. If unnoted, reference is Fineberg et al. lo

TABLE 4. Life e x p e c t a n c y Life Expectancy

SD

Reference

7.0

*

10

7.0

*

10

Uncomplicated myocardial infarction

10.7

*

10

Population adjusted for age and sex

18.0

*

10

Ventricular fibrillation survivor Block survivor

Mean

•The Weibull distribution's variance is a function of the mean and the annual rate of increase in mortality rate (about 8% per year).

uations w h e r e the s t r a t e g y u n d e r consideration reflected current practice (eg, s e n d i n g a v e r y l o w - r i s k patient home) but allowed it to be considered when the strategy differed from current practice (eg, admitting a high-risk patient to a nonmonitored bed).

The Model The o u t c o m e for any strategy in this model can be expressed graphically as a tree (Figure 1). Because the three outcome measures are not directly c o m p a r a b l e , t h e r e m u s t be three such trees for each of the four strategies, or 12 trees in the entire model. The expected utility for each tree is determine~., by multiplying the utility for each branch by the branch probability a n d s u m m i n g all t h e branch utilities for a given node; 48 this process is r e p e a t e d f r o m the 'qeaves" back to the "root" (Figure 2, which shows the algebraic expres18:9 September 1989

sion for the expected utility for the general subtree in Figure 1). The input variables for each tree were d e t e r m i n e d f r o m the clinical l i t e r a t u r e w h e r e v e r possible. T h e values used in the s i m u l a t i o n are presented (Tables 1 through 4). Because t h e s e v a l u e s are e s t i m a t e s , there is a certain a m o u n t of uncert a i n t y about the " t r u e " value of a variable, w h i c h m u s t be accounted for in the model. This u n c e r t a i n t y was expressed by assigning each variable a variance estimated from the literature (see Appendix). Because the costs of care are incurred in the present but the benefits of increased life expectancy are realized in the future, years of life were discounted at a rate of 3% per a n n u m . 47 The cost of future m e d i c a l care incurred by the group of patients whose lives were saved was not included in the model. The outcome utility for each of the 12 subtrees was determined for all Annals of Emergency Medicine

probabilities of AMI between 0.0 and 1.0 in increments of .02, and the difference in u t i l i t y a m o n g strategies was determined. Because there are six possible p a i r w i s e c o m p a r i s o n s among the four strategies, these comparisons produced six marginal utilities for each outcome as a function of PMI, each representing the gain or loss in utility expected from moving from a less intensive to a more intensive strategy. The marginal utilities then were used to calculate cost-effectiveness ratios, w h i c h reflect the additional c o s t i n c u r r e d per u n i t of b e n e f i t gained for changes in strategy at a given PMI. Thus, there are two costeffectiveness ratios for each comparison, having units of (additional) dollars spent per (additional) life saved and (additional) dollars spent per (additional) year of life saved. Because cost-effectiveness ratios are not addit i v e or t r a n s i t i v e (ie, t h e c o s t effectiveness ratio comparing strategy 1 with strategy 3 is not equal to t h e s u m of t h e r a t i o s c o m p a r i n g strategies 1 with 2 and 2 with 3), we examined all of the six possible pairwise c o m p a r i s o n s a m o n g the four strategies, although for the purpose of analyzing the decision to admit or discharge, only the comparisons of the three i n p a t i e n t strategies w i t h the outpatient strategy are relevant. It should be noted that these ratios should not be interpreted as placing a value on a life or year of life; they have no intrinsic m e a n i n g taken in isolation.48, 62 Rather, they should be used in c o m p a r i n g strategies or in c o n j u n c t i o n w i t h similar i n f o r m a tion about resources expended per unit of benefit gained to make more rational decisions about the allocation of limited resources. We i m p l e m e n t e d this model as a Monte Carlo simulation (see Appendix) and used the results to estimate the desired threshold probabilities.

RESULTS At any value of PMI of .02 or more, the more intensive strategies provide less mortality and more life expectancy than do less intensive strategies and, w i t h one e x c e p t i o n , are more costly. The exception (Figure 3) is t h a t the cost of the o u t p a t i e n t strategy exceeds that of the routine care strategy for AMI probabilities of more than .36; because virtually all patients are admitted at this level, 34 957/99

ADMITTING STRATEGIES Wears et al

TABLE 5. Cost effectiveness ratios for admissiorl versus discharge PMI

.00

INT vs OUT

CCU vs OUT

-42,093,671 ± 5,011,344

ROUT vs OUT

- 37,483,215 _+ 4,218,257

Undefined 1,420,122 ± 15,955

.02

3,984,602 ±

52,181

3,820,933 ±

54,318

.04

1,893,796 ±

17,355

1,822 891 _+

17,767

676,169 _+

7,932

.06

1,250,851 ±

10,865

1,189 575 ±

10,410

425,669 ±

5,376

.08

928,043 ±

8,199

888 473 +

8,817

302,748 _+

4,191

.10

738,999 ±

6,870

701 278 _+

6,996

224,289 ±

3,415

.12

619,330 ±

6,540

582 154 ±

6,323

174,380 ±

2,971

.14

524,599 ±

5,901

496 903 ±

6,079

136,522 ±

2,685

.16

462,108 ±

5,477

439 422 ±

5,705

112,088 _+

2,472

.18

405,293 ±

5,088

383 286 ±

5,518

87,284 _+

2,320

.20

367,183 ±

5,002

343 663 _+

5,154

72,865 ±

2,211 2,112

.22

331,625 ±

4,803

310,979 _+

5,062

57,454 ±

.24

305,823 ±

4,927

283,736 ±

4,783

45,486 ±

2,052

.26

281,027 ±

4,760

262,805 ±

4,779

37,257 ±

1,977

.28

258,284 ±

4,456

238,838 _+

4,626

27,027 _+

1,937

4,406

224,866 _+

4,681

18,162 ±

1,906

.30

242,944 ±

.32

225,675 ±

4,487

206,945 _+

4,454

11,527 _+

1,886

.34

212,476 ±

4,230

196,174 ±

4,600

4,952 ±

1,844

.36

199,400 ±

4,233

186,015 _+

4,445

413 ±

1,813

Values are dollars per life saved _~ 95% confidence interval.

it is not very useful as a threshold. Traditional s e n s i t i v i t y analysis 48 was unable to identify additional thresholds for any plausible values of efficacy, c o m p l i c a t i o n r a t e s , or c o s t . Thus, there is no strategy w i t h i n the realm of clinical p l a u s i b i l i t y that is s i m u l t a n e o u s l y less costly and m o r e beneficial. The expected gain or loss in u t i l i t y incurred by m o v i n g from a m o r e int e n s i v e to a less i n t e n s i v e s t r a t e g y was e x a m i n e d for e a c h of t h e six c o m p a r i s o n s to see w h e t h e r it differed from zero for any probability of AMI. A l l of t h e s e differences were significantly different from zero (overall P < .05, t test for paired samples w i t h Bonferroni a d j u s t m e n t 68 for m u l t i p l e comparisons), thus agreeing w i t h the result from traditional sens i t i v i t y analysis. (Full tables of the o u t c o m e utilities, the differences in u t i l i t y , the cost-effectiveness ratios, a n d t h e i r s t a n d a r d c~eviations are available on request.) T h e cost-effectiveness ratios calculated from differences b e t w e e n inpat i e n t and o u t p a t i e n t s t r a t e g i e s are s h o w n (Figures 4 and 5). For example, at an AMI probability of .10, choos L 100/958

ing C C U o v e r O U T w i l l d e c r e a s e m o r t a l i t y by .0060296 + .00002555 ( m e a n ± 95% c o n f i d e n c e i n t e r v a l ) and i n c r e a s e c o s t s by $4,455.88 ± 22.541 p e r p a t i e n t ; t h e r e f o r e , t h e cost-effectiveness ratio at this p o i n t is $4,455.88/.0060296 or $738,997 ± 6,87069 per life saved. Shown are the three coste f f e c t i v e n e s s r a t i o s r e l e v a n t to t h e a d m i t or discharge decision (_+ 95% c o n f i d e n c e i n t e r v a l 69) for PMI from .02 to .36 (Table 5) a n d t h e c o s t effectiveness ratios comparing the three i n p a t i e n t strategies w i t h one another (Table 6). N o t e that despite the m a n y u n c e r t a i n t i e s in the i n p u t v a r i a b l e s , t h e c o s t - e f f e c t i v e n e s s ratios show a reasonably narrow 95% c o n f i d e n c e i n t e r v a l . ( N o t e t h a t for p a t i e n t s w h o s e PMIs are zero, t h e cost-effectiveness ratios are negative; this m e a n s t h a t a d m i s s i o n of these p a t i e n t s costs m o r e m o n e y and results in higher mortality.) Finally, we l o o k e d at the proportion of t i m e s t h a t a given strategy was s u p e r i o r to another. T h i s is an i m p o r t a n t m e a s u r e in t h e c o m p a r i son of strategies because a high probability of even a m o d e s t increase in Annals of Emergency Medicine

benefit m i g h t lead to selection of a certain strategy, whereas a small p r o b a b i l i t y of a t t a i n i n g a large increase in benefit m i g h t not. Despite the degree of u n c e r t a i n t y in the input variables, the distribution of the differences in u t i l i t y a m o n g strategies w a s r a t h e r n a r r o w . D i f f e r e n c e s in costs s h o w e d the largest variation, which reflects the greater uncert a i n t y in t h e c o s t i n p u t v a r i a b l e s . The probabilities of incurring an increase in cost of m o r e than $1,000, a decrease in m o r t a l i t y of m o r e t h a n .0025, and an increase in life expectancy of m o r e than 22 days for moving from O U T to I N T as a function of PMI are shown (Figure 6). A t PMI of .08, use of the C C U strategy will result in a 99.3% chance of some decrease in m o r t a l i t y (no m a t t e r h o w small) compared with OUT and a 95.2% chance of a decrease of m o r e than .0025. Conversely, there will be a 99.9% chance of incurring s o m e increase in cost (no m a t t e r h o w small) than w i t h O U T and a 99.8% chance that the increase will be m o r e than $1,000. Our analysis revealed that for any of t h e six c o m p a r i s o n s , t h e p r o b a 18:9 September 1989

TABLE 6. Cost effectiveness ratios for inpatient strategies PMI

CCU vs INT

CCU vs ROUT

INT vs ROUT

.00

-7,393,386 ± 20,024

.02

482,265 ± 90,481

-35,162,137 ± 4,170,699 639,063 _+

21,192

-30,260,354 + 3,388,265 685,759 _+

.04

214,915 _+ 21,776

283,014 _+

5,761

303,151 ±

6,664

.06

152,311 ±

12,923

186,614 _+

3,316

196,269 _+

3,762

.08

108,748 _+

9,091

138,194 _+

2,362

146,471 _+

2,897

.10

90,857 _+

6,716

109,904 _+

1,884

115,188 -_+

2,096

.12

80,063 +_

6,518

93,112 _+

1,723

96,696 _+

1 830

.14

66,000 +

5,949

80,189 ±

1,529

84,021 _+

1 693

.16

56,298 ~.

5,199

70,617 _+

1,364

74,523 -±

1 534

.18

50,718 _+

4,922

62,072 _+

1,231

65,121 -+

1 437

.20

48,870 _~

4,808

57,325 _~

1,201

59,597 _+

1 337

.22

43,599 _+

4,622

51,699 _+

1,117

53,842 -+

1 264

.24

42,099 ±

4,550

48,368 _+

1,114

50,073 _+

1 194

.26

37,714 _+

4,580

44,537 ±

1,062

46,315 _+

1,161

.28

36,210 +

4,238

41,650 ±

998

43,123 -+

1,128

.30

34,097 _+

4,345

39,527 _+

971

40,966 _+

1,118

.32

32,771 _+

4,255

36,814 _+

967

37,890 _+

1,053

.34

29,847 --

4,118

35,050 _+

910

36,440 _+

1,071

.36

26,898 _+

4,116

33,398 _+

907

35,138 -+

1,044

27,082

Values are dollars per life saved ± 95% confidence interval.

bility of a m o r e i n t e n s i v e s t r a t e g y having a lower m o r t a l i t y than a less intensive one was relatively constant for any PMI of .02 or more, and it generally r a n g e d a r o u n d 95%. T h e probability of i n c u r r i n g an i n c r e a s e in cost, however, was a decreasing function of PMI, ranging from 80% to 60% as PMI increased from .02 to .50 (comparing C C U w i t h INT); for strategies further apart in cost, such as CCU and OUT, the probability of the c o s t of C C U b e i n g g r e a t e r showed a s i m i l a r d e c l i n e b u t never fell below 85%. Similarly, the probability of a m o r e i n t e n s i v e s t r a t e g y p r o d u c i n g great'er l i f e e x p e c t a n c y than a less i n t e n s i v e one was relatively constant over the range of PMI and v a r i e d f r o m 80% to 85% for CCU compared w i t h I N T to 96% to 99% for C C U compared w i t h OUT.

Analysis Our r e s u l t s i n d i c a t e ~ t h a t at a n y level of PMI of .02 or more, the m o r e intensive care is highly l i k e l y to produce an overall i n c r e a s e in benefit but at i n c r e a s e d cost. Even t h o u g h some of t h e e x p e c t e d b e n e f i t increases a r e s m a l l , t h e r e is a h i g h 18:9September1989

probability of obtaining an improvem e n t i n o u t c o m e if t h e s u p e r i o r strategy is chosen. Because t h e m o d e l d e m o n s t r a t e s t h a t there is no s t r a t e g y t h a t is sim u l t a n e o u s l y less c o s t l y and m o r e beneficial, a choice a m o n g strategies m u s t be m a d e by identifying a point at w h i c h the a d d i t i o n a l benefit offered by the more intensive strategy is no longer w o r t h the additional cost that w o u l d be incurred. This p o i n t is called a threshold; if the choice were b e t w e e n an i n p a t i e n t and an outpatient strategy, the p o i n t w o u l d be referred to as an a d m i s s i o n threshold. The choice of a threshold should be m a d e in the light of c o m p e t i n g uses for the same resources and in reference to other similar trade-off decisions. In t h i s s e t t i n g , d e c i s i o n - m a k e r s can act in one of two ways. T h e y can choose a cutoff level of cost per u n i t benefit and then choose the strategy that produces the greatest benefit as long as it is available at or below the cutoff level. Alternatively, they m a y b e subject to a fixed budgetary constraint, so t h e y c h o o s e the l o w e s t cost strategy u n t i l all resources alloAnnals of Emergency Medicine

cated to that purpose are exhausted; if resources are still available after e v e r y o n e has r e c e i v e d t h e l o w e s t cost strategy, the next lowest-cost strategy is used u n t i l resources are exhausted, and so on. It is not clear w h i c h p a r a d i g m h o l d s in c u r r e n t h e a l t h policy or w h e t h e r some interm e d i a t e f o r m exists. H o w e v e r , because the total resource allocable to t r e a t m e n t of A M I is u n k n o w n , we have chosen to structure our analysis along the first alternative. In s i m i l a r trade-off decisions, implicit cutoff values can be discerned. For example, an analysis 7° of 132 federal regulatory decisions w i t h respect to e n v i r o n m e n t a l carcinogens has revealed a c o n s i s t e n t , i m p l i c i t cutoff r a t i o of a b o u t $2 m i l l i o n p e r life saved. U s i n g H o w a r d ' s s m a l l r i s k m e t h o d 79 and adjusting for the average age of patients a d m i t t e d to rule out infarction, we have estimated a c u t o f f v a l u e of $1.5 m i l l i o n p e r life saved in the small risk of death (~ 10 3) range obtained here. Once a cutoff value is determined, a d e c i s i o n a m o n g strategies can be m a d e by u s i n g t h e m o s t e f f e c t i v e s t r a t e g y (CCU) for t h e h i g h e s t risk 959/101

ADMIqq-ING STRATEGIES Wears et al

patients and continuing to use that strategy in progressively lower risk patients until the additional cost per additional life saved reaches the cutoff. At that point (the C C U - I N T threshold), the next most effective strategy (INT) is used for progressively lower risk patients until the costeffectiveness ratio approaches the cutoff; then a still tess expensive strategy is selected, and so on. The admission threshold probability will vary with the treatment s t r a t e g y selected. The a d m i t t i n g threshold probabilities for cutoffs of $1 million and $2 million per life saved and the comparison of each inpatient strategy against OUT (these are the only comparisons that make sense in the admission compared with discharge decision) are shown (Table 7). Note that wide variation in the choice of a cutoff does not produce m u c h v a r i a t i o n in the r e s u l t i n g threshold, especially when one considers that these differences are probably much smaller than clinicians or decision rules can reliably detect. Thus, the admission threshold is not very sensitive to the choice of a cutoff criterion. These results suggest that patients whose probability of AMI is more than 0.08 s h o u l d be a d m i t t e d to CCU; patients whose probability of AMI is more than 0.03 but less than 0.07 should be admitted to ROUT; those whose probability of AMI is less than 0.03 should be given outpat i e n t follow-up. D i f f e r e n t cutoff values will, of course, produce different thresholds. However, it seems reasonable to assume that the cutoff could range between $1 and $2 million per life saved, thus bounding the range of admission threshold values to at least between 0.02 and 0.08. (A similar argument can be made for cutoff values of cost per additional year of life gained. However, the use of life expectancy as the unit of benefit creates other problems because it will produce different thresholds for patients of different ages. This raises an ethical question beyond the scope of this analysis.) Finally, the risk ratios for CCU, INT, and ROUT compared with OUT shed additional light on the appropriate admission threshold (Figure 7). Note that the relative risk of death is a p p r o x i m a t e l y c o n s t a n t for each strategy (compared with OUT) and is 102/960

TABLE 7. Cutoff value ($ per life saved) $1 Million

$2 Million

CCU vs OUT

.08

.04

INT vs OUT

.07

.04

ROUT vs OUT

.03

.02

roughly five times greater for a discharged patient compared with one admitted to CCU until the probability of AMI falls to around 10%. Below this point, the expected benefits start to decrease, again suggesting that at this point, a change to a less intensive, less expensive strategy might be appropriate. To translate our results into true(or false-) positive admission rates, the distribution of the probability of AMI in the population at risk must be known. Using Tierney's results on the distribution of the clinically estimated probability of AMI in patients with and without AM134 and using the procedure outlined above, we can show that if the admitting threshold probability were 0.05, only about 20% of patients admitted to CCU (and presenting without arrhythmias, failure, or continuing pain) would ultimately prove to have suffered an AMI; about 33% would rule in for a threshold probability of 0.10. Because only one estimate of the distribution of PMI was available, we were not able to rest the robustness of these conclusions against changes in the distribution of PMI. Further research on the distribution of estimates of PMI would allow more confident direct computation of the average falsepositive rate that should be experienced for a given level of care. DISCUSSION "In Medicine one must pay attention not to plausible theorizing but to experience and reason together ... I agree that theorizing is to be approved, provided that it is based on facts, and systematically makes its deductions from what is observed ... But conclusions drawn from unaided reason can hardly be serviceable; only those drawn from observed fact." Hippocrates, Precepts We are well aware of the dangers of "thought experiments" in medicine; ho@ever, it does not seem likely that we will ever be able to study this Annals of Emergency Medicine

problem by direct observation, much less by experimental intervention. This model is an abstraction of a highly complex mix of events and uncertainties and required many assumptions in its development. The Monte Carlo process allowed us to obtain stable estimates of the expected gains (or losses) in utilities while simultaneously adjusting for the uncertainties in the input variables. However, factors that are not as objective have necessarily been left out; this by no means reduces their importance but suggests that a degree of caution is appropriate in the interpretation of our results. Our fundamental results, the calculated utilities, are averages expected for an entire population at risk; individual patients may value utilities differently or have differing attitudes regarding risk such that population values are not applicable to them. The charges that we used in our model may not accurately reflect the true economic costs of the strategies. Because there is extensive crosssubsidizationZl, z2 of more sick by less-sick patients in CCUs, simply removing some of the less sick may not reduce actual costs by as much as we have estimated, leading us to overestimate the cost-effectiveness ratio. Further r e f i n e m e n t of the model to reflect actual resource cons u m p t i o n more closely is anticipated. Also, we did not include induced costs in our model. Every life saved by a more intensive strategy will inevitably produce additional medical care costs in the future. This effect (which should be appropriately discounted over time) would cause us to underestimate the cost-effectiveness ratios. Finally, we excluded from our model patients who would have warranted admission due to conditions such as shock, congestive failure, or arrhythmias. Because this group of patients is likely to have a high pro18:9 September 1989

portion of true-positive admissions for A M I , if t h e y a r e i n c l u d e d i n calculating the overall false-positive admission rate, it should be lower than that c a l c u l a t e d h e r e . S i m i l a r l y , m a n y patients admitted for unstable angina tend to be true-positives in a selffulfilling m a n n e r ; if t h e y h a d h i s t o r i cal c r i t e r i a f o r u n s t a b l e a n g i n a s u f f i cient to g e t a d m i t t e d , t h e y v i r t u a l l y by d e f i n i t i o n h a v e u n s t a b l e a n g i n a confirmed at discharge• Thus, our estimate of the appropriate false-positive a d m i s s i o n r a t e c a n b e c o n s i d e r e d a lower bound on the global falsepositive rate calculated on all patients admitted for heart problems.

out AMI may be as high as 70% to 80% (with false-negative releases from the ED 2% to 5%); lower rates could indicate excessively restrictive a d m i t t i n g p o l i c i e s . B e c a u s e t h e s e results are close to clinicians' current p e r f o r m a n c e s , 4 , s a n y a t t e m p t s t o restrict the exercise of clinical judgm e n t i n t h e s e d e c i s i o n s m u s t b e rigorously tested. Finally, further research in the distribution of estimates of PMI in ED patient populations is needed to translate the marginal probability admitting thresholds into global false-positive and false-negative rates.

CONCLUSION The simulation of a complex clinical decision has allowed us to estimate a reasonable false-positive admission rate and permitted explicit identification of the trade-off between health benefits and economic resources as a function of the probability of AMI. Our model shows that cost-effective strategies, in the precise meaning z3 of the phrase (having an additional benefit worth the additional cost), may be devised by making explicit judgments about the appropriate ratio of benefits to costs. Because there is at least some evidence that reduction in in-hospital deaths has been an important factor in the overall decline in m o r t a l i t y from coronary heart disease since 1968, TM it is important that proposed changes in admitting practices be carefully analyzed before they are adopted because they m a y produce significant changes in benefits. In conclusion, we have four suggestions. First, opinions about the appropriate rate of false-positive admissions to rule out AMI should be based on explicit, objective analysis. Second, future attempts to rank patients by risk should concentrate on the appropriate range of threshold probabilities estimated by objective analysis, tt is not helpful to stratify patients unless they can be accurately assigned to one or the other side of the threshold probability.tO,46 No currently available m e t h o d for as-

T h e a u t h o r s w o u l d be glad to s u p p l y m o r e detailed i n f o r m a t i o n to i n t e r e s t e d parties.

18. L'Abbate A, Carpeggiani C, Testa R, et ah In-hospital myocardial infarction: Preinfarction features and their correlation with short-term prognosis. Bur Heart J 1986;7:53-61.

REFERENCES

19. Lee TH, Cook El:, Weisberg M, et al: Acute chest pain in the emergency room. Arch Intern Med 1985;145:65-69.

signing patients to risk categories app e a r s to h a v e the d i s c r i m i n a t o r y

power that this analysis suggests is necessary. Third, given current discriminati n g ability, the acceptable proportion

of false-positive admissions to rule. 18:9September1989

1. Pozen MW, D'Agostino RB, Selker HP, et al: A predictive instrument to improve coronary care unit admission practices in acute ischemic heart disease. N Engl J Med 1984;310:1273-1278. 2. Pozen MW, D'Agostino RB, Mitchell JB, et ah The usefulness of a predictive instrument to reduce inappropriate admissions to the coronary care unit. Ann Intern Med 1980;92:238-242. 3. Behar S, 8chor S, Kariv I, et ah Evaluation of the electrocardiogram in the emergency room as a decision-making tool. Chest 1977;71:486-491. 4. Goldman I, Weinberg M, Weisberg M, et ah A computer-derived protocol to aid in the diagnosis of emergency room patients with acute chest pain. N Engl J Med 1982;307:588-596. 5. Lee TH, Rouan GW, Weisberg MC, et al: Clinical characteristics and natural history of patients with acute myocardial infarction sent home from the emergency room. Am J Cardiol 1987;60:219-224. 6. Wennberg JE, McPherson K, Caper P: Will payment based on diagnosis-related groups control hospital costs? N Engl J Med 1984;311: 295-300. 7. Brush JE, Brand DA, Acamphora D, et al: Use of the initial electrocardiogram to predict inhospital complication of acute myocardial infarction. N Engl J Med 1985;312:1137-1141. 8. Detsky AS, Stricker SC, Mulley AG, et ah Prognosis, survival, and the expenditure of hospital resources for patients in an intensive-care unit. N Engl J Med 1981;305:667-672. 9. Eisenberg JM, Horowitz LN, Busch R, et al: Diagnosis of acute myocardial infarction in the emergency room. J Community Health 1979;3: 190-197. 10. Fineberg HV, Scadden D, Goldman L: Care of patients with a low probability of acute myocardial infarction. N Engl J Med 1984;310: 1301-1307. 11. Fuchs R, Scheidt S: Improved criteria for admission to cardiac care units. JAMA 1981;246: 2037-2041. 12. Goldman L: Acute chest pain: Emergency room evaluation. Hosp Pract 1986;2h94A-94T. 13. Greenberg JA, Kropf R: The economic impact of alternative cardiac care unit treatment

Annals of Emergency Medicine

practices. Health Serv Res 1982;17:45-59. 14. Harrison DC, Irwin WG, Rapaport E, et ah Workshop II: Cost containment issues in acute myocardial infarction and coronary care mana g e m e n t - Points of view. A m J Cardiol 1985;56:65C-71C. 15. Harrison DC: Cost containment in medicine: W h y cardiology? A m J Cardiol 1985; 56:10C-15C. 16. Hedges JR, Rouan GW, Toltzis R, et ah Use of cardiac enzymes identifies patients with acute myocardial infarction otherwise unrecognized in the emergency department. Ann Emerg Med 1987;16:248-252. 17. Hoffman JR, Igarashi E: Influence of electrocardiographic findings on admission decisions in patients with acute chest pain. A m J Med 1985; 79:699-707.

20. Lee TH, Weisberg MC, Cook F, et al: Evaluation of creatine kinase and creatine kinaseMB for diagnosing myocardial infarction: Clinical impact in the emergency room. Arch Intern Med 1987;I47:115-121. 21. Lee TH, Weisberg M, Daley K, et ah Clinical impact of rapid availability of CK and quantitative CK-MB levels for diagnosis of myocardial infarction in the emergency room. Clin Res 1985;33:257A. 22. Miller DH, Kligfield P, Schreiber TL, et ah Relationship of prior myocardial infarction to false-positive electrocardiographic diagnosis of acute injury in patients with chest pain. Arch Intern Med 1982;147:257-261. 23. Mulley AG, Thibault GE, Hughes RA, et ah The course of patients with suspected myocardial infarction: The identification of low-risk patients for early transfer from intensive care. N Engl J Med 1980;302:833-838. 24. Plomick GD, Fisher ML: Risk stratification: A cost-effective approach to the treatment of patients with chest pain. Arch Intern IVied 1985; 145:41-42. 25. Poretsky L, Leibowitz IH, Friedman SA: The diagnosis of myocardial infarction by computer-derived protocol in a municipal h o s pital. Angiology 1985;36:165-170. 26. Rouan G, Goldstein B, Hedges J, et ah A system of rapid followup for early detection of myocardial infarction in patients with chest pain initially evaluated in the emergency department. Clin Res 1985;33:731A. 27. Rude RE, Poole WK, Muller JE, et ah Electrocardiographic and clinical criteria for recognition of acute myocardial infarction based on analysis of 3,697 patients. Am J Cardiol 1983; 52:936-942. 28. 8chor S, Behar 8, Modan B, et ah Disposition of presumed coronary patients from an emergency room. JAMA 1976;236:941-943. 29. Schreck DM, Ng L, Schreck BS, et ah Nonlinear transformation of the resting electrocardiogram in the diagnosis of coronary artery disease. Ann Emerg Med 1986;15:897-900. 30. Schroeder JS, Lamb IH, Harrison DC: Pa-

961/103

ADMITTING STRATEGIES Wears eI al

tients admitted to the coronary care unit for chest pain: High risk subgroup for subsequent cardiovascular death. A m / Cardiol 1977; 39:829-832. 31. 8elker HP, Griffith JL, Dorey FJ, et al: How do physicians adapt when the coronary care unit is full? JAMA 1987;257:1181-1185. 32. Selker HP: Electrocardiograms and decision aids in coronary care: The truth, but not the whole truth. / Gen Intern Med 1987;2:67-70. 33. Thibault GE: Making the coronary care unit cost-effective. Am / Cordial 1985;56:35C-39C. 34. Tierney WM, Fitzgerald J, MeHenry R, et al: Physicians estimates of the probability of myocardial infarction in emergency room patients with chest pain. Med Decis Making 1986; 6:12-17. 35. Tierney WM, Roth BJ, Psaty B, et al: Predictors of myocardial infarction in emergency room patients. Crit Care Med 1985;13:526-531. 36. van der Wieken LR, Kan G, Belier AJ, eta]: Thallium-201 scanning to decide CCU admission in patients with non-diagnostic electrocardiograms. Int / CardioI 1983;4:285-295. 37. Wears RL, Kamens DR: Predictive instrument to improve coronary-care-unit admission practices (letter). N Eng] / Med 1984;311: 1253-1284. 38. Weinstein MC: Cost-effectiveness of interventions to prevent or treat coronary heart disease• Ann Bey Public Health I985;6:41-63. 39• Wiegert HT, Wiegert O: Acute ischemic heart disease: An empiric trial in the use of the Pozen and D'Agostino formula. / Kentucky Med Assoc 1983;81:834-836. 40. Wrenn KD: Protocols in the emergency room evaluation of chest pain: Do they fail to diagnose lateral wall myocardial infarction? [ Gen Intern Med 1987;Z:66-67. 41. Yusuf S, Pearson M, Sterry H, et al: The entry ECG in the early diagnosis and prognostic stratification of patients with suspected acute myocardial infarction. Eur Heart /.1984; 5:690-696. 42. Goldman L, Batsford WP: Risk-benefit stratification as a guide to lidocaine prophylaxis of primary ventricular fibrillation in acute myocardial infarction. Yale [ BioI Med 1979~ 52:455-466. 43. Slater DK, Hlatky MA, Mark DB, et al: Outcome in suspected acute myocardial infarction with normal or minimally abnormal admission electrocardiographic findings. Am J Cordial 1987;60:766-770. 44. Pauker SJ, Kassirer JP: The threshold approach to clinical decision making• N Engl J Med 1980;302:1110-1116. 45. Creches RA, Cain KC, Begg CB: Patientoriented performance measures of diagnostic tests: Tools for prospective evaluation of test order decisions. Med Decis Making 1984;4:7-16. 46. Greenes RA, Begg CB, Cain KC, et ah Patient-oriented performance measures of diagnostic tests: Assignment potential and assignment strength. Med Decis Ma~king 1984;4:17-32. 47. Cain KC, Greenes RA, Begg CB: Patient ~ oriented performance measures of diagnostic tests: U-factor. Med Decis Making 1984;4: 33-46. 48. Weinstein MC, Fineberg HV: Clinical Decision Analysis. Philadelphia, WB Saunders Co,.

104/962

1980, p 12-75. • 49. Lie KI, Wellens HJ, van Capelle FJ, eta]: Lidocaine in the prevention of primary ventricular fibrillation: A double-blind, randomized study of 212 consecutive patients. N Eng] J Med 1974;291:1324-1326. 50. Lie KI, Liem KL, Schuilenburg RM, et al: Early identification of patients developing late in-hospital ventricular fibrillation after dischal\ge from the coronary care unit. Am f Cardial 1978;41:674-677. 51. Lowrie DW, Higgins MR, Godman MJ, et a]: Ventricular fibrillation complicating myocardial infarction. Lancet 1968;2:523-528. 52. DeSilva RA, Hennekens CH, Lawn B, et ah Lignocaine prophylaxis in acute myocardial infarction: An evaluation of randomised trials. Lancet 1981;2:855-858. 53. Hindman MC, Wagner GS, JaRo M, et ah The clinical significance of bundle branch block complicating acute myocardial infarction: I. Clinical characteristics, hospital mortality, and one-year follow-up• Circulation 1978~58: 679-688. 54. Hindman MC, Wagner GS, JaRo M, et al: The clinical significance of bundle branch block complicating acute myocardial infarction: 2. Indications for temporary and permanent pacemaker insertion• Circulation 1978;58:689-699. 55. Valentine PA, Frew JL, Mashford ML, et al: Lidocaine in the prevention of sudden death in the pre-hospital phase of acute infarction: A double-blind study. N Engl J Med 1974;291: 1327-1331. 56. Eisenberg MS, Copass MK, Hallstrom AP, et al: Treatment of out-of-hospital cardiac arrests with rapid defibrillation by emergency medical technicians. N Engl / Med 1980;302:1379-1382. 57. Crampton RS, Aldrich RF, Gascho JA, et al: Reduction of prehospital ambulance and community coronary death rates by the communitywide emergency cardiac care system. Am /Med 1975;58:151-165. 58. Chatterjee K, Harris A, Leatharn A: The risk of pacing after infarction and current recommendations. Lancet 1969;2:1061-1064. 59. Kertes P, Hunt D: Prophylaxis of primary ventricular fibrillation in acute myocardial infarction: The case against lignocaine. Br Heart ] 1984;52:214-217. 60. Mulley AG, Thibauh GE, Hughes RA, et at: The course of patients with suspected myocardial infarction: The identification of low-risk patients for early transfer from intensive care. N Engl J Med 1980;302:833-838. 61. Thibault GE, Mulley AG, Barnett GO, et al: Medical intensive care: Indications, intervention G and outcomes. N Engl [ Med 1980; 302:938-942. 62. Weinstein MC, Stason WB: Foundations of cost-effectiveness analysis for health and medical practices. N EngI / Med 1977;296:716-721. 63. Schwartz WB: Decision analysis: A look at the chief complaints. N Engl J Med 1979; 300: 556-559. 64. Harris JM Jr: The hazards of bedside Bayes. [AMA 1981~246:2602-2605. 65. Critchfield GC, Willard KE: Probabilistic analysis of decision trees using Monte Carlo . simulation• Med Decis Making 1986;6:85-92. 66. Rifkin RD: Classical statistical considera-

Annals of Emergency Medicine

tions in medical decision models. Med Decis ML~king 1983;3:197-214. 67. Willard KE, Critchfield GC: Probabilistic analysis of decision trees using symbolic algebra. Med Decis Making 1986;6:93-100. 68. Wilkinson L: 8YSTAT: The System for Statistics. Evanston, Illinois, SYSTAT, 1988, p 490-491. 69. Pollard JH: A Handbook of Numerical and Statistical Techniques. Cambridge, UK, Carw bridge University Press, 1977, p 15. 70. Travis CC, Richter SA, Crouch EAC, et al: Cancer risk management: A review of 132 federal regulatory decisions. Environ Sci Technoiogy 1987;21:415-420. 71. Wagner DP, WineIand TD, Knaus WA: The hidden costs of treating severely ill patients: Charges and resource consumption in an intensive care unit. Health Care Finan Rev 1983; 5:81-86. 72. Finkler 8A: The distinction between cost and charges• Ann Intern Med 1982;96:102-109. 73. Doubilet P, Weinstein MC, McNeil BJ: Use and misuse of the term "cost-effective" in medicine. N Engl J Med 1986;314:253-255. 74. Gomez-Martin O, Folsom AR, Kottke TE, et ah Improvement in long-term survival among patients hospitalized with acute myocardial infarction, 1970 to 1980. N Engl J Med 1987;316: I353-I3591 75. Cooper G, McGitlem C: Probabilistic Methods of Signal and System Analysis. New York, Holt, Rinehart & Winston, 197i, p 227. 76. Hogg RV, Tunis EA: Probability and Statistical Inference. New York, Macmillan, 1983, p 319. 77. Rubinstein RY: Simulation and the Monte Carlo Method. New York, John Wiley & Sons, 1981, p 26-92. 78. Trivedi KS: Probability and Statistics With RelJability, Queuing and Computer Science Applications. Englewood Cliffs, New Jersey, Prentice-Hall, 1982, p I30, 214. 79. Howard RA: On making life and death decisions, in Schwing RC, Albers WA Jr (eds): Societal Risk Assessment. New York, Plenum Publishing Company, 1980, p 483-506. APPENDIX Monte Carlo Process A major objection to analytic dec i s i o n t e c h n i q u e s 63,64 h a s b e e n t h a t large u n c e r t a i n t i e s in the inp u t probabilities and utilities m u s t inevitably produce large uncertainties in the output. Recent w o r k 65-67 has s h o w n M o n t e Carlo s i m u l a t i o n to be an effective m e t h o d of handling these uncertainties. M o n t e Carlo s i m u l a t i o n is an iterative process in w h i c h a value is calculated for each variable in the decision tree based on its mean, its variance, and a d i s t r i b u t i o n function (such as the n o r m a l distribution); this allows each i t e m to vary o v e r a r a n g e of p l a u s i b l e v a l u e s . The o u t c o m e utility t h e n is evaluated and the entire process repeated u n t i l the average of t h e e x p e c t e d

18:9 September 1989

utilities c o n v e r g e s o n a s t a b l e value. At the end of each iteration, pairwise differences between strategies can be d e t e r m i n e d and t h e distribution of the differences can be accumulated. Monte Carlo simulation has the advantage of not requiring any underlying assumptions about the nature of the distribution of the difference in utilities but the disadvantage of requiring large amounts of computer time to achieve stability in the estimates.

Variables We h a v e e s t i m a t e d v a l u e s for some of our variables in such a m a n n e r as to i n c l u d e c e r t a i n , otherwise unmodelled effects. The risk of lidocaine toxicity was used to represent not only direct, lifethreatening t o x i c i t y b u t also the small but real possibility of other life-threatening therapeutic misadventures because bad t h i n g s can happen to normal people placed in an intensive care environment. The disurility of a false-positive admission was expressed by assuming that the days spent in the hospital r e p r e s e n t e d a " l o s s of life" equal to 50% of t h e a c t u a l inhospital time; for the O U T strategy, the time lost in returning for follow-up e X a m i n a t i o n s was also modelled as a "loss of life" equal to 25% of the time a patient was in the outpatient follow-up stage. Recall that life expectancies were discounted for time; because t h e s e "losses" were immediate rather than occurring in the future, they were hardly decreased at all by the discounting process. Because the raw litigation cost estimates reflect current practice, they were adjusted for PMI as follows. First, for p a t i e n t s who suffered an AMI, litigation estimates were multiplied by a factor equal to .5 + PMI/2. Second, for p a t i e n t s who ruled out, the adjustment factor was set equal to .5 - PMI/2.

These a d j u s t m e n t s have the effect of increasing the litigation cost in patients suffering infarction who were t r e a t e d w i t h less i n t e n s i v e strategies and in noninfarction patients suffering complications secondary to a d m i s s i o n . It was ass u m e d t h a t l i t i g a t i o n came o n l y from cases resulting in death.

Variance The variances were calculated as b i n o m i a l v a r i a n c e s [pq/n) w h e n only a single study was available; w h e n m u l t i p l e reports were used, t h e range t r a n s f o r m a t i o n 7s was used to estimate the variance. The values for m e a n and v a r i a n c e of each variable used in this analysis and their sources are shown (Tables 1 through 4). For variables such as defense costs, where no variance e s t i m a t e was available, we arbitrarily assigned one w i t h a variat i o n c o e f f i c i e n t of 10%; t h i s is e q u i v a l e n t to a s s e r t i n g t h a t t h e v a l u e of t h e m e a n is k n o w n to w i t h i n _+ 20% w i t h 95% confidence. Implementation of Model We implemented a Monte Carlo simulation of this model in the C p r o g r a m m i n g language microcomputer equipped with a numeric coprocessor. P r o b a b i l i t i e s were m o d e l l e d by beta d i s t r i b u t i o n s , w h i c h are defined on the interval 0 to 1.0. Alt h o u g h the d i s t r i b u t i o n f u n c t i o n for costs is unknown, there is evid e n c e s u g g e s t i n g t h a t it is positively skewed; 61 therefore, costs were modelled by lognormal distributions, w h i c h also are positively skewed. Life expectancy was modelled by the Weibull distribution, 76 which can be considered a generalization of the actuarial G o m p e r t z Law. The expected increase in annual m o r t a l i t y rates (a c o m p o u n d rate of 8% per yearlO, 76) was incor-

porated into the Weibull distribution's parameters. Random-number generators were obtained by making function calls on the Lattice Scientific Subroutine Package (v 3.0) for the u n i f o r m , n o r m a l , beta, and n e g a t i v e exponential distributions and were implemented directly for the Weibull and l o g n o r m a l distributions.Th z8 All r a n d o m - n u m b e r g e n e r a t o r s were validated by the X2 goodness of fit, runs, and a u t o c o r r e l a t i o n tests over the proposed range of application. (Full i m p l e m e n t a t i o n details, including source code and executable files, are available on request.) Trial s i m u l a t i o n runs indicated that the differences in utilities for t h e p a i r w i s e c o m p a r i s o n s converged after about 1,000 iterations. For probability ranges of particular interest (as determined by the trial simulation), runs over that range at 15,000 iterations were performed, requiring anywhere from five to 20 hours of computer time.

Converting Thresholds to False-Positive Rates It is important to remember that the threshold probabilities that we have determined in this model are marginal, n o t average, probabilities. A marginal threshold probability of, for example, 0.05 means that one should rank patients in order of risk, assign the subgroup of p a t i e n t s w i t h the highest risk to the m o s t effective strategy, and c o n t i n u e d o w n t h r o u g h progressively lower and lower risk subgroups until, for a given subgroup, its probability of AMI is 0.05. The average probability in the groups already assigned will, therefore, be more than 0.05.95% confidence intervals for cost-effectiveness ratios were approximated from the 95% confidence i n t e r v a l s for cost and utility. 6s

See r e l a t e d e d i t o r i a l , p 1014

18:9 September 1989

Annals of Emergency Medicine

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