Dose-Response Curves and Competing Risks

Dose-Response Curves and Competing RisksAuthor(s): Peter G. GroerSource: Proceedings of the National Academy of Sciences of the United States of America,Vol. 75, No. 9 (Sep., 1978), pp. 4087-4091Published by: National Academy of SciencesStable URL: http://www.jstor.org/stable/68492 .

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Proc. Natl. Acad. Sci. USA Vol. 75, No. 9, pp. 4087-4091, September 1978 Applied Biology

Dose-response curves and competing risks (censored data/dependent risks/linear hypothesis)

PETER G. GROER

Institute for Energy Analysis, Oak Ridge Associated Universities, Oak Ridge, Tennessee 37830

Communicated by Alvin M. Weinberg, June 1, 1978

ABSTRACT Points of the underlying dose-response curve of a lethal response or group of lethal responses induced by varying doses of a toxicant in a homogeneous population can be estimated from knowledge of the time of occurrence for all responses if the response(s) of interest is (are) statistically independent from the other competing responses (risks). In the case of statistical dependence, only tight upper and lower bounds can be estabished within which the points of the dose-response curve have to lie. These bounds for the response(s) of interest are far apart if the frequency of occurrence of the competing response(s) is large. In such situations, the shape of the underlying dose-response curve is only suggested by the imaginary band connecting the estimated bounds. The estimation procedures for both cases are illustrated with data from an experiment in which beagles received injections of

The shape of so-called dose-response (D-R) curves, particularly for human and animal populations exposed to ionizing radiation, continues to be the subject of worldwide controversy. A bewildering variety of "shapes" has been reported in the lit- erature (1, 2) and many different mechanisms have been pos- tulated and described, either verbally or mathematically, to explain them (3, 4). Some explanatory remarks to clarify the terms used in this paper may be helpful at this point.

D-R curves as they are derived from mathematical models of pathogenesis are continuous functions that describe the change in the "frequency of occurrence" (to be defined below) of a particular "response of interest" as the dose of toxicant administered to the subjects under study changes. The term "response of interest" describes here any event that is of interest to a particular investigator. Subsequently, we will sometimes simply call it "response (R)." Some examples are leukemia, lens opacification, and bone sarcomas. These examples and this paper emphasize diseases as responses of interest.

Pathogenesis develops in time, and the frequency of occurrence of the R for different Ds therefore has to be evaluated at some given time. This time has to be the same for different values of the administered D. The disease process will then have equal time to progress at the different Ds, thus permitting a fair comparison of the influence of D on the development of the disease. In mathematical models, this common time is often set equal to infinity. If experimental data are analyzed, one clearly has to settle for some shorter "long" time. What is long and what point in time has to be chosen for a meaningful comparison will become clear below.

It is obvious that the whole function whose graph is the D-R curve can never be observed. We will call this graph the underlying D-R curve. We will only be able to estimate some of its points. To obtain such estimates, it is customary to plot a fraction, number of subjects showing a particular R/total number of subjects exposed, versus the D to which the subjects

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were exposed. Usually the points estimated in this way are then connected by straight lines, or a straight line or curve is fitted to these points by other procedures. This suggests a shape for the estimated D-R curve. Frequently, mathematical models are then constructed that are supposed to explain the mechanism that produced the "discovered" shape. This custom of estimating the incidence with fractions is justified in situations in which the response of interest is the only observed R in the population at risk ("uncensored" case).

In the uncensored case, fractions as defined above represent estimates of the proportion of subjects who show the response of interest before or at a certain time t; therefore, in statistical terminology, these fractions are values of the so-called empirical distribution function (EDF) (5). The values of the EDF for different times give the estimated incidence of the response of interest as a function of time and are estimates for the underlying "true" frequency of occurrence. The EDF is the "best" estimate (6) for the underlying time distribution of the observed Rs. Subsequently, we will call this underlying distribution the "R-t distribution."

If other competing Rs are present ("censored" case) these fractions become meaningless as estimates for values of the R-t distribution of interest. They retain significance only as a measure of the predominance of the response of interest over all the other competing responses. Arley (3) clearly recognized the problem of obtaining a meaningful incidence estimate to determine D-R curves from censored data but suggested no solution.

This paper is organized as follows. The definition of incidence as the value of the EDF at some given time is extended to the censored case, in which incidence is again defined as the value of a distribution function at a fixed point in time. In the censored case the distribution function cannot be estimated by the simple fractions defined earlier, however, and more sophisticated procedures have to be used to estimate the underlying R-t distribution. Plotting the incidences (as defined above) versus the Ds corresponding to the different dose levels leads to a meaningful definition of a D-R curve for censored data. I then discuss what can be estimated in the case of dependent Rs. The same set of experimental data from an experiment with beagles that were injected with 239Pu is used throughout the paper to illustrate the statistical procedures in the different cases. This particular example was selected, from the small set of available data, because the response of interest (osteosarcomas) becomes completely dominant at higher Ds and because the number of subjects is small enough to permit an easy verification of the calculation of the various estimators. Throughout this paper, R, risk, and cause of death are synonymous.

Abbreviations: D, dose; R, response; EDF, empirical distribution function; R-t distribution, response-time distribution; K-M, Kaplan- Meier.

4087



4088 Applied Biology: Groer Proc. Natl. Acad. Sci. USA 75(1978)

CENSORED CASE

In a typical study designed to obtain an estimate for the D-R curve of a toxicant (e.g., ionizing radiation) for a population of humans or animals, the total number of subjects, N, is divided into n groups exposed to D1, D2, . . , Dn (Di+ 1 > Di) doses of the toxic agent. Let the first group, with D1 = 0, be the control group. Each member of these n groups may die from one of several different causes of death that compete among each other for the lives of the group members. Some of these causes are induced by the toxicant; the occurrence of others is independent of it. These different causes of death and the risk that the study is terminated before all members have died or that a subject is lost before its death occured are collected in a set of risks R1, R2, ... Bk. At different times after the administration of the toxicant the risks R1, R2, . Rk will take the "lives" of the members of the n groups. Throughout this paper it is assumed that two or more causes of death cannot occur simultaneously in one individual.

It is instructive first to consider briefly the uncensored case in which only one response, R1, is caused by the toxicant and in which there is only one dose group, say Di > 0. R1 will cause the death of the mi group members at time ti, t2, . . ., tTn and the cumulative response R-t distribution, assumed to be continuous, can be estimated by a step function with steps of magnitude 1/mr at times tk (k = 1, 2,. .i n m1). This gives an estimate of the incidence of R1 as a function of time. An example for an EDF is shown in Fig. 1, which is based on the data (7) in the third row of Table 1.

The estimation of the R-t distribution of one or a group of responses of interest in the censored case, in which other Rs (so-called losses) are present, was described by Kaplan and Meier (8) in a now classical paper. They assumed statistically independent risks and gave the following nonparametric estimator (8) of the distribution function: that is, they made no assumptions about the form of the underlying distribution:

rmax

t1i(t) =i1- 11 (mir-r)/(m-r + 1) [1] r=rmi,n

in which Fli(t) is R-t distribution for R1 in dose group Di (the hat in Eq. 1 indicates estimation), mi is number of subjects at risk in the dose group Di, and r is the rank of a R in the sequence of the tks (k = k, 2, . .. , mi) ordered according to increasing magnitude. rmax(rmin) is the maximal (minimal) rank of a R time for which trmax(trmin) < t.

1.0

0.8

., 0.6

C

0.4

0.2

1 1.5 2 Time ( Days x 10-3)

FIG. 1. Empirical distribution function for osteosarcomas in beagles injected with 239Pu (level 3.0).

It is evident from Eq. 1 that the Kaplan-Meier (K-M) estimates depend on the number and the rank of the events of interest in the event sequence. Therefore, the K-M estimates use more "information" about the R-t distribution than do the customary "fractional" estimates. Because the estimator in Eq. 1 is nonparametric, no "predictions" of the value of the distribution function for times larger than the maximum time of occurrence of a loss (maximal loss time) can be made. The incidence at larger times is only defined to the extent that it has to lie between the value at this maximal loss time and 1.0. This fact, and the observation that the maximal loss times will vary from group to group, poses the following problem: At what comon time should the incidences in the different groups be evaluated? It is clear-as pointed out in the introduction-that the K-M incidence estimates for the different D groups should be performed at the same time. This must be done to give a fair comparison of the influence of varying D levels on the mechanism of induction of a R as characterized by the R-t distributions for the different Ds. Without extrapolation, all incidences in the different D groups can be estimated by the K-M procedure up to the minimum of all maximal loss times that conclude an event sequence in the different groups. A problem may arise if the life-spans are drastically reduced at the higher D levels and if the last events in the sequence happen to be losses, leaving the K-M estimator undefined. Then, only an "early" D-R curve uncharacteristic of the full development of the R in the lower D groups can be estimated. This suggests extrapolation of the K-M estimator to a common "long" time for all D groups. Such an extrapolation could be accomplished by assuming a func- tional form for the R-t distribution and estimating the unknown parameters by analytical methods. However, in many instances, a graphical procedure-hazard plotting (9), which is frequently used in reliability analysis-may suffice. This extrapolation procedure is described further in refs. 9 and 10.

I will again use the data in Table 1 to illustrate the K-M procedure. The response of interest is osteosarcomas, and the K-M estimator for the R-t distribution F(t) at injection level 1.0 in Table 1 is shown in Fig. 2. The maximal loss time for this level is 5277 days. The K-M estimator for F(t) is therefore undefined for t > 5277 days. The minimum of the maximal loss times occurs at 4375 days in row 5 of Table 1. In all other rows (1-4) the sequence of events ends with a response of interest (osteosarcoma) and the K-M estimator is therefore well defined (namely, equal to 1) for all times larger than the maximal response times in these rows. The common time at which the incidences for all groups can be estimated by the K-M procedure is therefore 4375 days, the maximal loss time in row 5. A plot of these incidences versus dose is shown in Fig. 3. Approximate standard deviations of the K-M estimates were calculated by using the asymptotic formula for the variance given in ref. 8. Instead of using the so-called mean skeletal D (7, 11), I used the injected amount in ,uCi/kg of body weight as a measure of D, because this quantity is time invariant.

All the loss times for the D groups in which no osteosarcomas have yet been observed (7) are not given in Table 1. Only the maximal loss time for these groups is shown in Table 1. The corresponding points are shown in Fig. 3 at the accompanying Ds with zero incidence (see also footnote t in Table 1). For comparison, the points corresponding to the "fractional estimates" are also given in Fig. 3.

It seems worth repeating that these fractions do have meaning as estimates of the predominance of a R over losses in a given population. But the effects of the same or a different toxicant on different populations or D groups have to be com- pared with the K-M estimates because the set of losses, the corresponding censoring distributions, and the resulting event



Applied Biology: Groer Proc. Natl. Acad. Sci. USA 75 (1978) 4089

Table 1. Response times for osteosarcomas* in beagles injected with 239PU

Osteo- sar- Mean

Dogs, comas, dose, Level no. no. MCi/kg Response time, dayst

5.0 9 7 2.88 9 8 7 6 5 4 3 2 1 499 1145 1192 1194 1324 1491 1562 1576 2059

4.0 12 12 0.909 12 11 10 9 8 7 6 5 4 3 2 1 1066 1157 1198 1198 1241 1245 1288 1343 1357 1463 1556 1724

3.0 12 12 0.296 12 11 10 9 8 7 6 5 4 3 2 1 1198 1476 1504 1547 1604 1617 1627 1659 1770 1894 1947 1950

2.0 12 10 0.0951 12 11 10 9 8 7 6 5 4 3 2 1 1617 1761 2014 2093 2284 2423 2780 2912 2947 2948 2985 3185

1.7 14 9 0.0477 14 13 12 11 10 9 8 7 6 5 4 3 2 1 467 2221 2500 2659 2777 2973 3025 3282 3312 3353 3430 3430 4214 4375

1.0 26 4 0.0156 26 25 24 23 22 21 20 19 18 17 16 15 14 13 1331 1539 2257 2374 2381 2381 2736 2765 2765 2765 2765 2809 2809 2809 12 11 10 9 8 7 6 5 4 3 2 1

2809 3367 3649 3764 3981 4292 4292 4549 4572 4810 5160 5277 0.7 38 0 0.103 2809t (maximal loss time) 0.5 24 1 0.0054 4

3829* 0.2 10 0 0.0019 4542t 0.1 13 0 0.0007 4503t (maximal loss time) 0.0 29 0 0.0000 5362t

* The same classification of tumors as in ref. 7 was used. The cutoff date for the data in ref. 7 is Mar. 31, 1977. After this date, two osteosarcomas occurred at the 0.5 level. Only one at 3829 days was included in the analysis because the second tumor occurred in a "special assignment" dog (see ref. 7).

t Boldface indicates presence of osteosarcoma. The number above the response and loss times indicates the reverse rank. It is given because the K-M estimator in Eq. 1 can conveniently be expressed as a function of the reverse rank. For levels 0.0-0.2 and 0.7, only the maximal loss times among all adult beagles that were not used in a special study are given to reduce the size of the table. All the loss times can be found in ref. 7. It has to be emphasized that, in the lower D groups, most beagles are still alive and that some osteosarcomas may occur during the next several years. Such additional tumors will make the corresponding incidences greater than z.Prn All hpno,lp' in levels 1.7-5.0 and 14 at level 1 0 have dipd so far

sequences are not exactly the same under the varying circum- stances. For example, consider that one set may contain more virulent diseases than another, or the set of losses may be dose-dependent. Similarly, mathematical models for a disease process (e.g., carcinogenesis) should not be fitted to D-R curves based on these fractional estimates because the mechanism as characterized by the R-t distribution is often confounded by varying sets of competing responses or different event sequences.

A few words about the methods of comparing "treatment" with varying Ds seem appropriate at this point. D-R curves utilize incidence at a fixed time to compare "naively," without

1.0

0.8 -

0.6

5277 days

- 0.4 4375 days

0.2 -

0 2 3 4 5

Time (Days x 10-3)

FIG. 2. K-M estimate of the R-t distribution for osteosarcomas in beagles injected with 239Pu (level 1.0).

a statistical procedure, the effects of different Ds on the proportion of subjects ultimately succumbing to a particular risk. This approach has intuitive appeal, but other ways of comparing treatment effects do, of course, exist. For example, a nonparametric way to compare two R-t distributions censored by different mechanisms was given by Efron (12). Such a comparison of estimated distributions is of special interest in studies of the efficiency of treatments for certain diseases. Here, interest focuses on the distribution of recurrence times of the treated diseases and of the occurrence times of lethal side-effects associated with a particular treatment.

1.0 1 x ? x

0.8 0

0.6-

c0.4-

x x 0.2 16

0 x x II I ' I I - 104 10-3 10-2 0.1 l.0 5.0

Mean amount injected (fCi/kg)

FIG. 3. D-R curve for osteosarcomas induced by 239Pu in beagles. X, K-M estimates at 4375 days; 0, fractional estimates; ?, coincidence of the two estimates. Vertical line shows SD.



4090 Applied Biology: Groer Proc. Natl. Acad. Sci. USA 75 (1978)

DEPENDENT COMPETING RISKS Up to this point we have assumed statistical independence of the responses of interest and the losses. If the risks are dependent and the estimation of a R-t distribution is performed under the assumption of independence, serious errors can be made (13-15). Peterson (14) showed-analogous to results reported by Frechet (16) and Hoeffding (17)-that, without the assumption of independence, only asymptotically tight upper and lower bounds for the R-t distribution can be estimated from the knowledge of the time of occurrence and of the risk for each subject. Estimated bounds for the unknown "true" incidence in a particular D group Di are given by (see ref. 14):

Estimated lower bound, p ?-i?(tmi); 2 estimated upper bound, 1 - S?(tmin)-Si(t min).

The risks are here divided into two sets with at least one member. Sl and S are the estimates for the so-called subsur- vival functions (14) for RI and 12 in the dose group Di. The Sk(t) are defined by

Sl(t) = P(XI > t, X1 < X2)

Sil(t) = P(X2 > t, X2 < X1)

in which the Xks (k = 1, 2) are, respectively, the R-t and loss time and Pt = P(XI < X2). If Pl = 1 (i.e., if only one risk is present), the bounds collapse (14). The bounds in relation 2 simplify to 1 and 51 for the D group with tmin as the maximal loss time.

Fig. 4 shows the points of the D-R curve from Fig. 3 together with the Peterson bounds. The data in Table 1 show that osteosarcomas become more and more dominant over competing causes of death as the injected D increases. Simultaneously, the difference between the upper and lower bounds decreases; they finally collapse for those D levels at which osteosarcoma was the sole cause of death. This illustrates the general result (14) that the difference function between upper and lower bounds tends to the zero function for pI -1 1 (i.e., in the limit of total predominance of one risk). As stated in ref. 14 and proven in ref. 18, it is not possible to demonstrate the independence of the observed risks by using the information usually available in a competing-risk situation (i.e., knowledge of the type of risk and when it occurred). However, sometimes additional knowledge about the etiology of a disease may justify the assumption of independence.

UB UB UB UB + + + +

1.0 - -X- - X-

0.8 - UB X

t 0?

UB -0- 0.6 ~~~~~U LBL p LB 0.8 T LB=p1~~~-0 LB p,

0 . LB =pi

C 0.4

x X 0.2 -I P

LB =p1 i LB

0 4L L..I....I. I.. I I I b.

104 10-3 10-2 0.1 1.0 5.0 Mean amount injected (,uCi/kg)

FIG. 4. Peterson bounds for the D-R curve for osteosarcomas induced by 239Pu in beagles. x, K-M estimates; 0, fractional estimates; ? coincidence of the two estimates. Vertical line, bounds (UB, upper; LB, lower). Arrows indicate the collapse of bounds with estimated points or with each other.

CONCLUSION AND OUTLOOK Before closing, I want both to point out the problems of competing risk theory, which stem from the assumptions inherent in any abstract theory conceived as a description of observable phenomena, and to defend its evident virtues. The first problem originates in the assumption that a cause of death can be as- signed unambiguously to each subject. This is not possible according to testimony from biologists and pathologists [see the papers by Fry and Staffeldt (19) and Holland (20)]. Further statistical research could illuminate the estimation procedure if this assumption is dropped and more than one cause of death is ascribed to a subject. The second problem centers on the assumption of independence. Peterson's bounds are a big step forward but leave the nagging question, Where between the bounds does reality lie? (14). An alternate, more sophisticated approach to describe dependent diseases, the Markov illness- death model, was originated by Neyman (21). This model gives a more realistic description of the complex interaction of diseases, but the large number of involved parameters presents a difficult estimation problem. A recent paper by Clifford (22) illustrates the importance of serial sacrifice for testing independence of diseases in such a model and discusses the re- maining nonidentifiability problems.

Both these problems also induce unsettling distrust of the validity of some conclusions drawn from the information in multi- and single-decrement life tables. Clearly, more information than cause and time of death is needed to describe the interdependence of different causes, but what kind, how much, and the possibility of obtaining it have not been established yet. However, despite these unanswered questions, competing-risk theory has the virtue that it is a step closer to the complexity of reality and represents the best way in survival and D-R analysis to estimate the underlying time distributions and D-R curves from the existing data.

The custom of estimating the incidence of a R and with it the points of the D-R curve by the fractions defined earlier con- stitutes an inconsistent methodology in a competing-risk situation and is, in my opinion, responsible for the tremendous confusion surrounding the question of the "true" shape of D-R curves. The K-M estimator, if independence is assumed, and Peterson's bounds, if this assumption is dropped, provide con- sistent (18) estimation of the incidences. Consideration of competing risks by paying attention to the number and order of the R times in the event sequence has to affect the shape of the D-R curve as suggested by the estimated points.

In the given example (Fig. 3), a sigmoid shape emerges from the noise of losses. The fact that, in the past, D-R curves from censored data were not estimated with the K-M procedure poses the vexing question of whether D-R curves for mammals, reported as linear over a D-range of several orders of magnitude, would remain so if analyzed properly. I conjecture that this will not be the case, although a linear approximation may still be adequate for small Ds. Support for this guess comes from an analysis of two other examples (23) in which again, a sigmoid shape appeared and from the following heuristic argument. The points of a D-R curve result from a cut through the cumulative R-t distributions for the different D levels at a common time. It is difficult to see how increases in D will always correspond to proportional increases of the values of the R-t distribution with the same proportionality factor if one considers the rapidly changing slope of these time distributions (see e.g., Figs. 1 and 2). Analysis of more experimental data might prove this conjecture.

In particular, the data on the atomic bomb survivors in Hi- roshima and Nagasaki should be reanalyzed with consideration



Applied Biology: Groer Proc. Natl. Acad. Sri. USA 75 (1978) 4091

of competing risks, because they are of pivotal importance for the establishment of radiation exposure standards. Every analysis of leukemia incidence in atomic bomb survivors, in- cluding the BEIR report of the National Academy of Sciences (24), has completely disregarded the competing-risk problem. Neyman, referring to these data, noted and deplored (25) that "the methodology used to solve the problem of competing risks is not described in this volume" (i.e., in ref. 25).

Nowadays, ionizing radiation and other toxicants in our en- vironment cause us to ask, with increasing frequency, what influence pollution might have on the predominance or, in the time frame, acceleration of certain causes of death. Ironically, such questions are in a sense just the reverse of the ones Daniel Bernoulli asked when he tried to establish quantitatively the benefits of inoculation against smallpox and reported his efforts to the Academie Royale des Sciences (26). For this purpose, he eliminated the force of mortality of smallpox from a life table and originated competing-risk theory over 200 years ago.

I thank Prof. Jerzy Neyman for opening my eyes to competing-risk theory, Dr. Charles W. Mays and his group for providing an update of the data, Drs. R. Uppuluri, B. Sivazlian, and J. R. Totter for stimu- lating discussions and, with Ms. Lisa Ford, for the translation of Fre- chet's paper, Drs. A. M. Weinberg and A. M. Perry for their con- structive criticism, Mrs. Frances Yaste for her care and expertise in typing the final manuscript, and Ms. Nancy Burwell for her cathartic editing. This work was supported by the Division for Biomedical and Environmental Research of the U.S. Department of Energy.

1. Brown, J. M. (1976) Health Phys. 31, 231-245. 2. Mays, C. W., Spiess, H., Taylor, G. N., Lloyd, R. D., Jee, W. S.

S., McFarland, S. S., Taysum, D. W., Brammer, T. W., Brammer, D. & Pollard, T. A. (1976) in The Health Effects of Plutonium and Radium, ed. Jee, W. S. S. (J. W. Press, Salt Lake City, UT), pp. 343-362.

3. Arley, N. (1961) in Proc. 4th Berkeley Symp. Math. Stat. Prob., ed. Neyman, J. (Univ. Calif. Press, Berkeley, CA), Vol. 4, pp. 1-18.

4. Kellerer, A. M. & Rossi, H. H. (1972) Curr. Top. Radiat. Res. 8, 85-158.

5. Fisz, M. (1965) Probability Theory and Mathematical Statistics (Wiley, New York).

6. Grenander, U. (1956) Skand. Aktuarietidsk. 39,71-153. 7. Jee, W. S. S. (1977) Radiobiology Lab. Annual Rep., No. COO-

119-252 (Univ. Utah, Salt Lake City, UT), pp. 26-45. 8. Kaplan, E. L. & Meier, P. (1958) J. Am. Stat. Assoc. 53, 457-

481. 9. Nelson, W. (1972) Technometrics 14,945-966.

10. Aalen, 0. (1976) Scand. J. Stat. 3, 15-27. 11. Mays, C. W. & Lloyd, R. D. (1972) in Radiobiology of Plutonium,

eds. Stover, B. J. & Jee, W. S. S. (J. W. Press, Salt Lake City, UT), pp. 409-430.

12. Efron, B. (1967) in Proc. 5th Berkeley Symp. Math Stat. Prob., eds. LeCam, L. & Neyman, J. (Univ. Calif. Press, Berkeley, CA), Vol. 4, pp. 831-853.

13. Tsiatis, A. (1975) Proc. Natl. Acad. Sci. USA 72,20-22. 14. Peterson, A. V. (1976) Proc. Natl. Acad. Sci. USA 73, 11-13. 15. Miller, D. R. (1977) Ann. Stat. 5,576-579. 16. Frechet, M. (1951) Ann. Univ. Lyon Sci. Sect. A., Ser. 3 14,

53-77. 17. Hoeffding, W. (1940) Schr. Math. Inst. Univ. Berlin 5, 181-

283. 18. Peterson, A. V. (1975) Stanford Univ. Tech. Rep., No. 73. 19. Fry, R. J. M. & Staffeldt, E. (1978) Proc. Symp. Evaluation of

Environmental Biological Hazards and Competing Risks, eds. Groer, P. G., Fry, R. J. M., Moeschberger, M. L. & Uppluluri, V. R. R. (Institute for Energy Analysis, Oak Ridge, TN), in press.

20. Holland, J. M. (1978) Proc. Symp. Evaluation of Environmental Biological Hazards and Competing Risks, eds. Groer, P. G., Fry, R. J. M., Moeschberger, M. L. & Uppluluri, V. R. R. (Institute for Energy Analysis, Oak Ridge, TN), in press.

21. Neyman, J. (1950) First Course in Probability and Statistics (Holt, New York).

22. Clifford, P. (1977) Proc. Natl. Acad. Sci. USA 74, 1388-1340. 23. Groer, P. G. (1978) Proc. Int. Symp. Late Biol. Effects Ionizing

Rad. (IAEA, March 1978, Vienna, Austria), in press. 24. National Academy of Sciences (1972) Report of the Adv. Comm.

Biol. Effects Ionizing Rad. (Natl. Acad. Sci. and Natl. Res. Council, Washington, DC).

25. Neyman, J. (1972) in Proc. 6th Berkeley Symp. Math Stat. Prob., eds. Le Cam, L., Neyman, J. & Scott, E. L. (Univ. Calif. Press, Berkeley, CA), Vol. 6, pp. 575-587.

26. Bernoulli, D. (1776) Memoires de Mathematique et de Physique de l'Academie Royale des Sciences 1-45.



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Dose-Response Curves and Competing Risks