A comparison of statistical methods for low dose extrapolation utilizing time-to-tumor data

FUNDAMENI"AL AND APPLIED TOXICOLOGY 3:140-160 ~tO~3)

A Comparison of Statistical Methods for Low Dose Extrapolation Utilizing Time-To-Tumor Data D. KREWSK] ^, K.S. CRUMP L~,J. FARMERc, D.W. GAYLOR I~, R. HOWE", C. PORTIER ~:. O. SALSBURG ~', R.L. SIELKEN~;and I. VAN RYZIN H ^Environmental Health Directorate, Health & Welfare Canada, Ottawa, Ontario, Canada KIA OL2; l~Sctence Research Systems, Inc., 1201 Gaine S t , Ruston, LA 71720; C.Hq TCATA, M&A Directorate, West Fort Hood,,TX 76544; i~National Center for Toxicological Research, Jefferson, AR 72079; ~:Nationa] Institute of Environmental Health Sciences, P.O. Box 12233, Research Triangle Park, NC 27709; ~'Dept. of Clinical Research, Pfizer Inc., Eastern Point, Rd., Groton, C'I" 06340; "Institute of Statistics, Texas A&M University, College Station, TX 77843; m'Sandoz, Inc., East Hanover, NJ 07936

ABSTRACT

"[he assessment o f health risks due to low levels o f e x p o sure to potential e n v i r o n m e n t a l hazards based on the results of toxicological experimen~[s necessarily involves e x t r a p o l a tion of results o b t a i n e d aj[ relatively high doses to the low dose region of interest. In this paper, different statistical extrapolation procedures which take into account both timeIo-response and the presence of c o m p e t i n g risks are c o m pared using a large simulated d a t a base. The study was designed to cover a range o f plausible dose response models as well as to assess the effects of competing risks, b a c k g r o u n d response, latency a n d e x p e r i m e n t a l design on the performance of the different e x t r a p o l a t i o n procedures. It was found that point estimates o f risk in the low,dose region m a y differ from the a c t u a l risk by a factor of 1000 or more in certain situations, even when precise information on the time o f occurrence o f the p a r t i c u l a r lesion o f interest is available. Although linearized upper confidence limits on risk can be highly conservative when the underlying dose

response curve is sublinear in the low dose region, they were found not to exceed the actual risk in the low dose region by more than a factor o f 10 in those cases where lhe underlying dose response curve was linear at low doses., 1. I N T R O D U C T I O N Concern over the possible adverse health effects of potentially hazardous substances in the environment has led to increased efforts to determine the extent of risk posed to man. One of the most frequently employed methods for assessing risk is the long-term animal bioassay (OTA, 1981), in which laboratory animals are exposed to the risk factor of interest throughout the major portion of their lifespan. Because such studies are necessarily conducted using doses far in excess of anticipated human exposure levels in order to obtain observable rates of response, it is necessary to extrapolate these results downward to the low dose region of interest.

~(i;d)

":" EL

I-O.

0

n,-

0.5 "900

0

a.

[days)

0

Dose d (fraction of MTD.)

0

FIG. 1. Probability of Response P(t;d) as a Function of q'ime t and Dose d. CopytJgh! I9~3, 5octely of Taxl¢o|~gy

140

Fundam. App/. ToxicoL {3)

3lay/,/une, 1983

BIOLOGICALAND STATISTICALIMPLICATIONSOF'THE ED{,~STUDY TABLE ] Possible Experimental Outcomes in a Cancer Bioassay Outcome

Cause of Death"

Tumor Absent (0) or Present (I)

Available Response

at Necropsy

Times~"

Condition

> T,~,T, > T.

!

S

0

T6

T:.

2 3 4

S C C

1 0 1

Th T~ T~ T,, T4

5

T

I

T. T~

T.,, > T,~ -t TI T, > T,, T,; ~" T4 T,~ > "h, "r~ ~ T, ~ T, T4 - T,~, T,; .~, T.a

"S : 5aclifice, C : Competing Risk, T = Tumor. bAs.,,umin[4the time-tt,-tum¢lr "[', is observable. The problem of low dose extrapolation requires the use of a statistical model relating the probability of response of interest occurring to the administered dose level (Cornfield, 1977; Krewski and Van Ryzin, 1981). Since the time at which an adverse effect occurs is also an important consideration in quantitative risk assessment, it is important that the statistical models used for low dose extrapolation take into account the effects of time as well as dose (Crumpet el., 1981; Farmer et al., 1982; Van Ryzin, 1981). Such temporal data is not only useful in the extrapolation process, but also provides for characterizations of risk which may better reflect longitudinal effects (Sielken, 1981 ). In order to study the behavior of different statistical extrapelation procedures in the low dose region of interest, a large data base of simulated bioassay data was created and analyzed using existing statistical techniques. The advantage of the simulation approach is that the shape of the underlying dose response curve used to generate the data is known and provides a yardstick against which to gauge the performance of the different extrapolation procedures used. An important feature of this study is that information on time-to-occurrence of the response of interest and the subsequent time,to-death is available for use in tl~e extrapolation process.

The results in th~lbaper were originally preserved on September 14, 1981, at the Workshop on Biological and ~tatistical Implications of the EDo~ Study and Related Data B~ses sponsored by the Society of Toxicology and the National Cehter for Toxicological Research. Prior to the meeting, D. Krew~ki distributed 46 sets of computer-generated bioassay data to ~ Crump and R. Howe, D. Gaylor and J. Farmer, C. Portier, D. Salsburg, and R. Sielken, who analyzed the data independently and presented their findings at the Workshop. Although the present report was drafted by D. Krewski and J. Van Ryzin after the meeting, all of the participants in this study have since reviewed its contents. The underlying structure of time-to-response data and the statistical models used in generating the simulated data base are described in section 2, with the overall design and objectives of the simulation study outlined in Section 3. The statistical procedures for low dose extrapolation used in analyzing the simulated data are described in section 4. (The reader uninterested in the technical details involved in model fitting need refer only briefly to section 4.2.) The results of this analysis are presented in section 5, with our main conclusions summarized in section 6.

06

0.5 t/) tO ¢3,

"

i

~alt.

~ 0.3

23 o

0-2

{3,.

O.I

~

, -

: ~talt=9001

/

1

*

~

0"0 :" O.O' O.2

=7001 i

. "

lt 5001

"l :. ! .I

A

Comneting Risks .absent / Competing Risks P r e s e n t /

0.4

q3 Q:

O e,,

9001

---

.

I

O-4 L: O.6 0 . 8

., I-O

B

. . :.

o.o 0.2 • !0-4.

"

:.'

-

i

-

:

o.a

//

dll.900}

i:o

Dose d'.(froction of:MTD) FIC,:-Z. Dose Response Curves Eor:Experiinent .i"in ii~e!Absence andPresence of Com0etin¢ Risks. Fundamental end AppliedrTOXtCOJOgy

(3) 5.6/83

141

TABLE 2, Possible Error Distributions for the Log-Linear Model Distribution of Response Times Lognurmal Log-l(~gistic Weibull

Error Distribution

Density' f(wl

Distribution F(w)

Normal Lol4istic Extreme value

f(w) = 'I"(w) e'*l(l + e')" e" "

d'(w) e=l(l + e') ]-e "

2. STATISTICAL MODELS FOR TIME, TO-RESPONSE DA TA 2. I Time-to-response data Let P(t;d) = Pr I T _< t Id t denote the probability that the random time T to occurrence of particular response is less than t at dose d and let p(t,;d) = dP(t;d)/dt. The probability of not observing a response by time t is then given by the survivor function Q ( t ; d ) = 1 - P(t;d) while the time-specific incidence rate is given bythe hazard function Mr;d) = -d log Q(t;d)/dt. At a given dose d, the hazard function specifies the rate of response among individuals not affected at time t. As illustrated in Figure 1, P(t;d) may be represented as a response surface depending on both time t and dose d, with P(t;d) increasing in both t and d, (At a given time t, the curve defined by P(d It) = P(t;d) defines the dose response relationship for the response of interest.) Thus, the risk P(t;d) may be reduced by decreasing either the exposure time t or the dose d. I n the analysis of time-to-tumor data, several different responses are of interest. We let T; denote the time from the start of the study until a histologically detectable tumor occurs, T~ denote the consequent time until death, and Ts denote the time from the start o f t h e s t u d y until death asa result of tumor occurrence. We will also let T4 denote the time from the start of the study until death from other causes (competing risks) such as natural mortality, Ts = min(Ts, T~) denote the actual survival time and T~ denote the scheduled sacrifice time. The distribution, density, survivor and hazard functions associated w i t h the i th response time Ti will be denoted by P, p, Q, and kl, respectively (i = 1 ..... 6). Not all of the latent failure times T~..... T~ are of course observable in any one case. If T4 < T , for example, the timeto-tumor T= is censored due to an earlier death occurring at time T4 as a result of some competing risk. Nonetheless, it will be helpful to consider these quantities at least in conceptual terms in order to describe the generation of the simulated data base. Within this framework, it is possible for an animal to die either as a result of a scheduled sacrifice, a competing risk or the occurrence of a tumor. In addition, an animal may or may not have developed a tumor by the time of death, The possible experimental outcomes defined in terms of the cause of death and the presence or absence of a tumor at necropsy are summarized in Table 1. Spontaneously occurring tumors may be assumed to occur either independently o f those induced as a resu!t of exposure to the test compound or additively in a mechanistic manner (Hoel, 1980), Letting T,, and Tt-, denote the respective induction t i m e s for theSe;two types of :tumor, w e have" Tx.= min(TmTt2).The application of this two-mechanism framework in thecase of the remaininglatent failure times is described in detaii in Appendix A. We note for.future reference thatwhile~Pt(t;d) specifies the probabiiit~ of a ¢'umor occurring'by time : t in the abSenCe Of comde~ing risks~:the probabiii'tY:ofactually observing a-tumor 14"2

in an animal dying during the course of the study or killed in the terminal sacrifice at time T~ = t is given by PE (t;d) = P1 (t;d) -~' P~ (u;d} !:h (u;d) du

(2J)

in the case of independent competing r;sks. More generally, since Pr(t;d)= P r l T 1 _ < T h T ; < _ t l < P r I T ; - < t l

= P;(t;d),

(2.2)

the probability of observing a tumor in the presence of competing risks is necessarily less than that in the absence of competing risks (Figure 2). 2.2 Time-to-tumor models Statistical models for time-to-occurrence data may be classified into several broad categories, including log-linear, multi-event, compartmental and proportional hazards models (Kalbfleisch eta/., 1983). The particular models selected for use in generating the simulated data bast =, are described in general terms below, w i t h the actual model parametrizations detailed in Appendix B. Log-linear models Under the log-linear model, the logarithm of the response time T is assumed to follow the linear model log (T-A) = n + .B log d + o W .

(2.3)

Here, A > 0 denotes the minimum possible response time; ~x, /3 <_ 0 and o ,> 0 are constants; and W i s a random error term w i t h cumulative distribution F(w). Letting z(t;d) = [Iog(t-A)-(n + /Ylogd) ]/o, the corresponding distribution and hazard func'dons are given by P(t;d) = ~" F(z(!;d)) I, O

(t>A) (t <-- ~)

(2.4)

and 1 = ~ err.A) ~.(t;d) l 0

f(z(t;d)) z(t;d)

(t > A) (2.S) (t _< A ) ,

where f(w) = F' (w) denotes the error density function. Such tog-linear models have been previously applied in tl~e analysis . of failure time data (Kalbfle!sch and Prentice, i 980). Whe n the distribution of the error term W is normal, mor'eover, t h e log,linear m o d e l corresponds to that Considered by Dihc!ffey (1967). In this case, the resP0nsetime T !s assumed to be Iog-n0rmally distributed w i t h median m satisfi/ing the Di'dJtki'eY:equation m ~ d ~ c > O , w h e r e k>_.l and the geometric standard deviation a is independent of the dose d: This corresponds t0 (Z3) With ~r-k=log(c) and ,8 ~- -1/k. Other posSible ~error :diStributions i a r e g i v e n in :Table 2 '(Chand :and Hoel,.~1974). Fundam. AppL ToxlcoL (3)

May~June, 1983

BIOLOGICAL A N D STATISTICAL IMPLICATIONS OF THE EDo~ STUDY

The general product model Ur~der the general product model

where g(d) = -i-o ~id i and H{t) = v~l flit i with ai, ,8, > 0 and ~/~, = 1. The corresponding hazard function is given by Mt;d) = g(d)h(t)

(t > tL) (t < A)

1-exp I -g(d)H(t)l

P(t;d) =

0

(2.6)

(;LT)

where h(t) = H'(t).

TABLE 3 S u m m a r y of Experiment S i t u a t i o n s S i m u l a t e d

Experiment

Shape of Dose-Respuase Curve at Low Doses

Leading T e r m -;n Hazard

Tumor Lethalily ~

Curvature in DoseResponse Curve b

M

Function t

M

[

1

R

3

[

t '~

M M

I

Sublinear

Design t

Time-to-Tumor ModeP

D~

GP

I

z

#

5

Linear

t

o

[

1

I~

7

j

f"

M

9

Sublinear

M

DI

GI?

Linear

t t~ ta

M

10

R

M

Dz

Gp

lI

Linear

M

L

Dt

GP

12 13

Linear Sublinear

1

J

:

1

1

14

Linear

M

DI

PH (Weibu!l baseline)

Sublinear

t t

M

15

M

M

Dl

LL (W~ibull)

::

l

1

1

l

l

LL (Log:normal) LL (LogiLogislicl

18

Linear

t

M

M

D~

LL (Weibull)

19

Linear

ta

M

M

D,

GP

20

Linear

M

M

21

Linear.

t t

M

M

D~ D~

GP GP

zz

1

1

1

1

D,

23

Linear

t

M

M

D~,

GP

"M = Moderate (median = 130 days) R = Rapid (median = 30 days) bL = Low M = Moderate H --- High ~Dz = .5 equally spaced doses, balanced allocation D~ = 5 equally spaced doses, un~alance'd allocation Da 3 equally spaced doses, near-balanced allocation D4 3 equally spaced doses~ un~alanced~allocation D~ = 5 equally spaced doses, haLinced allocation, 25% sacrificed at 500 days. dGP = General product PH = Proportional hazards LL = Log-linear Notes: (i) Moderate ,.::ompeting ri~ks (50% ,{tad 90% mortality at 500 or 900 days, respectively) forlall but E~pt. 9. (ii) Backg,ound tun, or ral~ is S%.,t '~00 days for a" but E~pt. ~o,~nd EXpiZ ~6 and 17. Background t u m o r rate i~, ,1%f°r]Expt" 10 and 6.8% and 6.3,°,%for Expts. 16'and 17, reso'ectively. (This is due to the fact t ha~tthe'err'or dist ributions in Expt s. 16 and 17 Were stand,ardized ~ fis to hav~ the sa me mean variance as Expt. I~) Backgr.ound is additivefor ~xpt. IS.and independent otherwise. (ill) Minimum t u m o r indui:tion ~Jme i s z e r o ~or all btit Expt. 19. M i n i m u m tutnoi" induction t}meis 300.500 days for E~:pt. t9. l

Fundame ntal and Applied Toxicology

~3) 5.6/8"3

1,13

Summary

of Main Comparisons

TABLE 4 M a d e in A n a l y z i n g t h e S i m u l a t e d D a t a B a s e C o n t r a s t A m o n g Experiments C = v~.~~,E;a.

Comparison I. 2, 3 4. .5. o.

Dose response curve (linear vs. subiint.ar at low dose) Hazard function (steep vs, shallow) T u m o r lethality (rapid vs. moderate) C o m p e t i n g risks (dose-dependent vs. mt~derate) Bat'kground (8% vs. I% in absence of competing risks) C u r v a t u r e in dose response curve [linear trend: low-high) (nonlinear trend: Low-medium-high) T i m e - t o - t u m o r model (LL Weibull vs. PEt Weibull) (LL Weibull vs. LL Probit) ILL Probit vs. LL Logit) Background (additive vs. independent) M i n i m u m induction time (thre,~h,J!d vs. nonthreshold) Numb,'r ",;f ;.h,~e groups (five vs: !hret.) Allocation (unbalanced vs. balanced) Sacrifice (interim vs. terminal only)

7. 8. 9. I0. 11. 12, 13. 14.

+ E: + E,) - (El + E.~ + E~ * E~)]I4 [(E3 * E~ + E: * E,) - (El + E; ", E.~ + E~)I/4 [(E= * E~ + E, * E,) - (El ÷ E., + E~ + E;)]/4 E;.- E~ E,- E,~ En - E~:~ [ZEj= - (E. + Ej:0]I2 Em.~- E, Et;.- El, E~,.. - Et7 Et, - El,, E=,- Et2 [(E5 * E;,,) - (E~ + E;-~)}/2 [(E=, * Ez=) - (E~ ÷ E=0I/2 I:.._~- E~,

"El denotes the value o{ some summary statistic averaged over the two replicates of experiment i= I ..... 23. bSE of .1 con t rast i~ given by 5 v ~ (h,12)J :t, w h e r e S" = ~'~:~(A~12)123and A, = E,-E,z d e n o t e s the difference in I he value of the s u m m a r y statistic between replicates.

Hartley. ez a/.. (1981) provide two biological foundations for this model. (The term general product refer/, to the facl that the hazard function in (2.7) factors into separate functions of dose any time.) The first formulation reflects a dynamic or compartment-analytic framework in which each compartment involves a large number of cells and certain restrictions are imposed on the deterministic transfer of the carcinogen from compartment to compartment and on the Slochastic transformation of a normal cell to a cancerous cell. In the ,~econd formula;ion the administration of the carcinogen leads to a series of attacks on the target tissue. The proportional hazards model The hazard function for the proportional hazards model considered by Cox (1972) is given by ,k(t;d} = ~'~.(t)expl~z~S} L 0

(t > A) (t _< A)

(2.X)

where ,%(t).'>0 is an unspecified baseline hazard function ~8 = (,B,..... ,8,)" is a vector of constants and Z = (z~..... zt,) is a vector of regression variables which are functions only of dose (e.g., z = (d, log d)). Since z is independent of time, A(t;d=) is proportional to ,~(t;d2). The corresponding response time distribution is given by P(t;d) =

{

I-O~(t) expl~'~S}

(t > A) (2.9)

0

(t ~-~A)

where do(t) = exp{-./o=,\ (u)du }

3. DESIGN OF THE SIMULATION STUDY Th ~ generatio'n of simulated experimenta! data o n t i m e - t o tumor necessarily involves,making certain ' " assumptions " .concermng the underlying mathemat=cal model used to describe the .carcinogenic process: Slml ar assumptions are .a so req u ired: to.il.accommodate: .m¢~talitv du,o/.to.con~petlng!.dSkS. Because:the biologicalrnechanisms go;ernirig tumorigenesis ..

144

•

.

•

.

•

.

;

"¢=

. . . .

. . . . .

are not adequately understood (Toxicology Forum, 1982). it could be argued that specific assumptions may be unwarranted. Nevertheless, we feel that the models selected for use in our study are not implausiblo. As will be seen later, moreover, the major conclusions derived from this study are not likely to be critically dependent on the actual models employed in generating the simulated experimental data. Another major consideration in the generation of the simulated data base is the variety of different experimental situations covered and the number of replicates of each experiment analyzed. Different experimental situations are necessary in order to reflect a variety of plausible biological effects, such as linearity and sublinearity of the dose response relationship in the low dose region or different rates of tumor occurrence with time. Replication of the same experimental situation, on the other hand, provides a means of assessing the random experimental error resulting from the fact that data for only a sample of test animals is analyzed in each replicate. In order to keep the data base of manageable size,, we elected to simulate 23 carefully selected "experimerltal situations and generate two replicate data sets for each of these situations. Thi~, permitted coverage of a fairly broad range of plausible experimental conditions and yet allowed some assessment of experimental error. The 23 experimental situations considered are summarized in Table 3. The time-to-tumor models include the general product, Iog-linea r and proportional hazards models discussed in Section 2. These models may. be linear or sublinear at low doses and exhibit low, moderate o/• high curvature at highor doses, The corresponding hazard function increased in acco(dance with something between the'first and fifth power of time.The subsequent time~.to:cleath Wasassumed to f0tlow a Weibull m0del'thr0ugh0ut, with ~the med an survival:after tumor onset being either 30 0r.130 days. The:spontaneousresp0nse~rate Pitt;of in the absence of competing risks Was'fixed at 8%at:t ~; 900:days~ except in the Case :0f experiments i O, 1 6 i a n d . 1 7 ~ (16experfment 10,i the Fundam. AppL ~Toxicol. (3)

l~fay/Jun¢,' 1983

BIOLOGICAL AND STATISTICAL I M P L I C A T I O N S OF THE ED,,= STUDY

background rate of re.~ponse .at 9CaPdays was taken to be 1%. In experiments 16 and 17, the spontaneous rates were 6.8% and 6.3%, respectively, due to the fact that the error distributions in these two cases w e r e standardized so as to have the same mean a n d v a r i a n c e as in experiment 15.) Independent background was assumed throughout, except in experiment 19. The expected response rate P~(t;d) in the absence of competing risks at the maximum tolerable dose or MTD (d = 1) was 60% at t = 900 days in all cases, except experiments 16 and 17. (Here, the error distribution for both spontaneous and induced tumors were standardized to have the same mean and variance as those in experiment 15. This resulted in response rates of 64.6% and 65.9% at the MTD in these two cases, respectively.) The minimum tumor induction time was to be zero, except in experiment 19 where A ranged from 300-500 days. Deaths due to competing risks w e r e assumed to follow an independent Weibull model throughout, with expected mortality values of 50% and 90% in all cases, except experiment 9. There, competing risks were taken to be dose-dependent with 75% and 99% mortality expected at 500 and 900 days, respectively, at the highest dose.

Simulated Time-to-Tumor Dose 0

114

l/2

3/4

Cause of Death"

Tumor Absent (0) Present (1)

In most cases~the experimental design used involved five equally-spaced dose levels (d : 0 ; 1 / 4 , 1/2, 3 / 4 and 1)with 48 ~nimals per dose}~Such experiments would generally be consldered relatively i~arge by current standards (tLSI, 1983). In view of the recent interest in developing improved experimental\designs for bioassay studies (Krewski et al., 1983; Portier andtHoel, t 982). however, other designs involving fewer dose levels unbalanced allocations and interim sacr=f=ces were also considered (see experiments 20-23). The 23 experimental situati0ns studied were selected so as to permit the comparisons listed in Table 4. The results for replicat/es one and two of experiment 1 ( s h o w . in Tobies 5A and 5B, respectively) illustrate the type of data generated for analysis. =In the t'ir~t replicate of experiment 1, for example, only two animals at the h=gi~os! dose survived to the terminal sacrifice at 900 days; one of these animals had a tumor which had developed after 878 days on I tloth ;t t i s l i l l / a n d coriipiilcr lapc ill l h e c o m p l c l c ~ilnulaled data lllise ;ire ;l~ oilllhl¢ ill i~oM lln i'!21tiil~M Irltrli; i)r, l)~lnicl gr¢~ski, t~nlilOelllCnlal tlcalih I)irccioral¢. t lc,illh lind Vi~'lhlll: ('illKIdlt, OllaVla.,Onlarhl, ('all;trill I ~ l A (11."l,

TABLE 5A D a t a f r o m R e p l i c a t e " I of E x p e r i m e n t i

N u m b e r of Observations

S

0

7

C

0

39

C T

l l

1 l

Time-to-Tumor, T i m e - t o - D e a t h 000 900 o00 OO0 o00 900 000 404 280 189 854 315 083 470 1.50 580 384 580 205 493 164 302 398 708 385 788 701 288 743 544 050 828 300 512 262 27 51.5 882 75o 304 287 042 406 81o 442 411 6711707 3301475

S

0

5

900 9O0 900 900 900

S

I

l

8051900

C

0

40

787 S t 0 487 850 671 425 380 058 161 505 739 .191 395 613 814 753 300 432 305 459 42 018 478 495 419 339 227 453 452 267 022 387 Io2 512 345 248 318 584 464 146

T

t

2

1801254 380/556

5

0

4

900 900 900 900

C

0

42

C

1

l

577/708

T

1

1

185t307 900 900 900

t 2 9 693 258 219 314 501 100 261 207 46 867 l b b 813 437 875 o00 496-320 418 485 607 482 183.0o7 875 458 354 616 583 235 566 604 702 209"633 657 530 266 172 44 232 246

S

0

3

C

o

41

C

l

l

136/216

T

l

3

430/653 639/728 337/814

S

o

I

9O0

S

1

1

C

o

36

C T

1 1

4 6

618 569 198 468 438 235 422 499 3o4 739 364 243 238 103 795 598 a0t 675 228 461 48Z 4z7 8 7 5 S 2 o 72t 88o 37z 8 7 6 5 0 t 369 340 778 34z 225 248 437 576 2 3 1 3 9 7 259 5Z5

8781900 266 454 98 2 5 8 8 2 4 $17 875 349 506 472,471 260 81g 393 3 8 4 7 0 8 666 425 313"- 8 3 " 4 5 2 : 7 0 1 . 7 2 5 311 1"90 216"496 170 4 4 2 . 3 6 6 2 5 8 " 1 9 3 " 3 4 7 : 1 6 8 ; 4 2 1 296 3481426 239/337 Z941315 s l g / a a s

32sl466:497i569]2i0)3465331706

49!1546:5781679

"S =.Sa~i'ifice C,= Competing~Zi~k T = T u m o r Fundamental and Applied Toxicology

(3) 5-6/83

145

TABLE 5B

Simulated Time-to-Tumor Data from Replicate Z of Experiment 1 Dose

C a u s e of Death"

Tumor A b s e n t (0)

N u m b e r of Observations

T i m e - t o . T u m o r , "i~Ime-to-Death

P r e s e n t (l) 0

114

112

o00 o00 000

S

0

3

C

0

41

C

T

1 1

1 3

5011721 4 3 0 1 5 1 5 1011331 2 0 5 / 3 1 4 gO0

303 374 203 143 678 586 8o.I 747 237 554 602 287 076 340 571 258 876 601 434 175 209 482 273 625 742 318 502 545 854 031 843 51o 700 051 300 832 532 208 .173 525 329

S

0

I

C

0

46

C

I

l

5071002

S

0

3,

000 900 ooo

C

o

39

227 o81" 028 876 347 741 . 8 . 458 256 230 6 4 0 498 390 609 753 5"10 485 706 105 ~07 380 482 318 341 262 312 429 3 2 0 7 1 9 058 3~8 200 544 355 074 431 531 720 409 040 703 233" 280 451 4 o l 440

201 817 ~g2 654 6 1 2 484 306 304 57~ 2q2 441 170 391 220 555 238 310 291 •137 552 441 665 737 328 741 323 68 ° 230 491 555 755 451 729 341

505 550

451 200 810

31.1

1

"S = Sacrifice

C

I

2

1901339 2 4 0 1 3 2 9

"l"

1

4

1491310

o00 900 O00

393i481 4321478 5191038

S

0

3

C

0

35

C

I

2

T

1

8

C

o

34

430 24q 317 523 580 533 500 356 475 540 432 407 321 330 397 595 350 308 305 325 301 355 586 535 077 408 177 256 378 88 243 202 390 402

C T

I 1

4 I0

1041217 4 9 2 1 5 2 7 4 4 3 1 5 2 6 1051231 3521.142 3791425 5 0 4 1 0 4 2 5 7 0 / 7 0 o 4271O31

C = C o m p e t i n g Risk

630 035 150 401 351 5 1 0 704 105 378 502 456 036 105 158 587 310 007 165 ol o 307 517 200 33,1 414 047 423 608 207 518 780 104 172 4QO 389 2~5 4201472

680/o94

12/201 3571512 2 6 8 1 3 0 8 3 2 3 1 4 0 0 3 2 2 1 4 7 0 4511581

3.11172 2 4 5 1 4 2 8 3 8 1 / 5 1 4 6 6 3 / 7 3 0 7 2 4 1 8 1 2

T = Tumor

test. Thirty-six animals died of competing causes between 83 and 824 days on test without developing tumors. Four animals who had developed tumors subsequently died of competing causes while six animals died due to tumor occurrence. In total, 2/48, 3/48, 2/48, 4 / 4 8 and 11/48 animals developed tumorsduringthecourseofthestudyatdosesd = 0 , 1 / 4 , 1 / 2 , 3/4, and 1, respectively. The dose response curves Pddlt) -- Pdt;d) in the absence of competing risks for experiment 1 are shown in Figure 2A for t r. 500,700 and 900 days. These curves are sublinear at low doses a n d e x h i b i t only moderate curvature at high doses. Because the hazardfunctlon is.increasing only.in proportion to t, moreover, t h e s h a p e o f t h e dose response curve does not change, m a r k e d l y w i t h l time. The dose response curve P~' (dlt) --PT (t;d) !0 tl~.e presence0f compet.ing:risks is shown in Figure 2B for.lt=: 900 days..Note that Pr. ( d l t ) < Pi (dlt) due to the faetth'fit'ii3 t~e f0rff(er case manv animals wil! be removed from the p'0'pulat On at rlsk as a r~si~ t"0f C6mpeting r.isks prior tO t u:m0r dev'ei0~ment •

4. A NA LYSIS OF)THE SIMULA TED.DA TA BASE In the absence of a population.threshold dose below which a tUmorwill not occurat any.t,me) a v,rtually.safedoseo .VS .do at time t maybedefmed.s,mphc=tI.Lby 146

1491262 4 9 0 1 5 0 8

rr,, :

II (dolt),

(4.t)

where [ l ( d l t ) : P=(dll)- PdO]t) denotes the excess risk over background at dose d and n'o denotes some suitably small increment in the spontaneous response rate PdO]t)(see Figure 3). Given an estimate P l (dlt) of P=(dlt), a corresponding estimate ~'o of the VSD d= can be obtained as in (4.1). In thissection, a number of existing statistical methods fo r estimating virtually safe levels of exposure are discussed. Included are procedures which utilize information on thetimet0-response as well as those based simply on theproportion of animals responding by the end of the study.

4.1 Statistical models In its simplest form, the H~rtley-Sielken mode/(Hartley and Sieikon, 1977)coi'responds lo the genel:al i~r(>duct model P(t;d} = 1 - exp{rg(d}H(t)l

f4.2~

with g(d) = ~'ald i and H(t) = ~ flit! (~i, fly>---O, ~fli .= 1) i~

i=l

~eing polynomial functions of dose and.time ~respectively. The ~iultistage m0de (crumpeta/, 1976)-is:of the~same general [orm P(t;d).=:-I rexp{-g(d) A.(t)} Fundam; ApDl..ToxleOl. (3)

(4.3) A,fay/June, 1983

BIOLOCICAL AND STATISTICAL IMPLICATIONS OF THE ED,~ISTUDY ((~, -'_:O), w i t h ,%it) simply absorbed into each cr, as a multiplicalive constant (Crump, 1979}. The simpler linear-quadratic

model P(dlt) = 1 - expl-(~o * uld + (~d")l ,

PrOb00.hty of Response P, (d.t)

(.I.6)

w i t h no constraints on the n, w a s also considered.

4.2 Model filling: point estimates and confidence limits

{ BOC k(~round

~CCeDIObte Zncrcmenl 1I +n Re$oonse

. +-

~

ill.~ .....+ ,

.... t V,,~uol~y Safe Dose d,,

Dose d

FIG. 3. Determination of a Virtually Safe Du~,e at Time t m the Presence of Backgn,und Rub,ponce. In the nonparametric multistage model (Dafter et al., 1980). .x(t) is un a t b i i r a r y increasing f+jnr.tion, wherea¢ +n the multictage Weibullmodel ( C r u m p e t el., 1981) .X(t) is assumed to have the Weibull form .%it) = (t-A)", (~, _> O./:1 > 0). The NCTH hnear extrapolation procedure given by Gaylor and Kodell (I 980) and modified by Farmer et al(1982)was also employed. This procedure involves extrapolating linearly from an upper confidence limit on the excess t u m o r risk using the multistage model in (4.5) b e l o w fitted to time-adjusted t u m o r incidence data discussed in section 4.2. The extrapolation is done either from the lowest experimental dose or the dose estimated to produce an excess risk of 1% (whichever is higher), and is intended only to provide an upper limit on risk for dose response curves w h i c h are either linear or sublinear =n the low dose region. In addffion to these time-to-response models, several quantal response models w e r e fit to the tumor incidence data in the simulated data base. A general class of quanlal response models is given by P(dlt) -- -! * (1-7)F(a + .~log(d+6))

(4.4)

(o ~3, < t , /3>0, 6>-O). w h e r e 8 = 0 or -y= O in the case of purely independent or purely additive background, respectively, and F is any cumulative distribution function (Krewski and Van Ryzin, 1981). In particular, Fix)= ,I,(x) and 1-expl-e'l for the probit and Weibu// models, respectively. A linear extrapolation procedure w h i c h involves extrapolating linearly from the 1% or 10% excess risk point on the fitted We/bull curve w a s also applied (Van Ryzin, 1980). The multistage model in the quantal case has the form =. i P(dJt) -- 1 - exp I- ,~0 ~,d I (4.5)

The H~rtley-Sielken model w a s fit by m a x i m u m likelihood using a computer program called MRS. T. (Hartley and Sielken. 1978a). In analyzing the simulated data, a and b w e r e set equal to 4 (one less than the number of dose levels), except in experiments 21 and 22, w h e r e a w a s set equal to 2 because only three doses w e r e used. In order to construct a 95% l o w e r confidence limit for the VSD d~. the original data for each simulated experiment w e r e sampled with replacement G- 1 = 4 times to form a total of G -- 5 (I;,tn .~et=_ cf the 3--.me ge¢,,~sal ~.tructure. Letting'y', = log d't. w h e r e ~+ denotes the e.stimaled VSD based on the original data and ~'~..... ~t; t l - corresponding estm=ates ba~od on the G-1 additional data sets, an approx=mate 95% Iov~er confidence lhnit on d~ is given by ~t. = e x p l ~ j - Stn ,] .

(4.?I

~ ( y^, - ..K., y ) - / ( G - f ; . ~ = ~ ' ~ , / G a n d t . . , denotes the where S" = (; i*| i,l 95th percentage point of th,+ t distribution w i t h G- 1 degrees of freedom. An empirical j u s ' d i c a t i o n for this m e t h o d is given in Hartley'and Sielken (197".b). M a x i m u m likelihood e.=timates of the parameters in the multistage Weibull model w~,re obtained us!ng RANK 81 (Crumpet a/., 1981) w i t h a = 4 .n all experiments except 21 and 22 w h e r e a -" 2. Lower conhdunce limits on the VSD w e r e obtained by finding an upper limit on the linear term ~ in dose consistent w i t h the likelihood of the observed data (Crump, 1981 ). Estimates for the nonparametric multistage model are based on the procedures described b;, Daffer et el. (19BO). using a stepwise approach to choosing k in order to obtain lower confidence limits ( C r u m p e t al., 1981). M a x i m u m likelihood estimators of the parameters in the probit and We/bull quantal response models w e r e obtained using RISK 81 (Kovar and Krewski0 1981), w i t h l o w e r confidence limits on tt~e VSD obtained based on the asymptotic distribution of log ~o (Krewski and Van Ryzin, 1981 ). This same procedure w a s used to obtain both point estimates and lower confidence limits on the VSD using linear extrapolation from the 1% and 10% excess risk points of the fitted Weibull model. Point estimates for the multistage model w e r e obtained using GLOBAL 79 w i t h lower confidence limits on the VSD again based on an upper lirn'it on ot'(Crump0 1981).

TABLE 6 Likelihood Construction for Different Analyses of Time-to-Tumor Data Cause

Tumor

Contributions to Likelihood

of Death

Absent (0) Present it)

AI. Time-to-Tumor" (observable)

Az. Time-to-Death from Tumor

A,. Time-to-Death with Tumor

A+. Time-to-Tumor (unobservable)

S S C C T

0 1 0 I I

Q(T;;d) piThd) Q(T~;d) p(T~;d) p(Tt;d)

QtT.;;d) Q(T~,;d) QtT.';d) Q(T;;d) p(T~;d)

QtT~-d) p(T;;d) Q(T~;d) p(T;;d) p(T~;d)

Q(T~;d) PCW+;d) Q(T.C;d) I'(T~;d) P(T~;d)

"Ts = Observed time-to-tumor, T~ = min(TsoT=) "- Observed Ume-to-de,~th or s,~crif,(e.

Fundlment,I Ind Applied Toxlcololly

(3)5-6/83

147

TABLE 7 V i r t u a l l y Safe D o s e s at .% = I O s in t h e P r e s e n c e a n d A b s e n c e of C o m p e t i n g R i s k s (fraction of MTD). Competing Risks Absent"

t = 500

700

900 days

Competing Risks Present (t = 900 days)

|,2o0 3.4 5,o,20-23 7,B

tLo-3 2.0.2 1.o-4

I.o-3

6.4-3 1,0-2. 8.3-5 2,2-4

5.1-3 ,';. ! -3 5.2-5 5.2-5

B.O-3 t. t -2 1.3- I 2.5.4

I0 I1 12 13 14 1.5 *J~, ~7 18

t.b-3 1.3-4 1.3-3

2.2-4 3.,I-5 3.4-4

2.4-4 5,0-5 4.0-4 8.4-2 t .3-4 t .o-3 4.0-2

Io

Experiment

I,I-I

7.0-2

1.7-4 2.0-3 5.3-2 o.8-S

&7-5 1.5-3 3.8-2 7.8-3 5.1-5

.I.O-5 ! .3-5 1.3-4 6.0-2 5.4-S t .2-3 3,0-2 ~..t 3 3,2-5

0.0-3

o.I-4

t.3-4

t.l-2

9.4-3 7.8-5

0.0-4

"8.q-3meansS.o X lO:I 1he simple linear quadratic model in (4.6) was fit to the quantal data by using the bootstrap method from Efron (1982). In this approach, the observed data are first used to establish an empirical probability distribution function. The computer then samples at random from that distribution function and the parameters ¢~t~,cYi and ~z are estimated for each sample. A total of 1001 samples were generated w i t h a VSD computed for each sample. After ordering these VSD's, the median was taken as the bootstrap estimate of the VSD with the 95% lower confidence limit given by ~he appropriate order statistic of the ordered bootstrap samples. Rough estimates of n,, n~ and ~z for each bootstrap sample were obtained by linearizing the cumulative hazard function and solving by ordinary least squares. This involved solving the set of equations -In(l-p) = vr.a+ o,~d + n~d

(4.8i

(d = 0, I / 4 , 1 / 2 , 3 / 4 , 1) for n0, n~ and ~.~. Here p = [x(t;d)+ O.24]/[n(d}+O.24] and x(t:d) denotes the number of animals with tumors at dose d and time t of the n{d) animals started on test. (The constant 0.24 is included to avoid problems with the log transformation w h e n x(,t;d) = O.) These same quantal response models w e r e also fitted with the number of animals at risk adjusted so as to take into account the elfects of premature deaths. (The actual method of adjustment is described in the next section.)The linear extrapolation procedure of Farmer eta/., ( ! 982) was also applied to the adjusted tumor incidence data in order to obtain lower confidence limits or, the VSD using the multistage model and the procedure described by Crumper al. (1977).

4.3 Methods o/analysis While the method of maximum likelihood is used to estimate the unknown parameters in the time-to-resp0nse models discussed i/~ section 4.2, the actual form of the likelihond depends on the type o f analysi s being carried out (Kalbf!eiscl'~et al., 1983):As described below, the typeof analysis performed will depend on the degree o f information available concerning w h e ' n t h e tfina'or~actually occurred and the Cause el death: 148

A~. Time-to-tumor (observable) Ideally, the time~ at which all tumors developing during the course of an experiment would be available for analysis, Such data could conceivably be obtained in the case of visible, palpable or rapidly lethal tumors, although such information seems rarely available in practice. When the actual time-totumor is observable, however, it is possible to estimate the time-to-tumor distribution using the likelihood for analysis Az in Table 6. A2. Time-to-death from tumor If it can be determined w h e t h e r or not an animal died from a tumor, then an analysis of the time-to-death for those animals dying from tumors is possible (Table 6). Because the endpoint under consideration i3 now time-to-death from' tumor rather than time-to-tumor itself, it should be recognized that the estimates obtained apply to the former rather than the latter measure of time-to-response. This analysis would of course be similar to A~ if the tumors under study were rapidly lethal.

A~. Time-t0-death with tumor Since the cause of death cannot always be determined, an analysis similar to A2 may be carried out based on the times-todeath for those animals found to have a tumor at necropsy. regardless of whether or not the tumor was responsible for the death of the animal. The likelihood in this case is given in column 3 of Table 6. A~. Time-t0-tum0r (unobservable) When the actual time-to-tumor is unknown, the observation of a tumor at necropsy indicates that the tumor occurred at some point in time prior to the death or sacrifice of the animal. Utilization of this information in the' construction of the likelihood function (Table 6) will provide a valid estimate of the VSD whenever all tumors are incidental necropsy findings unrelated to the death of the animal. This analysis will of bourse not be strictly valid for [ffeqhreatening tumors. It is clear from the above discussion that analyses At .: A~ address stightlydifferent e ndpoints and that appropriate estimates of Pl(dlt) in (4.1) w i l l generally be ave.liable only When Fundam. AppL Toxieol. O)

May/June. 198.7

BIOLOGICAL

AND

STATISTICAL

IMPLICATIONS

O F T H E ED~, S T U D Y ~-" oa e- C,

C

,~

-

iill ......

~

i

H

oo

~Z~ ~k'=z

-

E .X~ ~ ,r~ ~. j:: ~

o

o

~.o

V/

~

¢,"~

o '~

E: ,!

t

t

L.__.L.-- ' - z l ~

t~

(%)

,~3uilnbeJ.-I

- ,J:l

I

I

I

I

G~

i/

l-

E (~

_

1

,.. ~,,

41

q

|

.... t

!

I_

J::

]

'

#

t

"

~Ca

.J

aAqOl~E~

,-

I",-

~

I

I

I

" o o ~:_~

mt

V/

/'~

--

A: E C~

•

I

•~ ., : EE -,:-

._El

E

I--

~T~

w.-i

F~

_ --

",=I

C

(X

=

'

•

,,,1

~

.......

1

I

1%)

.. I

,~uanb~Jq

~.~.

I

m

a^!IOl~E}

O~ m

0

I_ ~

1

~JD

~" -"1

~ I

......

t ~ -• - "l 1

L~

~ ~ ...._

I

,

I....... I

I_

E

O

E

=."2-

. 0

0 V/ ('~

0 A

" (..3

" ¢J

(M

~.,

.~ c ~'~ ,.C:

~r~ =,J3

F-

6

gT. ol

LE I O

,,t O w)

1 O (M

1 O --

_

a_j

l

r~

~"

Ji_~ C.,.

!

•

~

.

71

-~ 0

,9, i ....... I O O ~" 14)

• O N

. "t O --

t O .7"

O I¢)

l O N

O

--

_ !

|

0

0

NO

0

~>';

G

~ 8

._~-~ E"' o ta.. , ~ , . _ L3 , . . ' ~

(%)

Fund-menial

and Applied Toxicololy

,~::)uonb~J~

(3) 5.~/83

a^!lOla~J

149

in comparison with the variation in the estimates across experiments, w e will thus take as the target parameter the VSD defined in terms of (4.1) regardless of the method of analysis. 40

5. RESULTS The performance of the different procedures for low dose extrapolation will be assessed using the criteria

30 20 i0

CI = Iog,,(~./do)

Multistage I...... eJ

Linear - Quadratic I

>

o (ll rlr

and

~: = Iogj,,lll(~,,)/ll(d,,)l

¢:: ¢1

40

30 20 tO I'""|

l

I

i

l

!

l

•-3-2-1 0 I 2 3

I

!

l

!

!

I

-3-2-1 0 I 2 3

C2

Cz

Oc2~<0

C= provides a measure of the relative error of the estimator ~,, on an order of magnitude basis. C~ = 1 for example, w o u l d indicate that the VSD has been overestimated by a factor of 10, while Ct = -1 would correspond to an underestimate of the same order of magnitude. C= provides a measure of relat'ive error on the risk rather than dose scale. C~ = 2, for example, would indicate that the actual risk at the estimated VSD ~,, is in fact 100 times the acceptable risk tr,,.

5. 1 Analysis of time-to-response data Our comparisons of the different methods At - A4 of analyzing time-to-tumor data discussed in section 4 are based on the multistage Weibutl model (Figure 4z). In the ideal case where JI h r o u ~ z h o u l I-i~.r¢~ 4-13. all ('l and C., ~aluc,, e x c e e d i n g II1CIUIICLI ill I.IIL' ~.'alC~lilllC% 011 Iht' c~.l~rcII1c ICll ;irld i I ~ h l .

[] C z > O FIG. 7, Frequency Distributions of the Values of Cz for Point Estimates of the VSD Based on the l)robit, Weibull, Multistage and Lim..ar-Quadratic Models.

time-to-tumor is observable tAd. Similarly, the quantal response models provide estimates of the dose response crave P~(dit) in the presence of competing risks. Because of this, the observed incidence data x(t;d)/n(d) was also analyzed using the quantal response models with n(d) adjusted for the presence of competing risks. When estimating the probability of tumor occurrence Pitt;d). the effective number of animals at risk over the period (Off) was estimated by ^n (t;d) --' x(t;d)

dud)

(4.9)

in analogy w i t h At, w h e r e P dr;d) is the nonparametric es~matar of Pdt;d) given by Kodell eta/. (1982). (When x(t;d) = 0, n (t;d) is set equal to half the number of animals started on test.) When estimating the probability of death from tumor occurrence P~(t;d), the effective number of animals at risk is given by ,.~ P ~(t;d)

(4.10)

150

~aluc arc

Weibu,

I

Addilive Rockgreund J

40 30

IO

41

U r(13 3 O"

Lineo"r Extrapoiationi Weibull (1%) I

u.

Linear Extrapolation 1 Weibull (10%) J

40 30

!,,[~,'-I-"

I0

/%

in analogy with A.,, w h e r e P3(t;d) is the Kaplan-Meier (1958) nonparamevic estimator of the distribution of times-to-death from tumor P,.,(t,d): Although the different analyses discussed above in effect provid e esti.ma!es" of different endpoints, the differences in the target VSD are not expected to be great. As shown inTable 7, for example, th e difference between the true VSO's defined in terms of thedose response curve P; {dli) were at most a factor of five greater~tha~.those define d in ~erms of thed0se respons ~ curve P~' (dlt) in the presence of.compeUng risks in all cases considered, since,thi-~ differenceWill be seen to be negligible

,~ ill , I h , , o l u t c

[

20 n (lid) "-- x(t;d)

(5.1)

"t

-3-2-10

!

J

f

23

I

I

I

-3-2-10 I 23

C2

C2

r-i c ~ o

[]

>o

FIG. 8. Frequency Distributions 0 f . t h e .Values of C2 {or Point Estimates of the VSD Based o n the Weibull Model With Independent and Additive Background and LinearExtrapolation from the 1% and i 0 % E>:cessRisk Points on' the,Fitted Weibuil.Model,

Fundam. AppL Toxicol. (3)

May/June, I983

BIOLOGICAL AND STATISTICAL IMPLICATIONS OF THE EDo~STUDY time-to-tumor is observable (Aft, 2 0 / 4 6 or 43% of the point estimates of the VSD at t ---.900 days were within an order of magnitude below the true VSD (-1 < Cz -<0), with 43% of the estimates also being greater than the true VSD (C ~> 0). In some cases, however, values of both Cz and C2 in excess of 2 and even 3 were encountered. The results for analyses A2 - A4 based on less information than required in Az are similar, although the percentage of cases in which Ct or C= is greater than unity exceeds that found with Ax. Possible systematic differences among the four methods of analysis were assessed using the sign test (P < .05). This analysis confirmed that A~ tended to yield higher estimates of the VSD than did A,~ which ,~n turn appeared to give higher results than AI and A~. (These last two analyses did not differ appreciab|y.) Although the different methods of analysis did produce estimates of the VSD which differed by more than three orders of magnitud e in some experiments, the differences were generally less extreme. (Considering all possible pairwise comparisons among methods, more than 85% of the C= values were within an order of magnitude of each other.) This same analysis of the Cz values provided similar findings. Frequency distributions of the criteria Cj and Cz with point estimates of the VSL) replaced by 95% lower confidence limits are shown in Figure 5, With analysis A~ based on the actual times of tumor occurrence, none of the 46 data sets resulted in values C~ or C~ greater than zero. Although a few results were more than 1000 times lower than the actual VSD, more than half of the lower confidence limits on the VSD were within an order of magnitude of the actual VSD. Although one of the 46 data sets resulted in values of C= and C= greater than

[Multistoge I

°°t 3O

IO t.-. aJD

JLineor

t L!ne°r~-C~uO'drolicI ~J 0

n,-

Exfrop'oiol:ionJ

_ Weibull (1%), ]

4o 30

20 IO

-1 -3-2-1 0 I 2 3

":3-2-I 0 1 2 3 Cz

Cz

[]

Cz~O

[] c2>o FIG, 9. Frequency Distributions ot: the Va|ues of C2 for Lower C0hfidence Limits on the VSD Based on Four Quantal Response Extrapolation Procedures. Fundamental and Anplled Toxicology

(.3)5.6183

POINT

ESTPdATES

1Unadjus'edl 40 30 A

~

IO i

U "m O"

LOWER CONFIDENCE LIMITS

h

lUnadjusted--]

Q.) Q

rr

40

lAaiusteal

30 20 lO I

I

l

I

|

I

I

!

I

I

I

I

I

I

-3-2-1 0 I 2 3

-3-2-1 0 I 2 3

C2

C2

I

r-] C2~< 0

F'] C 2 > 0 FIG. 10. Frequency Distributions of the Values of C2 for Point Estimates and Lower Confidence Limits on the VSD Based on the Weibull Model and Quantal Response Data Both Adjusted and Unadjusted for Competing Risks. zero using analysis A2. the results for both analyses Az and A,~ are similar to those for A~. Analysis A~, however, provided values of C~ and C= greater than zero in 15% of the data sets analyzed, in excess of the 5% that might be expected at the nominal 95% confidence level. (All of the CI and Czvalues were, however, less than 1.) Consideration of pairwise differences for confidence limits revealed similar trends as with the point estimates, with sign tests indicating that A4 gave larger C1 values than A2, and that A~ gave larger Cl values than A~. (A1 and A2 did not differ appreciably.) For all but one of the simulated data sets, however. the different methods yielded lower confidence limits on the VSD which differed by less than an order of magnitude. Again. the findings baseq on an examination of the C~ values were similar. Further analyses carried out at t = 500 and 700 days on test led to results similar to those discussed above. In what follows, w e w i l l t h u s continue to present detailed results only at t = 900 days, the termination time of the simulated experiments. Because of the general concordance in the findings based on either criteria C~or C2, moreover, we will, for the most parL focus our attention on the latter Criterion in What follows. For, brevity, we will also confine ourselves to a discussion of analysis A,.

5,2 Comparisons among time-to*rs¢ponse models Frequency distributions of the C~ Values f0t: both point 'estimates 'and lower Confidence limits-~n t h e V S D ;for the multi: StageWeibull~'imultistage nor~parametric and Hartley-SielEen 151

The tower confidence limits on the VSD based on the NCTR linear extrapolation procedure were higher than those based on the mulUstage Weibull model in nearly 87% of the cases, althoug'h the two estimates were within an order of magnitude of each other in over 91% of the data sets analyzed. (The median ratio of the confidence limits was about 1.7.)

Of

60

40

5.3 Comparisons among quantal response models

20 l"'t

uo

ILower Confid-ence

t.i;nits i

8O

llv

a:

6O

4O 2O l

I

I

I

I

-3-2-! 0 t 2 3 C= (Multistage WeibulI-Multistage)

!

i

!

I

I

i

For several models, point estimates based on the quantal data alone were frequently in excess of the actual VSO. tending to give values of C= exceeding zero (Figure 7). (Unless otherwise indicated, all quantal response models were fitted undel" the assumption of independent background.) The probit model performed particularly poorly, with risk at the estimated VSD in most cases being more than 100 times the acceptable risk. The Weibull, multistage and linear-quadratic models performed progressively better, with the distribution of point estimates based on the last method similar tothose in Fig~,'e 6 for the models utilizing the time-to-t~Jmor data.

I

POINT ESTIMATES

-3-2-I 0 I 2 3

~ge.

Cz (Multistage WeibulI-Multisfage)

[]

MultistageWeibulI-Multistage ~ 0

60 50

[~]

Multistage Weibull- Multistage > 0

40 20 tO

Quantal Response Data.

~1n lhL'~¢, 1~c;~s. Ihe¢ot'arianccm~tr~xof theparameter¢~llmalc.~wast,in~utarand could not heinverted. '~Con;*VJ'$cnceprobtem~v,'crccncouractcdin oneof the4bdata:,cts. 15,!

Weibu,l

30

Fit]. ! l. Frequency ~istributions of the Difference (Multistage Weibull.Multist.tge) of C1 and C= \;alues for Point Estimates and Confidence Limits on the VSD Ba;,ed (~n Time-to-Response and models are shown in FigurG 6, along with the confidence limits provided by the NCTR linear extrapolation procedure. As noted earlier with the multistage Weibull model, point estimates of the VSO based on both the multistage nonparametric and Hmtley-Sielken models can result in risks more than 1O0 fold greater than the desired risk. None of the lower confidence limits based on the multistage Weibull or multistage nonparametric models resulted ;n risks in excess of the desired value, though confidence limits based on the latter model could not be calculated for 15 of the 46 data sets analyzeds and are thus not represented in the frequency distribution Shown in that case. Lower confidence limits based on the HuTtley-Sielken random 9roup method resulted in risks in excess of the target risk in 7% of the cases analyzed, close to the nominal 5% level. With the NCTR linear extrapolation procedure, fewer highly conservative results were obtained tha.n with other procedures, although 6/454 or 13% of the lower confidence limits resulted in risks in excess of the target risk. The multistage Weibull model tended to give higher estimates of the VSD than the Hartley-Sielken model, with the median ratio of multistage Weibull estimates to HartlaySielken estimates being 2.4 and 2.5 for point estimates and confidence limits, respectively. The multistag e nonpararnetri c model was very similar.to the multistage .Weibull, with the median ratio being 0.9B for point estimatesand 1.06 for confidence limits. I n addition, the estimates (both point estimates and !0~er confidence limits) based 0n.these two models were w i t h i n arl order o f m a g n i t u d e of"each,other in all cases cor~sidered.

L O W E R CONFIDENCE LIMITS

[Harf!ey 40 30

20 I0 =1 ¢T

{N~R

u.

Linear Exi,rapolelionJ

50-

>= "~

[]

Cz(Rept-Rep2) =;0

3O

n,-

[]

C2(Rept-Rep2) > 0

20 I0

r"l 60 50 40 30 20 I0 I

I

|

|

|

|~---

-~-2-1 0 I 2 3

J~t

F

I

1

II

-3-2-1 0 I 2 3

C2 (Repl ~- Rep2) Cz (Rep ! - Rep2) FIG: I2. Frequency Distributions of the [,lifJ~eren~:e(Replicate ] Repllcatez) of C2 V,~lues forPoint Estimates and Confidence Limits On the VSD, Based On Four Extrapolation Procedures. Fundam. Appl. Toxicol. (3)

31a),/June, 1983

BIOLOC;ICAL AND STATISTICAL IMPLICATIONS OF THE EDo~STUDY Linear extrapolation from the 1% and 10% excess risk points of the fitted Weibull model no;ably reduced the degree of underestimStion of risk m the tow dose region (Figure 8). In the la~t~:r case, however, 1 6 / 4 6 (35%) of the estimates were quite conservatwe with C.,, <_ -3. The type of background response a,csume~J made an appreciable difference on the estimates. With the We~b~tl model, for example, assuming additive background led-to f0"t,'er underestimates of risk, with the distribution of esthnat~s being somewhat similar to thai fur linear extrapolation tr~m the fitted 1% point. Although lower confidence limits for the VSD based on the quantal data are necessarily lower than the point estimates. most of the procedures considered were subject to error rates somewhat in excess of the 5% nominal value (Figure 9). Under the Weibull model, point estimates of the VSD based rm the quantal response data adjusted for competing risks as in (4.9) were somewhat similar to those based on the original unadjusted data (Figure 10). Although the error rate for lower confidence limits was reduced somewhat by this adjustment. the nominal 5% ~evel was still notably exceeded.

5,4 Comparison of time-to.response and ~uantal response models In an attempt to assess the extent to which the use of time-to-response data represents an improvement over the use of only quantal response data in low dose extrapolation, estimates based on the multistage Weibull model using the time-to-tumor data (At) were compared with those based on the multistage model using only the incidence of tumors observed up to t = 900 days (Figure 11 ). For both criteria Cz and C~, point estimates based on these two procedures were within an order of magnitude of each other in more than 80% of t.he cases. Similar results were found in comparing lower confidence limits. On the whole, these results suggest that the use of precise information on time-to-response will not result in estimates of risk in the low dose region that are substantially more precise than those based on quantal data alone.

5. 5 Variation between replicates The frequency distributions discussed in sections 5.1 - 5.3 include variation due to differences between experimental conditions as well as variability between replicates generated under identical experimental conditions. In order to examine the magnitude of the latter error term in isolation, we also prepared frequency distributions of the difference between the value of C2 in replicate one and that in replicate two (Figure 12). Although the variation between replicates is generally somewhat less than tile total variation depicted in the previous histograms, the difference between replicates in the values of C2 was found to be well in excess of 3 in some experiments. (The values of C=were, however, somewhat less variable.) Confidence limits based on both the multistage WeibuHand multistage models varied considerably less than did the corresponding point estimates, reflecting the fact that these confidence limits on the VSD are necessarily linear at low doses whereas the corresponding point o-stimates may be either linear or sublinear. The confidence limits based on these two models as well as the NCTR linear extrapolation procedure also varied less between replicates than did those based on the HartleySielken model. Fundamental snd ADolled Toxicolo[y

(3) 5..6/83

5.6 Comparison of experimental conditions The comparisons among the specific experimental situations involved in the design of the simulation study (see Table 4) are summarized in Table 8 for the multistage Weibull and Hartley-Sielken models as well gs the NCTR linear extrapolation procedure.'~Point estimates based on the multistage Weibull model were unaffected by most of the underlying experimental conditions. The only significant contrasts were those relating to independent versus additive background and the presence or absence of an interim sacrifice, as well as the low dose linearity-tumor lethality interaction. Experiment 18 with additive background gave larger values of C= than did experiment 15 with independent background due to the low dose linearity imposed by addil,ivity. Experiment 23, with an interim sacrifice, gave larger values of C~ than did experiment 5 without such a sacrifice. The fact that interaction between low dose linearity and tumor lethality is significant while the corresponding main effects are not suggests that these two factors somehow compensate each other. (Larger values of C~ were associated with sublinear dose response curves with moderate lethality and low dose linear dose response curves with rapid lethality.) More factors appeared to affect the performance of multistage Weibull confidence limits. (This may in part, however, be attributed to the lower variation between replicates for these estimates.) Low dose linearity (as well as the low dose linearity-tumor lethality interaction) is highly significant for the confidence limits as expected, with the sublinear dose response models giving somewhat lower values of C~. The overall curvature of the dose response also had an effect, with the values of C~ decreasing with increasing curva~ur,=. There were also differences due to the underlying time-to-tumor model, with the proportional hazards model giving higher values of C~ than the log-linear mc,del. Within the tog-linear framework, the Weibull and log-logistic models gave higher values than the log-normal. The comparison between independent and additive background was again significant, although the interim sacrifice did not seem to have much effect on the confidence limits. Similar observations can be made for the Hartley-Sielken model, with a few notable differences. For the point es!imates, the overall curvature in the dose response curve had a significant ~mpact on the values of C~, as did the shape of the hazard function. (Shallow hazards resulted in somewhat lower values of C= than did steep ones.) The difference between independent and additive background was again significant, although unlike the multistage Weibull model, the presence of an interim sacrifice had little effect. Confidence limits for the HartleySielken model were affected by essentially the same factors as the point estimates, except that the effect of additive background was not significant. (This may be explained in partby the fact that the variation between replicates for confidence limits is comparable to that for point estimates for this model, whereas the confidence limits are Considerably mo;'e st'able than the point estimates for the multistage Weibull model.) The most notable effects on the confidence limits produced by the NCTR linear extrapolation procedure were due to the ~Thi~ analys~s is ha~cd on Ct rathcr tha'n Cz bccau~ of the sreatcr stability of zhc former statistic in selms of ~,~riation bctv,,ccn replicatcs. 153

p,

.,,.

a

Minimum induction time (threshold vs. nonthreshold) N u m b e r of dose groups (five vs. three)

Allocation (unbalanced vs. balanced) Sacrifice (interim vs. terminal only)

1 I. 12.

13. 14.

-0.1 3,1""

1.7"" -0.Z -0.I

0.9 2.1" 0.5 -1.3

1.2 -0.6 2.7 "° 0.4 -O.l

-0.2 -0.9 0.3 0.6 3.9""

0.9

0.3

3.4""

1.1

0.2

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-O.3

-1.2

-0.1 -0.2

0.1 0.4

0.6 -0.0 1.5 1.7

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0.2

0.1 0,4

-0.8""

1.5"'"

-1,2""

0.8""

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1.2"" 1.3""

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2.7"

0.4

0.I 10

1.3""

1,4""

Confidence Limits on VSD Multistage Weibull Hartley-Sielken

Value of Contrast

0.4

"Significant interaction (') with tumor lethality for all estimates except Hartley-Sieiken confidence limits. "Significantly different from zero p < .05 "'Significantly different from zero p < .01 ""Significantly different from zero p < .001

Error T e r m S

(LI~ Probit vs. LL Logit) Background (additive vs. independent)

(LL Weibull vs. LL Probit)

T i m e - t o - t u m o r model (LL Weibull vs. PH Weibull)

(nonlinear trend: low-medium-high)

Dose response curve (linear vs. sublinear at low doses)' Hazard function (steep vs, shallow) T u m o r lethality (rapid vs. moderate.)" Competing risks (dose-dependent vs, moderate) Background (8% vs. 1% in absence of competing risks) C u r v a t u r e in dose response curve (linear-trend: low-high)

9. 10.

8.

7.

1. ::. 3. 4. 5. 6.

Point Estimates of VSD Multistage Weibull Hartley-Sielken

TABLE 8 C o m p a r i s o n s B e t w e e n E x p e r i m e n t s in t h e E s t i m a t e d V S D (C0

0.5

0.3 0.2

-0.8 0.3

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3.4 °'"

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BIOLOGICAL AND STATISTICAL IMI'LICATIONS OF"THE EDu~STUDY mathematical assumptions underlying the generation of this data will be in concordance with the biological mechanisms to which they apply, the major findings of this investigation would not appear to be crucially dependent on the particular assumptions made.

I Mulfistocje weibut!l 50 40 30

I!

20

!

]

I0

r-I

(J, r-

=0 u..

An immediate consequence of the incorporation of the time parameter into the extrapolation process is that the virtually safe dose will depend not only on the acceptable risk level, but also on the period of exposure. The simulation of time-toresponse data as discussed in section 2 also focuses attention on the distinction between the dose response curve in the presence and absence of competing risks. Although it is not immediately clear which curve is most relevant for purposes of quantitative risk assessment, we have elected to define the target VSD's in terms of the latter dose response curve. (In the 23 experiments considered here, the VSD in the absence of competing risks was less than that in the presence of competing risks, although the latter values exceeded the former by at most a factor of about five. The VSD's based on the dose response cu(ve adjusted for competing risks did. however, differ by a factor of up to 70 between 500 and 900 days on test.)

[NCTR' Linear

Extrap~iation I

a,l

o

n-

50 40

M

30 20 t0 I

!

I

I

-3-2-I 0 1 2 3

I

I

-3-2-10

Ct

I

I

I 23

C2

Shape of Dose Response Curve at Low Doses [--1 Linear

[]

Sublineor

[]

C I ,C2>0

FIG. 13. Frequency Distributions of the Values of CI and Cz for Lower Confidence Limits on the VSD Based on the Muhi~tage Weibull and NCTR Linear Extrapolation Pro;:edures for Linear and Sublinear Dose Response Curves. shape of the dose response curve in both the observable and unobservable response range. (Both of these factors also influenced the confidence limits based on the multistage Weibull and Hartley-Sielken models.)

6. DISCUSSION The generation and analysis of the simulated data base discussed in this paper will hopefully serve to focus attention on several issues relating to the use of time-to-response data in low dose extrapolation as well as offer some empirical guidance on the accuracy with which risks at low doses may be estimated. The generation of such complex time-to-response data not only requires specification o f underlying dose response models for the underlying carcinogenic and competing risk processes and the experimental design, but also illustrates clearly several,important differences between the assessment of time-to-response and quantal response data. The exhaustive analysis of the" simulated data in which the actual risk in the low dose region is known also offers an opportunity to assess the error associatedwith the different extrapolation procedures 'employed, an opportunity riot afforded by the analysis of actual experimental'data no matter how extensive. Although there' is no guararitee~:that the Fundamental and Applied Toxicnlogy

(3) 5.6/83

The availability of information on time-to-response also offers the possibility of performing one or more of the four analyses outlined in section 4,3. Although valid inferences concerning the time.to-tumor distribution are possible when the actual time-to-tumor is observable, or when the induced tumors are incidental necropsy findings unrelated to the cause of death, neither situation iS likely to occur frequently in practice. Thus, an analysis of the death times for those animals dying from tumors, or of the death times of those animals dying with tumors should the cause of death be indeterminate, may be considered. Our comparisons of these different methods of analysis using one particular model (the multistage Weibull), however, suggest that they may generally yield somewhat similar results. Analysis of the time-to-tumor data using the multistage Weibull, multistage nonparametric and Hartley-Sielken madels indicated that point estimates of risk in the low dose region were highly variable, with the actual risk at the estimated VSD often being a factor of 1000 or more greater than the target risk of 10 ~ while all of the lower conf=dence limits on the VSD based on the multistage Weibull and multistage nonparametric models were below the true VSD, the Hartley-Sielken random group method and NC~R linear extrapolation procedures resulted in error rates of 7% and 13%, respectively, the latter value being somewhat in excess of the nominal 5% level. Point estimates using the quantal response data based on the probit, VVeibull, multistage and linear-quadratic models were also highly variable and tended to result in a somewhat greater degree of overestimation of the true VSD than did estimates based on the time-to-response data. Linear extrapolation from the 1% excess risk point of the fitted Weibull model reduced the degree of overestimation to some extent as did direct extrapolation using this model under the assu'mptionof additive rather than independent background. Lower confidence limits Were of course more conser~,ative than the point estimates, alth0ugh'none of the procedures considered resulted in'error rates as 10Was the desired 5% level. Adjustment of.the quantel' response .data for the presence "of competing risks did not appear to improve markedly the performance of the extrapolation procedures~ 155

Although precise data on time-to-tumor occurrence and timeto-death from competing causes clearly contains much more information than simple proportions of animals w i t h tumors in each dose group at the end of the study, the value of this additional information in low dose extrapolation is to a certain extent questionable, at least in cases w h e r e mortality is not highly dose-dependent, In most cases, the actual risk at the estimated VSD's based on the multistage Weibull model fitted to the t i m e - t o - t u m o r data and the multistage model fitted to the quantal response data w e r e in fact w i t h i n an order of magnitude of each other. Similar results were also obtained w i t h l o w e r confidence limits on the VSD using these same t w o models. Although point estimates of risk in the tow dose region are subject to considerable error, the variation between estimates based on replicate data sets generated under identical experimental conditions w a s less extreme. (Factors identified as contributing to the former source of error may be the modelling of a background response as either additive or independent and both the overall shape of the dose response curve end its behavior in the l o w dose region.) Nonetheless, replicate estimates w e r e found to vary by as much as a factor of 1000 or more in some experiments. The replicate-to-replicate variation in confidence limits obtained using linearized confidence limit procedures w a s notably less. The results of this study clearly indicate that point estimates of risk in the l o w dose region based on purely statistical extrapolation procedures are subject to considerable uncertainty even w h e n detailed information on the time of tumor occurrence is available for analysis. L o w e r confidence limits on the VSD calculated using some form of linearized confidence limit procedure such as that for the m u l t i s t a g e W e i b u l l model or NCTR linear extrapolation procedure can, however, be useful w h e n the underlying dose response curve is linear in the low dose region (Figure 13), When the dose response curve is actually sublinear at l o w doses, however, these same procedures tend to be quite conservative.

A CKNO WLEDGEMENTS In carrying out a study of this magnitude, w e necessarily received a greet deal of help from many people, M. Bickis. C. Clark. J. Nong and R. Stapley all provided invaluable assistance in both the generation of the simulated data base and the s u m m a r i z a t i o n of the statistical analyses performed by the different analysts. ,We are particularly grateful to Robert Stapley for his excellent work in this regard. We are also grateful to Lyn More and Gill Murray for their patience in preparing several preliminary drafts of this manuscript. REFERENCES

Beck~.r, F.F. (1979). Cancer Res. 36:2503-2566. gerenblum, I. and Shubik, P. (1947). A New, Quantitative Approach to the Study of the Stages of Chemical Carcinogenesis in the Mouse's Skin. ltr. J. Cancer 1:383. Brown, K.G. and Heel, D.G. Modeling Time-to-Tumor Data: Analysis of the ED,~ Study. Accepted for publication in Fundam. AppL To.vitaL

Brown, K.G. and Heel, D.G. Multistage Prediction of Cancer in Serially Dosed Animals with Application to the EDo~ Study. Accepted for publication in Fundam. AppL To.ricoL Bruce, R.D., Carlton, W.W., Ferber, K.H., Hughes, D.H., Quest, JAr., 5atsbu'rg, D.S., Smith, I.M., and Clayson, D. (198]). ReExamination of the EDol Study. Fundam. AppL Toxicol. 1:27-128. 156

Bryan, W.R. and Shimkin, M.B. (194l). Quantitative Analysis of Dose-Response Data Obtained with Carcinogenic Hydrocarbons. J. A'all. Catwer Dist. 1-807-833. Chand, N. and Heel, D. (1974). A Comparison of Models for Determining Safe Levels of Environmental Agents. In: Reliabilityund Biometr.v(F. Proschan and RJ. Serfling, eds.), SIAM, Philadelphia, pp. 081-700. Cornfield, J. (1977). Carcinogenic Risk Assessment. Science 198:693-69q.

Cox, D.R. (1972). Regression Models and Life-Tables. Journal of the Re.ca! Stati.ttical Soch,ty B34:187-220. Crump, K. (1979). Dose Response Problems in Carcinogenesis. Biometrh'.~ 35:157-167. Crump, K., Heel, D,, Langley, C., and Pete, R. (1'970). Fundamental Carcinogenic Processes and Their Implications for Low Dose Risk Assessment. Catwer Research 36:2'973-2979. Crump, K.S. (lqal). An Improved Procedure for Low-Dose Carcinogenic Risk Assessment from Animal Data. Yournalofb.)svironmental PathohJg.v and Toxh'ohJg.l'. (In press). Crump, K.S., Guess, H.A., and Deal, K.L. (lO77). Confidt.nct, Intervals and Tests of Hypothesis Concerning Dose Response Relations Inferred from Animal Carcinogenicity Data. fliomctric~ 33:437-45 I. Crump, K.S., Howe, R.B., Masterman, M.D., and Watson, W.W. (1981). RANK ,~1:A Fortran Program for Risk Assessment Using Time-to-Occurrence Dose-Response Data. Prepared under NIEHS Contract No. NIH-ES-77-22. Daffer, P.Z., Crump, K.S., and Masterman,.M.D. (1980). Asymptotic Theory for Analyzing Dose Response Survival Data with Applicat ion t o the Low Dose Ex t rapola t ion Problem. Mathemath'ul 11io.~ch,/wcs50:207-230. Druckrey, H. (1067). Quantitative Aspects of Chemical Carcinogenesis. In: Potential Carotin)genie Ilacards front l)rug.~ (EvaluationofRi.~ks). (R. Truh;lut, ed.), VICC Monogr,,Jph Series, Vol. 7, Springer-Vertag, New York, pp. 60-78. Efron, B. (1982). The Jackkn(h,. the llootstrup and Other Re.~ampling Plans. SIAM, Philadelphia. lrarmer, I.H., Kodell, R.L., and Gaylor, D.W. (I'982). Estimation and Interpolation of Tumour Probabilities from a Mouse Bioassay with Survival/Sacrifice Components. Ri.~k Anal c.~is2:27-34. Gaylor, D.W,, Frith, C.H°, and Greenman, D.L. (1983). Urinary Bladder Neoplasms Induced in Balb/C Female Mice with Low Doses of 2-Acetylamino[luorene. Submitted to .I. Amcricatz Colh.ge of 7b.ricohJgj" fpr publication. Gaylor, D.W. and Kodell, R.L. (1980). Linear Extrapolation Algorithm for Low Dose Risk Assessment of Toxic Substances. Journal of Environmental Patholoh:v attd To.vicolog.r 4:305-3 ] 2. Harlley, H.O. and Sieiken, R.L. (1977a). Estimation of "Safe Doses" in Carcinogenic Experiments. 13iomi'trics 33:1-30. Hartley, H.O. and Sielken, R.L. (1977b). Estimation of "Safe Doses"in Carcinogenic Experiments. JournalofEnvironmental PatholoA:r and Ta.~colog.r 1:241-278. Hartley, H.O, and Sielken, R.L. (1978). MRS. T.: A Fortran Program to Analyze Experimental Data from Toxicology Experiments. Institute of Statistics, Tex,~s A&M University, TX. Her tley, H.O., Tolley, H.D., and Sielken, R.L. (1981). The Product Form of the Hazard Rate Model in Carcinogenic Testing. In: Statisth's and Related Topics (M. Cs~rg6, D. Dawson, J.N.K. Ran and E. Saleh, eds.), North-Holland, Amsterdam, pp. 185-200.

Heel, D.G. f1980), incorporation of Background in Dose-Response Models. Federation Proceedings 39:73-75. Heel, D.G. and Walburg, H.E. (1972). Statistical Analysis of Survival Experiments..L NatL Cancer Inst. 49:361-372. Holland, ].M., Gipson, L.C., Whitaker, M.J., Eisenhower, B.M., and Stephens, T.J. (1981a). Chrbnic Dermal Toxicity of Epoxy Resins, I. Skin Carcinogenicity Poten(:y and General Toxicity. ORNL Report 5762. Fundam. Appl. ToxicoL (3)

l~lay/June. 1983

BIOLOGICAL AND STATISTICAL IMPLICATIONS OF THE ED,,, STUDY Holland, I.M., Wolf, D.A., and Clark, B.R. (1081b). Relative I'otency Estimation for 5ynth.etic Petroleum Skin Carcinogens, Fnv. Ih'ahh Pro.v;. 38:149-155. Hueper, W.C., Wiley, F.II., and Wolfe, H.D. (1938). Experimental I'roduction of Bladder Tumors in Dogs by Administration of Beta-Naphthyl,'~mine../. h~chl.str. II.vg. 20:40. ILSI: International Life Sciences Institute (1983). The Selection of Doses in Chronic ToxicitylCarcinogenic.ity St udies. To appear. • • / J. (1 o 8,3). Dt~se Kalbflelsch, I., Krewskt," D., and Van Ryzm, Response Models for Time to Response Tox'~=it¥ Data. Cattadian Journal c~l Slutfitics. In press. Kalbfleisch, I.D. and Prentice, R.L. (1080). The Slalistit al Analy~s of Fail,re Time Data. John Wiley and Sons, N e w York• Kaplan, E.L. and Meier, P. (1058). Nonparametric Estim,ltion from Incomplete Observations. Journal o f the American Stati.tthal A.~xoc'httiott 53:457-.18, I. Kodell, R.L., Farmer, J.lt., Gaylor, D.W., and Cameron, A.M. (IO82). Influertee of Cause-of-Death Assignment on Time-loT u m o r A n a l y s e s in Animal Carcinogenesis Studies. J. A'at'l. Ca~uer ht.~t. 6o:o59-664. Kodell, R.L., Shaw, C.W.; and Iohnson, A.M. (l o82). Nonparametric Joint Estimators for Disease Resistance and Survival Functions in Survival/Sacrifice Experiments. Biometri~.x 38:43-58. Kovar, I. and Krewski, D. (IO81). RISKSI: A Computer Program for Low Dose Extrapolation of Quantal Response T~xicily Data. Heahh and Welfare Canada, Ottawa. Krewski, D., Kovar, J., and Bickis, M. (1983). Optimal Experimen• tal Designs ft~r Low Dose Extrapolation. In: 7"~qfics i~:/Ipldied Stati~ricx (T.W. Dwivedi, ed.). Marcel Dekker, New York. In press. Krewski, D.R. and Van Ryzin, J. ( t o81 ) Dose Response Models for Qt=anlal Toxicity Data In: Stati.wic.~ and Rehtted 7b/tics. (M.

Cs~irgd, D• Dawson, J.N.K. Rat, and E. Saleh, eds.). North Holl;,nd, Amslerdam, pp. 201-231. M i l l e r , E.C. and Miller, J.A. (1974). In: 71,. Molecular Iliohqo" of Cu,wer. Academic Press, p. 377. O T A : Office for Technology Assessment. (I q'tH). Assessment of Technologies for Determining Cancer Risks from the Environment. U.5. Government Printing Office, V,/ashington, D.C. Portier, C. and Heel, D.G. (1o82). Optimal Design of the Chronic Animal Bioassay. Submitted. SAS Institute Inc., Statistical Analysis System. (t 979). P.O. i]ox 8000, Cary, NC 2751 t. Shubik, I'. and Sice, J. (1o501. Chemical Carcinogenesis a~ a Chronic Toxicity Test. A Review. J. Ca,u-er Res. 1o:728-742. Sielken, R.L. (IO,ql). Re-Examination of the ED,,~ Study: Risk Assessment Using Time. I.'umlam. ,.tpIJL "l'oxic,~l. t :,q8-123. • Toxicology Forum. (1082). Proceedings r~" flw "l'o.vir,dogy I",rum. Annual Vv'inter Forum, Feb. 15-t7. Toxicology Forum, Washington, D.C. UIIrlch, R.L. and Storer, J.B. (1o7o). Influence of h radiation on the Development of Neoplastic Disease in Mice• Radiati, m Re.war¢h 80:303-342. Van Ryzin, [. (1980). Quantitative Risk Assessment. Journal o f ()c~'~qJadonal Medicim. 22:321-320. Van Ryzin, J,(Io81). Review of Statistits: The Need for Rt,alis~ic Statistical Models for Risk Assessment. Iqlndam. AI,I,L "l'oxicoL I:124-126. Walburg, H.E. (lO72). Meeting Reports: Workshop on Statistical Interpretation of Mammalian Survival Experi,nents. Ili,,Scie,ce 22:019-o20. Yamagiwa, K. and Ichikawa, K. (I o18). Experimental Study of the Pathogenesis of Carcinoma. J. Cam.er Re:,. 3:l.

APPENDIX A Modelling of Background Response The purpose of this A p p e n d i x is to describe h o w spontaneously occurring responses are handled in the t i m e - t o - r e s p o n s e m o d e l s discussed in section 2. F o l l o w i n g Heel (1980). w e a s s u m e that a s p o n t a n e o u s response m a y occur either indep e n d e n t l y of those induced by the test c o m p o u n d or additively in a d o s e - w i s e fashion. We w i l l t h u s let TH denote the induction t i m e for the f o r m e r type of t u m o r and take Pit(t), Q~=(t) and ,k=t(t) to be the corresponding distribution, survivor and hazard functions, respectively• (Note that since such t u m o r s occur i n d e p e n d e n t l y of those induced by the test c o m p o u n d , the d i s t r i b u t i o n , survivor and hazard f u n c t i o n s do not d e p e n d on the dose d.) Similarly. w e will let Tt~ denote the induction t i m e for the latter type of t u m o r (either s p o n t a n e o u s or induced) w i t h distribution, survivor and hazard functions P~2(t;d), OvJ(t;d) and A12(t;d), respectively• W i t h i n this t w o - m e c h a n i s m framework, w e have T1 = min(Tl),Tv~) w i t h

T12 I

T21

~ T22 J

t=0

T2

In general, h o w e v e r , w e w i l l have T~ = T3-T, w h e r e Ts = min (Ts~,T3.~)w i t h T3. =T,= +}T.,l and T,~2 = Tr, + Tzz• The actual distribu%on Pzof the survival t i m e Tz f o l l o w i n g the occurrence of a t u m o r may be expressed as

= f o P3(t÷u; d)pt (u;d)du

Qt(t:d) = Qn(t)Ov-,(t;d) , and kdt;d) = ~kll(t} + ~n(t;d)

(A.I)

The t i m e - t o - d e a t h after t u m o r o c c u r r e n c e T2 and the t i m e t o - d e a t h from c o m p e t i n g risks T, m a y be handled in an analogous fashion. It is i m p o r t a n t to note, h o w e v e r , t h a t w h i l e T4 = min(T4~,T4~) as in the case of T~. it is not in general true that T= = min(T2~.T22). In the f o l l o w i n g case, for example, w e have T= = (Tn + T=) - Tn• and Applied Toxicology

)

P2(t;d) = PrlT~ - T, --
Pdt;d) = 1 - Odt;d)

Fundamental

TII

(3) 5-6/83

(A.2)

The d i s t r i b u t i o n P3 = 1 -Q,~ of the t i m e Ts f r o m the start of the study until d e a t h f r o m a type one or type tWO t u m o r in (A•2) is Similarly d e t e r m i n e d by Q~(t;d) = (~(t)Q~2(t;d) , w h e r e "l

1 - Qal(t) = Pzdt) = Jo Pn(t-u)p2du;d)du

and

1 - Q~2(t;d) = P,~2(tid) = fot Pl2(t- u;d)p22(u;d)du

(A.3)

w i t h p2j(t;d) = dP2j(t;d)/dt (j = 1.2). 157

APPENDIX B Statistical Models Used in Generating the Simulated Data Base T h e p u r p o s e of t h i s A p p e n d i x is to describe t h e precise m o d e l p a r a m e t r i z a t i o n u s e d in g e n e r a t i n g t h e s i m u l a t e d d a t a base. In g e n e r a t i n g t i m e - t o - t u m o r d a t a T~ = min(T~,T~), U n d e r t h e l o g - l i n e a r model, w e t o o k Iog(Tn - A~) = n, + o~W~

( A. I I

Iog(Tr~ - Az) = n2 + /3log{d+',/) + ozWz ,

IA.2)

respectively. The hazard function for the proportional hazards m o d e l w a s t a k e n to be of t h e f o r m

Al(t;d)

and

Av.,[t,d)

I<_A, b.,

f[

[

~ ,~

~,(d+a)'][~-~ r,8~,(t-A2)'~] ,

t >~

(A.4)

r'~

t<-A~

0

.

TABLE Model

Experiment

Parametrization

Used

t > A (A.5)

t%A.

T h e a c t u a l m o d e l p a r a m e t e r s u s e d in e a c h o f t h e t h r e e m o d e l s are g i v e n ;n T a b l e B.1. T i m e - t o - d e a t h f r o m c o m p e t i n g r i sks w a s a s s u m e d to f o l l o w a W e ! b u l l m o d e l t h r o u g h o u t w i t h ctj = 6 . 3 9 a n d o~ = 0 . 4 9 w i t h t h e e x c e p t i o n of e x p e r i m e n t 9. (There. ,~ = cr~ = 6. 39, o= = o2 = 0 . 4 9 a n d / 3 = -2.0.) T h e t i m e t o - d e a t h a f t e r d e v e l o p m e n t of t u m o r w a s a s s u m e d to f o l l o w a W e i b u l l m o d e l w i t h ~l = ttz = 4 . 9 9 , el = oz = 0 . 3 6 f o r e x p e r i m e n t s (1, 3, 5. 7, 9, 1 1 - 1 3 , 15-23); n l = (~., = 3,49. el = o~ = 0 . 2 4 for e x p e r i m e n t s (2, 4. 6, 8. 10) a n d cr~ = 4 . 9 9 , o~ = 0 . 3 6 f o r e x p e r i m e n t 14. T h e v a l u e s of ~, a n d / % i n ( A . 1 ) a n d (A.2) w e r e t a k e n to be zero t h r o u g h o u t .

w i t h t h e e r r o r t e r m s W~ a n d VV= f o l l o w i n g t h e n o r m a l , l o g i s t i c or e x t r e m e v a l u e d i s t r i b u t i o n s as g i v e n in Table 2. The h a z a r d f u n c t i o n s f o r T~ a n d "F~zu n d e r t h e g e n e r a l p r o d u c t m o d e l are of the form

0

,~A.,(t)expl/&, + fi~dl / " 0

,=

B.1

in Generating

Time-to-Tumor

Model f o r Time-to-Tumor"

Data

Model Parameters ~'

|,2,o

GI'

~ln :

1 . 0 3 - 7 . c,2 : t~j -- 5 . 1 4 - 7 , fl~ = I

3,4

GP

o,~ :

1.57-Io.~1;¢ : c~.~ = 7 . 8 . | - [ 0 , ~

5.0,20-23

CI'

(t~ :

1 . 0 3 - 7 , nt : 2 . 5 t - 7 . o~ : 7 . 7 1 - 7 , g z :

7,8

CP

el(, : 1 . 5 7 - 1 0 , nl = 3 . o 2 - 1 o . ( ~ 1

: =

l0

G['

~¢~ = 1 . 8 o - 2 0 , ~

It

GP

on = 1 . 2 7 - ! 3 , n~ = ! . 2 7 - 1 2 , g , :

:

12

GP

~o : | . 2 7 - 1 3 , ¢ ~ j

13

(';P

n0 = 1 . 2 7 - 1 3 . n4 :

3.oZ.!O,o.,

= 1 . 2 7 - I 3 , n~ : 1 . 2 7 - 1 2 . fl= :

14

PIq

nt : 8 . 0 4 , 0 ,

15

LL

~, = 8 . 0 4 . ol ": 0 , 5 (VVeibull)

Io

LL

:

! !

1.18-18,1~6 : ' 1 1,18-18,

fl,; :

!

] t. I 4 - I 2 , . / / t

= !

l

: 0.5 ( W e i b u l l b a s e l i n e h a z a r d ) f l l : 2 . 4 0

o: " 0.91, o.~ : O.b, fl : -! (Weibull)

t~, = 7.70, el = 0.64 (L0gnormal) o;t = 0.57, o2 = 0.77,/3 = -1 (Lo,qnorma])

17

LL

otl=

7 . 7 0 , e l = 0.3.5 ( L o g - l o g i s t i c )

a'~ = 6.57, 02 = 0 . 4 2 , IJ : -1 ( L o g - l o g i s t i c )

IB

LL

a2 : 7.42, o2 : 0~5, fl : -1.22,3, : 0.o tWeibtdl)

tq

CI ~

~n = 3.26-12, ol = 0.43-13,~2 : 5.78-12, fl4 = 1 Ai = 5 0 0 , A2 = 3 0 0

"LL = Log-linear CP = General product PH = Proportional hazards ~'Unspecified parameters

a r e all z e r o .

¢ 1 . 0 3 - 7 means 1 . 0 3 X 10":

[58

Fundam. Appl. Toxi¢ol. (3)

May~June, 1983

ATTENDEES A 7 EDo= WORKSHOP

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......

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i59:

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~o

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FU,d~m:AnpI. ro~icoz fz~

Ma)/a~,.198z