A bivariate limiting distribution of tumor latency time

ELSEVIER

A Bivariate Limiting Distribution of Tumor Latency Time

SVETLOZAR T. RACHEV, CHUFANG WU, AND ANDREJ YU. YAKOVLEV* Department of Statistics and Applied Probability, University of California, Santa Barbara, California 93106

Received 5 April 1994; revised 11 July 1994

ABSTRACT

The model of radiation carcinogenesis, proposed earlier by Klebanov, Rachev, and Yakovlev [8] substantiates the employment of limiting forms of the latent time distribution at high dose values. Such distributions arise within the random minima framework, the two-parameter Weibull distribution being a special case. This model, in its present form, does not allow for carcinogenesis at multiple sites. As shown in the present paper, a natural two-dimensional generalization of the model appears in the form of a Weibull-Marshall-Olkin distribution. Similarly, the study of a randomized version of the model based on the negative binomial minima scheme results in a bivariate Pareto-Marshall-Olkin distribution. In the latter case, an estimate for the rate of convergence to the limiting distribution is given.

1. I N T R O D U C T I O N

It is generally agreed that tumorigenesis is a multistage process, clonal in origin, which proceeds through its early stages via growth of initiated cell clones. Once a malignant cell comes into existence, its growth is irreversible and the progression stage begins, resulting in a detectable tumor after a lapse of time. The progression stage of carcinogenesis is a large part of the period of tumor latency and its duration is thought of as a random variable characterized by a cumulative distribution function (c.d.f.) F(x). Some authors [11] identify the promotion and progression stages with the whole latent period of carcinogenesis. How- ever, the initiation mechanism determines much of the temporal organi- zation of tumor latency. There may be many initiated and promoted lesions but it is a single one that results in an observable tumor. In the construction of a mathematical model intended for the statistical analy-

* Department of Statistics, The Ohio State University, 1958 Neil Avenue, Colum- bus, OH 43210.

MATHEMATICAL BIOSCIENCES 127:127-147 (1995) © Elsevier Science Inc., 1995 655 Avenue of the Americas, New York, NY 10010

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128 s. T. RACHEV ET AL.

sis of time-to-tumor observations, it seems practical to combine the processes of initiation and promotion into a single stage [8, 9]. For the purposes of such an analysis, one needs a parametric form of the latent time distribution allowing both for the initiation of precancerous lesions and for their subsequent evolvement into a detectable tumor.

The crux of the difficulty is that there are still no theoretical grounds to specify the progression time distribution F(x). To overcome the difficulty, some attempts at a stochastic description of carcinogenesis on the basis of extreme value theory has been made with the aim to find a limiting form of the latent time distribution when the number of lesions is large in some sense. The first significant attempt of this kind is attributed to Pike [14] as early as 1966. The principal idea of this paper is to represent the tumor latency time U as the minimum of n independent random observations from a c.d.f. F(x). The nonrandom quantity n is interpreted as the total number of cells in a tissue. Every normal cell is subject to neoplastic transformation, and for any given cell the c.d.f, of the potential time to occurrence of cancer is F(x), which is characterized exhaustively by its asymptotic behavior in the neighbor- hood of x = 0. If n is very large, the asymptotic theory of extreme values may provide some guidance in a search for plausible distribution of time to tumor onset in a tissue. This simplistic reasoning leads, in particular, to a Weibull distribution with parameters a and )t, provided that

limx-XF(x) =a, F(O+) =0, a>O, A>I. (1) x -'-'~ 0

The two-parameter Weibull distribution of the tumor latency time often provides a good fit to animal carcinogenesis data (see [3] for a survey). The same distribution arises in the multistage model of carcinogenesis developed by Armitage and Doll [1].

As far as tumors induced by irradiation or chemical carcinogens are concerned, it is more realistic to proceed from a random minima scheme, assuming that it is the initial (random) number of altered cells u that contributes to the rate of tumor development. In the model of radiation carcinogenesis proposed by Klebanov, Rachev, and Yakovlev [8], it is postulated that the immediate effect of irradiation is the formation of precancerous lesions capable in the long run (if misre- paired and promoted) of producing an overt tumor. In accordance with the "hit and target" principle [5,19], the number of such lesions is thought of as a Poisson random variable (r.v.) with expectation 0 = OoD, where D is the radiation dose, and 00 is the expected number of lesions per unit dose. Other ways of parametrization of 0 with respect to D are

BIVARIATE DISTRIBUTION OF T U M O R LATENCY 129

also possible. It takes a random time X~, i = 1,2 . . . . for the ith lesion to produce a detectable tumor. The nonnegative r.v.'s Xi, referred to as potential progression times, are assumed to be independent and identi- cally distributed (i.i.d.) with a common c.d.f. F(x) . The latent period of carcinogenesis induced by a single-dose irradiation is defined as the r.v.

xi, i = 0

where A is the minimum symbol, and Pr{X 0 = + oo} = 1. If the r.v. v is independent of the sequence X = ( X 1 , X 2 . . . . ) of

progression times, the survivor function of U is given by

G ( t ) = e OF~t) (2)

This parametric family of improper distributions appears to be very flexible in applications to the carcinogenic risk estimation since it provides a description both of the rate of tumor progression and of the proportion of affected individuals in a population. With this approach limiting forms of the latent time distribution are naturally expected to arise at high doses of irradiation or other carcinogens. In particular, the Weibull distribution can be obtained under conditions (1) which define a fairly wide class of the progression time distributions.

In this communication, our interest is in extending the model to multiple tumors at different sites. To do this, the model should be formulated in terms of dependent competing risks. Confining ourselves to the two-dimensional case, we give such a formulation in the next section. The main result, given in Section 3, provides conditions for the corresponding bivariate limiting distribution to be represented in the form of a Weibull-Marshall-Olkin distribution. A randomized version of the model considered in Section 4 is based on the negative binomial random minima scheme [8]. This scheme yields a bivariate limiting distribution which can be called a Pareto-Marshal l-Olkin distribution. Section 5 presents results on the rate of convergence to this distribution.

2. THE MODEL AND PRELIMINARIES

Let /,(1) represent the number of precancerous lesions induced by a single-dose irradiation in tissue 1, and v ~2) the number of such lesions in tissue 2. In accordance with the hit and target principle the components of the vector v = (v (1), v ~2)) have the Poisson marginal distributions with parameters 01 and 0 2, respectively. Since we are concerned

130 S. T. RACHEV ET AL.

with large dose values, it is assumed that Pr(u = 0) = 0. The corresponding progression times (X/o), X/(2)) are i.i.d, random vectors (but the components X/0) and X/(2~ may be mutually dependent within each pair) with c.d.f. F and are assumed to be independent of u.

To study asymptotic behavior of the model, the vector of latent periods, U = (U (1), U (2)) is described as U = NY := (N(1)Y (1), N(2)y(2)), where

v(k)

y(k) = A Xi (1'), i=1

k = 1,2, and N = ( N (1), N (2)) is an appropriate normalizing vector. Let {Xn; n = l , 2 . . . . } be i.i.d, random vectors with the following

survivor function:

f f(x,y)=Pr(X(~l)> x,X(nZ)> y ), ( x , y ) ~ R 2.

ff is said to be in the min-domain of attraction of the survivor function G, i.e., ff ~ D(G), if there exist normalizing constants 0 (1) ,7 (2) ~ R, and w n , Wn

b~ ~ R + , n = 1,2,.. . , such that as n ~oo,

A n XO,_a(1) A n 1X/(2)-a(n 2)) i = 1 i i = ~ (y(1),y(2)),

b n ' bn

d where ---, stands for "convergence in distribution," and G is the c.d.f. of Y = (y(1), y(2)).

From [18] we infer the following statement:

L E M M A 1

If ff satisfies the regular var~tion condition

lim 1 - ff( tx, ty) = w(x , y ) > 0 x ,y > O, t-~o 1 - f f ( t , t )

with

w(cx,cy) =c~'w(x,y) , forallc>O,

then ff ~ D(G), where G--(x,y)= e -w(x'y).

B I V A R I A T E D I S T R I B U T I O N O F T U M O R L A T E N C Y 131

Remark 1. Note that in the particular case w ( x , y ) = Alx+ A2y+ A12 max(x,y), G ( x , y ) is the survivor function of the Marshall-Olkin distribution [12].

To study the same problem within the random minima framework, we need the following transfer theorem for min-infinite!y divisible random pairs.

THEOREM 1

Suppose for any n >t 1, (~,.n,~t,n 'd''(1) t:'(2))',>~1 are independent random pairs (with possibly dependent components) such that

(Ai=I 'i('12' i=l ~ ~i('22)d (g l ,g2) ,

Suppose also that

and

Then

with

as n ~ . (3)

V 1) 12 (2)) ~ ('I~l,'I~2) n ' n

P Vn" (1), Vn"(2) ___) 0% as n ~ ~ .

v c2)

A $:.(1)

ff{z.z:)(Yl,Y2) = ( f Fu'^u~t . . . . xff(~,-u:)+ JJ . 2 (YI'Y2)\Yl"r2] Ya (Y l ) R+

X K(u2-.,)+y2 (Y2) dF(n,,n2)(ul,u2),

(4)

~(J) = Xi (j) - a~J)

sufficient conditions for (3) to be valid are given by l_emma 1.

where (.)+ := max(.,0).

The proof of the theorem follows from the transfer theorem of Gnedenko [4] supplemented with arguments similar to those in [6]. In the particular (min-stable) case

132 S.T. RACHEV ET AL.

3. A LIMITING FORM OF THE LATENT TIME DISTRIBUTION

We deduce the sought-for limiting distribution from a general result given below.

Suppose v is a positive integer-valued random vector having location parameter 0: O -o R+, where the domain O is unbounded. We assume that v is independent of Xi's and our objective is to study the asymptotic behavior of the normalized minima

( v(~)(O) v(2)(O) ) [o(1) c(2)~ A x, (1,, A x F )

i=1 i=1

as 0 -~ oo. re(l) c(2) -~ The following three ways of normalizing the vector ~J~(1),~,~(2)j are

meaningful:

(i) random centering and random scaling; (ii) random centering and scaling by constants; (iii) centering and scaling by constants.

THEOREM 2

Suppose ff ~ D(Cr). In case (i): assume that as 0 ~ 0%

Then

P P lV(1)(O) ---> 0% /.,(2!(0) --) 00.

[ ~(1) ~(1) ,_h,,2,- u~(2, ) d (y(1),y(2)) / ~)/2(1) - - Up(l) ¢~(2) ~(2)

\ Up(l) tip(2)

In case (ii): assume that for some scaling function b: ® ~ R+ the weak convergence

by(i,(0) bv(2,(o) ) w 1/(2)) b( O) ' b( O) ._. (~qO),

takes place. Then

(~(1) (1) ¢,(2) (2) \ , )vo)- av(,) o r ( : ) - av(2) | d (y(1),rl(i),y(2).r/(2)) := (.Z(1),Z(2)) '

BIVARIATE DISTRIBUTION OF TUMOR LATENCY 133

where q is independent of Y. In case (iii): assume that a centering vectora: 0 + R* and scaling functions b,c,d: 0 + R, satisfy the followingconditions:

i

a:!, - a(l)( 0) a$ - aC2)( 0)

c(O) ’ c(e) 11 (X(l), X(2)) = x,

40)--+-Y1<w,

WV4~)

-+Y*<M,@)

and for some /cI > 0, as 8 +a,

a:” - aCk)( 13)dP-l(e).:“fN bt- bP(e) +O” k = 1,2.

Then

py(~) - a (I)( e> $4 - aC2)( e> d

4~) ’ WI+ Rx+ Y*Y,

where X and Y are independent.

Remark 2. This theorem was formulated in [17] without proof.

Proof of Theorem 2. We shall give the proof in case (i) only. The othercases can be treated in a similar way (see also [lo]>. Denote

k(j)

Q, = A xp, j=1,2,i = l

!

(1) (1)GCkcl,,kOjI( s, t) = Pr

Sk(l) - akO)(1) ’ s,

bk”,

and

G1,*(s,t) =Pr(Y(‘)>s,Y(*)> t).

134 S . T . ~ C ~ V E T ~ .

Then

a(b,(1), V(2')( S, t ) - G1,2(S, t )

~< E E P r ( v ( ' ) = k (1), p ( 2 ) = k(2)) ~(k(1),k(2)) ( $ , t ) - - a l , 2 ( s , t ) I k(l) k(2)

: = A + B + C + D ,

where

A = E E Pr(v°)=k(1), vCz)=kCz)) G(k'l),k¢2))(s,t)-G1,2(s, t) , k 0) < k o k (2) < k o

B= ]~ ~_~ Pr(v°)=k°),v(2)=k ~2)) G<ko,,kab(s,t)-G,,2(s,t ) , k (1) < ko k (2) ~ ko

C= Y~. E Pr(v(1)=k(l), v¢2)=k(2)) G(k,, ,ka,)(s,t)-G1,2(s, t) , k 0) >1 k o k (2) < k o

D= E E Pr(v(1)=k(1), v(2)=k(2)) G(k~',,ka')(s,t)-Gl,z(s,t) , k o)/> k o k (2) >/k o

for a fixed integer k o (to be de te rmined later). Since ff ~ D(G) , we have

G ( k 0 ) , k(2)) "~ a l , 2 as k(1), k (2) -~ oo.

Consequent ly, for e > 0 choose k o = ko (e ) such that

G ( k , , , k % ( s , t ) -- Gl.z( s , t ) ] < 6

for all k(1),k(2)>/ko, and all ( s , t ) ~ R 2. Thus D < e. Recal l that v(J)(O) ~ as 0 ~ ~, and so there exists 0 0 = O0(k o) such that

Pr (v( i ) < ko) < e , j = 1,2,

for all 0 >~ 0 o. Now we can get bounds for the te rms A, B, and C:

A<~ Y'~ Y'. Vr(v(1)=k(1),v(2)=kC2))<<.Pr(v(1)<ko)<~, k ( l)<k ok (2)<k o

B ~< Pr( v O' < k o , v '2' >i k o) <~ Pr( v O' < k o) < e ,

BIVARIATE DISTRIBUTION OF TUMOR LATENCY

and

C ~< Pr(p(1) ~ ko '/~,2) ( k o ) ~< Pr (v (2) < ko) < e.

135

U = ( N(1)Y (1), N(2)y(2)),

THEOREM 3

v(J)

Y<J) = A x/(j), j = 1,2. i=1

a- P(uxl/o ,uyl/O2) lim u--,0 1 - f f ( u , u ) = A l x + A z y + A 1 2 m a x ( x , y ) , x,y>~O,

for some positive al, a2, A1, h2, and A12. Let (v (1), v (z)) be a pair of jointly dependent Poisson random variables

with mar#nab

k

Pr (v (j)= k) /~)-e-~'J, > O. = k ! /zj

Suppose also that ( v (1), v (2)) is independent of (Xi)i>~ 1, and

i~ i = Oi n --~ 00 as n ~ ~,

for some constants 0 < 01 < 02. Then, as u ~ ~, the normalized random minima

l~ 1 V 2

A x<" "l/ct2 A X(2)] i ~/~2 I~t i | i=1 i=1 !

Suppose X i = (X/(1), X(2)) , i >/1, are i.i.d, random pairs with survivor function F satisfying the min-domain of attraction condition:

Summing up A, B, C, and D, the result is obvious. •

Next we formulate a version of Theorem 2 (i) characterizing the Weibull-Marshall-Olkin distribution as a limiting one for the vector of latent times


converges in distribution to a vector (W1, W 2) with Weibull- Marshall- Olkin distribution of the form

Pr(WI > Yl,W2 > Y2) = e x p { - AlY~I - ( A2 + A12(1-~z ) )Y~2

- A,2 max( y~',-~22Y,2) } (5)

for y 1 >1 O, Y2 >~ O.

Proof We can easily reduce the case of Weibull-Marshall-Olkin limiting distribution to the simpler case of Marshall-Olkin law by a monotone transformation of the form

#J ' : ( x ? ) = ' .

In fact, applying Lemma 1 and Theorem 3 (i), we have the following proposition:

PROPOSITION

Since ~ = ( sc~ O, ~(2)) n >/1 are i.i.d, random pairs with a survivor function F~(x, y ), satisfying

lim 1 - ff~(tx,ty) t-~o 1- f f~( t , t )

for x, y >1 O, then, as n ~ ~, .

with

= AlX + Azy + Alzmax(x,y )

W /.,(1) /.,(2) ) /x2 A N A L ( i=1 i=1

Pr(Z1 > Y1,Z2 > Y2)

= exp{ - A 101Yl - ( A20z + A12 ( 02 - 01 ) )Yz - A12 01max( Yl, Y2 )}-

Applying the claim together with the monotone transformation defined above, we complete the proof. •

4. A RANDOMIZED VERSION OF THE BIVARIATE LATENT TIME DISTRIBUTION AND ITS LIMITING FORM

Just as in the univariate case [8] we construct a randomized version of the model with two types of radiation-induced tumors by assuming


that the random vector v follows a bivariate counterpart of the negative binomial distribution. First we consider a simpler model with a bivariate geometric distribution of v.

DEFINITION The vector Y = (Y1, Y2) is said to be general bivariate geometric ( GBG)

distributed [13] if

Pr(Y l = m , Y 2 = n )

= Pr(Y 1 > m - l , Y 2 > n - 1 ) - P r ( Y 1 > m , Y 2 > n - l )

- P r ( Y 1> m - l , Y 2 > n ) + P r ( Y 1> m , Y 2 > n)

p ' ; ; - l (Pao+P,~) ' -"-~p~o(Pol+Poo) i f m > n

p ~ - i if m = n = P00

p~-l(Pol+Pll)n-m-lpol(Pxo+Po0) i f m < n .

The following theorem characterizes the minima stable distribution via bivariate geometric scheme.

THEOREM 4 Let {(X/(1),X/(2)), i>~ 1} have survivor function F: R 2 -~[0,1] and

- - Ot 2 Ot 2 OF(s 1 ,S 2 )/OSjls=o---(Ay + A12) , j = l , 2 . Suppose also that (Y1,Y2) follows GBG distribution with parameters (Poo,Pol,Plo,Pn), such that P01 = hi0, Plo = h20, Poo = A120, 0 > 0, and Poo + Pol + Plo + Pu = 1, Plo + Pll < 1, Pol + Pu < 1. Then the joint survivor function of

Y1 Y2 ) (Poo + Po , ) - ~' A x/(1),(poo + p lo) - ~2 A x ? )

i = 1 i = 1

converges weakly, as 0 ---, 0, to a so-called Pareto- Marshall- Olkin distribution with the following survivor function:

A + ( A 1 + A12)2t~ TM + ( A 2 + A12)2t21/a2

A2 A1 ) × I + ( A I + A12)t~ TM + I + ( A z + A,2)t21/~2 + A12 ,

(6)

where A = A 1 + A 2 + A12.

138

Remark 3. N o t e that function of the fo rm

S. T. RACHEV ET AL.

when t 2 = 0, (6) gives the marginal survivor

1

1 + ( A 1 + )tl2)t~/al '

which is the survivor function of a Pareto type distribution, also known as the log-logistic distribution.

Remark 4. One of the conditions used in Theorem 4 is that

C)ff( S?I's~2)CgSj s = 0 = --( ~j + /~12)' j = 1 ,2 .

In fact, it is just another way of setting restrictions over the hazard function since if(0, 0 ) = 1.

Proof of Theorem 4. We have

F((Po0 q-PoI) ~1 /~ YI iX/1),(poo +Plo) a2 A Y_21/:2))(tl' t 2 )

m - 1

= y" ~ p~l- l (P10+P11)m-n- 'PlO(1--P10--P,1) m=2 n = l

X ( f f ( ( P o 0 + Pol)~t l , (PlO + Poo)"2t2)) n

X(f f ( (Poo + Po1)a'tl,O)) m-n

o o

+ E P~-lpoo(ff((Poo + Po1)altl,(PlO + Poo)a2t2)) m m = l

oo oo

+ E E P~I - I (PoI+Pl l )n -m- lpoI (1 - -pOI- -P l l ) m = l n = m + l

× ( ((poo + p01)°'t , ,{pl0 + poo) °2,2)) m

X(ff(O,(Poo + PlO)~2t2)) n-m


Let us take the second term of the right-hand side of the above equation as an example to show how the approach is done:

o0 ~_, P~I-lpoo(ff((Poo + Pol)~Xtl,(Plo + Poo)a2t2)) m

m=l pooff((Po0 + Po1)altl,(Pxo + P00) a2t2)

1 -- Pll(1 -- (/~1 + '~12)(P00 + P01)t~ / al

-- ( /~2 + )k12) ( P00 + Plo)t~/~2 + o(0))

)tl20ff((A1 + A12)al0al t l , (A 2 + A12)a20a2t2) ) t 0 + ( A 1+ A12)2 0t~/a' + ( A 2 + A12)2 0t 1/a2 + O( O )

3.12 - , as 0 -=, 0 .

A + ( A 1 + A12)2t~/al + ( A 2 + A12)2t21/~2'

In a similar way, we can deal with the other two terms; finally we have the survivor function of Pareto-Marshal l-Olkin distribution as claimed in (6).

Now we assume that the random vector v is bivariate negative binomial distributed with parameters (r, po o + Pol,Poo + Pl0) and is independent of the sequence {Xi; i = 1 , 2 . . . . }. The explicit form for bivariate negative binomial distribution is not very clear, we create the random vector v in the following way:

Suppose ~ = (~(1), ffj(2)), and ~'s are i.i.d, random vectors with GBG distribution with parameters ((P00, P01, PI0,PI1 )" Define v as follows:

V=(/J(1),V(2))-~- ( ~ ~.(1), ~ ~j (2)) j=l j=l

so that v °) follows negative binomial distribution with parameters (r, Poo + Pox), and u (2) follows negative binomial distribution with parameters (r, po o +Pl0). Hence, v is said to have bivariate negative binomial distribution with parameters (r, P0o + P01, P00 + Pl0).

Considering only integer values of r, the random vector Y may be represented in the form

y = (y(1),y(2)) = A yj0), y/2) = yj , i=1 i i=1

where ~j(k)

yj(k)_~ A x(k), k = 1,2, i=1

140 S. T. RACHEV ET AL.

and $5 are i.i.d. random variables with common GBG distribution.Bearing in mind that Yj’s are i.i.d. random vectors, consider the follow-ing random vector

i

[ (1) 5 (2)

s = A xi”‘, A xj2) )

i=l i = l I

Which has the same distribution as each Yj; j = 1,2,. . . , r. Assuming theasymptotic behavior of F as in Theorem 3, and letting

iv’) = (PO0 + pm) - a’,N(2)=(Poo+plJu*,

We see (according to Theorem 3) that the limiting survivor function ofNS is as in (4).

Thus we state the weak convergence

NS+NT,

where T = (A/:: Z!", A!:: 2j2'), the sequence {Zi; i = 1,2,. . .) are i.i.d.copies of Z, and Z has a survivor function as in (41, the random vector 5has GBG distribution, and does not depend on {Zi; i = 1,2,. . .}.

Besides, we may write

where

So A;+ NTj has a survivor function equals to the rth power of (6).Our model appears to be “bivariate-Pareto stable,” i.e., if Xi havebivariate-Pareto distribution then NY and Xi are equally distributed. n

5. RATE OF CONVERGENCE FOR THE RANDOMIZEDMODEL

In the next theorem, we shall study the rate of convergence forminima stable distributions.


For simplicity, some notations are used:

[ ~,0) r(2) )

S=(S(1)'S(2))--" I A=lX(1)' A x(2) /=1

T=(TO) ,T(2) ) = Z! 1), =AIZ~2) ki=l i

W = ( W (1), W (2)) = ( N(I)S 0), N(2)S(2)),

( 1) Q = (QO),Q(2)) = 1(1) , W (2) ,

R = ( R ( 1 ) , R ( 2 , ) = ( 1 1 ) Z(1), Z(2) •

Based on the definitions of Kolmogorov and weighted Ko!mogorov distances [15], we define two other useful distances in terms of survivor functions:

and

~ ( x , v ) = sup I~x(X) - P~ (x ) I , x>O

D~(X,Y) = sup (x (1) V px(X)- I. x > O

THEOREM 5

Let Xiy be i.i.d, nonnega@e random vectors, and gj has GBG distribution, where Xij and gj are independent. Then for every s > 1 / a 1 and s > 1 / a 2

A xiy, A Ziy <~rCM1/(l+S)Bl/(l+s), j=l i=l j=l i=l

where

C = 1 + sup x2F~(,(x) + sup x2F~(~(x), x>0 x>0

M = ( Poo + P01) oqs - 1 q.. ( Poo + Plo) ' ~ - 1,

n = ps(X~I),z~ 1,) V Ps(g~2),Z~2)),

and v is the maximum symbol.

The proof of this theorem is given in the Appendix.


6. CONCLUDING REMARKS

In the univariate case, the limiting distribution associated with model (2) is the Weibull distribution [8]. Its bivariate version assumes the form of the Weibull-Marshall-Olkin distribution given by formula (5). To describe variations of radiosensitivity among individuals a randomized counterpart of model (2) can be introduced, assuming the parameter 0 is random. A simple randomized model arises if we take 0 to be gamma distributed with shape parameter r. In this case the model properties can be studied within the negative binomial minima framework [8]. The corresponding limiting distribution is a generalized log-logistic (or a Pareto type) distribution with the survivor function

1 G ( t )

(1 + at l /~) r '

whose bivariate version follows from Theorem 4. However, this version is not sufficiently flexible since both of its marginal distributions incor- porate the same parameter r. Along similar lines, a more general bivariate distribution can be obtained with the survivor function

1 1 >( + + (1 +(/~1 "t- ~12 ) /~ / / °~1 ) I'l'l

A2 A1 ) r, X l+(Al+A12)t~/a 1 + m+(A2+A12)t l /a 2 +)1,12 (7)

where r and m are positive integers. This expression defines a generalized Pareto - Marshall- Olkin distribution.

For statistical inferences based on the distributions considered above, the maximum likelihood method is a natural choice. The relevant likelihood function within the competing risks framework is considered in [2].

The usefulness of limiting forms of the latent time distribution is illustrated with the application of model (7) to experimental data on irradiated RFM strain mice [7]. In that study, 82 germ-free male mice aged 5-6 weeks were exposed to a single radiation dose of 3 Gy, and their lifetimes were followed for the two major causes of death, reticulum cell sarcoma and thymic lymphoma; all other causes were combined into a single group. The hypothesis: A12 = 0 appears to be consistent with the data. Shown in Figures 1 and 2 are the life-table nonparametric estimates of the survivor function for the two tumors, respectively. They are in close agreement with the corresponding parametric counterparts


L° i r

o,8!- L ~ .

l " ! \ , - - - % . . . . . . I

0.6 "'., i

\- O,q i ,~

i I 0.2

t I L ] ~ I ! I I ~ [ I

10 20 30 40 50 60 70 80 90 100 110

FIG. 1. Estimated survivor functions for the latent period of reticulum cell sarcoma induced by a single-dose irradiation of germ-free mice. The stepwise curve is the nonparametric life-table estimate; the solid curve is the corresponding parametric estimate. The unit time is 10 days.

plotted in the same figures. This is borne out by the goodness of fit test for censored observations [6], yielding a significance level of greater than 0.8 in both cases.

APPENDIX

Proof of Theorem 5. Section 5 are:

p ( Q , R) ~< (1+ supF~,l)(X) + supF~2)(x))L(Q,R), x > 0 x > 0

p ( Q , R ) = p W ' --- ~ (W,Z) ,

and

The relations among the distances introduced in

(A1)

(A2)

ps(Q,R ) >~ L ( Q , R ) I + s, for s > 0. (A3)

The last inequality is true since if L(Q,R) > E > 0 then there exists y ~ R2+ such that

I FQ(X) - FR(x) l > e,

for y ~< x ~ y + ~1.

144

1.01

0.8

0.6

0.4

S. T. RACHEV ET AL.

-'7..-!

o,2

t 1 I I I I I J I i I

10 20 30 40 50 60 70 80 90 100

FIG. 2. Estimated survivor functions for the latent period of thymic lymphoma induced by a single-dose irradiation of germ-free mice. The stepwise curve is the nonparametric life-table estimate; the solid curve is the corresponding parametric estimate. The unit time is 10 days.

Therefore ,

p s ( Q , R ) >~ sup ( x O) A x<2))s I FQ(X) - FR(X) I y~<x~<y+el

>~e s u p ( x(1) A x(2)) s <..~ ~ ( ~ A ~ ) s -~ ~s + l .

y~<x<y+~l

According to (A1), (A2), and (A3), the relation between p and Ps is clear:

~ ( W , Z ) p ( Q , R ) ~< C L ( Q , R ) ~< Cps(Q,R) 1/(1+ s) C - "W Z xl/(1 + s) = = PA , ) ,

where

C = 1 + s u p F/~,I}(X) + sup F~(~}(x). x>0 x>0


Suppose the conditions in Theorem 3 hold, then the marginals of the limiting distribution (4) are as follows:

Fz,l,( t l ) = 1 + ( A1 + A,2)t~/~' '

and

1

Pz,2,(tz) = I + ( A 2 + A,2)t 1 / ~ "

Since R (1) = 1 / Z °) and R = 1 / Z (2) we have

1 FR,,( tl)

I + ( A 1 + ) t l2) t l l / a ' '

and

FR,2,( t2) = I + ( A 2 + A,2)t~ 1/a2"

Therefore

sup F~,l,(x) = sup x > 0 x > 0

( t~1 "+" A12)(1 /Ot l )X -1 /~ ' -1

(1 q-- ( X 1 -{'- t~12)X-1/a') 2

Note that when al = a2 = 1, C = 1 + 1/(A 1 + )t12)+ 1 / ( h 2 + AI2). Hence the relation between ~,(S, Z) and b~(W,T) can be obtained as

follows:

~s(W,Z) = sup (y(l)V y(2))-s]ffw(y ) - Pz(Y) ] y > 0

= sup (NO)z O) V NC2)z C2)) -s[ ffs(Z ) _ fiT(Z) [ z>0

< sup (N (I) V N(2)) - S(z(1) V z (2)) SIFs(z ) - F,r(z)l z>0

= ( N ( " v n

146

We now study the bound of ~s(S,T):

S. T. RACHEV ET AL

Ps(S 'T) = E EP(~- (1 ) _ kl ' ~(2) = k2 ) sup ( x (1) V x (2)) -~ k 1 k2 x>O

X p( k' k2 ) A X/(1) > x(1), A x/(2) )" x(2)

i=1 i=1

- e A Z} 1) > X(1), A Z~ 2, > x(2) i=1 i=1

E EP(C(I)= kl,C(:)= ~) kl k2

X[ supx-skllP(X}l)> x)- P(Z~I) > x) l t x>0

+ supx-'k21P(X¢12)> x)- P(Z~2) > x)l ] x>0 l

E EP(~(1) = kl ' C(2)_- k 2 ) ( k l .]_ k2 ) kl k2

×(~(X}I),zI1)) v ~(X}Z), Z~2)))

~< (E~ "(1) + Esr '2))(~,(x}l), Zi 1)) V ~(X}2), Zi2)))

( 1 1 ) × B , Poo + Pol + Poo + Plo

where B := ps(X} 1), Z~ 1)) v ~s(X} 2), Z~2)). Therefore,

- s . . \1/(1 + s) ~(W,Z) < C(~,(W,Z)) 1/(1+ s)<~ C((NO) V N (2)) ~,(S,T))

(( <~ C (Poo + Pol)~" A(poo + plo) )

X POO + POl + Poo + p l o

<~ C((Poo + P o l ) • l s - I +(Poo + Pio)a2s-1) 1/(i+s)sl/(l+s)

Finally, we have

) A zij- ~< r~(S,T) = r~(W,Z) j = l i = = i = I

<~ rCM1/O +~)B1/(1 +s).

BIVARIATE DISTRIBUTION OF TUMOR LATENCY

w h e r e M .'= (P00 + P01) ~ls- 1 + (P00 + Pl0) ~2s- 1. This p r o o f o f the t h e o r e m .

comple t e s

147

the

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A bivariate limiting distribution of tumor latency time

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