Biom. J. 88 (1991) 6, 679-698 akademie Verlag

Confidence Intervals for the Relative Relapse Rate of Placebo VS 6-Mercaptopurine Group of Acute Leukemia Patients Under a Random Censorship Modell

B. RAJA RAO

SHEEU TALWALKEB

DEBASIB K ~ D U

University of Pittsburgh

Southwest Missouri State University

University of Texas a t Dallas

Summary

The present paper considers a random censorhip model in which the remission times of acute leu- kemia patients in a study are randomly censored according to a given probability distribution and constructs confidence intervals for the Relative Relapee Rate [RRR] of the placebo versus 6-Mer- captopurine group of acute leukemia patients. Two important models are discussed for the prob- abiIity distributions of the remission timee-the exponential and the Weibull distributions. The censoring distributions are also regarded as following exponential and Weibull distributions. Several methods are used to construct the confidence intervals including its maximum likelihood estimator and its approximate sampling distribution. Sprott'e and cox's methods a8 well 88 the Box-Cox traneformations are also utilized.

Thesemethodaareusedtoilluetrate the constructionofthe confidence intervals with the FREIBEICH et al. data (1964) involving a placebo and a 6-MP group of acute leukemia patients. The results of the preaent paper provide valuable alternatives to the usual parametric and nonptarametric tests for the treatment effect, that are discussed in the literature, such as Gehan's test, Logrank test. and the Mantel-Haenszel test.

Key worh and phrases: Remission times; Censoring mechanism ; Control and treatment groups ; Exponential and Weibull distributions.

1. Introduction

A common problem in epidemiologic research is to compare health-related data for astudypopulation with that of a standard or referent population. In particular, it is often of interest to examine, for a given cause of death, the association between mortality rate and the type, intensity, or duration of exposure to a suspected etiologic agent. For example, in the en&onmental setting, an in- vestigator may wish to examine the relationship between digestive system cancer mortality and the concentration of a carcinogen in the drinking water of a com- munity residing near a hazardous waste site.

Consider a biomedical or an epidemiologic study, in which the investigator is interested in comparing the mortality experience of a group of individuals, who

1 Premnted to Professor V. 5. HVZUBBAZAB on hie 70th birthday

580 B. R. RAO et al . : Relative Relapec Rate

are exposed to a carcinogen, such ae radiation or arsenic, with that of a second group of individuals, who are not exposed to the carcinogen. Mortality is inter- preted in a general sense. It could literally mean a death due to a particular cause, or it could mean in some cancer research problems, the appearance of a tumor in a previously healthy individual. In such studies, the life time of an individual or his time to cancer ia measured.

Let us consider the remiseion times of 42 patients with acute leukemia, reported by FREIREICH e t al. (1903) in a clinical trial undertaken to aaseaa the ability of 6-Mercaptopunne (6 -MP for short) to maintain remission. Each patient is random- ized to receive 0-MP or a placebo. The study wae terminated after one year. The following remission times, in weeks, are recorded.

Table 1

Remiseion times of 42 patiente ~~

6-MP: 6, 6, 6, 7 , 10, 13, 16. 22, 23. 8'. 9*, lo* , ll', 17*, 19'. 20.. 25*, 32,* 33'. 34'. 36*

Placebo: 1 , 1 , 2, 2. 3, 4, 4. 5. 6. 8, 8, 8. 8. 11, 11, 12, 12, 16, 17, 22, 23 ~~ ~ ~~~

denotea that the observation ia censored

The data from the placebo group are complete, that is, all the 21 patients in that group had a relapse by end of study. The data from the 6-MP group were progressively censored. Such data arise more frequently from clinical studies where patients enter the study at different times and the study laeta a predetsr- mined period of time. See LEE'S (1980) book for an excellent treatment of the topic. During t h e period of time, some patients will experience a relapse, some patients will be lost to follow-up or some patients will die due to other causes, whle some other patients remain alive by the end of the study. The remission times of all such patients are said to be censored and are denoted by an aaterisk (*) .

Thereis another type of a censoring mechanism that is also used in the literature. The study starts with n patients and the investigator decides in advance to terminate the study after a preepecified or predetermined number -7 of patients have experienced a relapse. The remaining n--7 patients are still in remission and their remiseion times are regarded aa censored observations.

In all them studme, the problem is to t a t whether the treatment has any effect to prolong the remimion time of a patient. Suppose that the remission time of a patient in the control group is denoted by a random variable X. Let us define hie survival function :

(1.1) &(t) = P (a patient's remission time zt) = P ( X n t ) .

This gives the probability that the patient is still in remission a t time t , or

Similarly let u8 denote the remiseion time of a patient in the 6-MP group by that the patient has not experienced a relapse by time 1.

Biom. J. 33 (lB9i) 5 58 1

the random variable Y . Let us define his survival function by the equation

(1.2) Sz(t)=P ( Y z t ) . A reasonable null hypothesis in the present context is

time of t or more, whether he receives a placebo or 6-MP. In symbols,

(1.3) HO * S~(t)=Sz(t), for all t . An alternative hypothesis is that the treatment has some effect, in symbols,

(1.4) HI - &(t) +Sz(t) for some t . Sometimes, an investigator may be interested in testing whether the treatment 6-Mp prolongs a patient’s remission time, that is, testing HO against a one-sided alternative hypothesis

(1.5) H2 - Sz(t)z&(t) for all t .

Ho: The treatment does not have an effect. That is, a patient has a remission

Under the hypothesis Hz, a 6-MP patient experiences a longer remission time than a patient in the placebo group.

The survival functions &(t) and &(t) can be estimated nonparametrically by the KAPLAN-MEIER estimator (1953) using the observed data. These can be graphed. If the estimated &(6) curve lies significantly above the estimated Sl(t) , i t is concluded that the treatment sigmficantly prolongs a patient’s remission time. Otherwise, the null hypothesis HO is accepted.

There are also some statistical tests for the null hypothesis HO against the two sided or one-sided alternative hypotheses (HI or Hz) . If there are no censored observations in the study, one can use the Mann-Whitney-Wilcoxon rank sum test, see BROWN and HOLLANDER (1977). If there are censored observations, there are severalnonparametric statistical tests, such &B GEEAN’S (1965) generaliz- ed Wilcoxon test, with M~NTEL’S (1967) modification of his procedure, or the Cox-Mantel test, Cox (1972), or PETO and €%TO’S (1972) logrank test which uses Altshuler’s estimate of the log survival function, or PETO and PETO’S (1972) generalization of Wilcoxon’s two-sample rank sum test , which uses the Kaplan- Meier estimate in its scoring procedure, or COX’S (1974) F test based on ordered scores from the exponential distribution or the well-known MANTEL and HDN- SZEL (1959) test if the survival data can be arranged in the standard life table form.

In the present paper, we have taken a different approach. We have defined three parameters 9, p and e for the two-sample situations involving censored data, such as the above, but under a random censoring model, in which the remis- sion times of leukemia patients in the treated and the untreated (control) groups are randomly censored on the right by independent censoring times following a given probability distribution. The parameters y , y and e have the value 1 under the null hypothesis Ho of no treatment effect. Also the parameters 9 and p have aome unsatisfactory feature and are therefore not discussed. The third 58 Blom. J.. 98 (1091) 6

582 B. R. RAO et el.: Relative Relapae Rate

parameter e has been called the remission-time-specific Relative Relapse Rate of the control versus the treatment group patients. Approximate confidence intervals are constructed for the parameter e , which are valid in large samples. These confidence intervals are evaluated for the FREIREICR et al. (1963) data involving the remission times of acute leukemia patients.

2. Some Parametric Tests in the Two-Sample Situation

The exponential distribution has been used by several writers. This plays an important role in lifetime studies analogous to that of the normal distribution in many fields of statistics. Many applications for this distribution in animal and human studies can be found in ZELEN (1966), FEIGL and ZELEN (1965), ZIPPIN and ARMITAGE (1966), and BYAR e t 81. (1974).

The exponential distribution is used to describe a purely random failure pattern and is well-known for its unique “lack of memory”, which says that the age of the animal or individual does not affect future survival. In terms of the remission times of a leukemia patient, this means that the conditional probability that a patient will experience a relapse in the interval (z, z+dz) given that he is in remission at time z does not depend on z, but is a constant 6. In such a case, the probability density function of the remission time X of a patient is

(2.1) f (z)=6 exp (-&), 6>0, zizo. The parameter 6 can be called the hazard rate or the relapse rate. The survival function of a patient can be shown to be (2-2) #( t )=P (Xzz )=exp (-8z). Assuming that the remission times X and Y of the patients in the control p u p and the treatment group follow exponential distributions with relapse rates 6 1 and 8 2 , statistical testa are discussed for the null hypothesis HO against HI and EZ2 in the literature, See’LawLEss (1982), GROSS and CL~RX (1975), LEE (1980) among others, both in the uncensored (complete) and censored cases. A number 711 in the control group and 712 in the treatment group of patients are followed until some prespecified numbers, say 7 1 and 72 patients experience a relapse. This is called a Type I1 censored data.

Type I censored data, also called the fixed censored case, haa been discussed by LAWLESS (1982) and LEE (1980). The patients in the study are censored at fixed times, if they are still in remission. For example if X I , XZ, ... , X n l , are the remission times of 711 patients in the control group and if L 1 , L 2 , ..., L n 1 are the fixed censoring times, the observed data consist of 2 1 , ZZ, ... , Z n l , where

ZC =XC if the patient i has relapsed at time X C ZC =Lc if the patient i L still in remission at time LC .

Confidence intervals for the ratio 6r/82 have also been constructed.

Biom. J. 88 (1991) 5 583

Similar tests and confidence intervals are also constmcted when the remission times follow the Weibull distribution or the gamma distribution , see BAIN (1 972, 1978), LEE (1980) and Lawmss (1982). The case of the lognormal distribution is also considered.

3. Parameters of Interest in the Biomedical Study

The objective of the study is to examine if the 6-MP treatment prolongs the remission time of patients over their counterparts in the placebo group. The following parameters describe the objective of the study in the present context :

Expected Remission Time in the 6-MP group (3.1) ' = Expected Remission Time in the placebo group

Probability of a relapse in the 6-MP group '=Probability of a relapse in the placebo group ( 3 4

Under the null hypothesis HO of no treatment effect, both parameters rp and y take the value 1. While cp describes how long, on the average, patients in the 6-BfF group are in remission over their counterparts in the placebo group, the parameter y describes the frequency of a relapse in the 6-MP group relative to that in the placebo group. Values of q significantly bigger than 1 and values of y significantly smaller than I are reasons for rejecting the null hypothesis HO in favor of the alternative Hz.

For example if cp = 3, we conclude that a patient receiving 6-MP has a remission time, on the average, three times as long as a patient receiving a placebo. Simi- larly, if y = 1/4, we may conclude that relapses occur about four times aa frequently in the placebo group as in the 6-MP group.

The parameters cp and y suffer from the following defects. These parameters compare the 6-Mp group aa a whole with the placebo group as a whole. rp takes the average of the remission times of all patients in a group. This is similar to comparing the mortality experience of two groups of individuals without reference to their ages. Any comparison must be age-specific. In the present context, our parameter must be remission-time specific.

The parameter y suffers from another deficiency. Under our random censorship model, the parameter y depends not only on the parameters of the remission time distributions, but also on the censoring time parameters. See Appendix I.

In view of these considerations, we remark here that the parameters rp and y are unsatisfactory for our purpose. As remarked earlier, we need a parameter, that is remission-time specific. We define the instantaneous remission-time- specific relapse rate of a patient aa the conditional probability of a relapse in the interval (z, z+dz) given that a patient is in remission at time z. In symbols, A(z) &= P {a relapse occurs in (2, z+&) remission at z} and A(%) is called the instantaneous remission-time-specific relapse rate of a patient. This is analogous 38.

584 B. R. RAO et at.: Relative Relapse Rate

to the age-specific death rate or mortality rate in epidemiologic studies or the hazard rate in industrial experiments.

It can be shown that in the placebo group the relapse rate is

(3.3) al(z) =tl(~)isl(~) , where S1(z) is the survival function and fl(z) is the p.d.f. of the remission time X of a patient,see eq. (1.1). Similarly the relapse rate in the treatment group is

In the present paper we have defined the parameter, called the Relative Relapse Rate (RRR) of a patient in the placebo group relative to a patient in the treatment group at remission time z as

This parameter is remission-time-specific. It compares patienta with remission time z in two groups. If the treatment haa no effect, then it will be seen that e = 1 for all z, i.e., if a patient haa remiseion time z, he ie equally likely to have a relapse in (2, z+dz) whether he receives 6-lHP or the placebo. Thus if HO is true, then e(z) = 1 for all z. On the other hand if e(z) = 3 for some z, we may infer that a placebo patient with remission time z will experience a relapse in (z, z+ dz) three times aa likely aa a 6-MP patient with remission time x.

This parameter e(z) appears to be a reaeonable measure to compare leukemia patienta with given remission time. Moreover, this does not depend on the para- meters of the censoring mechanism.

In Appendix 1, we have discussed several parametric forms of the remission time distributions in the placebo and 6-MP group, 88 well aa wme commonly used censoring distributions under our random censorship model. The forms of the three parameters 9, y and e(z) are derived. Some of their unsatidactory features are discussed.

4. A Random Censorship Model and the Maximum Likelihood Estimates of the Parameters

I n the present paper, let us consider a random censorship model, in which the remission times XI, XZ, ... , Xnl in the control group are independently censored on the right by a random censoring mechanism in such a way that the censoring times 6'1, Ca, ..., C,,, are not fixed constants (like in the type I cenaored data) but are random variables following some probability distribution. The observed data consist of the n1 pairs:

(21, w, (ZZ, az), ... Y Vn1, b,)

Biom. J. 88 (1991) 6 585

where the observation 21 is either a remmission time Xt or a censored time Cr, which ever is the smaller. In symbols,

Zt=min(Xr,Ct), i = 1 , 2 ,... ,w =Xt if patient i had a relapse at time Xg =Ct if patient i is still in remission at time Ct .

Similarly in the treatment group if the remission times TI, Yz, ..., Yn, are independently censored by the random censoring times D1, Dz, ..., D,,, according to an independent censoring mechanism, then the observed data consist of the 122 pairs

where (J+‘lJ d1)i (WZ, Az), ..., (J+’n,, An,)

Wj=min (PI, Dj), j= i , 2, ..., n2 = Yj if patient j had a relapse at time Yj =Dj if patient j is still in remission at time Dj .

We aasume that the censoring mechanism is non-informative, in the sense that if a patient’s remission time is censored at a point C in time, no information is available on this patient’s status after he has been withdrawn from the study, except that his remission time (or follow-up time) is the same as the time at which he is censored (LAGAKOS, 1979).

Two models are discussed in which the remission times in the two groups follow (i) the exponential and (ii) the Weibull distributions. The case when the remission times follow a Rayleigh distribution follows as a particular case. The censoring times are also regarded as random variables having (i) the exponential and (ii) the Weibull distributions, with the same scale parameter. As mentioned in Section 3, the parameters of interest in the study are cp, y and

the RRR parameter e(z). These parameters do not have closed form expressions for other distributions.

The maximum likelihood estimates of the parameters under the random censorship model are derived in Appendix 2 for exponential distributions and in Appendix 3 for the Weibull distributions. Their approximate variances are also indicated.

Approximate confidence intervals for the Relative Relapse Rate e(z) are constructed, which are valid in large samples, using several methods. These methods use the approximate sampling distributions of the m.l.e’s of the para- meters. Fieller’s theorem has been used. The Box-Cox transformation has also been used to transform the estimators to near normality. Sprott’s, Cox’s and several other methods are used to construct confidence intervals for the RRR parameter e(z4. These are presented in Appendix 4.

586 B. R. RAO et 01.: Relative Rehpse Rate

5. Confidence Intervals for the Relative Relapse Rate of Leukemia Patients for FBEI&EIcH et rtl. Data (1963)

5.1. Exponential remimion times and exponential censoring times

In FRE~EICH et a]. data (1963), there are n1= 21 patients in the placebo group, all of whom (&=21 patients) had a relapse by end of study. In the 6-MP group there are 712 = 21 patients and d2 = 9 relapses, The total remission times in the two groups are, see Appendix 1,

The parameter estimates are

This gives 81 R=-=4.61. 82

The confidence intervals for the RRR paranieter derived in Appendix 4 are computed for the Freireich et al. data and are presented in Table 2 for a confidence of 0.95.

The confidence intervals given by Box-Cox transformation (method 2) are evaluated for several values of the power A = 1/18, 2/18, 3/18, ..., 1. Their lengths are also shown in Table 3. It is seen that these confidence intervals start with a length of 7.8160 and get shorter and shorter aa A increases and are shortest for A= 14/18. The length increases again and reaches the value 7.1916 for A = 1, in which case, the confidence intervals reduce to those given by the Fieller’s theorem. A similar remark applies to the Weibull-Weibull case.

In the Wilson Hilferty Method (see Method 6)

2 2 2 2 9vl 387 9- 171

w=-- --=.0051, 02=-=-- - .0117

a1 = 1 - 01 = .9949, a2= .9883 . The equation (A.4.9) becomes

(a2 - 3.841 02) F2B - 2ala2PlB + (a1 - 3.841 01) = O i.0.

0.93F2‘3- 2.96F’B + 0.97 = 0 .

Biom. J. 33 (1991) 5

Table 2

Confidence Intervale for the RRR Parameter for Exponential and Weibull Remiseion Tim-

Confidence = 0.95

Method. Exponential-Exponential Weibull-Weibuil Confidence Intervals for Confidence Intervals for

587

I (2.1132, 10.05) (2.8781, 14.201) Length = 11.4229

3 (1.8680, 9.2217) (2.5442, 12.5599) Length = 10.0157

4 (1.0142, 8.2058) (1.3814, 11.766) Length = 9.7952

Length = 7.9368

Length = 7.3537

Length = 7.1916 Confidence =0.9R Length Confidence = .98 Length (1.8891, 12.365) 10.4659 (2.5848, 16.905) 11.3203

~ ___- 5

Confidence =0.90 Length Confidence =0.90 Length (2.4325, 9.0874) 6.6549 (3.3283. 12.4338) 9.1055

Length =0.0880 Length = 12.4338

__F~ -

6 (0, 9.088) (0, 12.4338)

- ~ _ _ _ F ~ ~

Confidence intervals by thc Method 2 (Box-Cox transformation) are shown in Table 3 for different choices of t he parameter 1.. Confidence intervnls by the Wilson-Hilferty method are not shown here. These are very wide and .ire not of any importance.

Table 3

Confidence Intervals for the RRR Paribmeter for Exponential and Weibull Remission Times (Box-Cox Method). cq. (A.4.2)

Confidence -0.95

1

1/18 2/18 3/18 4/10 511 8 6/18 7/18 811 8 9/18

lO/lS 11/18 12/18 13/18 14/18 15/18 16/18 17/18 I _I

Exponential-Exponential Weibttll-Weibull

C.I. for e Length of C.I. for e Length of Interval Interval

.. -

(2.0767, 9.8927) 7.8160 (2.8287, 13.4743) 10.6456 (2.0387, 9.7402) 7.7015 (2.7769, 13.2665) 10.4896 (1.9990, 9.5977) 7.5987 (2.7227, 13.0726) 10.3499 (1.9574, 9.4644) 7.5070 (2.6661, 12.8910) 10.2249 (1.9138, 9.3393) 7.4255 (2.6067, 12.7206) 10.1139 (1.8680, 9.2217) 7.3537 (2.5444, 12.5604) 10.0160 (1.8199, 9.1108) 7.2009 (2.4788, 12.4093) 9.9305 (1.7691, 9.0060) 7.2369 (2.4096, 12.2666) 9.8570 (1.7153, 8.9069) 7.1916 (2.3354, 12.1317) 9.7953 (1.6684, 8.8130) 7.1546 (2.2588, 12.0037) 9.7449 (1.5978, 8.7239) 7.1261 (2.17G3. 11.8822) 9.7059 (1.5330, 8.6390) 7.1060 (2.0681. 11.7667) 9.6786 (1.4637, 8.5583) 7.0946 (1.9926. 11.6567) 9.6631

(1.3079, 8.4078) 7.0999 (1.7814. 11.4517) 9.6703 (1.2195, 8.3375) 7.1180 (1.6611, 11.3560) 9.6949 (1.1223, 8.2702) 7.1479 (1.5287, 11.2644) 9.7357 (1.0142, 8.2058) 7.1916 (1.3814, 11.1766) 9.7952

(1.3889, 8.4813) 7.0924 (i.ag18. 11.5519) 9.6601

588 B. R. RAO e t 01.: Relative Relopee Rate

its roots are Flls=0.3709 and 2.8117. This gives gl=0.0510, g2=22.2283. The confidence intervala by Wilson-Hilferty method become

0.2283 s ,g ~99.5142 . These are too wide and are not shown in Table 2.

In Freireich et al. data of Table 1, we assume that the remission times of patients in the placebo group as well as in the 6-Mp group follow Weibull distributions with parameters (81, a) and (82, a) respectively. The censoring distributions in the two groups have also Weibull distributions with parametera (81,a) and (b2, a) see BAIN (1978). The r e w n for keeping the scale parameter a the same in all the distributions is explained in the appendix.

The m.l.e’s of the parrtmeters &,&, 6; and 8 2 are obtained as

n2-d~ 12 , p2=-=-* d2 9 B2=-=- T2 T2 T2 T2

The likelihood equation to solve for the parameter a is

The estimate is a= 1.575 .

q a ) = 777.7236

After (5 is found, we conipute

T2(B) = 2104.845 . This gives

21 - = 0.027 a,=-- dl Ti 777.7236

dz T2 2104.845

9 = 0.0043 B2=-=

and

81 8 2 - R=-=6.279.

Biom. J. 88 (1991) 6 589

Appendu 1

The form of the parameters Q, cp and e(x) for some particulnr choices of the remission and censoring distribution8

Example (1): Let the remission times in the placebo and the 6-MP group have the exponential distributions with the densities

(A.1.1) fr(z)=& exp (-61z), 61=-0, 2 2 0

The mean remission times are

f2(y)=&. exp (-thy), ~92>0, ymO.

1 1 E(X)=-* E ( Y ) = - 81 9 2

and the survival functions are

&(z) =exp ( -&z), 4 2 ) =exp ( - 6 2 2 ) . Then the parameter Q becomes

l/&. 81 1/61 62'

(A.1.2) cp=-=-

If the censoring times in the random cemrship model have also expnentk l distributions with censoring parameters and b2. then

(A.1.3) P (a relapse in the control group) = P (X s C)

=I exp (-@1z) & exp ( -612) dx -

0

u Similarly, the probability of a relapse in the 6-MP group is &/(a2 +pa) and the measure y be- comes

This mensure depends upon the censoring parameters fi1 and PZ.

are, see eq. (3.3) It is easy to see thnt the instantaneous remission-time-specific relapse rates in the two groups

~ ~ ( z ) =&l. a2(4 =62

in view of the lack of memory property of the exponential distribution. This gives the RRR para- meter

61 (A.1.5) e(z)=G, forallx.

This, of course, is Q. While cp is the inverse ratio of their mean remission times, e is the ratio of their instantaneous relapse rates given remission up to z.

Example (2): In the placebo group let the remission time X and the censoring time C be indepen- dent and have Weibull distributions with shape and scale parameters (&, el) end (&, a1) respecti- vely. Their p.d.f's ere

(A.l.6) f l (2) =cl&Zcl-1 e q (-&&a) , gi(y)=adW1-1 exp (-&pi), q > O , &>O, &>O, a1-0, ZZO, y S 0 .

590 B. R. Rao et al.: Relative Relapse Rote

Then the mean remhion time

(A.1.7) E ( X ) = - r l+- . the placebo group ia

9' ( :I) P (a relapse in the cun&d group) = P (X sC)

=/ exp ( -p1z"l) 61cp3-1 exp (-&z"l) dz . - 0

Thie integral cannot be evaluated in a closed form. h a particular case if the scale parameters a1 =el, then (A.1.8) P (a relapse in &he W r o I group)

= 61 I ~ 1 9 1 - 1 exp ( - (61 +Bl) *I} dz

=&/(el +&) *

- 0

Similerly if the remission time Y and the censoring time D are independent end hevc Weibull distribution with parameters (62 , a) and (b, a), then the parameters p and cp become

and

The parameter rp depends on the censoring parameters. The survival functions in the two group are

&(z) = P ( X z z ) =exp { -6&} &(z)=P (Ysz )=exp (-w}

and the instantaneous relapse rates become

Then the remission-time-specific RRR parameter becomes

h a particular case if the parameters q = q = c say, then

(A.1.12) V = ( : r

nnd 61 (AJ.13) e(z) =&.

It ia intereating to note that even under the restrictive Baeumption that c1 =ca=c, the parameter cp not only depends on 61 and 6z but also on the common c, while p(z) depends only on 61 and &, like in example (I).

The special case ei=ca=c--2 givea the results for the Rayleigh distribution. Example (3): Let the remhion time X and the censoring time C in the control group be inde-

pendently dietributed according to Gompertz distributione

(A.1.14) / (z)=6iF exp { -$ (tF- 1))

Biom. J. 88 (1991) 5 59 1

and

with the same parameter a.

do not exist and the psrameter cp, which is their ratio, does not have a simple form.

The RRR parameter becomes

Closed form expressions for the mean remiaeion times in the placebo and the treatment groups

We can show that the parameter 'p depend on the censoring parametera and p2.

Ae a particular case if cq =a2 then e(z) reduces t o 61/62. Remark (1): If the remiaeion time X and the censoring time Cof an individual in the control group

(or in the treatment group) have different distributions, such as exponential and Weibull or Weibull and exponential, etc., a closed form for the probability of a relapse in either group does not seem to be available. In such acasetp does not hnve a simple form. The parameters 6 snd e ( X ) can be computed.

Remark (2): If the remission times and the censoring times have some other distributions, such as the gamma, lognormal, etc.. the parameter cp can be calculated while rp and e(z) do not have clos- ed form expressions.

Appendix 2

Maximum Likelihood Estimates Exponential remission times and exporrential aensoriiig times

We shall assumc that remission times of n1 patients in the control group are XI, X2, ... X,,, which have the exponential distribution with p.d.f.

(A.2.1) f(z) =4 exp (-612). 61 SO, z r O . These remission times are independently censored by the censoring times Ci, C2, ... C,, with an exponential distribution

The ob&rved data consist of the n1 puira (21, 81), (22, 62), ..., (Znlr &I,) where

&=min (Xi, Ci), i=l, 2, ..., nl

(A.2.2) g(c) =A exp { - h c } , fll A, cmO.

and 6s is an indicator variable defined by 6s = 1 if 2' =Xi, i.e. 2s ie a remission time 6(=0 if Zi=Cs, Macensored time.

The density of any observation 2 when 6 = 1 and 6 = 0 are respectively

(A.2.3) h(z. 1)=& exp (-(&+/?I) z}. z Z 0 . h(z, 0) =& exp {-(&I +@I) z}, z ZO .

This gives the likelihood function of the observed data in the control group n1

~ 1 = conat. [61 exp { - (81 +&I Y } I ~ [Bi exp { - (61 +Pi) tr)I'-6, (-1

=const. +B;l -di exp {-(@I +&)Ti} where (8.2.4) dl =number of rdapec8

Ti =total o b s e d remission t i m a Sl

t-1 =z [4&+(1-&) Ctl.

592 B. R. RAO et 01.: Relative Relapse Rate

The likelihood equations to estimate 61 and are

Them give the m.1.e.s dl n1-dl Ti Tl *

(8.2.5) &=-, !I=-

The approximate variance of the m.1.e.e 61 is

Another approximation is simply 8’ di

(8.2.6) V(&) =r. Similorly in the treatment group the estimates ore

aa ns-ds Ta TZ (A.2.7) &=--, la=---

where (A.2.8) & =number of relqses

Ta =total observed remisawn time na

J-1 =z [AjYj+( l -d ) D j ] .

and approximately

Appendix 3

Maximum Likelihood Estimates Weibull remission times and Weibull cellsoring time8

Assume that the remiaaion times of the nl patients in the control group are Xi, Xa ... Xn, which have the Weibull distribution with p.d.f.:

(A.3.1) /(z)=u6120-1exp{-@12(1}, U>O, 61>0, 210.

These times are independently censored by the censoring linee Ci,Cz, ..., Cn, which oleo follow a Wei- bull distribution with p.d.f. :

(A.3.2) q(z)=&20-1 exp { -&z”}, u>O, &rO, Z Z O .

As in case (I), the observed data consist of (21, &), (22, 62). ..., (Znlr 6nr) where Zd=Xc when 6 ~ = 1

=CC when 6#=0 . The density of any observation 2 when 6 = 1 and 6 =O are

(8.3.3)

We can write down the likelihood funotion aa in Appendix 1 and find the m.1.e of the parametera 61, and u:

W(z, 1) = u ~ I z O - ~ exp { - (& + ,91) to}, t 1 0 W(z. 0) = u ~ i t ~ - 1 exp { -(&+@l),zO), ZSO .

Biom. J. 83 (1991) 5 593

(A.3.4)

where dl is, ae before the number of relapses in the control group and TI is defined by

After 61, and 81, are found, the likelihood function becomes a function of the scale parameter a. Its logarithm is

(A.3.6) log Ll(a)=const+nl log n+d1 log 61+(nl-di) log /h-(Si+/h) 2'1 +(a -1 ) Ul

=const+nl log a -n l log T l + ( a - 1 ) 9. The likelihood equation is

n1 nl aa a Tl

Vl+Ul. a

(A.3.7) - log L ~ ( u ) = - - -

The notation is n1

(A.3.8) u1=2 [dtlog Xf+(l-d:j log C'l (-1

Incidentally, we can show that the approximate variance of 61, is

8; V(&) =& * Incidentally, we can show that the approximate variance of 61, is

Similarly in the treatment group of 712 patients, let Y1, Yz, ... Y,,, be their remission times with the Weibull distribution

(A.3.9) Let these be independently censored by tho censoring times D1, Dz, ... D,,, with the Weibull distri- bution (8.3.10) The m.l.e's become

/( y) = a6zya-1 e rp { - &ya} .

g(y) =&@-I exp { -w} .

dz na-ds (A.3.11) &=-, Bz=- Ta Ta and the liketiidod equation to solve for a is

a n r n z a4 a Tz

(A.3.12) - log&(a)=--- Tz+ u2.

594 B. R. RAO et el.: Relative Relapse Rate

The notation is

The Scale parameter 4 can be estimated from the pooled sample of remission and cenaoring times. The equation is

(A.3.14) -- + Ul + u2 =o 4

Here, Ti = 2 + 2(2a) + 3a+2(4a) +2(5a) +2(8a) +2(11a) + 2(12a) 15a + 17"

+ 220 + 230 and

Ta =4(6") +?a+ 90 + 2(100) + 11a+ 13a+ 160 + 17a+ 19a + 200 + 220 +23@+26a+2(32a) +340+35a

The m.1.e of the scale parameter 4 is found as follows. We define u1=2(1Og 2 ) + b g 3+2(log 4)+2(log 6)+4(log 8)+2(log 11)+2(log 12)

+log 16+10g 17+log 22+log 23 =38.3276

&=4(lOg 8)+log 7+log 9+2(log lO)+log l t+ log 13+log 18+log 17+log 19 +log 20 +log 22 + log 23 + log 25 +2(log 32) +log 34 +log 35

(log 2)+3O(log 3)+2(4a) log 4+2(5a) log 5+4(8a) log 8

= 66.8827

+2(11O) log 11+2(12") log 12+15'J(log 15)+17a(log 17)+22"(log 22) + 23a( log 23)

+13°(log 13)+16"(log 16)+17a(log 17)+19a(log l9)+2oO(log 20) +22"(log 22) +23a(log 23) +26cr(log 25) +2(32@) (log 32) +34a(log 34) + 35a( log 36)

Va=4(sa) log 6+P(log 7)+9a(log 9)+2(100) (log lO)+lla(log 11)

After the parameter a has been estimated from equation (8.3.14). the estimated parameters are

where, from equations (8.3.6) and (A.3.12)

(A.3.16) Ti(B)=: [S,Xf+(l-bc) ($1 1-1

As in Appendix 2, we can show that the approximate variancee of the estimatea & and & are

The approixmate sempliug dietributione of & and & are univariak normal with meens 61 and & and these variances.

Biom. J. 83 (1991) 6 595

Appendix 4

Approximate Confidenoe Intervals for the Relative Belapse Bate

For exponential or Weibull remimion times, the Relative Relapse Rate (RRR) is

81

@=& *

We ahall denote its eatimate by

Method 1 The atatiatic log R=log 4 -log & is approximately normal with mean log e and variance:

1 1 dl a2

Var (log R ) = - + - .

95 % confidence intervala for log p are 1 112

logR-1.96 (L+-) ~ l o g p ~ l o g R + 1 . 9 6 4 a2

On exponentiation, we get confidence intervala for p:

B e x p ( - 1 . 9 6 ( ~ + ~ ~ ] r ~ s R e x p dl a2

If the number of relapses dl or d2 is zcro, i t has been auggested to replace these by 0.5. This is not required for Freireich'a data.

Method 2 (The Box-Cox Metbod, 1964) The distribution of log R may not be normal in small eamples, such na n = 20. To improve nor- mality, we consider the Box-Cox transformation (1964) for aome A:

=log R, 1-0.

The case d=O is diecuaaed in Method 1. Then R(A) ia more nearly normal with mean

Vnr {R@))) =p2(a-x) Var (R) and variance

This givea 95 % confidence intervale for p@).

Multiplying by 1 and adding 1 we get

A = 1/2 correaponda to the aquareroot tranaformation

Method 3 The choice A=1/3 haa been auggeated by SPBOTT (1973) and ANSCOMBE (1964). Thin givea

(A.4.2) R [ 1-0.663 (i -+- A ) m ] r s e s R 11 +0.653 (&+$)un]s.

596 B. R. RAO e t al.: Relative Relapee Rate

Method 4 The choice A = 1 ie approximately equivalent to using Fieller'e theorem (See ~ E Y , 1978). This gives [ ( 1 ly] [ ( I 1)1'2] R 1-1.96 -+- a e s R 1+1.06 -+- .

di da di da (A.4.3)

Table 3 gives the confidence intervale for the RRR parameter for different choices of A in the Box- Cox trannformotion (Method 2). I t will be Been that for 1=14/18. we have the ehortest confidence internale for e. Method 6 (lJalng Cor'e sullgeetloo, 1963)

Following COX (1963). we may aaume that in group 1 the random variable

hae approximately a chi-equared dietribution with ( 2 d l + 1) d.f.

with (2da+ l ) d.f. Similarly in group 2 the random variable 2&T2 ha8 approximately a chi-squared diatribution

Since the two groups are independent, the ratio

TI (2d2+1) -- 281 T1/(2di + 1) 262T2/(2d2+l)=e Ta(2d1+1)

has an F ditribution with (261 + 1,2d2 + 1) d.f. Them give 98 % (or 90 %) confidence intervals of the form

where F1 and Fz are the correeponding lower end upper probability pointa of an F distribution with (2d1+1, 2d2+1) d.f.

Metbod 6 98 yo one-sided confidence intervale for e = 81/6a are

where F is the upper 6 yo probability point of the F distribution with (2dl+ 1, @ + 1) d.f.

Method 7 (The Wilson-Rllferty Method) Let ue define

y=2d1+1, y = 2 a a + l ~ = 2 / 9 y , rq-2Ih2, a l = l - y , a 2 = l - c s .

Then following a suggestion due to Wileon-Rilferty, ae quoted by JOHNSON and KOTZ (1970). p. 83, i t i8 Besumed that the etatiatic

hae a etandard normal diatribution. Then the etaternant

P ( - 1 . 9 6 ~ 2 ~ 1 . 9 8 ) = 0 . 9 6 ie equivalent to the statement

P (20~3.841)=0.96,

This, in turn, in equivalent to P ( g ; / 3 s F ! $ , ~ @ ) =0,96,

Biom. J. 58 (1991) 5 597

where g1 and 8% are found as follows: The event DS3.841 in equation (3.13) gives

(a9'I3 -ar)2_s(ILleE*/3+(q) 3.841 i.e. (A.4.7) (ag-3.841m) P13-2a1a@l3+(a~ -3.841w1)SO. If 8113, and g113 are the roots of the above quadratic equation in W 3 , we get equation (A.4.8).

Using Cox's (1953) result, we may write

T i 2 d z + l Ta 2d1+ 1 FvX,.,=e - - (8.4.8)

as having an F distribution wi thy nnd v2d.f. Equation (3.16) in conjunction with equation (3.14) gives 95 yo confidence intervals for the Relative Riak Ratio parameter:

(A.4.9)

Acknowledgements

The nuthors wish to express their grateful thmnks to Professor C. R. Rao, Depnrtment of Mathe- matitics and Statistics, University of Pittsburgh, for many helpful discussions.

This work is supported by Contrnct AFSO-88-0030 of the Air Force of Scientific Research and Contract F33615-88-C-CO630. The United Stntes Government is authorized to reproduce and dis- tribute reprints for governmental purposes notwithstanding any copyright notation hereon.

The nuthors also wish to thank Mr. JASEM MOEAMB~ED ALHUMOUD for his great help in preparing this manuscript and Ma. RITA PATWABDXAN for her help with the computer programming.

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