Design of clinical trials for recurrent events with periodic monitoring

STATISTICS IN MEDICINE

Statist. Med. 18, 3005–3020 (1999)

DESIGN OF CLINICAL TRIALS FOR RECURRENT EVENTSWITH PERIODIC MONITORING

SHIGEYUKI MATSUI ∗ AND HIDEAKI MIYAGISHI

Department of Industrial Management and Engineering; Faculty of Engineering; Science University of Tokyo; 1-3Kagurazaka; Shinjuku-ku; Tokyo 162; Japan

SUMMARY

This paper presents a design for randomized clinical trials in which incomplete data are collected on theoccurrence of potentially recurrent events through periodic monitoring. In particular, events are assumed toarise according to a point process, but information is available at the times of monitoring only if one ormore events has occurred since the preceding monitoring point. The event process is modelled via a piecewisePoisson process, and a proportional rates model is introduced to represent the di�erence in event rates betweentreatment groups. The design was developed on the basis of a Wald-type test derived from the generalizedestimating equations of Liang and Zeger (Biometrika 73, 13–22 (1986)). Robusti�cation of the variance ofthe estimator of the treatment e�ect was considered under a random e�ects model with a semi-parametricmixture distribution. The design was adopted to address issues which arose in an osteoporosis trial conductedin Japan. Copyright ? 1999 John Wiley & Sons, Ltd.

1. INTRODUCTION

In many clinical trials, patients are monitored at successive scheduled times as part of follow-up to obtain responses associated with the occurrence of a recurrent event. In most such trials,which deal with individually distinguishable events if they occur, the observable responses maybe occurrence times or counts of events during successive monitoring periods. However, we couldencounter another type of response in practice because it is sometimes impossible or techni-cally di�cult to distinguish individual events; an example is new vertebral deformity in patientswho su�er from osteoporosis. As an illustration, we describe in the next section an osteoporo-sis trial conducted in Japan. For such events, we may be forced to observe only whether oneor more events occurred between successive monitoring points. This situation gives us intervaldata for a sequence of monitoring, which represent incomplete information about the occurrenceof the event; the more frequent the monitoring, the greater the amount of information we ob-tain about the event process. At the extreme, continuous monitoring would enable us to detectoccurrence of the event immediately after it occurs and envisage the event process completely.However, frequent monitoring is generally prohibited due to ethical, practical, economic and other

∗ Correspondence to: Shigeyuki Matsui, Health Informatics and Biostatistics, Department of Health Sciences, Oita Universityof Nursing and Health Sciences, 2944-9 Megusuno, Notsuharu-machi, Oita 870-1201, Japan. E-mail: [email protected]

CCC 0277–6715/99/223005–16$17.50 Received January 1997Copyright ? 1999 John Wiley & Sons, Ltd. Accepted January 1999

3006 S. MATSUI AND H. MIYAGISHI

reasons. The theme of this paper is to develop a design for randomized clinical trials with intervaldata.Although some research has been directed towards developing designs for randomized clini-

cal trials with recurrent events,1; 2 they concern only situations in which the type of responseis occurrence times or counts of event. Cook1 provided a design which assumed a homogenousPoisson process for patients in each treatment group, and the proportional rate model betweentreatment groups, that is, event rate � for the control group and � exp( ) for the treatment group.A Wald-type test statistic based on the maximum likelihood estimator of the treatment e�ect, ,was used to develop the design criteria. Since our situation with interval data comes not fromany stochastic properties of the event process, but from our being unable to observe responses inthe form of occurrence times or counts of events, the stochastic model with which Cook dealt isstill assumable and provides a starting model for developing a design. A piecewise Poisson pro-cess with monitoring-period-speci�c event rates would be an extension to re ect non-homogeneityof the event process. However, to accommodate interval data, the mean of a binary responseshould be linked with the parameter of the treatment e�ect, , using the complementary log-logfunction.For analysis of our interval data, we take into account that responses within a patient tend

to be correlated. In the randomized clinical trials with which we are concerned, however, themain interest is the di�erence in the population-averaged responses3 between treatment groups,while intra-patient correlation is considered a nuisance characteristic of the data. Among the widevariety of methods developed to analyse correlated data,4 the generalized estimating equations(GEE) method of Liang and Zeger5 is most appropriate for our situation.3; 6

At the design stage, we must take intra-patient correlation into consideration. In the frameworkof GEE, this is addressed by specifying the intra-patient correlation matrix.7; 8 With periodic mon-itoring, we should specify a correlation matrix that appropriately re ects the fact that the lengthof the interval between successive monitoring points may vary over the sequence of monitorings.One approach to obtaining such a correlation matrix is to regard the source of correlation asinter-patient variability of event rate and assume a random e�ects model to re ect such variability.A correlation matrix can then be obtained through evaluation of the �rst two marginal moments ofthe responses of each patient under the random e�ects model; this approach is intuitively appeal-ing. Although, in principle, both full parametric as well as semi-parametric mixture distributionsshould be considered, we rarely have su�cient information to specify the parametric form of themixture distribution at the design stage. Further, the interval, which could possibly group multipleoccurrences of the event into a binary response, would make the conjecture based on past relevantdata inaccurate. These points suggest using semi-parametric mixture distributions even though theevaluation of intra-patient correlation under such distributions is not straightforward and may re-quire some approximations. With these methods, we obtain a robusti�ed variance of the estimatorof the treatment e�ect and we suggest that the adequacy of this robusti�cation should be clari�edin practical use.In this paper, we begin by brie y describing the osteoporosis trial and the aims of the current

work in Section 2. We establish a test statistic derived from the estimating equations, assumingindependence of the responses in each patient in Section 3. With this test, we develop designcriteria in Section 4. In Section 5, we accommodate intra-patient correlation, and in Section 6we adopt the design to address the issues which arose in the osteoporosis trial. We describethe results from a simulation study which investigates the adequacy of the developed design inSection 7 and conclude with some discussion in Section 8.

Copyright ? 1999 John Wiley & Sons, Ltd. Statist. Med. 18, 3005–3020 (1999)

RECURRENT EVENTS WITH PERIODIC MONITORING 3007

2. A CLINICAL TRIAL IN OSTEOPOROSIS

In April 1996, the Ministry of Health and Welfare in Japan started a post-marketing trial directedtowards osteoporosis patients; its purposes were mainly: (i) to evaluate the e�cacy of Menate-trenone (vitamin K2) in preventing new vertebral deformity; (ii) to gather information necessary toprepare guidelines for clinical evaluation of treatments on osteoporosis patients in post-marketingtrials.9 Eligible patients were to be enrolled between April 1996 and March 2000, and were ran-domly assigned to either the control (calcium alone) or the treatment (calcium and Menatetrenone)group.The primary endpoint is the occurrence of new vertebral deformity, and to obtain information

about this the trial was designed to monitor patients with X-ray examinations at 12, 24 and 36months after randomization. Here, deformity is de�ned as deformity of any vertebra, irrespectiveof number. At each monitoring time, we observe whether or not one or more deformities haveoccurred since the preceding one, because the number of deformities is not always equal to (gen-erally less than) the number of deformed vertebrae, which could be identi�ed from the X-rayexaminations. Consequently, information about the occurrence of new deformity depends on thefrequency of monitoring or the interval between monitorings. Frequent monitoring is prohibited bylimitations on the exposure of patients to X-rays.Patients were divided into subgroups based on whether they had any deformity at the time

of enrolment. In those free of deformity at the time of enrolment, interest is naturally focusedon the time deformity-free, while in those who had deformity at the time of enrolment, interestfocuses on the occurrence of new deformity and its development. If a new deformity was detectedduring monitoring among the latter patients, they could be taken ‘o� protocol’ to receive othertreatments; this implies that information only about the �rst new deformity after randomizationis obtained. This arises because patients have a right to select an appropriate treatment fromthe several approved, and it is unethical to force them to receive a speci�c approved treatmentcontinuously. If we continued to administer the treatments under study to patients irrespective ofthe outcome of the X-ray examinations, we would obtain more information about the developmentof new deformity when comparing the two treatments under study.After discussing this trial with a statistician, we believed that some sensitivity analysis of its

power or sample size, particularly changing the design so that monitoring was more or less frequentand by allowing patients who had any deformity at enrolment to continue the treatments understudy irrespective of the outcome of the X-ray examinations, was necessary and would provideuseful information in preparing guidelines not only for post-marketing trials but also the relevantphase III trials. This belief led us to conduct the current work.

3. TWO-SAMPLE TESTING PROCEDURE

We consider a clinical trial in which n patients are randomly assigned to either the control (i=1)or the treatment group (i=2). Let ni be the total number of patients assigned to the ith treatmentgroup, so that n= n1 + n2. For each patient, let 0= t0¡t1¡ · · ·¡tm denote the successive moni-toring times since randomization with duration dk = tk − tk−1 (k =1; : : : ; m). The monitoring timesare predetermined and common for all patients. For the jth patient in the ith group, let Yijk be theresponse at tk , so that Yijk =1 if one or more events occurred between tk−1 and tk , and Yijk =0otherwise, and let Rijk denote a missing indicator for observation of the response Yijk , so thatRijk =1 if Yijk is observed, and Rijk =0 otherwise (i=1; 2; j=1; : : : ; ni; k =1; : : : ; m). Missing



observations would result from withdrawal from the trial or termination of follow-up. We assumedthe pattern of missing observations to be monotone so that once a patient leaves the trial thenreturn is not possible (that is, if Rijk =0 for some k then Rijl=0 for all l such as k6l6m), andthat the missing mechanism is missing completely at random in the sense of Rubin.10 For conciseexpression, we denote Yij =(Yij1; : : : ; Yijm)′ and Rij =diag(Rij1; : : : ; Rijm).In the stochastic model for the event process, we assumed that events arise according to a

piecewise Poisson process with a proportional rates model, that is, for a patient in the ith group,the event rate during the period between monitoring times tk−1 and tk is expressed as

�k exp{ (i − 1)} (1)

where �k and represent the baseline event rate and the treatment e�ect, respectively (i=1; 2;k =1; : : : ; m). Then, the marginal expectation of Yijk , �ik , can be expressed as

�ik =1− exp[−�k exp{ (i − 1)}dk ]: (2)

Note that equation (2) can be rewritten as log{− log(1 − �ik)}= log(dk) + log(�k) + (i − 1),where log(dk) represents an o�set term and where the complementary log-log function is used tolink �ik with the parameters �k and .Here, we assume that the responses within a patient are mutually independent. For a given Rij,

the independent estimating equations5 for parameters �= (�1; : : : ; �m)′ and are given by

2∑i=1

ni∑j=1

C(�i)′V−1i Ri j(Yi j − �i)= 0

and2∑

i=1

ni∑j=1

Di(�i)′V−1i Rij(Yi j − �i)= 0

respectively, where �i=(�i1; : : : ; �im)′; C(�i)= @�i=@�, with (k; k) element [C(�i)]k; k = − (1−�ik)log(1−�ik)=�k and (k; l); (k 6= l), element [C(�i)]k; l=0 (i=1; 2; k; l=1; : : : ; m); Di(�i)= @�i=@ with kth element [Di(�i)]k = − (i − 1)(1 − �ik) log(1 − �ik) (i=1; 2; k =1; : : : ; m); and Vi is theworking variance-covariance matrix, expressed as Vi=diag(�i1(1 − �i1); : : : ; �im(1 − �im)) in theindependence assumption (see Appendix I). Note that the same estimating equations can be derivedby maximum likelihood estimation. The estimators of �= (�1; : : : ; �m)′; , � = ( �1; : : : ; �m)′ and , can be found by solving the above estimating equations. Under the assumption that we speci�edthe correct model (2) and the independence assumption, we can obtain the asymptotic variance of expressed as

W =(� − � ��−1�� )−1

where the �’s are the expected value of the negative of the derivatives of the left-hand side ofthe estimating equations for the parameters � or , expressed respectively as

� =2∑

i=1

ni∑j=1

Di(�i)′V−1i RijDi(�i); � � =

2∑i=1

ni∑j=1

Di(�i)′V−1i RijC(�i);

�� =2∑

i=1

ni∑j=1

C(�i)′V−1i RijDi(�i); �� =

2∑i=1

ni∑j=1

C(�i)′V−1i RijC(�i):

An estimator of W , say W , can be obtained by replacing the parameters � and with theirestimators. Then, for testing the hypothesis H0 : = 0 versus H1: 6= 0, a Wald-type statistic Z



can be derived as Z =( − 0)=√W , which is asymptotically distributed as the standard normal

distribution under H0. If |Z |¿z�=2, we reject H0 in favour of H1 at the signi�cance level �, wherezq is the upper 100q percentile of the standard normal distribution.

4. DESIGN CRITERIA

Once the design of the trial is chosen, we need to determine the values of the design parameters,which dominate the power of hypothesis testing. The values of these parameters should be setto satisfy a given power requirement. Here, we shall test the null hypothesis, H0 : =0, versusthe alternative, H1: = R (6= 0), at the size, �, for the power requirement of at least 1 − �,using the Wald-type test derived in Section 3 with two-sided alternative. In this setting, the designparameters should satisfy the inequality

2R¿(z�=2 + z�)2W (3)

and will include a parameter for the missing mechanism of responses, the assignment scheme, aswell as parameters for the distribution of responses, that is, the baseline event rates, �k ’s, and thesize of the treatment e�ect, R. We let

�ik = Pr[an enrolled patient is assigned to the ith treatment group and their

response is observed at the monitoring time tk ]

for i=1; 2; k =1; : : : ; m, and we denote �i=diag(�i1; : : : ; �im) (i=1; 2). Note that under the mono-tone missing assumption in Section 3, �ik¿�il holds for such k and l where k¡l. Then, for agiven total number of patients, n, we can estimate the value of W as

W =(� − � ��−1�� )−1

where

� = n2∑

i=1Di(�i)

′V−1i �iDi(�i); � � = n

2∑i=1

Di(�i)′V

−1i �iC(�i);

�� = n2∑

i=1C(�i)

′V−1i �iDi(�i); �� = n

2∑i=1

C(�i)′V

−1i �iC(�i)

where these are evaluated at the speci�ed values of the design parameters. Thus, replacing W withW in inequality (3) provides the design criteria.

5. ACCOMMODATION OF INTRA-PATIENT CORRELATION

We regard the source of the intra-patient correlation as inter-patient variability of event rate, andassume a random e�ects model. We replace the independent working variance-covariance matrixused in the previous sections with the ‘dependent’ variance-covariance matrix evaluated under therandom e�ects model. With this replacement, the corresponding estimating equations are referredto as the generalized estimating equations (GEE).5

For the jth patient in the ith group, we assume that the conditional rate, given that the patient’srandom e�ect is zij, between the monitoring times tk−1 and tk is expressed as

zij�∗k exp{ ∗(i − 1)} (4)



where �∗k and ∗ represent the baseline event rate and treatment e�ect, respectively, of the jthpatient in the ith group (i=1; 2; k =1; : : : ; m), and where the random e�ects, Zij’s, are indepen-dently and identically distributed as a distribution with mean 1 and variance �. Note that theseparameters express e�ects on the patient-speci�c event rate and not on the population-averaged ormarginal event rate, in which we are interested. However, as � approaches 0, the patient-speci�cevent rate (4) approaches the marginal event rate (1) so that �∗k = �k and ∗= . We link theconditional mean of Yijk given the patient-speci�c random e�ect zij, say �ijk , with the parameters�∗k and ∗ such that

h(�ijk) = log(zij) + log(dk) + log(�∗k ) + ∗(i − 1)= log(zij) + �∗ik : (5)

The function, h, is the complementary log-log function as before.To work with the design criteria of Section 4, it is necessary to evaluate the �rst two marginal

moments of the responses within a patient as well as the marginal treatment e�ect under the currentsetting. In what follows, we evaluate them assuming, �rst, a parametric mixture distribution, and,second, a semi-parametric mixture distribution.

5.1. Parametric mixture

For a given parametric mixture distribution with density f, the marginal mean of the responseYijk ; �ik , is given by

�ik =∫

�ijkf(zij) dzij =1−∫exp(−zijaik)f(zij) dzij =1− L(aik) (6)

where aik = �∗k exp{ ∗(i−1)}dk (i=1; 2; k =1; : : : ; m) and L(s)=E{exp(−sz)}, the Laplace trans-formation of the distribution f.11; 12 Using the conditional argument,13 the marginal variance-covariance matrix of Yij is given by

cov(Yij)= cov[E(Yij | zij)] + E[cov(Yij | zij)] (7)

which reduces to

cov(Yijk ; Yijl)=L(aik + ail)− L(aik)L(ail) + I(k = l) {L(aik)− L(2aik)}for element (k; l) (k; l=1; : : : ; m), where I is the indicator function. We thus adopt this matrix as theworking variance-covariance matrix, Vi (see Appendix I). Note that if the interval between succes-sive monitoring points is the same for all monitoring points and the event process is a homogeneousPoisson process, an exchangeable marginal variance-covariance matrix is obtained. Also note thatif the density f is speci�ed as the gamma density expressed as (1=�)1=�z1=�−1 exp(−z=�)=�(1=�),then the Laplace transformation L(a) is explicitly expressed as (1+�a)−1=�,12 and we can obtainthe �rst two marginal moments of the responses within a patient in an explicit form.The formal way to evaluate the marginal treatment e�ect, , is, �rst, to generate all possi-

ble realizations of (Yij; Rij) weighted by their respective probabilities in the ith group (i=1; 2),and then, to solve the relevant GEE for all realizations, where the probability for a realizationyij =(yij1; : : : ; yijm)′; rij =diag(rij1; : : : ; rijm), is given by

�rij1i1 (1− �i1)1−rij1

[m∏

k=2

(�ik

�i; k−1

)rijk (1−

(�ik

�i; k−1

))1−rijk] [

m∏k=1

{�yijk

ik (1− �ik)1−yijk}rijk]:



5.2. Semi-parametric mixture

We assume a less restricted mixture distribution, that is, only the mean and variance of the mixturedistribution are speci�ed as 1 and �, respectively, while the form is unspeci�ed. In exchange forreducing the restriction on the mixture distribution, we must make some approximations on the�rst two marginal moments and the marginal treatment e�ect. Here, we adopt the strategy givenby Zeger et al.3 The �rst-order Taylor series expansion of �ijk of (5) around log(zij)= 0 yields

�ijk ≈ ��ijk = h−1(�∗ik) +@h−1(�∗ik)

@�∗iklog(zij) (8)

where @h−1(�∗ik)=@�∗ik = exp{�∗ik− exp(�∗ik)} (i=1; 2; k =1; : : : ; m). We use the approximation E[log(zij)] ≈ 0 with respect to the semi-parametric mixture distribution, and we thereby obtain themarginal mean to be approximately:

�ik ≈ ��ik = h−1(�∗ik)= 1− exp(−aik): (9)

Using the approximation (8), the marginal variance-covariance matrix of Yij is approximated by

cov(Yijk ; Yijl) ≈ cov( ��ijk ; ��ijl) + I(k = l)E[ ��ijk(1− ��ijk)]

≈ �@h−1(�∗ik)

@�∗ik@h−1(�∗il)

@�∗il+ I(k = l) ��ik(1− ��ik) (k; l=1; : : : ; m)

based on the conditional formula (7). The leading term in the second expression corresponds to thecorrelation caused by inter-patient variability. The second approximation includes var[log(zij)] ≈ �and E[ ��ijk(1− ��ijk)] ≈ ��ik(1− ��ik). We thus use the last matrix as the working variance-covariancematrix, Vi (see Appendix I). A robusti�ed variance of the estimator of the treatment e�ect, , isthen obtained using the approximated marginal mean and this working variance-covariance matrix.To evaluate the marginal treatment e�ect, , the approximation (8) leads to the approximation, ≈ ∗. The above approximations would work well if � is small.

6. APPLICATION TO THE OSTEOPOROSIS TRIAL

This section illustrates the application of our design to the osteoporosis trial introduced inSection 2. In particular, we address the design issue of how the required number of patientswho have any vertebral deformity at the time of enrolment varies over the duration of successivemonitoring points. As we indicated in Section 2, we assumed that patients receive subsequentmonitoring continuously after detection of the �rst new deformity.The maximum duration of the follow-up period is the same for all patients and �xed at 3 years.

For the frequency of monitoring, we shall consider the following �ve scenarios: S-2; S-3; S-4; S-6,and S-in�nity. That is: monitoring twice in three years, that is, every 1.5 year (S-2); monitoringthree times in three years, that is, annually (S-3); monitoring four times in three years with irregularspacing at 6 months, 1 year, 2 years and 3 years (S-4); monitoring six times in three years, thatis, every half-year (S-6); and continuous monitoring with in�nite monitoring times (S-in�nity).Scenario S-6 would represent the outer limit of the duration between monitoring times from theethical point of view as described in Section 2. The S-in�nity scenario re ects an ideal situationin which the exact time a new deformity occurs is observable. The design formula developedby Cook1 is applicable to the S-in�nity scenario. Although Cook originally considered a design



scheme where patients are enrolled sequentially over an accrual period and scheduled for follow-upuntil the end of the continuation period of follow-up,14 its adaptation to the current design schemeis straightforward (see Appendix II). For S-in�nity monitoring, we shall use this adapted designformula. We assume an exponential censoring mechanism with censoring rate �i for all patients inthe ith group (i=1; 2), which induces observed probabilities of �ik = 1

2 exp(−�itk) if patients areassigned to one of the two groups with equal probability. For illustration, we assume a commoncensoring rate, that is, �1 = �2 = �. If the value of � is set equal to 0·074 or 0·119 (=year), theobserved probability 3 years from the time of randomization would be 0·40 or 0·35, respectively.We assume a homogeneous Poisson process for new deformity, that is, �∗1 = · · · = �∗m= �∗,

for illustration. The values of the patient-speci�c baseline event rate, �∗, and reference treatmente�ect, ∗R , were roughly prospected from the results of the relevant completed studies; the valueof �∗ ranged between 0·1 and 0·5 (=year), and the value of exp( ∗R ) ranged between 0·6 and 0·8.On the other hand, less information was available about the variance of the mixture distribution,�. Therefore, we arbitrarily set the value as 0·00, 0·25 or 1·00, where �=0·00 represents thecase in which no inter-patient variability exists so that �∗= � and ∗= , while �=1·00 wouldre ect substantial inter-patient variability. For intra-patient correlation, the values 0·25 and 1·00 of� induce an (exchangeable) correlation of 0·06 and 0·19, respectively, for the control, and 0·04and 0·15, respectively, for the treatment group under the setting of �∗=0·3; exp( ∗R )= 0·7, thegamma mixture distribution, and the S-3 monitoring scenario. Notice that the correlation di�ersbetween treatment groups. The size of the test, �, was set to equal 0.1 and the power, 1− �, wasset to equal 0.8.We calculated the required number of patients under several con�gurations of the design pa-

rameters. These are summarized in Tables I and II by the value of the censoring rate. In thesetables, the required number of patients in the presence of inter-patient variability, �=0·25 and�=1·00, was calculated assuming the semi-parametric mixture distribution in Section 5.2. Thesetables indicate that the more frequent the monitoring, the smaller the number of patients required.S-in�nity scenario monitoring provides the lower limit of the number of patients required. Gen-erally, the largest di�erence in the required number of patients is seen between the S-2 and S-3monitoring scenarios, especially for larger baseline event rates, �∗ (the relative di�erence rangedbetween 7 and 14 per cent).The expected number of monitorings, that is, X-ray examinations, is another important index,

particularly from an ethical point of view. In the current setting of the common censoring rate, theexpected number of monitorings per patient is the same for all, which is given by

∑mk = 1 exp(−�tk).

Table III summarizes the expected number of monitorings per patient for each con�guration of� and the S-2, S-3, S-4 and S-6 monitoring scenarios. Tables IV and V summarize the expectedtotal number of monitorings, n

∑mk = 1 exp(−�tk), of all patients under the con�gurations noted in

Tables I and II, respectively. Apparently, the number of patients and the number of monitoringsare a trade-o�.

7. SIMULATION STUDY

This section evaluates the adequacy of the approximations employed in Section 5.2 when weassume a semi-parametric mixture distribution. The operating characteristics of the design formulaassuming the semi-parametric mixture distribution were compared with those assuming a parametricmixture distribution through a simulation study.



Table I. Required number of patients: � = 0·1; 1− � = 0·8; � = 0·074

�∗ � exp( ∗R ) Monitoring scenario∗

S-2 S-3 S-4 S-6 S-in�nity

0·1 0·00 0·6 526 506 499 488 4700·7 988 949 936 912 8780·8 2350 2254 2221 2164 2075

0·25 0·6 554 534 527 514 4930·7 1045 1006 992 968 9260·8 2496 2398 2364 2305 2203

1·00 0·6 638 616 608 595 5640·7 1217 1174 1159 1133 10720·8 2935 2829 2792 2728 2575

0·3 0·00 0·6 197 182 177 169 1570·7 375 344 334 317 2930·8 901 824 797 754 693

0·25 0·6 225 210 204 196 1800·7 432 401 390 372 3410·8 1047 967 940 895 816

1·00 0·6 309 292 286 277 2510·7 604 569 557 538 4870·8 1486 1398 1367 1319 1188

0·5 0·00 0·6 134 119 113 105 940·7 257 226 215 199 1750·8 625 543 516 474 415

0·25 0·6 162 146 141 132 1180·7 315 282 271 254 2240·8 772 687 658 615 539

1·00 0·6 246 228 222 213 1890·7 487 451 438 419 3700·8 1211 1118 1085 1038 913

∗ See Section 6

The conditions of the simulation were set to mimic realistic situations in the osteoporosis trial.We considered monitoring scenarios S-3 and S-6, and the common censoring rate to be �=0·119(=year). We assume a homogeneous Poisson process as in Section 6 and set the patient-speci�cbaseline event rate, �∗, to be 0·1, 0·3 or 0·5 (=year), and the reference treatment e�ect, ∗R , to bethose satisfying exp( ∗R )= 0·6 and exp( ∗R )= 0·8. For the mixture distribution, we considered twoparametric distributions, the gamma and the log-normal, and a semi-parametric distribution withmean 1 and variance �=0·25 or 1·00. The gamma mixture is a very common choice because ofits mathematically convenient properties described in Section 5.1. The log-normal mixture leads toa model in which a normally distributed random e�ect enters the linear predictor on a transformedscale by the link function (see (5)), which is a popular random e�ects model, as seen in theliterature (for example, Diggle et al.4). We restrict attention to the performance of the Wald-typetest with two-sided alternative using the exchangeable working variance-covariance structure, thecorrect structure under monitoring scenarios S-3 and S-6 and the homogeneous Poisson process.The size of the test, �, was set to equal 0·1 and the power, 1−�, was set to equal 0·8. Theresultant number of patients is shown in Tables VI and VII for each con�guration of the design



Table II. Required number of patients: � = 0·1; 1− � = 0·8; � = 0·119


S-2 S-3 S-4 S-6 S-in�nity

0·1 0·00 0·6 580 552 542 526 5010·7 1090 1036 1016 984 9360·8 2594 2460 2411 2333 2215

0·25 0·6 611 582 572 555 5250·7 1154 1097 1076 1044 9840·8 2756 2616 2566 2485 2338

1·00 0·6 704 672 660 642 5960·7 1343 1281 1258 1222 11300·8 3240 3087 3030 2942 2713

0·3 0·00 0·6 218 199 192 182 1670·7 414 376 362 342 3120·8 994 899 865 814 738

0·25 0·6 249 229 222 211 1910·7 477 437 423 402 3610·8 1156 1055 1019 966 863

1·00 0·6 341 319 310 298 2620·7 667 621 604 580 5060·8 1641 1526 1483 1422 1235

0·5 0·00 0·6 148 129 123 114 1000·7 284 246 233 214 1870·8 690 593 559 511 443

0·25 0·6 179 159 152 143 1240·7 347 307 293 274 2360·8 852 749 713 663 568

1·00 0·6 271 249 241 230 1950·7 537 492 475 452 3820·8 1337 1220 1176 1120 939

∗ See Section 6

Table III. Expected number ofmonitorings per patient

� Monitoring scenario∗

S-2 S-3 S-4 S-6

0·074 1·70 2·59 3·56 5·280·119 1·54 2·38 3·32 4·90∗ See Section 6

parameters. The numbers are very similar when inter-patient variability is small (�=0·25), butdi�er to some extent when the variability is substantial (�=1·00), especially when the baselineevent rate, �∗, is large. Notice that in the latter situations, the number of patients assuming thesemi-parametric mixture distribution is intermediate between those which assume the gamma andlog-normal parametric mixture distributions.



Table IV. Expected total number of monitorings: � = 0·1; 1− � = 0·8; � = 0·074


S-2 S-3 S-4 S-6

0·1 0·00 0·6 892 1312 1774 25770·7 1676 2460 3328 48170·8 3985 5842 7897 11430

0·25 0·6 940 1384 1874 27150·7 1772 2608 3527 51130·8 4233 6216 8406 12174

1·00 0·6 1082 1597 2162 31430·7 2064 3043 4121 59840·8 4977 7333 9927 14409

0·3 0·00 0·6 334 472 629 8930·7 636 892 1188 16740·8 1528 2136 2834 3982

0·25 0·6 382 544 725 10350·7 733 1039 1387 19650·8 1776 2506 3342 4727

1·00 0·6 524 757 1017 14630·7 1024 1475 1981 28420·8 2520 3624 4861 6967

0·5 0·00 0·6 227 308 402 5550·7 436 586 764 10510·8 1060 1407 1835 2504

0·25 0·6 275 378 501 6970·7 534 731 964 13420·8 1309 1781 2340 3248

1·00 0·6 417 591 789 11250·7 826 1169 1557 22130·8 2054 2898 3858 5482

∗ See Section 6

We simulated data for each con�guration with no treatment e�ect, that is, exp( ∗)= 1, toinvestigate the control of type I error rates, and treatment e�ects with exp( ∗)= 0·6 and 0·8,to determine if the power requirement was met. The underlying mixture distribution to simulatethe data was the gamma or the log-normal distribution. We performed 1000 simulations for eachcon�guration. The empirical type I error rate was generally very close to the nominal level forall con�gurations (data not shown). The empirical power for each con�guration is summarized inTables VI and VII. The empirical power when the number of patients was calculated assumingthe semi-parametric mixture, was close to the nominal level even in the presence of substantialinter-patient variability, under both the gamma and log-normal parametric mixture distributionsused to simulate the data. This demonstrates the adequacy of the approximations employed whenthe semi-parametric mixture distribution is assumed. The empirical power when the number ofpatients was calculated assuming a parametric mixture showed a tendency not to maintain thenominal level for con�gurations with larger �∗ and �, if the underlying mixture distribution tosimulate the data is di�erent from that assumed in the calculation of the number of patients. We



Table V. Expected total number of monitorings: � = 0·1; 1− � = 0·8; � = 0·119


S-2 S-3 S-4 S-6

0·1 0·00 0·6 891 1311 1798 25760·7 1675 2461 3371 48190·8 3985 5844 8000 11425

0·25 0·6 939 1383 1898 27180·7 1773 2606 3570 51130·8 4234 6215 8514 12170

1·00 0·6 1082 1597 2190 31440·7 2063 3043 4174 59840·8 4978 7334 10054 14408

0·3 0·00 0·6 335 473 637 8910·7 636 893 1201 16750·8 1527 2136 2870 3986

0·25 0·6 383 544 737 10330·7 733 1038 1404 19690·8 1776 2506 3381 4731

1·00 0·6 524 758 1029 14590·7 1025 1475 2004 28400·8 2521 3625 4921 6964

0·5 0·00 0·6 227 306 408 5580·7 436 584 773 10480·8 1060 1409 1855 2502

0·25 0·6 275 378 504 7000·7 533 729 972 13420·8 1309 1779 2366 3247

1·00 0·6 416 592 800 11260·7 825 1169 1576 22140·8 2054 2898 3902 5485

∗ See Section 6

therefore conclude that the formula assuming the semi-parametric mixture distribution is morerobust than that which assumes the parametric mixture distribution, to misspeci�cation of themixture distribution.

8. DISCUSSION

This paper presents a design for randomized clinical trials in which interval data on the occur-rence of potentially recurrent events are collected at a sequence of periodic monitoring points.We modelled the event process via a piecewise Poisson process with a random e�ects model toaccommodate non-homogeneity of the event rate over time and intra-patient correlation of the re-sponses within a patient. A formula can be developed even if the form of the mixture distributionis not fully speci�ed, although some approximations are required to evaluate the variance of theestimator of the treatment e�ect. The results of the simulation study demonstrated the adequacyof these approximations and its robustness on the operating characteristics to misspeci�cation of



Table VI. Empirical power when the underlying mixture distribution to simulate the data isgamma: � = 0·1; 1− � = 0·8; � = 0·119

exp( ∗R ) �∗ � Monitoring scenario∗ Assumed mixture distribution

Gamma Log-normal Semi-parametricn power n power n power

0·6 0·1 0·25 S-3 588 0·814 588 0·814 582 0·809S-6 558 0·807 557 0·809 555 0·809

1·00 S-3 697 0·802 683 0·804 672 0·789S-6 653 0·804 646 0·796 642 0·801

0·3 0·25 S-3 236 0·817 234 0·803 229 0·801S-6 214 0·806 213 0·809 211 0·800

1·00 S-3 351 0·822 320 0·782 319 0·780S-6 312 0·814 294 0·793 298 0·801

0·5 0·25 S-3 167 0·805 164 0·797 159 0·795S-6 146 0·800 144 0·801 143 0·804

1·00 S-3 290 0·806 247 0·758 249 0·760S-6 246 0·809 221 0·778 230 0·797

0·8 0·1 0·25 S-3 2646 0·810 2642 0·811 2616 0·794S-6 2499 0·811 2497 0·812 2485 0·804

1·00 S-3 3211 0·806 3133 0·793 3087 0·781S-6 2998 0·802 2955 0·807 2942 0·792

0·3 0·25 S-3 1091 0·805 1080 0·794 1055 0·784S-6 981 0·804 975 0·799 966 0·799

1·00 S-3 1698 0·810 1523 0·779 1526 0·771S-6 1492 0·803 1391 0·782 1422 0·787

0·5 0·25 S-3 793 0·804 775 0·801 749 0·794S-6 680 0·797 671 0·798 663 0·791

1·00 S-3 1446 0·811 1198 0·742 1220 0·750S-6 1207 0·812 1063 0·765 1120 0·785

∗ See Section 6

the mixture distribution. The formula would be of practical use for designing trials similar to theosteoporosis trial. In conclusion, we discuss the issue of speci�cation of the degree of inter-patientvariability, and how the formula can be extended and further generalized.At the design stage, a plausible value for the variance of the semi-parametric mixture distribu-

tion, �, should be speci�ed. Estimating � using data from relevant completed trials would be aformal approach. Two approaches are available for estimating �. The most suitable is the quasi-likelihood approach. The analytical strategy proposed by Zeger et al.3 is also applicable to thecurrent context, through which we would obtain a simple moment estimator of � as well as infor-mation about the size of the baseline event rates and treatment e�ects. If the baseline event ratesare considered to be su�ciently small so that the Poisson approximation works reasonably well,the method proposed by Thall and Vail15 for interval count data is applicable. The other approachis the random e�ects approach assuming a parametric mixture distribution. For a gamma mixturedistribution, we can construct the likelihood explicitly and obtain the usual maximum likelihoodestimator. Using a di�erent approach to derive the range of � may be an adequate strategy inpractice.



Table VII. Empirical power when the underlying mixture distribution to simulate the data is log-normal:� = 0·1; 1− � = 0·8; � = 0·119

exp( ∗R ) �∗ � Monitoring scenario∗ Assumed mixture distribution

Gamma Log-normal Semi-parametricn power n power n power

0·6 0·1 0·25 S-3 588 0·802 588 0·802 582 0·806S-6 558 0·808 557 0·812 555 0·813

1·00 S-3 697 0·814 683 0·806 672 0·793S-6 653 0·808 646 0·814 642 0·814

0·3 0·25 S-3 236 0·807 234 0·805 229 0·803S-6 214 0·819 213 0·816 211 0·801

1·00 S-3 351 0·841 320 0·809 319 0·812S-6 312 0·833 294 0·807 298 0·814

0·5 0·25 S-3 167 0·819 164 0·809 159 0·792S-6 146 0·823 144 0·807 143 0·796

1·00 S-3 290 0·864 247 0·810 249 0·814S-6 246 0·850 221 0·805 230 0·826

0·8 0·1 0·25 S-3 2646 0·798 2642 0·813 2616 0·796S-6 2499 0·804 2497 0·808 2485 0·807

1·00 S-3 3211 0·806 3133 0·803 3087 0·804S-6 2998 0·809 2955 0·806 2942 0·804

0·3 0·25 S-3 1091 0·806 1080 0·807 1055 0·790S-6 981 0·808 975 0·808 966 0·804

1·00 S-3 1698 0·835 1523 0·814 1526 0·813S-6 1492 0·831 1391 0·809 1422 0·824

0·5 0·25 S-3 793 0·813 775 0·798 749 0·785S-6 680 0·814 671 0·807 663 0·803

1·00 S-3 1446 0·864 1198 0·812 1220 0·819S-6 1207 0·847 1063 0·821 1120 0·822

∗ See Section 6

In developing the formula, we assumed that the pattern of missing observation is monotone.In reality, however, we would encounter non-monotone missing, which typically arises when pa-tients visit on an intermittent basis. Non-monotone missing brings us to interval-censored data,16

for which establishing a hypothesis testing procedure is complicated because we cannot identifythe monitoring period [tj−1; tj) during which one or more events occurred. One way to handleinterval censoring is to introduce new monitoring periods which follow the observed pattern ofnon-monotone missing. For a patient, provided that a response is observed at tj1 and tj2 , wherej2− j1¿1, and between which responses are missing, and that the occurrence of event is detectedat tj2 , we introduce a new additional period [tj1 ; tj2 ). For this patient, we assume that one or moreevents did occur within this new period, although the responses between tj1 and tj2 are allowed tobe missing. For our interval data with this device, we can adopt the Wald-type test for analysis.A design formula can also be developed although it would require more complicated assumptionsabout the missing mechanism.Although we assumed throughout the paper that the treatment e�ect is constant over all monitor-

ing periods, we can take into account time-varying treatment e�ects by introducing the parameter



of monitoring-period-speci�c treatment e�ects, that is, =( 1; : : : ; m)′. We can then develop thedesign formula on the basis of hypothesis testing on a speci�c pattern of treatment e�ects byintroducing a p×m contrast matrix, L say. For testing the hypothesis, L =0, we would establisha Wald-type statistic, (L )(LWL′)−1(L )′, to develop the design criteria. In particular, by takingL=(1; : : : ; 1)=m, we can develop design criteria concerning the average treatment e�ect, whichwould also be useful for investigating the sensitivity of the sample size to deviation from theassumption of constant treatment e�ects.Generalization of the developed formula to the general framework of the GEE would be

straightforward.8 The generalized formula is considered a counterpart of the formula developedby Liu and Liang,7 which is based on a score-type test derived from the GEE.

APPENDIX I: ELEMENTS OF THE WORKING VARIANCE-COVARIANCE MATRIX Vi

Following are the elements of the working variance-covariance matrix, Vi, for the three models inthis paper. For simplicity, we consider the case m=2 indexed by k and l.

Independence

Under the assumption of independence, Vi is expressed as

Vi=

[�ik(1− �ik) 0

0 �il(1− �il)

]

where �ik and �il are given by (2).

Parametric mixture distribution

Under the assumption of a parametric mixture, Vi is expressed as

Vi=

[�ik(1− �ik) L(aik + ail)− (1− �ik)(1− �il)

L(ail + aik)− (1− �il)(1− �ik) �il(1− �il)

]

where �ik and �il are given by (6).

Semi-parametric mixture distribution

Under the assumption of a semi-parametric mixture, Vi is expressed as

Vi=

��ik(1− ��ik) + �

(@h−1(�∗ik)

@�∗ik

)2�

@h−1(�∗ik)@�∗ik

@h−1(�∗il)@�∗il

�@h−1(�∗il)

@�∗il@h−1(�∗ik)

@�∗ik��il(1− ��il) + �

(@h−1(�∗il)

@�∗il

)2

where ��ik and ��il are given by (9).

APPENDIX II: ADAPTATION OF COOK’S FORMULA

We assume a homogenous Poisson process with the event rate, �i= � exp{ (i − 1)}, for theith group (i=1; 2). We denote the total number of observed events in the ith group under the



S-in�nity monitoring scenario as Xi. Then, the inequality (5) in Cook1 is expressed in the currentcontext as

2R(z�=2 + z�)2¿E{X1}−1 + E{X2}−1 + 4�=n

where

E{Xi}= n2

∞∑x= 1

xp(x; �i; �)

and

p(x; �i; �)=(

�i

�i + �

)x {Q(x; tm(�i + �))−

(�i

�i + �

)Q(x + 1; tm(�i + �))

}where Q(x; �)=

∑∞s= x exp(−�)�s=s!. The above inequality provides the design criteria for the

S-in�nity monitoring scenario.

ACKNOWLEDGEMENTS

The authors thank Dr. Tosiya Sato for many helpful discussions and comments on an early draft,and Professor Isao Yoshimura for his supervision. Thanks are also extended to two referees fortheir very constructive comments, which substantially improved an early draft.

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10. Rubin, D. B. ‘Inference and missing data’, Biometrika, 63, 581–592 (1976).11. Hougaard, P. ‘Life table methods for heterogeneous populations: distributions describing the

heterogeneity’, Biometrika, 71, 75–83 (1984).12. Aalen, O. O. and Husebye, E. ‘Statistical analysis of repeated events forming renewal processes’,

Statistics in Medicine, 10, 1227–1240 (1991).13. Ross, S. M. Introduction to Probability Models, Academic Press, Toronto, 1989.14. Rubinstein, L. V., Gail, M. H. and Santner, T. J. ‘Planning the duration of a clinical trial studying the

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Design of clinical trials for recurrent events with periodic monitoring