morbidity measures in the presence of recurrent composite endpoints

PHARMACEUTICAL STATISTICS

Pharmaceut. Statist. 2003; 2: 39–49 (DOI:10.1002/pst.020)

Morbidity measures in the presence of

recurrent composite endpointsz

Martin Scott1*y, Joachim M .oocks1, Sam Givens2, Walter K .oohler3,J .oorg Maurer4 and Michael Budde4

1Roche Pharma, Global Development, Statistics, Mannheim, Germany2Roche Pharma, Global Development, Statistics, Nutley, NJ, USA3Baseline, Statistical Solutions, Frankfurt/Main, Germany4Roche Pharma, Global Development, Statistics, Basel, Switzerland

The analysis of recurrent event data in clinical trials presents a number of difficulties. The statistician

is faced with issues of event dependency, composite endpoints, unbalanced follow-up times and

informative dropout. It is not unusual, therefore, for statisticians charged with responsibility for

providing reliable and valid analyses to need to derive new methods specific to the clinical indication

under investigation. One method is proposed that appears to have possible advantages over those that

are often used in the analysis of recurrent event data in clinical trials. Based on an approach that

counts periods of time with events instead of single event counts, the proposed method makes an

adjustment for patient time on study and incorporates heterogeneity by estimating an individual per-

patient risk of experiencing a morbid event. Monte Carlo simulations demonstrate that, with use of a

real clinical study data, the proposed method consistently outperforms other measures of

morbidity. Copyright # 2003 John Wiley & Sons, Ltd.

Keywords: morbidity rates; recurrent events; patient heterogeneity; period counts

1. INTRODUCTION

This paper discusses recurrent morbidity eventsand the issues surrounding their methods ofanalysis within the framework of an oncologytrial. It also begins to address the needs of theclinician and statistician regarding a reliable and

valid measure of morbidity, and will examinepotential candidate methods of analysis.

Recurrent events are frequently encountered inclinical trials where patients may experience morethan one event over the course of the trial.Examples include tumour recurrences, infectiousepisodes, asthma attacks, myocardial infarctions,injuries, migraines, epileptic seizures and hospitaladmissions [1]. This paper focuses on recurrentmetastatic bone events following cancer.

In many medical indications an event recurrenceis regarded as clinically meaningful and cannotbe simply neglected or ignored. It is thus a

Copyright # 2003 John Wiley & Sons, Ltd.

*Correspondence to: Martin Scott, Pharmacia, Dept. Statis-tics & Programming, Viale Pasteur 10, 20014 Nerviano (MI),Italy

ySince Jan 2002 with Pharmacia Corporation.zPresented at the PSI Annual Conference, Chester, UK, May

2001.

requirement that statistical methods are ableadequately to incorporate recurrent events intothe analysis of clinical trial data. However, suchmethods are not without their difficulties. Specifi-cally, such analyses need to account for thefollowing issues that arise with recurrent eventdata: different lengths of patient observation timein study; between–patient heterogeneity; andcomposite endpoints.

2. ANALYSIS ISSUES OFRECURRENT EVENTS

2.1. Different observation times

Many inferential methods for recurrent event datarequire comparable follow-up times between pa-tients and treatments. Such methods includesimple event rates such as those discussed byStukel et al. [2], Poisson regression, or methodsbased on the negative binomial distribution asdescribed by Glynn and Buring [3]. However, it isusually likely that due to premature dropouts anddeaths, follow-up times will not be comparablebetween patients, especially in clinical trialsinvestigating severe diseases such as cancer orheart disease. Further, treatment comparisons canbe rendered invalidated if the dropout process isinformative or selective in nature (as discussed byM .oocks et al. [4]).

A built-in adjustment of observation time isachieved by the time-to-first-event method. Oneignores the recurrent nature of the data and onlyconsiders an analysis of the first event. Theobvious disadvantage of this method is a loss ofinformation when only the first event is considered[1]. If a drug is expected to reduce the frequency ofevents and not just to prolong the time to firstevent, then the analysis of the time to first eventmay also be regarded as clinically inferior.Further, time-to-event analyses assume a homo-geneous risk among patients. In order to addressthe potential of heterogeneous risk betweenpatients, one needs to turn to frailty arguments.

2.2. Patient heterogeneity

The second issue is patient heterogeneity. Eachindividual patient has an individual patient risk ofexperiencing an event. It can be consideredunreasonable to assume that this risk is the samefor all patients as the risk is determined by thepatient’s ‘position’ in the progression of disease.For a hypothetical disease progression, it may beinappropriate to assume that patients have com-parable risk (hazard) of disease progression uponentry into a clinical trial–see Figure 1.

To address patient heterogeneity, regressionapproaches have been developed such as those ofthe marginal hazard model by Wei et al. [5],generalized estimating equation methods by Zeger

Figure 1. Hypothetical risk of disease progression.

Copyright # 2003 John Wiley & Sons, Ltd. Pharmaceut. Statist. 2003; 2: 39–49

40 M. Scott et al.

and Liang [6] and Albert and Follmann [7],negative binomial methods as described by Law-less [8] and random-effect Poisson regressions asrecently detailed by Cook and Major [9].

Many methods account for between-patientheterogeneity by considering an adjustment ofthe treatment risk by use of baseline covariates. Itis necessary to consider, however, whether adjust-ment using covariates can address all the variationbetween patients in baseline disease status. Indeed,unpredictable bias will result from relevant cov-ariates that were not observed and are conse-quently not considered in the analysis. It is alsonecessary to address comparability between trialsthat, although identical regarding their design,primary variable and study population, possiblyuse a different set of covariates in the primaryanalysis. This would result in non-comparableadjusted analyses.

2.3. Composite endpoints

The third issue is composite endpoints. It is usualfor components of the composite endpoints to becorrelated with each other. Indeed, each differentcomponent may actually refer to the same medicalevent. If the components are correlated theassumption of independent events is invalid. Ifthis is the case, methods based on simple eventcounts are difficult to interpret. This is usually therule and not the exception regarding events thatrecur over time within composite endpoints.

Addressing the issue of event dependency, onecan focus on counts of intervals of time with atleast one event (e.g. 4; 6; 8; . . . week periods) rather

than using the events themselves as units ofmeasurement. That is to say, a time window isdefined in which one believes events to be clinicallyrelated. This will have the effect of removing muchof the dependency as events occurring in differentperiods are more likely to represent independentmedical events. Indeed, it is often the case thatdiagnoses such as X-rays, and treatments such asradiotherapy and chemotherapy, are performedperiodically, which suggests that the approach ofcounting periods instead of single events isapplicable in a clinical trial setting.

Figure 2 illustrates the period approach. PatientX has 10 events in total. These events fall intothree periods. This patient thus has three periodswith events and four periods in total. With theassumption that each period is independent of theothers, each period takes the form of a Bernoullitrial with a binary response ‘event in period ¼ yesor no’.

3. MORBIDITY AND RECURRENTEVENTS

Morbidity is a term that is used to refer to themedical condition of a patient, which is oftenterminal. It is measured by the patient’s ‘morbidevents’ according to the indication under investi-gation. In the field of osteoporosis and metastaticbone disease in cancer, for example, morbidity ofthe skeleton is quantified by analysing the fre-quency of fractures [10, 11] and is often referred toas ‘skeletal morbidity’. The link between recurrentevents and morbidity is that many morbidity

Figure 2. Analysis with periods instead of events.


Morbidity measures 41

events are recurrent in nature – that is to say, thepatient experiences multiple morbid events.

The events reported or recorded in the clinicaltrial reflect the patient’s disease morbidity. Indeed,the recorded morbid events are the very clinicalsource of measurements of the target morbidity fora patient in a clinical trial. The goal of a drugunder development is to reduce the target morbid-ity. Therefore, it is necessary to quantify thereduction of morbid events.

3.1. The challenge for the statistician

The statistician is required to develop a measurethat validly and reliably describes the targetmorbidity. This measure must be standardized inorder to allow valid comparisons between patientsand treatment groups. Above all, the statisticianmust also ensure that the standardized measure isclinically meaningful and interpretable.

The first requirement of a morbidity measure isthat it should be able to provide an adjustment forunbalanced observation time in study, and itshould work equally well for balanced time.Secondly, the measure has to take into accountpotentially dependent events, especially depen-dency among components of composite endpoints.

Thirdly, it has to account for heterogeneity of risksbetween patients without initially resorting tomethods of adjustment by baseline covariates.Methods of covariate adjustment such as thoseintroduced by Koch et al. [12], can be used in theinferential analysis to reduce variance if necessary.To maintain simplicity and objectivity the measureof target morbidity should first be determinedwithout the use of covariates.

3.2. Measures of target morbidity – the morbidity

period rate

A measure that can meet all of the above criteria isone that summarizes the morbidity for eachindividual patient separately. The morbidity periodrate is defined to be the number of periods with atleast one morbidity event over the time in studyper patient. First, counting periods takes intoaccount the dependency of events. Secondly,calculating a rate per patient allows between-patient heterogeneity to be incorporated. Thirdly,there is a simple adjustment for observation timein study.

Referring to Figure 1 again, we see that anindividual patient’s observation time in study isrelatively short compared to the total time of the

Figure 3. Conservative estimate of drug effect per patient.


42 M. Scott et al.

disease progression. Figure 3 shows the diseaseprogression for a particular patient over the courseof treatment during a clinical trial. It makes thereasonable assumption that the hazard underplacebo fluctuates only a little, remaining relativelyconstant during the course of the trial. This figuredemonstrates that a per-patient estimate of riskremains conservative compared to the lower risk ofthe ultimate drug effect, as events reported in theinitial period of treatment, during which the drughas yet to have an effect, are counted and subse-quently contribute to the estimate of morbidity.

Four possible measures of target morbidity arenow considered.

Morbidity measure 1: MPR1ðy;TÞ ¼ y

First, consider y; the number of periods withevents. This is simply the number of periods withevents, without an adjustment for observationtime in study. Indeed, this measure is not a rate atall ðT ¼ 1Þ and is potentially severely misleading ifthe observation time in study is unbalancedbetween the treatment groups.

Figure 4 illustrates the target morbidity measureMPR1ðy;TÞ ¼ y: Each line in this graph represents

a certain number of periods with events. Asmentioned already, the main drawback with thismeasure is that there exists no adjustment forobservation time in study. That is to say, a patientexperiencing zero periods with events out of sixtotal periods, y ¼ 0;T ¼ 6; will receive the samemorbidity score MPR1ðy;TÞ ¼ 0 as a patient withzero periods with events out of one total period,y ¼ 0; T ¼ 1:

Morbidity measure 2: MPR2ðy;TÞ ¼ y=T

The second measures divides the number ofperiods with events, y; by the total number ofperiods a patient participated in the study, T ; andtherefore provides an adjustment for observationtime which yields a ratio in the interval ½0; 1�:Referred to in the indication of metastaticbone disease as the ‘skeletal morbidity rate’, ithas been used to test hypotheses of efficacy inseveral clinical oncology studies; see Berenson et al.[13] where the numerator, y; referred to a simpleevent count and was not based on a periodapproach.

This morbidity measure is illustrated inFigure 5. Again, each line represents a certain

Figure 4. Morbidity period rate MPR1ðy;TÞ ¼ y:



number of periods with events. Within each line anadjustment for time on study is made, wherebyincreasing time on study returns a lower morbidityscore. There are two main drawbacks with thismeasure. First, there is no adjustment for time onstudy for patients having zero periods with events,that is, when y ¼ 0: That is to say, if no morbidityevent occurs a value of MPR2ðy;TÞ ¼ 0 is returnedregardless of the length of time a patient hasparticipated in the study. This can be an importantdeficiency in many clinical settings where manypatients in clinical trials do not experience anevent, that is, y=T ¼ 0: The second problemregarding this measure is that when y ¼ T ; thatis, the number of event periods equals number ofperiods in the study, adjustment for the time instudy has no effect and MPR2ðy;TÞ ¼ 1: It wouldbe reasonable to expect that a patient with y ¼ 6periods with events out of T ¼ 6 total periods inthe study would be clinically more informativethan a patient with y ¼ 1 period with event in T ¼1 total period in the study and would thereforereceive a higher morbidity score. These twoimportant deficiencies thus suggest that there is aneed to improve on the simple ratio y=T :

Morbidity measure 3: MPR3ðy;TÞ ¼ ðyþ cÞ=T

The third measure introduces a small constant, c51; to the numerator of the ratio. This constant hasthe effect of returning MPR3ðy;TÞ > 0 when y ¼ 0:Figure 6 demonstrates this. The addition of c ¼ 0:1to the numerator achieves an adjustment forobservation time for patients experiencing noevents. However, patients with y ¼ T ¼ 1(one period with event out of one total period)receive a higher score than patients where y ¼ T ¼2 (two event periods out of two total periods).Indeed, the lowest score for patients when y ¼ T isassigned to those with the worst possible outcome,namely y ¼ T ¼ 6 (an event in every period in thestudy). The clinical justification of this is certainlyquestionable.

Morbidity measure 4: MPR4ðy;TÞ ¼ ðyþ cÞ=ðT þ dÞ

The fourth measure introduces small constants, cand d; to the numerator and denominatorrespectively, where 05c5d: A particular form ofthis estimator is that used in the analysis of

Figure 5. Morbidity period rate MPR2ðy;TÞ ¼ y=T :


44 M. Scott et al.

proportions, that is,

MPR4ðy;TÞ ¼yþ c

T þ d¼

yþ 0:5

T þ 1

This provides an improved estimate in particularwhen outcomes at the boundaries of 0 and 1 arefrequent, and, as with y=T ; provides an estimate ofthe risk of a patient experiencing a morbid event.

Figure 7 provides a graphical representation.Note that this estimator resembles that of y=T :However, unlike the y=T estimator, it provides anadjustment for time in study for those patientswithout an event. It also provides an adjustmentfor time in study for those patients wherey ¼ T : That is to say that a patient with y ¼ T ¼6 is considered worse than a patient withy ¼ T ¼ 1 and consequently receives a highermorbidity score: MPR4ð6; 6Þ > MPR4ð5; 5Þ > � � � >MPR4ð1; 1Þ: The highest or ‘worst’ possible scoreis consistently assigned to patients with anevent in every period ðy ¼ T ¼ 6Þ; whereas thelowest or ‘best’ score is assigned to patients notexperiencing any event during the entire studyðy ¼ 0;T ¼ 6Þ:

4. THEORETICAL LINKS

4.1. Early approaches

In the mid-1950s, Haldane [14] and Anscombe [15]independently proposed the MRP4 formula in thecontext of odds ratios. By adjusting the logittransformation as shown below, infinity could beavoided at extreme values of p ¼ 0 and p ¼ 1:

Solving the expression for p returns a rateestimator of the form ðyþ 0:5Þ=ðT þ 1Þ:

logp

1� p

� �ffi log

yþ 0:5

T � yþ 0:5

A decade later, Gart and Zweifel [16] went on toinvestigate the bias of various estimators of thelogit, and expanded these ideas in the context ofbioassays.

4.2. Bayesian approaches

The measure of ðyþ 0:5Þ=ðT þ 1Þ can also be seenin a Bayesian context. Assuming a binomialdistribution for an event period,

PðY ¼ yÞ / pyð1� pÞT�y

Figure 6. Morbidity period rate MPR3ðy;TÞ ¼ ðyþ 0:1Þ=T :



and a conjugate prior beta distribution of priorknowledge,

f ðpÞ / pa�1ð1� pÞb�1

with mean a=ðaþ bÞ; then after having observed yperiods with events out of T total periods, theposterior distribution takes the form of a betadistribution:

fT ;yðpÞ / paþy�1ð1� pÞbþðT�yÞ�1

The posterior mean is ðyþ aÞ=ðT þ aþ bÞ: For theparticular choice of a ¼ b ¼ 0:5; the proposedmorbidity measure of ðyþ 0:5Þ=ðT þ 1Þ is returnedas the posterior mean.

Figure 8 shows the prior distribution when a ¼b ¼ 0:5: It is evident that there is high mass for lowand high values of p; which might be sensible in aclinical study where many patients are expected tohave no events, or an event in every period, as isthe case in metastatic bone disease. Figure 9 showsthe posterior distribution when y ¼ 3 and T ¼ 6:

In many Bayesian approaches the uniformdistribution is taken as a non-informative priorwhich is of the beta family with parameters a ¼b ¼ 1: Combined with a binomial data distribu-tion, the use of a non-informative prior returns aposterior mean of ðyþ 1Þ=ðT þ 2Þ; another candi-

date for a measure of morbidity. In this case c ¼ 1and d ¼ 2; which seems to suggest that the bestchoices of constants c and d are those that satisfyd ¼ 2c: The choice of d ¼ 2c has the effect ofreturning higher estimates of morbidity withlonger observation time if the majority ofperiods contain events, for instance MPR4ð6; 6Þ >MPR4ð5; 5Þ > � � � > MPR4ð1; 1Þ: In contrast, longerobservation times return lower estimates ofmorbidity when the majority of periods areevent-free, for instance MPR4ð3; 9Þ5MPR4ð2; 6Þ5MPR4ð1; 3Þ; cf. Figure 7.

Written in the following form, it can be seen thatthe addition of 0.5 to the numerator of MPR4

introduces a penalty to patients with shorterobservation times:

MPR4ðy;TÞ ¼y

T þ 1þ

0:5

T þ 1

The addition of the non-specific penalty 0:5=ðT þ1Þ can be thought of as representing the baselineunderlying morbidity of the patient uponentering the trial. Further, the additional penaltywill cause relative risks based on MPR4 to tendtowards unity and can thus be regarded asconservative.

Figure 7. Morbidity period rate MPR4ðy;TÞ ¼ ðyþ 0:5Þ=ðT þ 1Þ:


46 M. Scott et al.

5. SIMULATIONS

Monte Carlo simulations were performed on eachof the morbidity measures to investigate theirbehaviour with data from an oncology study.First, by specifying a mean average risk ðpÞ and

variance, a realistic, normally distributed, true riskset was created. Risks for each patient wererandomly chosen from this risk set and subse-quently ranked. Once drawn, these figures servedas the true risk set and their corresponding rankorder was defined as the true rank order.

Figure 8. Beta prior distribution, a ¼ b ¼ 0:5:

Figure 9. Posterior beta distribution, a ¼ b ¼ 0:5; y ¼ 3; T ¼ 6:



Observation times were then randomly chosenfrom a realistic distribution for each patient.Bernoulli trials were then performed on eachpatient using the respective risk estimate fromthe true risk set and the patient’s observation time.The different skeletal morbidity measures werethen applied to the outcome of the Bernoulli trialsand then ranked across the sample. These rankswere subsequently compared with the ranks of thetrue risk set and a correlation coefficient of rankcorrelation was calculated for each morbiditymeasure. This was performed 2000 times. Eachrun compared all morbidity measures with eachother and assigned one to be the best based on thehighest correlation within that run. Finally, thefrequency with which each measure achieved bestmethod status ðN ¼ 2000Þ was calculated.

Table I shows that for three average risks in thesample ðp ¼ 0:25; 0:33; and 0.40) the morbiditymeasure of ðyþ 0:5Þ=ðT þ 1Þ consistently outper-formed the others.

6. PRACTICAL USE

It appears that ðyþ 0:5Þ=ðT þ 1Þ enjoys manytheoretical links to statistical standards and canbe clinically defended. Further, Monte Carlosimulations suggest that on real data it canoutperform the other estimators. The estimatoralso has practical use in clinical studies and followsthat of the standard rate estimator y=T : Inferential

testing can be performed using rank-sum tests suchas Wilcoxon, Jonckheere–Terpstra or Kruskal–Wallis. Relative risks can be produced andconfidence intervals can be calculated using theHodges–Lehmann method on the log-transformedvalues.

7. CONCLUSIONS

The morbidity period rate ðyþ 0:5Þ=ðT þ 1Þ pro-vides an estimate of the number of periods perpatient-year – a clinically interpretable and simplemeasure of morbidity. Further, it provides anadjustment for time in study across all theoutcome possibilities and appears to agree withclinical rationale. It also follows the statisticalapproaches of Haldane, Anscombe, and Gart andZweifel, and enjoys a nice relationship to Bayesianapproaches. By also taking patient heterogeneityinto account, it can be concluded that it provides avalid summary measure of target morbidity of apatient. Based on such a measure, treatmentcomparisons can be carried out, preferably usingnon-parametric methods, which could includeadjustment by covariates.

One potential criticism is the addition of aseemingly arbitrary constant to the numerator anddenominator. It can be assumed, however, that theeffect of adding any arbitrary constant is marginal,especially following the ranking of morbidityscores for use in subsequent treatment compar-

Table I. Monte Carlo simulations of morbidity period rates.

Average Morbidity measuretrue risk ðpÞ

MPR1 ¼ y MPR2 ¼ y=T MPR3 ¼ yþ c=T MPR4 ¼ yþ 1=2=T þ 1

% Best Method 0.25 10% 3% 29% 58%ðN ¼ 2000Þ 0.33 8% 6% 13% 73%

0.40 7% 6% 6% 81%

Mean correlation 0.25 0.31 (0.06) 0.32 (0.07) 0.34 (0.07) 0.35 (0.07)coefficient (SD) 0.33 0.26 (0.06) 0.29 (0.07) 0.30 (0.07) 0.31 (0.07)

0.40 0.24 (0.06) 0.28 (0.07) 0.28 (0.07) 0.30 (0.07)


48 M. Scott et al.

isons. Indeed, it was empirically checked whetherthe choice of constants ðc ¼ 1; d ¼ 2Þ and ðc ¼0:25; d ¼ 0:5Þ would seriously influence the actualranking order. The resulting rank correlation ofthe proposed measure ðc ¼ 0:5; d ¼ 1Þ with thosefrom alternative definitions was greater than 0.97.This result offers some reassurance that treatmentcomparisons based on ranking approaches arepractically unaffected when other constants areused in the definition of the morbidity measure.

A second criticism is that for patients notexperiencing an event, where y ¼ 0; the morbidityperiod rate ranks patients solely according to theirobservation time. It does not attempt to breakdown the reasons for drop-out and rank accord-ingly thereafter. Whether this is necessary or not isquestionable. The paper by M .oocks et al. [4]addresses this issue and looks into the generalproblem of drop-out in clinical studies.

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morbidity measures in the presence of recurrent composite endpoints

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