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On reporting results from randomized controlled trials with recurrent events Kuramoto, Lisa; Sobolev, Boris G; Donaldson, Meghan G May 30, 2008

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ralBMC Medical Research ssBioMed CentMethodologyOpen AcceResearch articleOn reporting results from randomized controlled trials with recurrent eventsLisa Kuramoto*1, Boris G Sobolev2 and Meghan G Donaldson3Address: 1Centre for Clinical Epidemiology and Evaluation, Vancouver Coastal Health Research Institute, Vancouver, Canada, 2Department of Health Care and Epidemiology, University of British Columbia, Vancouver, Canada and 3SF Coordinating Center, San Francisco, USAEmail: Lisa Kuramoto* - lisa.kuramoto@vch.ca; Boris G Sobolev - sobolev@interchange.ubc.ca; Meghan G Donaldson - mdonaldson@sfcc-cpmc.net* Corresponding author    AbstractBackground: Evidence-based medicine has been advanced by the use of standards for reportingthe design and methodology of randomized controlled trials (RCT). Indeed, without thisinformation it is difficult to assess the quality of evidence from an RCT. Although a variety ofstatistical methods are available for the analysis of recurrent events, reporting the effect of anintervention on outcomes that recur is an area that remains poorly understood in clinical research.The purpose of this paper is to outline guidelines for reporting results from RCTs where theoutcome of interest is a recurrent event.Methods: We used a simulation study to relate an event process and results from analyses of thegamma-Poisson, independent-increment, conditional, and marginal Cox models. We reviewed theutility of regression models for the rate of a recurrent event by articulating the associated studyquestions, preenting the risk sets, and interpreting the regression coefficients.Results: Based on a single data set produced by simulation, we reported and contrasted resultsfrom statistical methods for evaluating treatment effect from an RCT with a recurrent outcome.We showed that each model has different study questions, assumptions, risk sets, and rate ratiointerpretation, and so inferences should consider the appropriateness of the model for the RCT.Conclusion: Our guidelines for reporting results from an RCT involving a recurrent event suggestthat the study question and the objectives of the trial, such as assessing comparable groups andestimating effect size, should determine the statistical methods. The guidelines should allow clinicalresearchers to report appropriate measures from an RCT for understanding the effect ofintervention on the occurrence of a recurrent event.BackgroundEvidence-based medicine has been advanced by the use ofstandards for reporting the design and methodology ofthat submissions adhere to the Consolidated Standardsfor Reporting Trials (CONSORT) guidelines for improv-ing report quality [1]. However, there are not yet availablePublished: 30 May 2008BMC Medical Research Methodology 2008, 8:35 doi:10.1186/1471-2288-8-35Received: 11 December 2007Accepted: 30 May 2008This article is available from: http://www.biomedcentral.com/1471-2288/8/35© 2008 Kuramoto et al; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.Page 1 of 12(page number not for citation purposes)randomized controlled trials (RCT). Indeed, without thisinformation it is difficult to assess the quality of evidencefrom an RCT. An increasing number of journals demandguidelines for reporting results from RCTs in which thesubject may experience the same event multiple timesduring follow-up. Examples of recurrent events includeBMC Medical Research Methodology 2008, 8:35 http://www.biomedcentral.com/1471-2288/8/35falls, fractures, certain cancers, infections, chronic diseaseexacerbations, and hospitalizations [2-7]. Through a trial,clinical researchers attempt to determine whether thestudy outcome occurs more frequently in the interventiongroup than in the control group. In such trials cliniciansare interested in a variety of questions, such as "Howmany events does the intervention prevent, on average,compared to the control?"; "Does the interventiondecrease the event rate over the study period compared tothe control?"; "What is the effect of intervention on therate of subsequent event among those who experiencedthe preceding event?"; and "What is the protective effect ofintervention on the rate of higher-order events comparedto the control?"Although a variety of statistical methods are available forthe analysis of recurrent events, reporting the effect of anintervention on outcomes that recur is an area thatremains poorly understood in clinical research [8,9].Appropriate statistical techniques are not always used toanalyze RCTs on recurrent falls [9]. Extensive work involv-ing simulation studies based on varying event processesand case studies have compared recurrent event methodsto illustrate their strengths and weaknesses [10-13]. Suchmethods include the gamma-Poisson model, and severalextensions of the Cox proportional hazards model,including the independent-increment, marginal, and con-ditional models [14-20].The purpose of this paper is to outline guidelines forreporting results from a trial of treatment that prevents arecurrent event. As an example, we are using the rationaleof a randomized trial on falls prevention. Falls are themost common cause of injury among elderly people. Onein three persons over the age of 65 falls at least once eachyear and this proportion increases to one in two peopleover the age of 80 [21,22]. Almost half of those who fallexperience the event recurrently [23,24]. The goal of RCTsis to reduce the occurrence of falls with specific interven-tions strategies such as multi-factorial intervention,strength and balance retraining, medication rationaliza-tion and expedited cataract surgery.In the Methods section we review the utility of regressionmodels for the rate of a recurrent event by articulating theassociated study questions, presenting the risk sets, andinterpreting the regression coefficients. Based on a singledata set produced by simulation, we report and contrastresults from statistical methods for evaluating treatmenteffect from an RCT with a recurrent outcome in the Resultssection. Finally, we summarize our guidelines for report-ing evidence from RCTs on recurrent events.MethodsIn this section, we relate study questions of interest inRCTs to methods for modelling recurrent event data.Recurrent event models were developed to account forpotential dependence among observations within a sub-ject. One approach allows for unobserved heterogeneitywhich is unmeasured, intraclass correlation where sub-jects have constant but unequal probabilities of experienc-ing the event [25]. Three other models, which weredeveloped for the analysis of continuous time recurrentevent data, are extensions of the Cox proportional hazardsmodel. They first fit a Cox model that ignores dependenceand then use the empirical sandwich estimator to adjuststandard errors for the parameter estimates [17,18,20].Several authors argued for a conditional approach thatestimates the rate of kth event among those who havealready experienced (k - 1) events [18,26]. This approachaddresses the issue of constant susceptibility in a morenatural way than marginal models [18,27]: while the asso-ciation between event times remains unspecified, theevent-specific rate functions condition on having had pre-vious events.There are substantial differences among the modelsdescribed in this section, but all estimate the effect of fac-tors on the occurrence and time to event while accountingfor the dependence between observations. The methodsthat we review model the rate function, λ(t)-that is, theaverage intensity of a recurrent event at a certain time. Wehighlight differences in the model assumptions, risk sets,and rate ratio interpretation. The data structure requiredto fit each model is shown to illustrate the different risksets, indicating which patients are considered to be at riskfor events at certain times [25,28]. Examples of SAS code(SAS System version 9.1 for Windows, SAS Institute Incor-poration, Cary, NC, USA) to fit each model are also pre-sented.Mean cumulative function"How many events does the intervention prevent, on aver-age, compared to the control?" is one study question in anRCT on recurrent events that could be addressed using themean cumulative function (MCF). The MCF shows thepopulation mean number of recurrent events by certaintimes [29]:MCF(t) = E{N(t)}.where N(t) is a random variable for the number of eventsthat have occurred up to time t. The MCF curve changes asa function of time and its derivative gives the rate func-tion, that isPage 2 of 12(page number not for citation purposes)λ( ) ( ).t ddtt= MCFBMC Medical Research Methodology 2008, 8:35 http://www.biomedcentral.com/1471-2288/8/35The rate and intensity functions quantify different aspectsof the recurrent event process: the intensity is the instan-taneous risk of a recurrent event and the rate is the averageintensity at time t [25,30]λ(t)dt = E[dN(t)],where dN(t) denotes the number of events in a smallinterval [t, t + dt).We interpret the difference in MCFs between the interven-tion and control groups as an indicator of how manyevents the intervention would prevent, on average, by acertain time [31].Gamma-Poisson modelA common study question for an RCT on recurrent eventsis "Does the intervention decrease the event rate over thestudy period compared to the control?", for which thegamma-Poisson model has been used. The gamma-Pois-son model evaluates the relationship between the numberof recurrent events and factors of interest when the datadeviate from the Poisson model [15,16]. This modelallows variation of the event rate among subjects in thesame group according to an unobserved random variable,frailty, which defines how likely a subject is to experiencethe event compared to the average rate [16]. When thefrailty follows a gamma distribution and a time homoge-neous model is assumed then the marginal distribution ofthe total number of events is negative binomial [15].Suppose Ni(t) counts the number of events that haveoccurred up to time t for subject i. Under the time-homo-geneous, gamma-Poisson model, Ni(t) has a Poisson dis-tribution with rate functionλi(t) = μi exp{α0 + βxi}, (1)where μi come from a gamma distribution with densityfunctionIn model 1, α0 is the logarithm of the baseline rate for theevent, μi is the unobserved frailty for subject i, xi is a cov-ariate value for subject i, β is the regression coefficient,and t represents the time from start of observation.The expected value and variance of the frailty random var-iable is 1 and θ, respectively. Subjects with μi greater than1 are considered more "frail" or more likely to experiencethe event at a higher rate; whereas, those with μi less thanCompared to the Poisson model which assumes the meanand variance for the number of events are equal, thegamma-Poisson model has an additional parameterwhich allows for over-dispersion. For a given set of covari-ates, this model assumes the expected number of events ist exp(α0 + βxi) and the variance is t exp(α0 + βxi) + θt2exp(α0 + βxi)2 [32].The rate function of any event for subject i averaged overthe gamma-distribution isSubjects are at risk of an event until they are censored.Suppose xi is a binary indicator of group membership,with value 0 if subject i belongs to the control group and1 if the intervention group. Then, exp  from model 3estimates the common rate ratio of event in the interven-tion group relative to the control. We interpret rate ratiosless than 1 as indicating the overall rate of event, that isthe rate of any event, in the intervention group is 100 [1 -exp ]% lower than in the control.The data structure for this model requires one record foreach subject, regardless of the number events experienced.This record contains the total follow-up time and totalnumber of events per subject. The data structure requiredfor this model is illustrated through an example. Supposesubject 1 in the control group experiences a recurrentevent at day 126, 216, and 314 from study start and is fol-lowed up for 365 days. In addition, subject 2 in the inter-vention group, who was followed for the same period oftime, had events at day 42 and 350. Under the time-homogeneous gamma-Poisson model, the data for thesesubjects are represented as shown in Table 1. In this dataset, pid is the subject identifier, time is the total follow-uptime, nevent is the total number of events experienced, grpis the covariate for group membership, and logtime is thenatural logarithm of time.For these data, SAS can be used to fit a time-homogeneousgamma-Poisson model:PROC GENMOD;f( )/ exp( / )( / ) /.μ μθ μ θθ θ θ=−−1 11 1Γ(2)λ α β θ α βi i it x t x( ) exp{ }[ exp{ }] .= + + + −0 0 11(3)( )β( )βTable 1: Data structure for the time-homogeneous gamma-Poisson modelpid time nevent grp logtime1 365 3 0 5.899Page 3 of 12(page number not for citation purposes)1 are considered to experience the event at a lower rate[16].2 365 2 1 5.899BMC Medical Research Methodology 2008, 8:35 http://www.biomedcentral.com/1471-2288/8/35MODEL nevent = grp/LINK = LOG DIST = NEGBINOFFSET = logtime;RUN;A major limitation of the time-homogeneous gamma-Poisson model is it assumes that the recurrent event rate isconstant over time, which is unlikely to hold in practice.Extensions to this model have been made to relax theindependent increment assumption for recurrent eventsand the specification of the within subject correlationbetween recurrence times. For example, the general frailtymodel assumes that the counting process is a non-homo-geneous Poisson process given the frailty and covariates,where the frailty is not restricted to follow a gamma distri-bution [33]. The proportional mean and rate modelrelaxed the non-homogeneous Poisson assumption forthe counting process and directly models means and rates[17].Independent-increment modelThe study question "Does the intervention decrease theevent rate over the study period compared to the control?"is also addressed by Lin's independent-increment modelfor the rate of recurrent events [17]. Originally this modelwas developed by Andersen and Gill to specify the inten-sity of a counting process with a Cox-type link function[14]. Lin et al. provided a rigorous formalization of themarginal rate model, which relaxes the assumption thatthe event history, Fi(t), can be completely described bytime-dependent covariates, xi(t), that is, [17,30]E[dNi(t)|Fi(t)] = E[dNi(t)|xi(t)].In contrast to Cox's model where subjects are at risk of anevent until its occurrence or they are censored, in the inde-pendent-increment model subjects still remain at riskafter an event occurs. Unlike the gamma-Poisson model,the independent-increment model does not assume therecurrent event rate is constant over time. This modelassumes that the number of events in disjoint time inter-vals are independent [27].Under the independent-increment model, the rate func-tion, λi(t), of any event for subject i isλi(t) = Yi(t)λ0(t) exp{βxi(t)}, (4)whereariate value, which may be time-dependent but may notcontain elements of the event history, for subject i, β is theregression coefficient, and t represents the time from startof observation.From model 4 we observe that both the baseline rate func-tions, λ0, and regression parameters, β, are assumed to becommon across events.Subjects are at risk of the an event until they are censored.Suppose xi is a binary indicator of group membership,with value 0 if subject i belongs to a control group and 1if an intervention group. Then exp  estimates the com-mon rate ratio of event for the intervention group relativeto the control. The rate ratio is assumed to be constantover time and common across recurrent events. We inter-pret rate ratios less than 1 as indicating the overall rate ofevent in the intervention group is 100 [1 - exp ]%lower than in the control. This model has a similar inter-pretation to the gamma-Poisson model except we nolonger require the assumption of time-homogeneity orgamma distributed frailty.Under the independent-increment model, the data forthese subjects use the counting process format, whereeach subject is represented by a set of time intervals andevent indicators. We illustrate these data in Table 2 usingthe example described in the Gamma-Poisson model sub-section. In this data set, pid is the subject identifier, tstartis time of previous event or study start, tstop is time ofevent or censoring, status is an indicator of event, and grpis the covariate for group membership. Subject 1 experi-enced 3 events and then was censored at the end of fol-low-up, so there are 4 corresponding records for thissubject. In contrast, subject 2 experienced 2 events beforebeing censored, so there are only 3 records.The corresponding SAS code to fit an independent-incre-ment model is as follows:PROC PHREG COVM COVS(AGGREGATE);Yif subject  is under observation at time if suiti t( ),,=10 bject  is censored by time i t.⎧⎨⎩( )β( )βTable 2: Data structure for the independent-increment modelpid tstart tstop status grp1 0 126 1 01 126 216 1 01 216 314 1 01 314 365 0 02 0 42 1 12 42 350 1 1Page 4 of 12(page number not for citation purposes)In model 4, Yi is the at risk indicator of event for subject i,λ0(t) is the baseline rate function for the event, xi is a cov-2 350 365 0 1BMC Medical Research Methodology 2008, 8:35 http://www.biomedcentral.com/1471-2288/8/35WHERE (tstart < tstop);MODEL (tstart, tstop) * status(0) = grp/RISKLIMITS;ID pid;RUN;Conditional modelsRCTs on recurrent events provide insight into the studyquestion "What is the effect of intervention on the rate ofsubsequent event among those who experienced the pre-ceding event?", which a condtional model can address.Pepe and Cai proposed the conditional model for the rateof recurrent events, where subjects are not considered tobe at risk for event until all previous events have occurred[18].Under the total, follow-up time conditional model, therate function, λij(t), of the jth event for subject i isλij(t) = Yij(t) λ0j(t) exp{βjxi(t)}, (5)whereFrom model 5 we observe that both the baseline rate func-tions, λ0j(t), and regression parameters, βj, can vary acrossevents. The covariate xi may not contain elements of theevent history.In model 5, t represents the time from start of observation.The conditional model can also be formulated in terms of"gap time", the time from previous event:whereand  is the time of the event just prior to time t.In contrast to the marginal model, subjects are consideredat risk for an event at time t only if the previous eventoccurred before that time and they are still under observa-tion. Suppose xi is a binary indicator of group member-ship, with value 0 if subject i belongs to a control groupand 1 if an intervention group. Then, exp  fromevent from study start in the intervention group relative tothe control, conditional on experiencing the previousevents. The event-specific rate ratio for the jth event frommodel 6 represents the rate of the jth event from the timeof the previous event in the intervention group relative tothe control. We interpret rate ratios less than 1 as indicat-ing that among those who experienced j - 1 events, theintervention reduces the rate of the jth event by 100[1 -exp ]% compared to the control. While the condi-tional model using total follow-up time compares sub-jects who experienced the same number of events andhave the same follow-up from study start, the gap-timeconditional model compares subjects who have experi-enced the same number of events and have the same dura-tion since their previous event.Fitting these conditional models relies on creating theappropriate data sets. These data sets are illustratedthrough the example presented in Gamma-Poisson modelsubsection. Under the conditional model for total follow-up, the data set for these subjects follows the countingprocess format as shown in Table 3. Similar to the inde-pendent-increment model (equation 4), the number ofrecords representing each subject depends on the numberof events experienced. The data structure differs from thatof the independent-increment model since we have a var-iable for the event number.Assuming that the most number of events observed persubject was seven, the corresponding SAS code for fittinga conditional, total follow-up time model is as follows:PROC PHREG;MODEL (tstart, tstop) * status(0) = group1-group7/RISKLIMITS;group1 = grp * (event = 1);group2 = grp * (event = 2);group3 = grp * (event = 3);Yif th event occured by time  and th event ij tj t j( ), ( )=−1 1 has not for subject if otherwise or censored at time it0,  for subject i.⎧⎨⎩λ λ βij N t ij j N t j it T t t T x t( ) ( ) ( )exp{ ( )},( ) ( )− = −− −Y 0(6)Yif th event occured by time  and th event ij tj t j( ), ( )=−1 1 has not for subject if otherwise or censored at time it0,⎧⎨⎩TN t( )−( )β( )β jTable 3: Data structure for the conditional model for total follow-up timepid tstart tstop event status grp1 0 126 1 1 01 126 216 2 1 01 216 314 3 1 01 314 365 4 0 02 0 42 1 1 12 42 350 2 1 1Page 5 of 12(page number not for citation purposes)model 5 estimates the event-specific rate ratio of the jthj2 350 365 3 0 1BMC Medical Research Methodology 2008, 8:35 http://www.biomedcentral.com/1471-2288/8/35group4 = grp * (event = 4);group5 = grp * (event = 5);group6 = grp * (event = 6);group7 = grp * (event = 7);STRATA event;RUN;Under the conditional, gap time model, the data set forthese subjects requires times between adjacent events, asshown in Table 4. Again, the number of records per sub-ject depends on the number of events experienced. Asopposed to time intervals, times between subsequentevents are required.Assuming that the most number of events observed persubject was seven, the corresponding SAS code for fittinga conditional, gap time model is as follows:PROC PHREG;MODEL gaptime * status(0) = group1-group7/RISK-LIMITS;group1 = grp * (event = 1);group2 = grp * (event = 2);group3 = grp * (event = 3);group4 = grp * (event = 4);group5 = grp * (event = 5);group6 = grp * (event = 6);group7 = grp * (event = 7);STRATA event;RUN;In these conditional model data sets, pid is the subjectidentifier, tstart is time of previous event or study start,tstop is time of event or censoring, gaptime is the time toevent from previous event, event is the event number, sta-tus is an indicator of event, and grp is the covariate forgroup membership.Marginal model"What is the protective effect of intervention on the rate ofhigher-order events compared to the control?" is animportant study question to help decide whether to starttreatment. This question is addressed by the marginalmodel, proposed by Wei, Lin and Weissfeld, which allowsfor different effects on each subsequent event [20]. Thismodel treats the ordered event like an unordered compet-ing risk problem [27]. Estimates from the marginal modelhave a practically useful interpretation which allows com-parison between groups at treatment onset [34].Under the marginal model, the rate function, λij(t), of thejth event for subject i isλij(t) = Yij(t)λ0j(t) exp{βjxi(t)}, (7)whereIn model 7, Yij, is the at risk indicator of the jth event forsubject i, λ0j(t) is the baseline rate function for the jthevent, xi is a covariate value, which may be time-depend-ent, for subject i, βj is the regression coefficient for event j,and t represents the time from start of observation. Frommodel 7 we observe that both the baseline rate functions,λ0j, and regression parameters, βj, can vary across events.Subjects are at risk of the jth event until it occurs or theyare censored. Furthermore, subjects are considered to beat risk for the jth event even if they did not yet experiencethe (j - 1)th event. Suppose xi is a binary indicator of groupmembership, with value 0 if subject i belongs to a controlgroup and 1 if an intervention group. Then, exp  esti-mates the average event-number-specific rate ratio of thejth event in the intervention group relative to the control.We interpret rate ratios less than 1 as indicating the tran-sition rate from 0 to j events in the intervention group is100 [1 - exp ]% lower than in the control. The mar-ginal event-number-specific rate ratios indicate whethersubjects in the intervention group will have fewer higher-Yif th event has not occured by time  for subjecij tj t( ),=1 t if otherwise or censored at time  for subject it i0, .⎧⎨⎩( )β j( )β jTable 4: Data structure for the conditional model for gap timepid gaptime event status grp1 126 1 1 01 90 2 1 01 98 3 1 01 51 4 0 02 42 1 1 12 308 2 1 1Page 6 of 12(page number not for citation purposes)order events of a certain number from the time of treat-ment onset [34].2 15 3 0 1BMC Medical Research Methodology 2008, 8:35 http://www.biomedcentral.com/1471-2288/8/35The data structure required for this model is illustratedthrough the example presented in the Gamma-Poissonmodel subsection. We would like to study the effect ofintervention on the first four events. Under the marginalmodel, the data set for these subjects show times of eventfrom study start for all events under study, as shown inTable 5. In this data set, pid is the subject identifier, tstartis time of study start, tstop is time of event or censoring,event is the event number, status is an indicator of event,and grp is the covariate for group membership. Both sub-jects are represented by the same number of records,namely four since we are interested in the first four events.The corresponding SAS code to fit this marginal model isas follows:PROC PHREG COVS(AGGREGATE);MODEL tstop*status(0)=group1-group4/RISKLIMITS;group1 = grp * (event = 1);group2 = grp * (event = 2);group3 = grp * (event = 3);group4 = grp * (event = 4);STRATA event;ID pid;RUN;ResultsUsing available statistical instruments for recurrentevents, we report results from a simple simulation studyof falls prevention to illustrate the utility of the methods.Although each of the models being compared has alreadybeen studied via simulation, we contrast reporting resultsin the context of an RCT based on a single data set. Theinclude the common rate ratio, which compares the aver-age rate of event in the intervention group to the control,the conditional event-specific rate ratios, which summa-rize the effect of intervention on a specific event condi-tional on experiencing previous events, and the marginalevent-number-specific rate ratios, which summarize theintervention effect on the transition rate of experiencing acertain number of events from study start. In addition, wereport the event rate, a measure of the average number ofevent accrued per person-time, and the mean cumulativefunction (MCF), a measure of the average number ofevents experienced per subject within a certain time.We simulated recurrent falls in two groups, control andintervention, using Matlab Version 7 software (see Addi-tional file 1). Each group had 250 subjects, and all sub-jects were followed for 365 days. Fall rates were based onthose observed in an RCT [35]. Times between falls wereassumed to follow an exponential distribution with fallsrates specified for each fall. In the control group the fallrates for all falls were held constant at 7.7 falls per 1000person-days. In the intervention group the fall rate was 5.3falls per 1000 person-days for the first fall, and changed to3.3 for all subsequent falls. Dependence within subjectswas modelled using a gamma frailty distribution withdensity function given in equation 2 and variance θ =0.10. We report the effect of the first 4 falls only sincehigher-order event-specific estimates are unreliable whenthere are only a few subjects with a large number of falls[25,27].Event ratesAfter 1 year, the control group had 675 falls, nearly doublethat of the intervention group with 373 falls. The total fol-low-up time in each group was 91,250 person-days. Theaverage observed fall rates in the control and interventiongroups were 7.4 (95%CI 6.8–8.0) and 4.1 (95%CI 3.7–4.5) falls per 1000 person-days, respectively. Compared tothe control group, the rate of falls in the intervention wasalmost halved, a crude approximation of the anticipatedeffect size. This effect size can be used to design RCTs onrecurrent events, specifically for determining the numberof subjects.Mean cumulative function, MCFFigure 1 shows the MCF by group, estimated by a non-par-ametric estimator [36]:where ej is the number of events at time tj, nj-1 is theMCFn( ) ,{ | }te jn jj t tj=−≤∑1Table 5: Data structure for the marginal modelpid tstart tstop event status grp1 0 126 1 1 01 0 216 2 1 01 0 314 3 1 01 0 365 4 0 02 0 42 1 1 12 0 350 2 1 12 0 365 3 0 12 0 365 4 0 1Page 7 of 12(page number not for citation purposes)measures discussed are the rate ratios from the recurrentevent models described in the Methods section. Thesenumber of subjects at risk just beyond time tj-1, and jindexes the observed event times. A subject is at risk ofBMC Medical Research Methodology 2008, 8:35 http://www.biomedcentral.com/1471-2288/8/35event until the end of follow-up. At one year of follow-up,an average of 2.7 and 1.5 falls per subject were experi-enced in the control and intervention group, respectively.Both MCFs were approximately linear, which indicatesthat the rate of falls is relatively constant in each group[31,36]. The control group experienced more falls andhad a higher fall rate than the intervention group. Onaverage, the control group experienced 1 more additionalfall by 301 days (Figure 1). From the MCF difference, weobserved that 1.2 falls were prevented per year on averagefor each subject.Common rate ratiosThe time-homogeneous gamma-Poisson and independ-ent-increment gave similar common rate ratio estimatesof 0.55 (95% CI 0.48–0.63) and 0.55 (95% CI 0.48–0.62), respectively (Table 6). The gamma-Poisson andindependent-increment models both infer that the rate ofany fall in the intervention group is 45% lower in theintervention group than control. In practice the assump-tion of a constant recurrent event rate over time may notThese common rate ratios indicate that the interventionhad an impact on the risk of falls; however, it does notinform whether the effect changes for subsequent events.Conditional event-specific rate ratiosThe majority of the control group experienced two fallswithin 1 year of follow-up: 228, 180, 122, and 77 subjectshad fall 1, 2, 3, and 4, respectively. The number of falls inthe intervention group was lower: 202, 104, 45, and 18subjects had fall 1 to 4, respectively (Table 7). Higher-order events, up to 7 falls, were experienced by 38 subjectsin the control group; whereas, in the intervention group,only 4 subjects had the highest-order event of 5 falls. Inthe conditional model, the risk set for a subsequent fallEstimated mean cumulative function (MCF) of falls by group (upper panel), their difference (lower panel), and 95% confidence intervalsFigure 1Estimated mean cumulative function (MCF) of falls by group (upper panel), their difference (lower panel), and 95% confidence intervals.0 100 200 300 4000123ControlInterventionTIME  (DAYS)NUMBER  OF  FALLS0 100 200 300 40000.40.81.21.6TIME  (DAYS)MCF  DIFFERENCETable 6: Effect of intervention on recurrent falls, as measured by common rate ratios and 95% confidence intervalsEffect Gamma-Poisson Independent-incrementControl 1.00 1.00Intervention 0.55 (0.48, 0.63) 0.55 (0.48, 0.62)Page 8 of 12(page number not for citation purposes)hold, so the independent-increment model is preferredover the time-homogeneous gamma-Poisson model.BMC Medical Research Methodology 2008, 8:35 http://www.biomedcentral.com/1471-2288/8/35consisted of only subjects who experienced the previousfalls, and total follow-up time decreased for later events.The crude rate ratios indicate a similar intervention effecton falls 2 and 3.As expected, the rate ratios for the first fall from the con-ditional models give identical estimates, 0.68 (95% CI0.57–0.83), since the total follow-up time and gap time tofirst falls refer to the same period (Table 8). For subse-quent falls, the fall-specific rate ratios from the condi-tional models overlap and remain relatively constantranging from 0.46 (95% CI 0.36–0.59) to 0.53 (95% CI0.31–0.88). The rate ratio for fall 5, 0.38 (95% CI 0.13–1.07), may be unreliable due to the number at risk for thisevent, and effects could not be estimated for falls 6 or 7.Among subjects who experienced preceeding falls, theeffect of intervention on the rate of the first four recurrentfalls did not differ (Wald χ2 test = 6.6, df = 3, p = 0.08 fortotal follow-up time model, and Wald χ2 test = 6.7, df = 3,p = 0.08 for gap-time model).For recurrent falls, the rate ratios from the conditional,total follow-up time model indicate that conditional onexperiencing the previous fall, the rate of second, thirdand fourth falls from study start are 54%, 47% and 50%lower in intervention than control. The rates of falls fromthe time of previous fall are 54%, 47%, and 47% lower inintervention than control, as estimated from the condi-between the groups. The conditional fall-specific rateratios evaluate how the intervention affected the rate ofkth fall among those who experienced k - 1 falls.For both the conditional total follow-up time model andconditional gap time model, subjects are considered to beat risk for an event only if the previous event occurred, sosubjects at risk may not consist of all who were intiallyrandomized. The number of subjects at risk for subse-quent events should be reported to allow evaluation ofhow different the treatment groups are from the start ofthe study (Table 7).Marginal event-number-specific rate ratiosIn the marginal model, all subjects were considered to beat risk for the 1st, 2nd, 3rd, 4th, and higher-order fallsregardless of experiencing previous events (Table 7). Sub-jects are at risk for a specific fall until its occurrence or cen-soring, so the total follow-up time accumulates oversubsequent falls. The crude rate ratios decrease with fallevents.The fall-number-specific rate ratios decrease from 0.68(95% CI 0.57–0.83) for fall 1 to 0.20 (95% CI 0.12–0.34)for fall 4 (Table 8). For higher-order events, the rate ratiofor fall 5 was 0.10 (95% CI 0.03–0.27) and could not beestimated for falls 6 or 7. The marginal model indicatedthat there was a difference in the average effect of interven-Table 7: Fall-specific characteristics for total events, number of subjects at risk, total follow-up in days, and crude rate ratios, as indicated by the marginal and conditional total time modelsConditional modelControl InterventionEvent # events # at risk follow-up rate* # events # at risk follow-up rate* crude RR†fall 1 228 250 34,355 6.64 202 250 44,726 4.52 0.68fall 2 180 228 24,361 7.39 104 202 30,264 3.44 0.47fall 3 122 180 14,641 8.33 45 104 10,500 4.29 0.51fall 4 77 122 9,673 7.96 18 55 4,301 4.19 0.53Marginal modelControl InterventionEvent # events # at risk follow-up rate* # events # at risk follow-up rate* crude RR†fall 1 228 250 34,355 6.64 202 250 44,726 4.52 0.68fall 2 180 250 58,716 3.07 104 250 74,990 1.39 0.45fall 3 122 250 73,357 1.66 45 250 85,490 0.53 0.32fall 4 77 250 83,030 0.93 18 250 89,791 0.20 0.22* fall rate measured per 1000 person-days† RR = rate ratioPage 9 of 12(page number not for citation purposes)tional, gap time model. The conditional models provideevidence of the constant difference in recurrent fall ratestion on the first four falls (Wald χ2 test = 32.2, df = 3, p <0.0001). Rate ratios based on the marginal model indi-BMC Medical Research Methodology 2008, 8:35 http://www.biomedcentral.com/1471-2288/8/35cated that, on average, the transition rate from zero falls atthe start of treatment to one, two, three and four falls were32%, 58%, 70% and 80% lower, respectively, in the inter-vention group than the control. These rate ratios do notimply that the effect of intervention increased with recur-rent falls. Rather, the marginal fall-number-specific rateratios indicate that subjects in the intervention group willhave fewer events overall.Given an objective of an RCT is to compare groups whichare similar in all aspects except for the treatment of inter-est, it is appropriate to use the marginal model since allsubjects are considered to be at risk for each number-spe-cific event from study start. In contrast, the groups beingcompared to evaluate the effect of subsequent events inthe conditional models may not consist of all subjects ini-tially randomized.DiscussionRecurrent events arise in many contexts, such as falls inseniors considered in this paper. In evidence-based medi-cine there is increasing need for guidelines on what toreport in the analysis of recurrent events [8]. In the Resultssection we have outlined briefly statistical methods forevaluation of treatment effect from an RCT with a recur-rent outcome. These should allow clinical researchers toreport appropriate measures from an RCT for understand-ing the effect of intervention on the occurrence of a recur-rent event.We used a simulation study to relate an event process andresults from analyses of the gamma-Poisson, independ-ent-increment, conditional, and marginal Cox models[15-18,20]. We showed that each model has differentstudy questions, assumptions, risk sets, and rate ratiointerpretation, and so inferences should consider theappropriateness of the model for the RCT. The gamma-Poisson and independent-increment models compare thecommon event rates between groups, with the assump-tion of independence of the number of events across timerecurrent events, and conditions on having had previousevents. In contrast, the marginal model treats the events asunordered, and all subjects are at risk for any event. In dif-ferent trials the outcomes of interest and validity ofassumptions will differ. Our guidelines for reportingresults from an RCT involving a recurrent event suggeststatistical methods which correspond to the objectives ofthe trial, such as addressing the study question of interest,assessing comparable groups and estimating effect size.First, the average event rate by intervention group is ameasure of the average number of events accrued per per-son-time. These event rates serve an important role indetermining sample size and follow-up time for thedesign of future RCTs involving recurrent events [37]. Sec-ond, the MCF by intervention group provides a measureof the average number of events experienced per subjectwithin a certain time. The MCF allows us to determinehow many events per subject the intervention would pre-vent, on average, compared to the control group [31].Third, the common rate ratio, as measured by the gamma-Poisson and independent-increment models, quantifiesthe average rate of event in the intervention group relativeto the control group. This rate ratio provides an estimateof the common effect size, thereby indicating whether theintervention had an impact on the event occurrence.Fourth, conditional event-specific rate ratios, which quan-tify the rate of the kth event in the intervention relative tothe control, conditional on experiencing precedingevents, should be reported. These rate ratios allow us toevaluate how the effect of intervention changes, if at all,on subsequent events. Lastly, we suggest reporting themarginal event-number-specific rate ratios, which repre-sent the rate of transitioning to higher-order events fromthe start of treatment in the intervention group relative tothe control group. These rate ratios allow us to evaluatethe overall protective effect of intervention. For methodsused in the assessment of goodness of fit for each modelwe refer the reader to the corresponding papers [17,27].It has been argued that the average event rate might haveTable 8: Effect of intervention on recurrent falls, as measured by fall-specific rate ratios and 95% confidence intervalsEffect Conditional, total follow-up time* Condtional, gap time† Marginal‡Control 1.00 1.00 1.00Interventionfall 1 0.68 (0.57, 0.83) 0.68 (0.57, 0.83) 0.68 (0.57, 0.83)fall 2 0.46 (0.36, 0.59) 0.46 (0.36, 0.59) 0.42 (0.33, 0.54)fall 3 0.53 (0.38, 0.75) 0.53 (0.38, 0.75) 0.30 (0.21, 0.42)fall 4 0.50 (0.30, 0.85) 0.53 (0.31, 0.88) 0.20 (0.12, 0.34)*effects on recurrent falls were not different (χ2 = 6.6, df = 3, p = 0.08)†effects on recurrent falls were not different (χ2 = 6.7, df = 3, p = 0.08)‡effects on recurrent falls were different (χ2 = 32.2, df = 3, p < 0.0001)Page 10 of 12(page number not for citation purposes)intervals being required in the latter, but not the former.The conditional model distinguishes between first andlittle relevance in the context of recurrent events becausethis measure does not acknowledge dependence betweenBMC Medical Research Methodology 2008, 8:35 http://www.biomedcentral.com/1471-2288/8/35events experienced by a subject [38]. However, by apply-ing appropriate statistical methods for recurrent events wecan make valid inferences on rates. Extensive simulationstudies based on varying event processes and case studieshave compared recurrent event methods to determinetheir strengths and weaknesses [10-13].Regression methods for the analysis of recurrent events isnot limited to modelling the rate of event. The meannumber of recurrences can be modelled using semi-para-metric Cox models and parametric models [17,39]. Pro-portional rates and proportional means models areequivalent when the rate only depends on covariates thatdo not directly impact the occurrence of event, namelyexternal covariates [17,40]. Regression models for theintensity function, which condition on event history, arealso available [14,19]. However, in RCTs treatment mayaffect event history, so conditioning on the event historymay underestimate the treatment effect [41].ConclusionOur guidelines for reporting results from an RCT involv-ing a recurrent event suggest that the study question andthe objectives of the trial, such as assessing comparablegroups and estimating effect size, should determine thestatistical methods. Guidelines for reporting results froman RCT involving a recurrent event should allow clinicalresearchers to report appropriate measures for under-standing the effect of intervention on the occurrence of arecurrent event.Competing interestsThe authors declare that they have no competing interests.Authors' contributionsStudy concept and design: BGS, LK. Analysis and interpre-tation: LK, BGS, MGD. Drafting of the manuscript: LK,BGS, MGD.Additional materialAcknowledgementsWe are grateful to the reviewers for their insightful comments.References1. Moher D, Schulz KF, Altman DG: The CONSORT statement:2. 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Campbell AJ, Borrie MJ, Spears GF: Risk factors for falls in a com-munity-based prospective study of people 70 years andolder.  J Gerontol 1989, 44:M112-M117.25. Ezell ME, Land KG, Cohen LE: Modeling multiple failure timedata: A survey of variance-corrected proportional hazardmodels with empirical applications to arrest data.  SociologicalMethodology 2003, 33:111-167.Additional file 1Matlab code to simulate recurrent falls data used in Results sectionClick here for file[http://www.biomedcentral.com/content/supplementary/1471-2288-8-35-S1.txt]Page 11 of 12(page number not for citation purposes)revised recommendations for improving the quality ofreports of parallel-group randomised trials.  Lancet 2001,357:1191-1194.26. Klen JP, Goel PK, (Eds): Survival analysis: state of the art, Kluwer 1992chap. Frailty models for multiple event times .Publish with BioMed Central   and  every scientist can read your work free of charge"BioMed Central will be the most significant development for disseminating the results of biomedical research in our lifetime."Sir Paul Nurse, Cancer Research UKYour research papers will be:available free of charge to the entire biomedical communitypeer reviewed and published immediately upon acceptancecited in PubMed and archived on PubMed Central BMC Medical Research Methodology 2008, 8:35 http://www.biomedcentral.com/1471-2288/8/3527. Therneau TM, Grambsch PM: Modeling Survival Data: Extending the CoxModel Springer; 2000. 28. Kelly PJ, L-Y LL: Survival analysis for recurrent event data: anapplication to childhood infectious diseases.  Statistics in Medi-cine 2000, 19:13-33.29. Nelson WB: Recurrent events data analysis for product repairs, diseaserecurrences, and other applications 1st edition. ASA-SIAM; 2003. 30. 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Nelson WB: Confidence limits for recurrence data: applied tocost or number of repairs.  Technometrics 1995, 37:147-157.37. Cook RJ: The design and analysis of randomized trials withrecurrent events.  Statistics in Medicine 1995, 14:2081-2098.38. Windeler J, Lange S: Events per person year-a dubious concept.British Medical Journal 1995, 310:454-456.39. Lawless J: Introductory overview lecture.  2005.40. Kalbfleisch JD, Prentice RL: The statistical analysis of failure time data 1stedition. John Wiley & Sons; 1980. 41. Schaubel DE, Zeng D, Cai J: A semiparametric additive ratesmodel for recurrent event data.  Lifetime Data Analysis 2006,12:389-406.Pre-publication historyThe pre-publication history for this paper can be accessedhere:http://www.biomedcentral.com/1471-2288/8/35/prepubyours — you keep the copyrightSubmit your manuscript here:http://www.biomedcentral.com/info/publishing_adv.aspBioMedcentralPage 12 of 12(page number not for citation purposes)

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