exploratory method for summarizing concomitant medication data – the mean cumulative function

PHARMACEUTICAL STATISTICS

Pharmaceut. Statist. 9: 331–336 (2010)

Published online 10 August 2009 in Wiley Online Library

(wileyonlinelibrary.com) DOI: 10.1002/pst.395 RESEARCH ARTICLES

Exploratory method for summarizing

concomitant medication data – the

mean cumulative function

Chris Barker�,y

Statistical Planning and Analysis Services, Inc., San Carlos, CA, USA

Concomitant Medications are medications used by patients in a clinical trial, other than the

investigational drug. These data are routinely collected in clinical trials. The data are usually

collected in a longitudinal manner, for the duration of patients’ participation in the trial. The routine

summaries of this data are incidence-type, describing whether or not a medication was ever

administered during the study. The longitudinal aspect of the data is essentially ignored. The aim of

this article is to suggest exploratory methods for graphically displaying the longitudinal features of

the data using a well-established estimator called the ‘mean cumulative function’. This estimator

permits summary and a graphical display of the data, and preparation of some statistical tests to

compare between groups. This estimator may also incorporate information on censoring of patient

data. Copyright r 2009 John Wiley & Sons, Ltd.

Keywords: mean cumulative function; SPLIDA; concomitant medications

INTRODUCTION

This note presents a new application of an existingmethodology for exploring and summarizing thelongitudinal features of concomitant medicationdata collected in clinical trials. Two differentlongitudinal estimators with corresponding gra-phical displays are presented; the author recom-mends both. The first is a simple cumulativesummary and the second is the Mean Cumulative

Function (MCF). The MCF has multiple func-tions: (1) It estimates the cumulative number ofadministrations of concomitant medications; (2) Itaccounts for varying length of follow-up due topatient drop outs by censoring those data; (3) Itallows computation of pointwise confidencebands; (4) It also allows the difference betweentreatment-specific MCF estimates in a randomizedtrial to be computed along with a confidenceinterval as a way to assess statistical significance oflongitudinal changes.

The statistical tools and graphical displaypresented here are useful for the exploratoryanalysis of concomitant medication data in ayE-mail: [email protected]

*Correspondence to: Chris Barker, Chris Barker StatisticalPlanning and Analysis Services, Inc., 287 Chestnut Street,San Carlos, CA 94070, U.S.A.

Copyright r 2009 John Wiley & Sons, Ltd.

clinical trial. Additional suggested uses may be forcleaning and monitoring the accumulating con-comitant medication data during an on-goingclinical trial.

Concomitant medication data are routinelycollected in clinical trials. The collection ofconcomitant medication may occur throughoutthe time a patient participates in the clinical trial,and captures all medications used by the patientother than study drug. These medications may beself-reported by the patient, or recorded based onprescriptions from the study investigator. Inaddition to recording the verbatim name of themedication, the date, and possibly time themedication was started, and/or stopped, the dose,dose regimen, or changes in dose or regimen,possibly whether the medication was used to treatan adverse event may also be captured. In clinicaltrials of serious chronic illnesses, large amounts ofconcomitant medication data may be collected.This creates challenges to evaluating the role ofconcomitant medications on patient outcomes aswell as issues of identifying and correctingdiscrepancies in concomitant medication data.

A common method of summarizing concomitantmedication data is mapping the verbatim termsusing a thesaurus such as WhoDrug. A commontabular summary of the data presents the incidence,that is, whether or not the medication was everused during a study, similar to the summary of theincidence of adverse events. The simple incidencesummary style does not permit a data reviewer toobserve longitudinal patterns over time in theconcomitant medications. A summary of long-itudinal patterns should also account for patientdrop out due to censoring or death for clinical trialsinvolving chronic or life-threatening diseases.

In a clinical trial with a short duration, in whichall patients are followed for the same fixed periodof time after randomization, concomitant medica-tion data could be summarized using the cumula-tive number of administrations of the medicationduring the study. This could be plotted versus theday post-randomization, and by treatment arm.The cumulative number may also be divided by thenumber of patients enrolled, to produce aninterpretable ‘per patient’ value for the number

of concomitant medications. A bootstrap con-fidence interval could be prepared to provide amethod for making comparisons between groups.This simple cumulative estimate, without thebootstrap, is easy to prepare because it does notrequire specialized statistical software.

METHODS

For this note, concomitant medication and follow-up data are obtained from a randomized double-blind clinical trial in a post-surgical hospitalsetting, and the treatments groups are labeled as‘A’ and ‘B’. Patients were randomized to one ofthe treatments and followed for 30 days or untildeath. Study day is computed as day sincerandomization. The MCF is calculated andassociated graphs are prepared using the SPLUSV7 package ‘SPLIDA’ [1] (Splus for Life DataAnalysis). The concomitant medication data arecombined with information on death or the end ofthe study and summarized using the MCF. Anestimator of the difference between the MCF ofthe two treatment groups and graphical display arealso presented.

Several important simplifications were made inthis note for demonstrating this exploratory tooland these can be relaxed. The data in this exampleare from a completed clinical trial. The data aretreated as if there were only a single type ofconcomitant medication and all medication re-ports are used in the exploratory analysis, includ-ing, for example, reports of dose changes. Eachmedication report is treated as a new administra-tion. Ties in date–time of administration werebroken by adding a value drawn from a uniformdistribution and representing 1min within a 24 hday. No attempt was made to re-evaluate the datafor inconsistencies, for example, to determinewhether the concomitant medication dosing in-formation for a report (e.g. once, twice, or threetimes a day, etc.) is consistent with the number ofrecords for that medication in the data set.

The reader is referred Nelson’s book [2] forexamples of the calculations for the statistics used

Copyright r 2009 John Wiley & Sons, Ltd. Pharmaceut. Statist. 9: 331–336 (2010)DOI: 10.1002/pst

332 C. Barker

Figure 1. Simple summary. Simple cumulative summary by individual concomitant medications.

Exploratory method for summarizing concomitant medication data 333


in this note and presented in an elegant and simpleExcel spreadsheet style; the use of formulae arelimited. The estimator of the Mean CumulativeFunction at time t is denoted as M�(t). Thecalculations for the MCF variance estimatesand confidence intervals are computed usingthe methods described in Nelson [2] and aredenoted as

�Y � Kcðs2t =NÞ1=2 ¼M�ðtÞ � Kcfv½M�ðtÞ�g1=2

where Kc is the (1001C)/2 standard normalpercentile. An expression for the variance estimateof M�(t) is provided in the paper by Nelson [3].This estimate is valid for uncensored observations.Nelson [3] derives the interval when the datainclude censored observations, using the followingassumptions, (1) the population model is apopulation of uncensored cumulative functions,(2) these functions extend to any time of interest,(3) the distribution of the cumulative is assumed tohave a finite mean, (4) the Mc(t) is a continuousfunction, and (5) this function has a derivativem(t)5 dM(t)/dt where m(t) is the populationmean rate, (6) the sample trajectories are asimple random sample from some population,(7) the censoring ages are assumed to be given,(8) the trajectories are assumed to be independentof their censoring ages, and (9) the times ofrecurrences and ends of history are known exactlyand are distinct points on a continuous timescale. The estimate of the difference betweentwo MCF estimates and the confidence intervaluses the methodology of Doganaksoy andNelson [4].

The formula for the pointwise confidence limitsfor the difference are denoted

½M�1ðtÞ �M�

2ðtÞ� � ZcfV ½M�1 ðtÞ� þ V ½M�

2ðtÞ�g1=2

RESULTS

The data are 1330 concomitant medication reportsfrom 141 patients receiving Treatment A and 1548concomitant medication reports for 138 patientsreceiving Treatment B. The patients are initially inthe hospital; follow-up continues after discharge.

All concomitant medication data reported afterrandomization are used for these exploratoryanalyses.

The following sequence of steps are recom-mended by the author. Figure 1(a) is a simplecumulative summary by treatment group, countingevery report of medication administration. It iseasily prepared and may be a useful first step in anexploratory analysis because it does not requirespecial statistical software. It may also be useful inthe data cleaning and data monitoring stage of aclinical trial. A useful, additional exploratoryanalysis described below involves ‘patient trajec-tories’ – here defined as the simple cumulativesummary of concomitant medication administra-tion for an individual patient. Despite the fact thatthis simple methodology does not account forpatient drop out or censoring, it may provideinsights into the data. The plot suggests thatconcomitant medication use was higher in Treat-ment A. Figure 1(b) is a simple cumulativesummary by a subset of the individual medications,to explore if one or more medications may havebeen used more in Treatment A than in B. Theseplots suggest that Medication 1 may have a greaterincrease than other medications. The simplecumulative is not standardized in any way bythe number of patients, and the cumulativenumber of medications administered may notbe an easily understandable number. Also there

Figure 2. MCF estimates for concomitant medications

treatment A and B and 95% confidence limits.


334 C. Barker

are no statistical tests associated with this parti-cular method of counting.

The next step is the estimation of the MCFthat accounts for patient drop out and censoring.The MCF and the associated 95% confidencebands, estimated in SPLIDA, are graphicallydisplayed for the two treatments in Figure 2,again, counting every report of administration ofmedication. The interpretation of Figure 2,for example at Day 20, suggests that the Treat-ment A patients had a mean cumulative adminis-tration of approximately 11 medications versus 9in Treatment B.

Figure 3 presents the summary of the differencebetween the two MCF functions with a 95%confidence interval. The cumulative number ofconcomitant medications is higher on Treatment Athan Treatment B throughout the study period.For a period of time, beginning approximately onDay 11, the confidence interval excludes zero,which suggests a statistically significant increasefor Treatment A. Statistical significance, in thisexploratory analysis must be viewed with caution,particularly in light of the simplifying assump-tions. Further analyses, for example, explorationof individual medications, may be appropriate todetermine the reasons for the increase in Treat-ment A versus B.

Underlying the MCF are the individual patientcumulative summary, referred to as ‘trajectories’.Figure 4(a, b) present a simple summary calcula-tion within each treatment group, cumulate themedications for each patient, and displaysthe ‘trajectory’. The trajectory plots for eachpatient may be difficult to distinguish in thismid-size clinical trial. A small section of the plot ora small number of patient’s data could bedisplayed as ‘exploded’ in order to improvereadability. In addition, the patient trajectoryplot, possibly specialized to particular medicationscould be prepared during an ongoing trialand possibly used for data cleaning in additionto the simple cumulative summaries describedabove.

Figure 3. Difference of mean cumulative function and

95% confidence limits.

Figure 4. (a) Treatment A individual patient trajectories (b) Treatment B individual patient trajectories.

Exploratory method for summarizing concomitant medication data 335


A key observation of these patient level trajec-tory plots is that the patients vary in terms of thenumber of days they are administered medications,due in part to differences in follow-up.

DISCUSSION

The simple cumulative summary and some corres-ponding summaries at the patient level or at themedication preferred-term level may be useful forexploratory graphical analysis of concomitantmedication data and for checking for unusual dataduring data cleaning of an on-going clinical trial.The simple summary does not require specializedstatistical software. In a prospectively designed andrandomized clinical trial, the MCF may be usefulfor extending the analyses of longitudinal con-comitant medication data, for example by prepar-ing statistical tests and unbiased estimates ofdifferences between treatments in the cumulativeadministration of concomitant medications.

ACKNOWLEDGEMENTS

The author thanks the anonymous reviewerswho provided very helpful suggestions. The authoralso thanks several colleagues for helpful discus-sions and comments during the preparation ofthis manuscript, in particular, Mei Cheng, AnnOlmsted, Robert Hoop and Whedy Wang.

REFERENCES

1. SPLIDA. Available from William Meekers website.http://www.public.iastate.edu/�splida.

2. Nelson W. Recurrent events data analysis for productrepairs, disease recurrences, and other applications.SIAM Press: Philadelphia, 2003.

3. Nelson W. Confidence limits for recurrence data,applied to cost or number of product repairs.Technometrics 1995; 37:147–157.

4. Doganaksoy N, Nelson W. A method to comparetwo samples of recurrence data. Lifetime DataAnalysis 1998; 4:5–63.


336 C. Barker

exploratory method for summarizing concomitant medication data – the mean cumulative function

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