We consider a two-group randomized clinical trial, where mortality affects the assessment of a follow-up continuous outcome. Using the worst-rank composite endpoint, we develop a weighted Wilcoxon–Mann–Whitney test statistic to analyze the data. We determine the optimal weights for the Wilcoxon–Mann–Whitney test statistic that maximize its power. We derive a formula for its power and demonstrate its accuracy in simulations. Finally, we apply the method to data from an acute ischemic stroke clinical trial of normobaric oxygen therapy.
SinghalAB. A review of oxygen therapy in ischemic stroke. Neurol Res2007; 29: 173–183.
3.
LittleRJRubinDB. Statistical analysis with missing data, Hoboken, New Jersey: Wiley, 2002.
4.
LachinJ. Worst-rank score analysis with informatively missing observations in clinical trials. Control Clin Trials1999; 20: 408–422.
5.
McMahonRHarrellFJr. Power calculation for clinical trials when the outcome is a composite ranking of survival and a nonfatal outcome. Control Clin Trials2000; 21: 305–312.
6.
MatsouakaRABetenskyRA. Power and sample size calculations for the Wilcoxon–Mann–Whitney test in the presence of death-censored observations. Stat Med2015; 34: 406–431.
7.
FelkerGMMaiselAS. A global rank end point for clinical trials in acute heart failure. Circulation2010; 3: 643–646.
8.
FollmannDWittesJCutlerJA. The use of subjective rankings in clinical trials with an application to cardiovascular disease. Stat Med1992; 11: 427–437.
9.
BakalJAWesterhoutCMArmstrongPW. Impact of weighted composite compared to traditional composite endpoints for the design of randomized controlled trials. Stat Med Med Res2012; 24: 980–988.
10.
HallstromALitwinPDouglas WeaverW. A method of assigning scores to the components of a composite outcome: an example from the MITI trial. Control Clin Trials1992; 13: 148–155.
11.
NeatonJGrayGZuckermanBet al.Key issues in end point selection for heart failure trials: composite end points. J Cardiac Fail2005; 11: 567–575.
12.
CaliffRDeMetsD. Principles from clinical trials relevant to clinical practice: part I. Circulation2002; 106: 1015–1015.
13.
BraunwaldECannonCMcCabeC. An approach to evaluating thrombolytic therapy in acute myocardial infarction. The ‘unsatisfactory outcome’ end point. Circulation1992; 86: 683–683.
14.
MoyéL. Multiple analyses in clinical trials: fundamentals for investigators, New York City, New York: Springer Verlag, 2003.
15.
HuangPTilleyBCWoolsonRFet al.Adjusting O’Brien’s test to control type i error for the generalized nonparametric behrens–fisher problem. Biometrics2005; 61: 532–539.
16.
HäberleLPfahlbergAGefellerO. Assessment of multiple ordinal endpoints. Biometrical J2009; 51: 217–226.
17.
O’BrienPC. Procedures for comparing samples with multiple endpoints. Biometrics1984; 40: 1079–1087.
FinkelsteinDSchoenfeldD. Combining mortality and longitudinal measures in clinical trials. Stat Med1999; 18: 1341–1354.
20.
MoyéLDavisBHawkinsC. Analysis of a clinical trial involving a combined mortality and adherence dependent interval censored endpoint. Stat Med1992; 11: 1705–1717.
21.
MoyéLALaiDJingKet al.Combining censored and uncensored data in a u-statistic: design and sample size implications for cell therapy research. Int J Biostat2011; 7: 1–29.
22.
SampsonUKMetcalfeCPfefferMAet al.Composite outcomes: weighting component events according to severity assisted interpretation but reduced statistical power. J Clin Epidemiol2010; 63: 1156–1158.
23.
AhmadYNijjerSCookCMet al.A new method of applying randomised control study data to the individual patient: a novel quantitative patient-centred approach to interpreting composite end points. Int J Cardiol2015; 195: 216–224.
24.
WilsonRFBergerAK. Are all end points created equal? The case for weighting. J Am Coll Cardiol2011; 57: 546–548.
25.
ArmstrongPWWesterhoutCMVan de WerfFet al.Refining clinical trial composite outcomes: An application to the assessment of the safety and efficacy of a new thrombolytic–3 (assent-3) trial. Am Heart J2011; 161: 848–854.
26.
MinasGRigatFNicholsTEet al.A hybrid procedure for detecting global treatment effects in multivariate clinical trials: theory and applications to fMRI studies. Stat Med2012; 31: 253–268.
27.
FisherLD. Self-designing clinical trials. Stat Med1998; 17: 1551–1562.
28.
RamchandaniRSchoenfeldDAFinkelsteinDM. Global rank tests for multiple, possibly censored, outcomes. Biometrics2016; 72: s1–s10.
29.
LachinJMBebuI. Application of the wei–lachin multivariate one-directional test to multiple event-time outcomes. ClinTrials2015; 12: 627–633.
30.
SamsonK. News from the AAN annual meeting: why a trial of normobaric oxygen in acute ischemic stroke was halted early. Neurol Today2013; 13: 34–35.
31.
FreemantleNCalvertMWoodJet al.Composite outcomes in randomized trials: greater precision but with greater uncertainty?JAMA2003; 289: 2554–2554.
32.
CordobaGSchwartzLWoloshinSet al.Definition, reporting, and interpretation of composite outcomes in clinical trials: systematic review. Br Med J2010; 341: c3920–c3920.
33.
TomlinsonGDetskyAS. Composite end points in randomized trials: there is no free lunch. JAMA2010; 303: 267–268.
34.
Ferreira-GonzalezIPermanyer-MiraldaGBusseJet al.Composite outcomes can distort the nature and magnitude of treatment benefits in clinical trials. Ann Intern Med2009; 150: 566–566.
35.
Ferreira-GonzálezIPermanyer-MiraldaGBusseJWet al.Methodologic discussions for using and interpreting composite endpoints are limited, but still identify major concerns. J Clin Epidemiol2007; 60: 651–657.
36.
Ferreira-GonzálezIPermanyer-MiraldaGDomingo-SalvanyAet al.Problems with use of composite end points in cardiovascular trials: systematic review of randomised controlled trials. BMJ2007; 334: 786–786.
37.
LubsenJJustHHjalmarssonAet al.Effect of pimobendan on exercise capacity in patients with heart failure: main results from the Pimobendan in Congestive Heart Failure (PICO) trial. Heart1996; 76: 223–223.
38.
LubsenJKirwanBA. Combined endpoints: can we use them?Stat Med2002; 21: 2959–2970.
39.
HuqueMFAloshMBhoreR. Addressing multiplicity issues of a composite endpoint and its components in clinical trials. J Biopharm Stat2011; 21: 610–634.
40.
MaschaEJTuranA. Joint hypothesis testing and gatekeeping procedures for studies with multiple endpoints. Anesth Anal2012; 114: 1304–1317.
41.
DmitrienkoAD’AgostinoRBHuqueMF. Key multiplicity issues in clinical drug development. Stat Med2013; 32: 1079–1111.
42.
SankohAJLiHD’AgostinoRB. Use of composite endpoints in clinical trials. Stat Med2014; 33: 4709–4714.
43.
LoganBTamhaneA. Superiority inferences on individual endpoints following noninferiority testing in clinical trials. Biometrical J2008; 50: 693–703.
44.
RöhmelJGerlingerCBendaNet al.On testing simultaneously non-inferiority in two multiple primary endpoints and superiority in at least one of them. Biometrical J2006; 48: 916–933.
45.
GómezGLagakosSW. Statistical considerations when using a composite endpoint for comparing treatment groups. Stat Med2013; 32: 719–738.
46.
GehanEA. A generalized wilcoxon test for comparing arbitrarily singly-censored samples. Biometrika1965; 52: 203–223.
47.
BraunwaldEAntmanEMBeasleyJWet al.ACC/AHA 2002 guideline update for the management of patients with unstable angina and non–st-segment elevation myocardial infarctionsummary article: a report of the American College of Cardiology/American Heart Association Task force on practice guidelines (committee on the management of patients with unstable angina). J Am Coll Cardiol2002; 40: 1366–1374.
National Asthma Education and Prevention Program (National Heart, Lung, and Blood Institute) Third Expert Panel on the Management of Asthma. Expert panel report 3: guidelines for the diagnosis and management of asthma. NIH Publication: US Department of Health and Human Services, National Institutes of Health, National Heart, Lung, and Blood Institute, 2007.
50.
Van ElterenP. On the combination of independent two-sample tests of Wilcoxon. Bull Int Stat Inst1960; 37: 351–361.
51.
ZhaoY. Sample size estimation for the van Elteren test – a stratified Wilcoxon–Mann–Whitney test. Stat Med2006; 25: 2675–2687.
52.
SennS. Change from baseline and analysis of covariance revisited. Stat Med2006; 25: 4334–4344.
53.
FitzmauriceG. A conundrum in the analysis of change. Nutrition2001; 17: 360–361.
54.
van BreukelenGJ. Ancova versus change from baseline in nonrandomized studies: the difference. Multivariate Behav Res2013; 48: 895–922.
Pearl J. Lord’s paradox revisited–(oh lord! kumbaya!). Technical report, Citeseer, 2014.
57.
OakesJMFeldmanHA. Statistical power for nonequivalent pretest-posttest designs the impact of change-score versus ancova models. Eval Rev2001; 25: 3–28.
58.
WillettJB. Questions and answers in the measurement of change. Rev Res Edu1988; 15: 345–422.
CampbellDTKennyDA. A primer on regression artifacts, New York City, New York: Guilford Publications, 1999.
61.
YoungFBWeirCJLeesKRet al.Comparison of the national institutes of health stroke scale with disability outcome measures in acute stroke trials. Stroke2005; 36: 2187–2192.
62.
AdamsHJrDavisPLeiraEet al.Baseline NIH Stroke Scale score strongly predicts outcome after stroke: a report of the Trial of Org 10172 in Acute Stroke Treatment (TOAST). Neurology1999; 53: 126–126.
63.
BrunoASahaCWilliamsLS. Using change in the national institutes of health stroke scale to measure treatment effect in acute stroke trials. Stroke2006; 37: 920–921.
64.
ParsonsMSprattNBivardAet al.A randomized trial of tenecteplase versus alteplase for acute ischemic stroke. N Engl J Med2012; 366: 1099–1107.
65.
BrittainEPalenskyJBloodJet al.Blinded subjective rankings as a method of assessing treatment effect: a large sample example from the systolic hypertension in the elderly program (SHEP). Stat Med1997; 16: 681–693.
66.
FelkerGAnstromKRogersJ. A global ranking approach to end points in trials of mechanical circulatory support devices. J Cardiac Fail2008; 14: 368–372.
67.
AllenLAHernandezAFO’ConnorCMet al.End points for clinical trials in acute heart failure syndromes. J Am Coll Cardiol2009; 53: 2248–2258.
68.
SunHDavisonBACotterGet al.Evaluating treatment efficacy by multiple endpoints in phase ii acute heart failure clinical trials: analyzing data using a global method. Circulation2012; 5: 742–749.
69.
SubherwalSAnstromKJJonesWSet al.Use of alternative methodologies for evaluation of composite end points in trials of therapies for critical limb ischemia. Am Heart J2012; 164: 277–277.
70.
BerryJDMillerRMooreDHet al.The combined assessment of function and survival (cafs): a new endpoint for als clinical trials. Amyotroph Lateral Scler Frontotemp Degen2013; 14: 162–168.
71.
RosenbaumPR. Comment: the place of death in the quality of life. Stat Sci2006; 21: 313–316.
72.
RubinDB. Rejoinder:causal inference through potential outcomes and principal stratification: Application to studies with “censoring” due to death. Stat Sci2006; 21: 319–321.
73.
FayMPProschanMA. Wilcoxon-Mann-Whitney or t-test? on assumptions for hypothesis tests and multiple interpretations of decision rules. Stat Surv2010; 4: 1–1.
74.
GailMHMarkSDCarrollRJet al.On design considerations and randomization-based inference for community intervention trials. Stat Med1996; 15: 1069–1092.
75.
PrattJW. Robustness of some procedures for the two-sample location problem. J Am Stat Assoc1964; 59: 650–665.
76.
ChungERomanoJP. Asymptotically valid and exact permutation tests based on two-sample U-statistics. J Stat Plann Infer2016; 168: 97–105.
77.
BrunnerEMunzelU. The nonparametric behrens-fisher problem: asymptotic theory and a small-sample approximation. Biometrical J2000; 42: 17–25.
