UNTIED WORST-RANK SCORE ANALYSIS

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1 Paper PO16 UNTIED WORST-RANK SCORE ANALYSIS William F. McCarthy, Maryland Medical Research Institute, Baltime, MD Nan Guo, Maryland Medical Research Institute, Baltime, MD ABSTRACT When the non-fatal outcome measures are missing completely at random, then the analysis of the non-missing data is unbiased (Rubin, 1976; Little, 1976; and Little and Rubin, 1987). This implies that a subset of measurements actually observed provides an unbiased description of the treatment effect in the entire population. When the nonfatal outcome measures are infmatively missing and an analysis is based only on the subset of measurements actually observed, the description of the treatment effect may be biased (Lachin, 1999). This paper describes a method f an unbiased description of treatment effect when the non-fatal outcome measures are infmatively missing. In addition, we present a SAS Program to calculate the power of the untied wst-rank sce analysis and a SAS Program to perfm the untied wst-rank sce analysis. KEY WORD missing data; non-fatal outcomes; unbiased description of efficacy; SAS Program INTRODUCTION In some randomized clinical trials, patients are scheduled to undergo some assessment of a non-fatal outcome at a fixed time ( times) after the initiation of treatment. Often, these follow-up assessments may be missing f some patients because a disease-related event occurred pri to the time of the follow-up assessment. In such a situation, these follow-up assessments are infmatively missing because the disease-related event and the non-fatal outcome both indicate progression of the underlying disease. F example, a study of congestive heart failure may schedule patients to undergo exercise testing at 12 weeks, but this measurement may be missing f those patients who died of heart disease pri to the exercise testing. When the non-fatal outcome measures are missing completely at random, then the analysis of the non-missing data is unbiased (Rubin, 1976; Little, 1976; and Little and Rubin, 1987). This implies that a subset of measurements actually observed provides an unbiased description of the treatment effect in the entire population. When the non-fatal outcome measures are infmatively missing and an analysis is based only on the subset of measurements actually observed, the description of the treatment effect may be biased (Lachin, 1999). PROCEDURE FOR THE UNTIED WORST-RANK SCORE ANALYSIS Let i denote the study group (1=placebo, 2= drug). Let j denote a patient (1,2,,n i ), where n i is the number of patients in group i. Let * ij denote an indicat variable, which indicates whether the infmative event (death) occurs in the ijth patient pri to the end of the study (1= Yes, 0 = No). * ij = I(t ij T); T is the fixed time of the follow-up measurement; and t ij denotes the survival time of the ijth patient. All patients will receive a value f t ij (t ij T), regardless of their value f * ij. Let x ij denote the observed primary outcome measure f the ijth patient. If a higher value of x indicates a better outcome, then use the approach outlined below (When a higher value of x indicates a wse outcome, use the approach outlined in Appendix A). Patients who die will be ranked on their time to death (t ij ), and will receive a rank sce cresponding to a value that is wse (lower) than any actually observed in the surviving patient population. These imputed ranks will reflect the relative dering of the event times, with the shtest time to death ranked wst (lowest). Surviving patients will be

2 ranked by their x ij with those having the highest value of x ij receiving the best (highest) ranking. The data structure will look as follows: t min,, t max, x min,, x max. F the ijth patient who has died pri to the follow-up measurement, use the value f x ij is used f several reasons: 1) to identify those patients who have no follow-up measurement, therefe no valid x ij and 2) to allow f the calculation of Q ij ; if the standard missing value of. is used, Q ij cannot be calculated is a value far removed from any valid value expected f x ij. Let Q ij denote the value to use in the rank analysis f the ijth patient. Q ij = (1 - * ij )x ij + * ij (0 + t ij ) [1] where 0 is a negative constant such that (0 + T) < >. > is equal to the wst possible valid value f x ij ; (0 + T) < > allows one to distinguish surviving patients from those who have died; thus set 0 to A rank analysis based on the {Q ij } provides an unbiased test of the joint null hypothesis [2] against the restricted alternative hypothesis [3]; Lachin, H 0 : [G 1 (x) = G 2 (x) and K 1 (t) = K 2 (t)] (0 < t T) [2] H 1 : [G 1 (x) G 2 (x) and K 1 (t) K 2 (t)] [3] [G 1 (x) G 2 (x) and K 1 (t) = K 2 (t)] [G 1 (x) = G 2 (x) and K 1 (t) K 2 (t)] G i (x) is the cumulative probability distribution of the observable values of x f all event-free members of the ith group observed at time T; i.e., G i (x) = Pr(x ij x t > T). K i (t) is the cumulative distribution of the infmative event times t in the ith group. The notation G 1 (x) G 2 (x) indicates that G 1 (x) is shifted to the left of G 2 (x); i.e., the observable values in group 1 (placebo) tend to be less than those of group 2 (drug). This indicates that there is a difference in fav of group 2 since higher values of x are better. The notation K 1 (x) K 2 (x) indicates that K 1 (x) is shifted to the left of K 2 (x); i.e., the infmative event times in group 1 (placebo) tend to be less than those of group 2 (drug). This indicates that there is a difference in fav of group 2 since higher values of t are better. Test the joint null hypothesis [2] that the two study groups do not differ with respect to the survival times and the distributions of the observable measurements. Use the restricted alternative hypothesis [3], which indicates that the drug group tends to have higher values of x and/ t while not having lower values f either. APPENDIX A When higher values of x are wse, patients who are infmatively censed early thus have higher rank sces than those infmatively censed late. The data structure will look as follows: x min,, x max, t max,, t min. F the ijth patient who has died pri to the follow-up measurement, use the value 9999 f x ij is used f several reasons: 1) to identify those patients who have no follow-up measurement, therefe no valid x ij and 2) to allow f the calculation of Q ij ; if the standard missing value of. is used, Q ij cannot be calculated is a value far removed from any valid value expected f x ij.

3 Let Q ij denote the value to use in the rank analysis f the ijth patient. Q ij = (1 - * ij )x ij + * ij (0 - t ij ) [1 ] where 0 is a positive constant such that (0 - T) > >. > is equal to the wst possible valid value f x ij ; (0 - T) > > allows one to distinguish surviving patients from those who have died; thus set 0 to A rank analysis based on the {Q ij } provides an unbiased test of the joint null hypothesis [2 ] against the restricted alternative hypothesis [3 ]; Lachin, H 0 : [G 1 (x) = G 2 (x) and K 1 (t) = K 2 (t)] (0 < t T) [2 ] H 1 : [G 1 (x) G 2 (x) and K 1 (t) K 2 (t)] [3 ] [G 1 (x) G 2 (x) and K 1 (t) = K 2 (t)] [G 1 (x) = G 2 (x) and K 1 (t) K 2 (t)] The notation G 1 (x) G 2 (x) indicates that G 1 (x) is shifted to the right of G 2 (x); i.e., the observable values in group 1 (placebo) tend to be me than those of group 2 (drug). This indicates that there is a difference in fav of group 2 since lower values of x are better. The notation K 1 (x) K 2 (x) indicates that K 1 (x) is shifted to the left of K 2 (x); i.e., the infmative event times in group 1 (placebo) tend to be less than those of group 2 (drug). This indicates that there is a difference in fav of group 2 since higher values of t are better. REFERENCES Rubin D (1976). Inference and missing data. Biometrika, 63: Little RJA (1976). Comments on inference and missing data by Rubin DB. Biometrika, 63: Little RJA and Rubin DB (1987). Statistical analysis with missing data. Wiley, New Yk, NY. Lachin JM (1999). Wst-rank sce analysis with infmatively missing observations in clinical trials. Controlled Clinical Trials 20: McMahon RP (1989). Analysis of nonfatal outcomes in clinical trials in which mtality is present. The Institute of Statistics, University of Nth Carolina at Chapel Hill, Mimeo series, No. 1867T. McMahon RP and Harrell FE (2000). Power calculation f clinical trials when the outcome is a composite ranking of survival and a nonfatal outcome. Controlled Clinical Trials 21: POWER CALCULATION FOR UNTIED WORST-RANK SCORE ANALYSIS /* PROGRAM TO COMPUTE THE POWER OF AN UNTIED WORST-RANK SCORE ANALYSIS OUTCOME IS A COMPOSITE RANKING OF SURVIVAL AND A NONFATAL OUTCOME [Lachin JM (1999). Wst-rank sce analysis with infmatively missing observations in clinical trials. Controlled Clinical trials 20: ] USES THE CONSERVATIVE ASSUMPTION THAT THE TREATMENT HAS A BENEFICIAL EFFECT ON THE NONFATAL OUTCOME BUT NO EFFECT ON MORTALITY */ ASSUMES A NORMALLY DISTRIBUTED NONFATAL OUTCOME **** Power Calculation based on: McMahon and Harrell (2000). Power calculation f clinical trials when the outcome is a composite ranking of survival and a nonfatal outcome. Controlled Clinical Trials 21: ; **** WRITTEN BY WF MCCARTHY 10/14/03;

4 *$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$; *$ calpha: $; *$ critical values f 2-sided alphas $; *$ alpha=0.05 calpha=1.96 $; *$ alpha=0.01 calpha=2.58 $; *$ alpha=0.001 calpha=3.29 $; *$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$; *$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$; *$ calpha: $; *$ critical values f 1-sided alphas $; *$ alpha=0.05 calpha=1.65 $; *$ alpha=0.01 calpha=2.33 $; *$ alpha=0.001 calpha=3.09 $; *$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$; *# YOU INPUT THE FOLLOWING: #; *# arm_n: the number of patients in each study arm (requires an equal sample size in each study arm) #; *# alpha: type I err probability. E.g., 5 % = 0.05 #; *# calpha: select appropriate critical value from above tables. E.g., if alpha=0.05 (2-sided) select calpha=1.96 #; *# survival: the survival probability of the patients in each study arm at the end of the study, #; *# the survival probability is required to be the same in both study arms. E.g., 90 % =0.90 #; *# effect: the absolute treatment effect. E.g., 6 % =0.06 #; *# sigma: the common standard deviation. E.g., 10 % = 0.10 #; *# #; *# NOTE: x = the value of the nonfatal outcome and t = the value of the survival time #; *# THE PROGRAM COMPUTES THE FOLLOWING: #; *# ste: the standardized treatment effect #; *# py12: Pr(x1i > x2j t1i and t2j >= T), the conditional probability that patient i #; *# (given treatment 1)has a "better" nonfatal outcome than patient j (given treatment 2), #; *# given both patients have survived to time T #; *# py112: Pr(x1i > x2j and x1i' > x2j all of t1i, t1i', and t2j >= T), i' not equal to i #; *# py122 : Pr(x1i > x2j and x1i > x2j' all of t1i, and t2j, t2j' >= T), j' not equal to j #; *# pr12: Pr(r1i > r2j), the probability that patient i (given treatment 1) has a #; *# "better" rank than patient j (given treatment 2) #; *# vpr12: the variance of pr12 #; data temp; input arm_n alpha calpha survival effect sigma; cards; ; data temp; set temp; ste=effect/(sigma*sqrt(2)); py12= *ste; py112= *py12;

5 py122=py112; pr12=((0.5)*(1-(survival)**2))+((py12)*(survival)**2); vpr12=((2/3)*(1-(survival)**3)+(py112+py122)*(survival)**3-2*((0.5)*(1-(survival)**2)+(py12)*(survival)**2)**2)/(arm_n); power= 1-probnm((calpha)+((0.5-pr12)/sqrt(vpr12))); title "Power of the Untied Wst-Rank Sce Analysis"; proc print data=temp noobs; var arm_n alpha calpha survival effect sigma ste py12 py112 vpr12 power; Power of the Untied Wst-Rank Sce Analysis arm_n alpha calpha survival effect sigma ste py12 py112 vpr12 power **** Example SAS Program f the Implementation of the Procedure f the Untied Wst-Rank Sce Analysis; **** group=1 (placebo), group=2 (Drug A); **** x is delta EF (%) -9999; **** t is survival time (months); **** d is indicat variable f death (1=yes 0=no); **** z is the value to use in the rank analysis; data test; input patient group x t d; cards; ;

6 data test; set test; z = (1-d)*x + d*( t); proc st; by group z; proc npar1way wilcoxon; class group; var z; **** Ranking of z in each Group; Obs patient group x t d z

7 The NPAR1WAY Procedure Wilcoxon Sces (Rank Sums) f Variable z Classified by Variable group Sum of Expected Std Dev Mean group N Sces Under H0 Under H0 Sce ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Average sces were used f ties. Wilcoxon Two-Sample Test Statistic Nmal Approximation Z One-Sided Pr < Z Two-Sided Pr > Z t Approximation One-Sided Pr < Z Two-Sided Pr > Z Z includes a continuity crection of 0.5. We will use the z value and two-sided nmal approximation p-value generated by the NPAR1WAY Procedure f the moniting of the primary study outcome. CONTACT INFORMATION William F. McCarthy Principal Statistician Direct of Clinical Trial Statistics and SAS Programming Maryland Medical Research Institute 600 Wyndhurst Avenue Baltime, Maryland (410) wmccarthy@mmri.g Nan Guo Seni Lead SAS Programmer SAS Certified Advanced Programmer f SAS9 Maryland Medical Research Institute 600 Wyndhurst Avenue Baltime, Maryland (410) nguo@mmri.g SAS and all other SAS Institute Inc. product service names are registered trademarks trademarks of SAS Institute Inc. in the USA and other countries. indicates USA registration. Other brand and product names are trademarks of their respective companies.