Evaluation of sample size and power for multi-armsurvival trials allowing for non-proportional hazards, loss

to follow-up and cross-over

F M-S Barthel, A Babiker, P Royston, M K B Parmar

15. - 19. August 2004

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Outline

(1) Introduction(2) The general method(3) Extensions(4) Performance of the method(5) Final remarks

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1 Introduction

� Logrank test commonly used in analysis of phase III clinical trialswith survival time outcome

� Sequential accrual followed by period of follow-up� Administrative censoring� Multiple arms� Loss to follow-up and cross-over� Schoenfeld (1983):� two survival distributions under logrank test, administrativecensoring

� Lachin & Foulkes (1986):� loss to follow-up using exponential distribution

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2 The general method

� H0 : lnf�2=�1g = 0� H1 : lnf�2=�1g 6= 0� Logrank statistic

U =

mXj=1

[observed(events)� expected(events)]

� Variance V� Test statistic Q = U 2=V � �21

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2 The general method

� Under H1 non-centrality parameter

� = n(ln2f�2�1g)p(1� p)

� Assume local alternatives� Total sample size required

n =(z1��=2 + z�)

2

(ln2f�2�1g) p(1� p)

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3 Extensions

� General framework: total trial time split into periods of equal length) examine number of patients at risk and occurrence of events inall groups separately for each period

� Length of periods depend on knowledge available about patientbehaviour at start of trial

� Allows to take into account:� staggered patient entry� loss to follow-up� cross-over� non-proportional hazards

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3.1 Staggered patient entry and loss to follow-up

� T as the total number of periods in a trial� Accrual over periods 0 to R� Accrual piecewise truncated exponential or uniform� Point mass at zero� Survivor function of loss to follow-up and of failure times� Survivor functions piecewise exponentials� Probabilities of failure and loss to follow-up over duration of trial) probability of not being censored

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3.2 Multi-arm trials

� n patients randomised to one of K treatments� K treatments to be compared globally, i.e.

H0 : �1(t) = �2(t) = ::: = �K(t)

H1 : �k(t) 6= �k�1(t) for at least one k� Test statistic under H0

Q = U 0(V (0))�1U � �2K�1

� Under H1 non-centrality parameter� = nE(U jH1)

0(V (0))�1E(U jH1)

� Important since heuristic approach does not take into accountmultiple comparisons during analysis

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3.3 Cross-over

� A patient who changes from designated treatment group into anotherbut remains available for follow-up

� C as time at which cross-over occurs� Hazard function �i(t) of cross-over� Calculate proportion of non-compliant patients� Readjust n

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4 Performance of the method

� Assess overall performance of sample size calculations as imple-mented in STATA 8

� Compare with software of similar scope, i.e. SIZE developed byJoanna Shih based on Lakatos (1988)

� Lakatos: loss to follow-up and withdrawal from allocated treatmentunder nonstationary Markov model

� Different states:� adhere to treatment� lost to follow-up� cross-over� experience event

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4.1 Simulation results

� Performed in STATA 8� 2 years accrual, 2 years follow-up� Median survival 1 year� Equal allocation� Uniform accrual and exponential survival� 90% power and 5% signi�cance level� 5000 simulated trials� Standard error 0.4%, i.e. CI from 89.2 to 90.8% power� Adjusted and unadjusted sample size calculation run� Analysed under intention to treat

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4.1.1 Loss to follow-up

Parameters Adjusted * Unadjusted SIZE *HR �L1 �

L2 n Power n Power % diff n n Power % diff n

0.7 5 20 424 89.9 408 88.9 - 3.9 n/a n/a n/a0.7 20 5 424 90.3 408 88.9 - 3.9 n/a n/a n/a0.7 5 5 414 89.6 408 89.1 - 1.5 423 90.9 + 2.20.7 20 20 433 90.8 408 87.7 - 6.1 447 90.5 + 3.20.7 30 30 448 89.7 408 87.4 - 9.8 466 90.6 + 4.00.7 40 40 466 90.3 408 85.1 - 14.2 489 91.9 + 4.90.7 50 50 487 89.7 408 83.3 - 19.4 516 91.2 + 6.00.8 40 40 1155 90.0 1015 86.7 - 13.8 1209 91.0 + 4.70.8 50 50 1206 89.6 1015 83.5 - 18.8 1276 91.0 + 5.8

* sample size adjusted for loss to follow-up

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4.1.2 Non-proportional hazards

Parameters Adjusted * Unadjusted SIZE *HR1 HR2 n Power n Power % diff n n Power % diff n0.6 0.9 274 89.9 206 80.9 - 33.0 281 90.1 + 2.60.6 0.8 249 90.1 206 85.3 - 20.9 255 89.9 + 2.40.6 0.7 227 90.1 206 87.0 - 10.2 232 90.8 + 2.20.7 0.8 458 90.2 408 85.6 - 12.3 466 90.5 + 1.80.8 0.7 869 89.3 1015 93.7 + 16.8 882 90.5 + 1.50.8 0.6 749 89.9 1015 96.9 + 35.5 761 90.0 + 1.6

* sample size adjusted for non-proportional hazards

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4.1.3 Cross-over

Parameters Adjusted * Unadjusted SIZE *HR �C1 �

C2 n Power n Power % diff n n Power % diff n

0.6 0 5 212 90.0 206 88.8 - 2.9 217 91.2 + 2.40.6 0 10 218 89.9 206 88.2 - 5.8 224 90.8 + 2.80.6 0 20 232 90.1 206 86.1 - 12.6 238 90.4 + 2.60.6 0 30 248 90.7 206 84.2 - 20.4 256 90.7 + 3.20.7 0 30 489 90.5 408 83.6 - 19.9 502 90.9 + 2.70.7 10 10 458 90.0 408 87.0 - 12.3 467 90.5 + 2.00.7 20 20 522 90.9 408 84.1 - 27.9 530 92.2 + 1.50.7 30 30 606 91.7 408 78.1 - 48.5 609 91.4 + 0.5

* sample size adjusted for cross-over

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4.2 Relationship between power and sample size

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020

040

060

080

010

00sa

mpl

e si

ze

50 60 70 80 90 100power

adjusted sample size

HR1 = 0.7, HR2 = 0.8, accrual = 2, follow-up = 2proportion lost to follow-up = 30% in both armscross-over = 30% from both arms

1

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4.3 Trial example: Optima

� Trial currently run in UK, Canada and USA� Management of patients with HIV infection and failure of 1st and2nd line HAART

� 4.5 years accrual and 1 year follow-up� Standard arm event rate in year 1 23%, 25% annual increasethereafter

� Cross-over from mega to standard 5% in year 1, 50% decrease everyyear thereafter

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4.3 Trial example: Optima

� Drop-in from standard to mega ART 1% in year 1 (increasing by10% thereafter)

� Hazard ratio 0.7� Loss to follow-up at 5.5 years 5%� Signi�cance level 5% and power 80%

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4.3 Trial example: Optima

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4.3 Trial example: Optima

� For 80% power:� Adjusted sample size 825, 287 events� Unadjusted sample size 711, 250 events

� For 90% power:� Adjusted sample size 1105, 384 events� Unadjusted sample size 951, 334 events

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5 Final remarks

� Adjustments for non-proportional hazards and cross-over substantialin terms of power

� Main differences with SIZE:� Sample size up to 5% higher with SIZE� Different loss to follow-up distributions in treatment arms� More than two treatment arms� Non-uniform accrual� User friendly interface� Convergence problems with SIZE (time)

� STATA program allows for distant alternatives from H0 but onlymarginal improvements in accuracy

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5 Final remarks

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87.5

9092

.5po

wer

.2 .3 .4 .5 .6 .7 .8 .9HR

Local alternatives

87.5

9092

.5po

wer

.2 .3 .4 .5 .6 .7 .8 .9HR

Non-local alternatives

equal allocation to both treatment arms, accrual = 2, follow-up = 2solid lines give 95% CI around 90% power

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5 References

Schoenfeld. Sample-size formula for the proportional-hazards regression model. Biometrics 1983; 39:499-503.

Lachin, Foulkes. Evaluation of sample size and power for analyses of survival with allowance for nonuniform entry, losses to

follow-up, noncompliance, and strati�cation. Biometrics 1986; 42:507-519.

Shih. Sample size calculation for complex clinical trials with survival endpoints. Controlled Clinical Trials 1995; 16:395-407.

Lakatos. Sample sizes based on the log-rank statistic in complex clinical trials. Biometrics 1988; 44:229-241.

Royston, Babiker. A menu-driven facility for complex sample size calculation in randomized controlled trials with a survival

or a binary outcome. The Stata Journal 2002; 2:151-163.

Barthel et al.. A menu-driven facility for complex sample size calculation in randomized controlled trials with a survival or a

binary outcome: update. The Stata Journal submitted.

Barthel et al.. Evaluation of sample size and power for multi-arm survival trials allowing for non-proportional hazards, loss to

follow-up and cross-over. Stat Med in preparation.

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