COMPUTER INTENSIVE AND RE-RANDOMIZATION TESTS IN CLINICAL TRIALS

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COMPUTER INTENSIVE AND RE-RANDOMIZATION TESTS IN
CLINICAL TRIALS

• Thomas Hammerstrom, Ph.D.
• USFDA, Division of Biometrics
• The opinions expressed are those of the author
  and do not necessarily reflect those of the FDA.

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OBJECTIVE OF TALK

• Discuss role of randomization and deliberate
  balancing in experimental design.
• Compare standard and computer intensive tests to
  examine robustness of level and power of common
  tests with deliberately balanced assignments when
  assumed distribution of responses is not correct.

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OUTLINE OF TALK

• I. Testing with Deliberately Balanced Assignment
• II. Common Mistakes in Views on Randomization and
  Balance
• III. Robustness Studies on Inference in
  Deliberately Balanced Designs

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I. TESTING WITH DYNAMIC ALLOCATION
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DYNAMIC ASSIGNMENTS

• 1. Identify several relevant, discrete
  covariates, e.g., age, sex, CD4 count
• 2. Change randomization probabilities at each
  assignment to get each level of each covariate
  split nearly 50-50 between arms

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• 3. Assign new subject randomly if all covariates
  are balanced
• assign deterministically or with unequal
  probabilities to move toward marginal balance if
  not currently balanced

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ISSUES WITH DYNAMIC ASSIGNMENTS

• 1. Why bother with this elaborate procedure?
• 2. Are the levels of tests for treatment effect
  preserved when standard tests are used with
  dynamic (minimization) assignments?
• 3. Does the use of minimization increase power
  in the presence of both treatment and covariate
  effects?

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II. COMMON MISTAKES IN ANALYSIS OF BASELINE
COVARIATES

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• Mistake 1. Purpose of Randomization is to Create
  Balance in Baseline Covariates
• Fact Purpose of Randomization is to Guarantee
  Distributional Assumptions of Test Statistics and
  Estimators

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• Mistake 2. It is good practice in a randomized
  trial to test for equality between arms of a
  baseline covariate.
• Fact All observed differences between arms in
  baseline covariates are known with certainty to
  be due to chance. There is no alternative
  hypothesis whose truth can be supported by such a
  test.

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• Mistake 3. If a test for equality between arms of
  a baseline covariate is significant, then one
  should worry.
• Fact Such test statistics are not even good
  descriptive statistics since p-values depend on
  sample size, not just the magnitude of the
  difference.

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• Mistake 4. Observed Imbalances in Baseline
  Covariates cast Doubt on the Reality of
  Statistically Significant Findings in the Primary
  Analysis.
• Fact The standard error of the primary statistic
  is large enough to insure that such imbalances
  create significant treatment effects no more
  frequently than the nominal level of the test.

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• Mistake 5. Type I Errors can be Reduced by
  Replacing the Primary Analysis with one Based on
  Stratifying on Baseline Covariates Observed Post
  Facto to be Unbalanced.
• Fact The Operating Characteristics of Procedures
  Selected on the Basis of Observation of the Data
  are not generally Quantifiable.

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• If the Agency approved of Post Hoc Fixing of Type
  I Errors by Adding New Covariates to the Analysis
  (or by other Adjustments to Fix Randomization
  Failures),
• Then it should also Approve of Similar Post Hoc
  Fixing of Type II Errors when Failure of
  Randomization Leads to Imbalance in Favor of the
  Control Arm.

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• Mistake 6. If the same Random Assignment Method
  gave more even Balance in Trial A than in Trial
  B, then one should place more trust in a
  Rejection of the Null Hypothesis from Trial A.
• Fact Balance on Baseline Covariates Decreases
  the Variance of Test Statistics and Estimators.
  It Increases the Power of Tests when the
  Alternative Hypothesis is True. It has no Effect
  on Type I Error.

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• Mistake 7. Balance on Baseline Covariates Leads
  to Important Reductions in Variances.
• Fact Even without Balance, the Variance of Test
  Statistics and Estimators are of size O(1/N)
  where N sample size.
• Balancing on p Baseline Covariates Decreases
  these variances by Subtracting a Term of size
  O(p/N2)

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• Typical model for Continuous Response
• Yik mi g1x1ik gpxpik eik
• where eik N(0, s2)
• mi treatment effect,
• Xik (x1ik,,xpik) vector of covariates
• g1 ,, gp unknown vector of covariate effects

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• s2 Precision of Estimate of (m1-m0 )
• N/2 - ZZ
• where N number per arm,
• Z V-1(X1. - X0.),
• V2 matrix of cross-products of X/2N, and
• randomization distribution of
• (X1. - X0.) N( 0, V2), of Z N(0, Ip),
• of ZZ Chi-square(p)
• Precision with Balance N/2,
• E(Precision without Balance) N/2 - O(p)

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III. ROBUSTNESS STUDIES ON INFERENCE IN
DELIBERATELY BALANCED DESIGNS

• A. MODELS USED TO COMPARE METHODS

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METHODS COMPARED

• 1. Dynamic Allocation analyzed by F-statistic
  from ANCOVA based on arm and covariates
• 2. Dynamic Allocation analyzed by
  re-randomization test, using difference in means
• 3. Randomized Pairs, analyzed by F-statistic from
  ANCOVA using arm and covariates

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BASIC FORM OF SIMULATED DATA

• 1. Control test arms, N subjects randomized
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• 2. X1j, , X7j binary covariates for subject j
• 3. ej unobserved error for subject j
• 4. Yj observed response for subject j
• 5. I1j 1 if subject j in arm 1, test arm
• 6. Yj mj I1j ej d Sk17Xkj

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MODELS FOR ERRORS

• 1. ej N( 0 , 1 )
  Normal
• 2. ej exp( N( 0 , 1 ))
  Lognormal
• 3. ej N( 4j/N , 1 )
  Trend
• 4. ej .9 N( 0 , 1) .1 N( 0, 25 ) Mixed
• 5. ej N( 0 , 4j/N )
  Hetero
• 6. ej N( cos(2pj/N) , 1 ) Sine wave
  
• 7. ej N( 0 , 1 ) if jltJ
• N(4, 1) if jgtJ
  Step

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MODELS FOR COVARIATES

• X1j, , X7j are
• 1. independent with p1, , p7 constant in j
• 2. correlated with p1, , p7 constant
• 3. independent with p1, , p7 monotone in j
• 4. independent with p1, , p7 sinusoid in j
• Coefficient d 1 or 0

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MODELS FOR TREATMENT

• 1. Treatment effect mj m, constant over j
• 2. Treatment effect mj m (4j/N), increasing
  over j

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COMPARISONS

• 1. Select one of the models
• 2. Generate 200 sets of covariates and unobserved
  errors
• 3. For each set, construct I1j once by dynamic
  once by randomized pairs
• 4. Compute the 200 p-values for different tests
  and assignment methods

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SIMULATED DATA FOR COX REGRESSION

• 1. Control test arms, N subjects randomized
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• 2. X1j, , X7j binary covariates for subject j
• 3. YLj potentially observed failure time for
  subject j on arm L 0 or 1
• 4. YLi / dL( 1 Sk17Xkj ) FL, L 0 or 1
• 5. FL Exponential or Weibull
• 6. Censoring Exp with scale large or small

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RESULTS WITH COX REGRESSION

• 1. Assign subjects by dynamic allocation.
• 2. Estimate treatment effect by proportional
  hazards regression
• 3. Re-randomize and compute new ph reg estimates
  many times.
• 4. Compare parametric p-value with percentile of
  real estimate among all rerandomized treatment
  estimates

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III. ROBUSTNESS STUDIES ON INFERENCE IN
DELIBERATELY BALANCED DESIGNS

• B. RESULTS OF SIMULATIONS

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SIMULATION RESULTS

• 1. In most cases considered, the gold standard
  but computer intensive re-randomization test gave
  the same power curve as the standard ANCOVA
  F-test for the dynamic allocation. Both level,
  when H0 was true, and power, otherwise, were the
  same.

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SIMULATION RESULTS

• 2. In most cases considered, the ANCOVA F-test
  gave the same power curve whether the subjects
  were assigned by dynamic allocation or randomized
  pairs. Deliberate balance on baseline covariates
  gave no improvement in power.

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SIMULATION RESULTS

• 3. There was one clear exception to the above
  findings. When untreated responses showed a
  trend with time of enrollment, the ANCOVA F-test
  for treatment gave incorrectly low power.

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SIMULATION RESULTS

• 4. In most cases considered with time to event
  data with dynamic allocation, the
  re-randomization test gave the same results as
  the Cox regression.

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SUMMARY

• 1. Modifying a Randomization Method to Achieve
  Deliberate Balance Serves Mainly Cosmetic
  Purposes Should be Discouraged
• 2. Balance on Covariates Reduces Variance of Test
  Stats Estimators but Only by Small Amounts
• Var( trt effect) O(1/N) when balanced
• When unbalanced , Var is larger by a term
  O(p/N2)

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SUMMARY

• 3. Rerandomization analyses based on Finite
  Population Models are gold standard for
  randomized trials
• 4. IID Error models are only approximations
• 5. Approximation is adequate for level with
  common minimization allocations under a wide
  variety of potential violations of the
  assumptions.

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SUMMARY

• 6. Deliberate Balance Allocations and Simple
  Tests Require Belief that God is Randomizing Your
  Subjects Responses.
• Randomization and Finite Population Based Tests
  Protect You if the Devil is Determining the Order
  of Your Subjects Responses