Should the Analysis of a MultiCenter Clinical Trial be Guided by the Study Design Basic ideas and in

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Should the Analysis of a Multi-Center Clinical
Trial be Guided by the Study Design?Basic ideas
and insights

• Marvin Zelen
• Harvard University

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My Collaborators
Summer Zheng, Ph.D. and Max.
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Basic Philosophy and Practice of Analysis of
Trials

• Philosophy
• Analysis should be guided by the design of the
  study and account for significant factors
  affecting trial outcomes .
• Also important is the kind of inference to be
  made.
• Practise
• Most analyses of multi center trials neglect
  center variation and ignore the design of the
  study e.g permuted blocks. Many multi-center
  trials may have a large number of centers
  entering a relatively small number of patients.

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Local vs Global Inference

• Local Inference Conclusions apply only to the
  patients entered on trial.
• Global Inference Conclusions of trial apply to
  the population with disease.
• Under what circumstances do these different kinds
  of inferences apply?

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Idealized Clinical Trial Process

• Population with Disease

Global Inference requires a random sample of
patients
Random Sample of Patients
Local inference only requires randomization
Randomization
A
B
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How is best treatment defined?

• The best treatment for a disease may be defined
  as having the most favorable outcome for a
  randomly chosen patient being treated in a
  randomly chosen hospital.
• Hence in reporting the benefit of treatment,
  there will be two sources of variation
  corresponding to the patient and institutional
  variation.

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Four Models

• Patient Population Institute Population
• Collection Collection
• Random sample Collection
• Collection Random sample
• Random sample Random sample

A collection refers to a group which is not a
random sample. 1 . Conclusions only apply to the
patients entered on the trial Local
inference 2. Conclusions apply to the population
conditional on each institution. 4. Conclusions
apply to the population with disease Global
inference
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Random Sample of Patients ???

• Nearly all clinical trials do not have a random
  sample of patients .
• Only a local inference can be made unless
  additional assumptions are assumed.
• The randomization process can serve as the basis
  for making a local inference.
• In most situations power will be increased as the
  inference is more narrow.

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Some Background

• Conditioning on Sample Size
• Permuted Blocks

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Two Methods of Randomization(Two treatments)

• Method I Place two balls labeled A and B in urn
  . Sample balls with replacement.
• Method II Place n balls in urn labeled A and n
  balls in urn labeled B. Sample without
  replacement.
• Note Method I may have unequal sample sizes for
  A and B. These sample sizes are random variables.
  Method II has equal sample sizes .

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Notation

• ?i 1 if ith patient is randomized to A and
  zero otherwise for i 1,2,,N
• Yi outcome for ith patient
• Sa ? ?i Yi Sum of outcomes for treatment A.
• Sa Sb S
• ? ?i na no. assigned to A,
• N na nb

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Two Methods of Randomization
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Analysis Sample sizes fixed or random ?

• Since Sa Sb S, comparing

is equivalent to comparing Sa with its expected
value.
Although the sample sizes in Method I are random
variables, the analysis should condition on the
actual sample size observed as this will result
in a smaller variance for comparing A with B.

Var Sa ?Yi2/4 (Method I) N pq ?
(Yi Y)2 /(N 1) (conditioning)
Where p na/N , q 1-p
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Permuted Blocks

• Arrange the randomization so that it is done in
  blocks of size N. Within the block, all
  treatments are represented exactly the same
  number of times.
• Example Suppose there are two treatments
  (designated by A and B) and N 4. Then each
  treatment appears twice in the block. After every
  4th patient there are an equal number of patients
  on each of the two treatments e.g.
• aabb baba baab abba etc.
• ?j ?j1 ?j2 ?j3 2 for j 1,2, . .
  .

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Permuted Blocks and Multi Center Trials

• Permuted Blocks is the most common way to plan
  multi-center trials. The use of permuted blocks
  takes account of changes in the patient
  population over time and the possibility that
  benefit may change with more experience with
  the treatments.
• If randomization is made by permuted blocks ,
• the number of patients assigned to each
  treatment,
• within an institution, is a random variable.
  These
• sample sizes are ancillary statistics.

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Analysis Based on Randomization

• Suppose permuted blocks of size N are used for
  randomization.
• Define Sa(j) Sum of outcomes for A in block j
• Sa ?j Sa(j) Overall sum of outcomes for A.
• The distribution of Sa , based on randomization,
  should be conditional on both the permuted blocks
  and the ancillary statistics ( sample size
  assigned to A within each institution by the
  permuted blocks).
• The first two moments of Sa can be calculated
  exactly which enables the calculation of the
  conditional asymptotic distribution of Sa.

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Investigations of Local Inference Power

• Outcome variables continuous, binomial,
  survival
• Randomized multi center trials (two treatments)
• Methodology conditions on the number of patients
  randomized within a center to the treatments
  (conditions on ancillary statistic)
• Study design is permuted blocks of size four
• Sample size for each treatment is equalized for
  every fourth patient entered on trial.
• Variation between centers is present.

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Conclusions

• Nearly all clinical trials do not have a random
  sample of patients. In order to use standard
  statistical techniques it is necessary to assume
  a random sample of patients.
• Realistically only a local inference can be made.
  
• Conditioning on the ancillary statistics tends to
  eliminate center variability without modeling. It
  results in greater power.