STATISTICS 542 Introduction to Clinical Trials RANDOMIZATION METHODS

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STATISTICS 542Introduction to Clinical
TrialsRANDOMIZATION METHODS
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RANDOMIZATION

• Why randomize
• What a random series is
• How to randomize

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Randomization (1)

• Rationale
• Reference Byar et al (1976) NEJM 27474-80.
• Best way to find out which therapy is best
• Reduce risk of current and future patients of
  being on harmful treatment
• Example Retrolental Fibroplasia
• (Silverman Scientific American 236100-107,
  1977)

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Randomization (2)

• Basic Benefits of Randomization
• Eliminates assignment basis
• Tends to produce comparable groups
• Produces valid statistical tests
• Basic Methods
• Ref Zelen JCD 27365-375, 1974.
• Pocock Biometrics 35183-197, 1979

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• Goal Achieve Comparable Groups to Allow
  Unbiased Estimate of Treatment
• Beta-Blocker Heart Attack Trial
• Baseline Comparisons
• Propranolol Placebo
• (N-1,916) (N-1,921)
• Average Age (yrs) 55.2 55.5
• Male () 83.8 85.2
• White () 89.3 88.4
• Systolic BP 112.3 111.7
• Diastolic BP 72.6 72.3
• Heart rate 76.2 75.7
• Cholesterol 212.7 213.6
• Current smoker () 57.3 56.8

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Randomization Basis forTests of Hypotheses

• The use of randomization provides a basis for an
  assumption-free statistical test of the equality
  of treatments
• Such tests were originally proposed by Fisher and
  are known as randomization or permutation tests
• With sample sizes na and nb , the conditional
  reference set Wc consists of n choose na possible
  combinations of na patients on treatment a out of
  n patients

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Statistical Properties of Randomization
A. Sampling-Based Population Model
B. Randomization Model
Population a yG(y?a)
Population b yG(y?b)
Study Sample
Sample at Random
Sample at Random
nb patients ybjG(y?b)
na patients yajG(y?a)
n nb nb patients
Randomization
na patients
nb patients
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Nature of Random Numbers and Randomness

• A completely random sequence of digits is a
  mathematicalidealization
• Each digit occurs equally frequently in the
  whole sequence
• Adjacent (set of) digits are completely
  independent of one another
• Moderately long sections of the whole show
  substantial regularity
• A table of random digits
• Produced by a process which will give results
  closely approximating
• to the mathematical idealization
• Tested to check that it behaves as a finite
  section from a
• completely random series should
• Randomness is a property of the table as a whole
• Different numbers in the table are independent

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Table of Random Numbers
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Allocation Procedures to Achieve Balance

• Simple randomization
• Biased coin randomization
• Permuted block randomization
• Balanced permuted block randomization
• Stratified randomization
• Minimization method

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Treatment Imbalance Statistical Properties of
Randomization
75-25 split reduces power 90 80 66-33 split
reduces power 90 87
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Simple Random Allocation

• A specified probability, usually equal, of
  patients assigned to each treatment arm, remains
  constant or may change but not a function of
  covariates or response
• a. Fixed Random Allocation
• n known in advance, exactly
• n/2 selected at random assigned to Trt A, rest
  to Trt B
• b. Complete Randomization (most common)
• n not exactly known
• marginal and conditional probability of
  assignment 1/2
• analogous to a coin flip (random digits)

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Simple Randomization

• Advantage simple and easy to implement
• Disadvantage At any point in time, there may be
  an imbalance in the number of subjects on
  each treatment
• With n 20 on two treatments A and B, the
  chance
• of a 128 split or worse is approximately 0.19
• With n 100, the chance of a 6040 split or
  worse is approximately 0.025
• Balance improves as the sample size n increases
• Thus desirable to restrict randomization to
  ensure balance throughout the trial

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Randomization Balance (1)Coin Flip

• n 100 tosses of a coin
• p ½ for a fair coin
• s heads
• E(s) n p 50
• V(s) npq 100 ½ ½ 25
• Probability of a 6040 split

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Randomization Balance (2)

• n 20
• p ½
• E(s) np 10
• V(s) npq 20/4 5
• Probability of a 128 split or worse
• 0.19

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Restricted Randomization

• Simple randomization does not guarantee balance
  over time in each realization
• Patient characteristics can change during
  recruitment (e.g. early pts sicker than later)
• Restricted randomizations guarantee balance
• 1. Permuted-block
• 2. Biased coin (Efron)
• 3. Urn design (LJ Wei)

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Permuted-Block Randomization (1)

• Simple randomization does not guarantee balance
  in numbers during trial
• If patient characteristics change with time,
  early imbalances
• can't be corrected
• Need to avoid runs in Trt assignment
• Permuted Block insures balance over time
• Basic Idea
• Divide potential patients into B groups or blocks
  of size 2m
• Randomize each block such that m patients are
  allocated to A and m to B
• Total sample size of 2m B
• For each block, there are 2mCm possible
  realizations
• (assuming 2 treatments, A B)
• Maximum imbalance at any time 2m/2 m

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Permuted-Block Randomization (2)

• Method 1 Example
• Block size 2m 4
• 2 Trts A,B ? 4C2 6 possible
• Write down all possible assignments
• For each block, randomly choose one of the six
  possible arrangements
• AABB, ABAB, BAAB, BABA, BBAA, ABBA
• ABAB BABA ......
• Pts 1 2 3 4 5 6 7 8 9 10
  11 12

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Permuted-Block Randomization (3)

• Method 2 In each block, generate a uniform
  random number for each treatment (Trt), then rank
  the treatments in order Trt in Random
  Trt in any order Number Rank
  rank order A 0.07 1 A A
  0.73 3 B B 0.87 4 A B 0.31 2 B

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Permuted-Block Randomization (4)

• Concerns
• - If blocking is not masked, the sequence become
• somewhat predictable (e.g. 2m 4)
• A B A B B A B ? Must be A.
• A A Must be B B.
• - This could lead to selection bias
• Simple Solution to Selection Bias
• Do not reveal blocking mechanism
• Use random block sizes
• If treatment is double blind, no selection bias

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Biased Coin Design (BCD)Efron (1971) Biometrika

• Allocation probability to Treatment A changes to
  keep balance in each group nearly equal
• BCD (p)
• Assume two treatments A B
• D nA -nB "running difference" n nA nB
• Define p prob of assigning Trt gt 1/2
• e.g. PA prob of assigning Trt A
• If D 0, PA 1/2
• D gt 0, PA 1 - p Excess A's
• D lt 0, PA p Excess B's
• Efron suggests p2/3
• D gt 0 PA 1/3 D lt 0 PA 2/3

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Urn RandomizationWei Lachin Controlled
Clinical Trials, 1988

• A generalization of Biased Coin Designs
• BCD correction probability (e.g. 2/3) remains
  constant regardless of the degree of imbalance
• Urn design modifies p as a function of the degree
  of imbalance
• U(?, ?) two Trts (A,B)
• 0. Urn with ? white, ? red balls to start
• 1. Ball is drawn at random replaced
• 2. If red, assign B
• If white, assign A
• 3. Add ? balls of opposite color
• (e.g. If red, add ? white)
• 4. Go to 1.
• Permutational tests are available, but software
  not as easy.

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Analysis Inference

• Most analyses do not incorporate blocking
• Need to consider effects of ignoring blocks
• Actually, most important question is whether we
  should use complete randomization and take a
  chance of imbalance or use permuted-block and
  ignore blocks
• Homogeneous or Heterogeneous Time Pop. Model
• Homogeneous in Time
• Blocking probably not needed, but if blocking
  ignored, no problem
• Heterogeneoous in Time
• Blocking useful, intrablock correlations induced
• Ignoring blocking most likely conservative
• Model based inferences not affected by treatment
  allocation scheme. Ref Begg Kalish
  (Biometrics, 1984)

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Kalish Begg Controlled Clinical Trials, 1985

• Time Trend
• Impact of typical time trends (based on ECOG pts)
  on nominal p-values likely to be negligible
• A very strong time trend can have non-negligible
  effect on p-value
• If time trends cause a wide range of response
  rates, adjust for time strata as a co-variate.
  This variation likely to be noticed during
  interim analysis.

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Balancing on Baseline Covariates

• Stratified Randomization
• Covariate Adaptive
• Minimization
• Pocock Simon

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Stratified Randomization (1)

• May desire to have treatment groups balanced with
  respect to prognostic or risk factors
  (co-variates)
• For large studies, randomization tends to give
  balance
• For smaller studies a better guarantee may be
  needed
• Divide each risk factor into discrete categories
• Number of strata
• f risk factors
• li number of categories in factor i
• Randomize within each stratum
• For stratified randomization, randomization must
  be restricted! Otherwise, (if CRD was used), no
  balance is guaranteed despite the effort.

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Example Sex (M,F) and Risk (H,L)
1 2 Factors X 2 2 Levels in each ? 4
Strata 3 4
H
L
M
H
F
L
For stratified randomization, randomization must
be restricted! Otherwise, (if CRD was used), no
balance is guaranteed despite the effort!
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Stratified Randomization (2)

• Define strata
• Randomization is performed within each stratum
  and is usually blocked
• Example Age, lt 40, 41-60, gt60 Sex, M, FTotal
  number of strata 3 x 2 6 Age Male
  Female 40 ABBA, BAAB, BABA, BAAB, ...
  41-60 BBAA, ABAB, ... ABAB, BBAA, ... gt60
  AABB, ABBA, ... BAAB, ABAB, ..

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Stratified Randomization (3)

• The block size should be relative small to
  maintain balance in small strata, and to insure
  that the overall imbalance is not too great
• With several strata, predictability should not be
  a problem
• Increased number of stratification variables or
  increased number of levels within strata lead to
  fewer patients per stratum
• In small sample size studies, sparse data in many
  cells defeats the purpose of stratification
• Stratification factors should be used in the
  analysis
• Otherwise, the overall test will be conservative

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Comment

• For multicenter trials, clinic should be a factor
• Gives replication of same experiment.
• Strictly speaking, analysis should take the
  particular randomization process into account
  usually ignored (especially blocking) thereby
  losing some sensitivity.
• Stratification can be used only to a limited
  extent, especially for small trials where it's
  the most useful
• i.e. many empty or partly filled strata.
• If stratification is used, restricted
  randomization within strata must be used.

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Minimization Method (1)

• An attempt to resolve the problem of empty strata
  when trying to balance on many factors with a
  small number of subjects
• Balances Trt assignment simultaneously over many
  strata
• Used when the number of strata is large relative
  to sample size as stratified randomization would
  yield sparse strata
• A multiple risk factors need to be incorporated
  into a score for degree of imbalance
• Need to keep a running total of allocation by
  strata
• Also known as the dynamic allocation
• Logistically more complicated
• Does not balance within cross-classified stratum
  cells
• balances over the marginal totals of each
  stratum, separately

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Example Minimization Method (a)

• Three stratification factors Sex (2 levels),
• age (3 levels), and disease stage (3 levels)
• Suppose there are 50 patients enrolled and the
  51st patient is male, age 63, and stage III
• Trt A Trt B
• Sex Male 16 14
• Female 10 10
• Age lt 40 13 12
• 41-60 9 6
• gt 60 4 6
• Disease Stage I 6 4 Stage II 13 16
• Stage III 7 4
• Total 26 24

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Example Minimization Method (b)

• Method Keep a current list of the total patients
  on each treatment for each stratification factor
  level
• Consider the lines from the table above for that
  patient's stratification levels only Sign of
  
• Trt A Trt B Difference
• Male 16 14
• Age gt 60 4 6 -
• Stage III 7 4
• Total 27 24 2 s and 1 -

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Example Minimization Method (c)

• Two possible criteria
• Count only the direction (sign) of the difference
  in each category. Trt A is ahead in two
  categories out of three, so assign the patient to
  Trt B
• Add the total overall categories (27 As vs 24
  Bs).
• Since Trt A is ahead, assign the patient to
  Trt B

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Minimization Method (2)

• These two criteria will usually agree, but not
  always
• Choose one of the two criteria to be used for the
  entire study
• Both criteria will lead to reasonable balance
• When there is a tie, use simple randomization
• Generalization is possible
• Balance by margins does not guarantee overall
  treatment balance, or balance within stratum cells

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Covariate Adaptive Allocation(Sequential
Balanced Stratification)Pocock Simon,
Biometrics, 1975 Efron, Biometrika, 1971

• Goal is to balance on a number of factors but
  with "small" numbers of subjects
• In a simple case, if at some point Trt A has more
  older patients that Trt B, next few older
  patients should more likely be given Trt B until
  "balance" is achieved
• Several risk factors can be incorporated into a
  score for degree of imbalance B(t) for placing
  next patient on treatment t (A or B)
• Place patient on treatment with probability p gt
  1/2 which causes the smallest B(t), or the least
  imbalance
• More complicated to implement - usually requires
  a small "desk top" computer

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Example Baseline Adaptive Randomization

• Assume 2 treatments (1 2)
• 2 prognostic factors (1 2) (Gender Risk
  Group)
• Factor 1 - 2 levels (M F)
• Factor 2 - 3 levels (High, Medium
  Low Risk)
• Let B(t) Wi Range (xit1, xit2) wi weight
  for each factor
• e.g. w1 3 w1/w2 1.5
• w2 2
• xij number of patients in ith factor and jth
  treatment
• xitj change in xij if next patient assigned
  treatment t
• Let P 2/3 for smallest B(t) Pi (2/3, 1/3)
• Assume we have already randomized 50 patients
• Now 51st pt.
• Male (1st level, factor 1)
• Low Risk (3rd level, factor 2)

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• Now determine B(1) and B(2) for patient 51.
• If assigned Treatment 1 (t 1)
• (a) Calculate B(t) (Assign Pt 51 to trt 1) t
  1
• (1) Factor 1, Level 1
• (Male)
• Now Proposed
• K X1K ? X11K
• Trt Group 1 16 17
• 2 14 14
• Range 17-14 3

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(a) Calculate B(t) (Assign Pt 51 to trt 1)
t1 (2) Factor 2, Level 3 (Low Risk)
K X2K X12K Trt Group 1 4 ? 5
2 6 6 Range 5-6, 1 B(1) 3(3) 2(1)
11
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(b) Calculate B(2) (Assign Pt 51 to trt 2)
t2 (1) Factor 1, Level 1 (Male)
K X1K X21K Group 1 16 16
2 14 15 Range 16-15 1 (2) Factor
2, Level 3 (Low Risk) K X2k X22k Group 1 4 4
2 6 7 Range 4-7 3 B(2) 3(1) 2(3)
9
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(c) Rank B(1) and B(2), measures of
imbalance Assign t t B(t) with
probability 2 9 2/3 1 11 1/3 Note
minimization would assign treatment 2 for sure
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Response Adaptive Allocation Procedures

• Use outcome data obtained during trial to
  influence allocation of patient to treatment
• Data-driven
• i.e. dependent on outcome of previous patients
• Assumes patient response known before next
  patient
• The goal is to allocate as few patients as
  possible toa seemingly inferior treatment
• Issues of proper analyses quite complicated
• Not widely used though much written about
• Very controversial

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Play-the-Winner Rule Zelen (1969)

• Treatment assignment depends on the outcome of
  previous patients
• Response adaptive assignment
• When response is determined quickly
• 1st subject toss a coin, H Trt A, T Trt B
• On subsequent subjects, assign previous treatment
  if it was successful
• Otherwise, switch treatment assignment for next
  patient
• Advantage Potentially more patients receive the
  better treatment
• Disadvantage Investigator knows the next
  assignment

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Response Adaptive Randomization

• Example
• "Play-the-winner Zelen (1969) JASA
• TRT A S S F S S S F
• TRT B S F
• Patient 1 2 3 4 5 6 7 8 9 ......

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Two-armed Bandit or Randomized Play-the-Winner
Rule

• Treatment assignment probabilities depend on
  observed success probabilities at each time point
• Example ECMO trial
• Advantage Attempts to maximize the number of
  subjects on the superior treatment
• Disadvantage When unequal treatment numbers
  result, there is loss of statistical power in the
  treatment comparison

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ECMO Example

• References
• Michigan
• 1a. Bartlett R., Roloff D., et al. Pediatrics
  (1985)
• 1b. Begg C. Biometrika (1990)
• Harvard
• 2a. ORourke P., Crone R., et al. Pediatrics
  (1989)
• 2b. Ware J. Statistical Science (1989)
• 2c. Royall R. Statistical Science (1991)
• Extracoporeal Membrane Oxygenator(ECMO)
• treat newborn infants with respiratory failure or
  hypertension
• ECMO vs. conventional care

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Michigan ECMO Trial

• Bartlett Pediatrics (1985)
• Modified play-the-winner
• Urn model
• A ball ? ECMO
• B ball ? Standard control
• If success on A, add another A ball .
• Wei Durham JASA (1978)
• Randomized Consent Design
• Results
• sickest patient
• P-Values, depending on method, values ranged
• .001 6 .05 6 .28

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Harvard ECMO Trial (1)

• ORourke, et al. Pediatrics (1989)
• ECMO for pulmonary hypertension
• Background
• Controversy of Michigan Trial
• Harvard experience of standard
• 11/13 died
• Randomized Consent Design
• Two stage
• 1st Randomization (permuted block) switch to
• superior treatment after 4 deaths in worst arm
• 2nd Stay with best treatment

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Harvard ECMO Trial (2)

• Results
• Survival
• less severe patients
• P .054 (Fisher)

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Multi-institutional Trials

• Often in multi-institutional trials, there is a
  marked institution effect on outcome measures
• Using permuted blocks within strata, adding
  institution as yet another stratification factor
  will probably lead to sparse cells (and
  potentially more cells than patients!)
• Use permuted block randomization balanced within
  institutions
• Or use the minimization method, using institution
  as a stratification factor

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Mechanics of Randomization (1)

• Basic Principle - Analyze What is Randomized
• Timing
• Actual randomization should be delayed until just
  prior to initiation of therapy
• Example
• Alprenolol Trial, Ahlmark et al (1976)
• 393 patients randomized two weeks before therapy
• Only 162 patients treated, 69 alprenolol 93
  placebo

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Mechanics of Randomization (2)

• Operational
• 1. Sequenced sealed envelopes (prone to
  tampering!)
• 2. Sequenced bottles/packets
• 3. Phone call to central location
• - Live response
• - Voice Response System
• 4. One site PC system
• 5. Web based
• Best plans can easily be messed up in the
  implementation

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Example of Previous Methods (1)

• 20 subjects, treatment A or B, risk H or L
• Subject Risk
• 1 H Randomize Using
• 2 L
• 3 L 1. Simple
• 4 H
• 5 L 2. Blocked (Size4)
• 6 L
• 7 L 3. Stratify by risk use simple
• 8 L
• 9 H 4. Stratify by risk block
• 10 L
• 11 H
• 12 H For each compute
• 13 H
• 14 H 1. Percent pts on A
• 15 L
• 16 L 2. For each risk group, percent of pts on
  A

10 subjects with H 10 subjects with L
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Example of Previous Methods (2)

• 1. Simple 1st Try 2nd Try
• (a) 9/20 A's 7/20 A's OVERALL BY
• (b) H 5/10 A's 3/10 A's SUBGROUP
• L 4/10 A's 4/10 A's
• 2. Blocked (No stratification)
• (a) 10 A's 10 B's
• (b) H 4 A's 6 B's
• L 6 A's 4 B's
• 3. Stratified with simple randomization
• (a) 5 A's 15 B's
• (b) H 1 A 9 B's
• L 4 A's 6 B's
• 4. Stratified with blocking
• (a) 10 A's 10 B's MUST BLOCK TO MAKE
• STRATIFICATION PAY
• (b) H 5 A's 5 B's OFF

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