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Randomization: Too Important to Gamble with

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Randomization: Too Important to Gamble with

A Presentation for the Delaware Chapter of the ASAOct 18, 2012

Dennis Sweitzer, Ph.D., Principal Biostatistician

Medidata Randomization Center of Excellence

- Randomized Controlled Trials
- Basics
- Balance
- Randomization methods
- Complete Randomization
- Strict Minimization
- Permuted Block
- Dynamic Allocation (Covariate-adaptive, not Response-Adaptive)
- Randomization Metrics
- Balance
- Predictability
- Loss of Power /Loss of Efficiency
- Secondary Imbalance: drop-outs
- Simulations comparing methods
- Confounding site & treatment effects (small sites)
- Overall performance
- Discontinuing patients
- Weighting stratification factors
- Meta-Balance

Why randomize anyway?

Some basic principles

Randomized Controlled Trial

- Trial: Prospective & Specific
- Controlled:
- Comparison with Control group
- (placebo or active)

- Controlled procedures ⇒ Only Test Treatment Varies

- Comparison with Control group
- Randomization: Minimizes biases
- Allocation bias
- Selection bias
- Permits blinding

¿ The Fact of bias ?

- (conscious, unconscious, or instinctive)
¿ The Question of bias ?

- Always 2nd guessing
- Critics will think of unanticipated things
¡ Solution!

- Treat it as a game
- 1 statistician vs N clinicians
- Statistician generates a random sequence
- Clinicians sequential guess at each assignment
- Statistician wins if clinician guesses are no better than chance (NB: 75% wrong is just as bad as 75% right)

What do we want in a randomization sequence or system?

Randomness Unpredictable

⟶ Reduce Allocation Bias (All studies)

⟶ Reduce Selection Bias (All studies)

⟶ Reduce placebo effects (Blinded studies)

Balance “Loss of Efficiency”

⟶ Maximizes statistical power

⟶ Minimize Confounding

⟶ Enhance Credibility (Face Validity)

Balancing

- Equal allocation between treatment arms
- Maximizes Statistical Power

Test

Control

- Statistical power limited by smallest arm
- 36 subject simulation with Complete Randomization
- ⟶
- average loss ≈ 1 subject10% lose ≥2 subject
- Can add 2 to compensate
- BUT only large imbalances have much effect on statistical power

- Severe Imbalances are rare in large studies
- Pr{worse than 60:40 split} for:
- n=25 ⟶ <42% n=100 ⟶ <4.4% n=400 ⟶ 0.006%

Resulting in light weight results….

1:1 randomization maximizes power per patient

But there are other considerations

- Utility:
- Need 100 patients on drug to monitor safety
- Study only requires 60 (30/arm)
- 2:1 randomization ⟶ 100 Test & 50 Placebo

- Motivation:
- Better enrollment if 75% chance of Test drug (3:1)

- Ethics:
- 85 Placebo + 255 Test vs. 125 Placebo + 125 Test

- Overall balance
- Only an issue for small studies

- Subgroup Balance
- Fixed size studies can have variable sized subgroups ⟶ Increased risk of underpowered subgroups

More than half of N=36 studies effectively “lost” 2 to 6 subjects because of imbalances

Males

Females

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Test

Test

Con

Effective Loss = Reduction of Power as Reduction in Sample Size

Simulations of:

36 and 18 subjects,

males as strata at 33% of population,

randomized 1:1

(complete randomization)

Males

Females

Leads to conversations like:

Higher estrogen levels in patients on Test Treatment ??

ANCOVA showed no differences in estrogen levels due to treatment

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Credibility…..

Test

Test

Hmm…?

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Treatment Imbalances within factors ⟶ spurious findings…..

Randomization Methods

(See Animated Powerpoint Slides…)

4 methods

Complete Randomization (classic approach)

Strict Minimization

Permuted Block (frequently used)

Dynamic Allocation (gaining in popularity)

Every assignment

Same probability for each assignment

Ignore Treatment Imbalances

No restrictions on treatment assignments

Advantages:

Simple

Robust against selection & accidental bias

Maximum Unpredictability

Disadvantage

High likelihood of imbalances (smaller samples)

.

Strict Minimization randomizes to the imbalanced arm

- Strict Minimization rebalances the Arms
- BUT at a cost in predictability
- Random only when treatments are currently balanced

Blocks of Patients (1, 2, or 3 per treatment)Here: 2:2 Allocation

T T

P P

T P

P T

P P

T T

T P

P ?

T P

T P

P T

T P

P T

P T

T P

P T

- Some Predictability

(Unless Incomplete Blocks:More strata ⟶ More incomplete)

T P

P *

- Balanced

- Biases Randomization to the imbalanced arm
- Unpredictable
- Almost Balanced

- Complete Randomization
- Optimizes Unpredictability
- Ignores Balance
- Strict Minimization
- Optimizes Balance
- Ignores Predictability
- Dynamic Allocation
- 2nd Best Probability Parameter
- Controls Balance vs. Predictability
- Tradeoff

2nd Best Probability= 0

⟶Strict Minimization

2nd Best Probability= 0.5

⟶Complete Randomization

(for 2 treatment arms)

Stratification Factors

Factors ≣ Main Effects

Strata ≣ 1st Order Interactions

Randomizing a 25 yo Male:

To PLA ⟶ Worsens Male balance

To Test ⟶ Worsens 18-35yo balance

Over both sexes

Balance w/in 6 Strata?

Males

Females

18-35 yo

35-65 yo

Marginal Balance

>65 yo

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Marginal Balance

Test

Test

Test

Test

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Over all Ages:

Test

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Test

Test

Test

Test

Test

Overall Balance

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- Only balances within strata
- Most strata will have incomplete blocks
- Imbalances accumulate at margins

P P * *

T P

x x

Over both sexes

Males

Females

T P

P T

P *

* *

T T

* *

T *

* *

18-35 yo

T P

T *

T P

P *

P T

* *

T P T *

35-65 yo

P T

T *

P T

P *

T T

P *

P*

* *

>65 yo

Pla

Pla

T T

P P

P P

T T

Test

Test

Pla

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Test

Test

Over all Ages:

Test

Test

T C

T C

P T

T P

P T

P T

Pla

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- * Only balances on margins
- * Useful if too many strata, e.g.:
- * Appropriate for a main effects analysis (ie, no interactions)
- *
- *

Over both sexes

Balance w/in 6 Strata?

Males

Females

18-35 yo

35-65 yo

Marginal Balance

>65 yo

Pla

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Marginal Balance

Test

Test

Test

Test

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Over all Ages:

Test

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Test

Test

Test

Overall Balance

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- DA: uses weighted combination of
- Overall balance
- Marginal balances
- Strata balance
- ⇒ Flexible

Over both sexes

Males

Females

Balance w/in 6 Strata?

18-35 yo

35-65 yo

Marginal Balance

>65 yo

Pla

Pla

Pla

Pla

Marginal Balance

Test

Test

Test

Test

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Over all Ages:

Test

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Test

Test

Test

Test

Overall Balance

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Site as a Special Subgroup

(Max 2 lines, 35 characters)

- Overall balance
- Only an issue for small studies

- Subgroup Balance
- Fixed size studies can have variable sized subgroups ⟶ Increased risk of underpowered subgroups

- Site as special case of subgroup
- Small sites ⟶ Increased risk of "monotherapy” at site ⟶ Confounding site & treatment effects ⟶ Effectively non-informative/”lost” patients
- Actual vs Assumed distribution of site size

- Data Sample
- 13 Studies
- 7.7 mo Average Enrollment period
- 3953 Obs.Pts
- 460 Listed Sites
- 372 Active.Sites

Size Categories:{0, 1, 2, 3, 4-7, 8-11, 12-15, 16-19, 20-29, 30-39, 40-49, 50-59, 60-79, 80-99, 100-149, 150-199, ≥200 }

- Data Sample
- 13 Studies
- 7.7 mo Average Enrollment period
- 3953 Obs.Pts
- 460 Listed Sites
- 372 Active.Sites

# Sites per Size Category {0, 1, 2, 3, 4-7, 8-11, 12-15, 16-19, 20-29, 30-39, 40-49, 50-59, 60-79, 80-99, 100-149, 150-199, ≥200 }

Simulation based on Observations

- 4 mo Enrollment Period
- Enrollment ~ Poisson distribution
μ = Obs. Pts/mo (active sites) or

μ ≈ 0.5 / Enrollment period (non-active sites)

- Randomize using CR, PB(2:2), or DA(0.15).
- Confounded Pts ≣ Patients at centers with only one treatment
⇒ treatment & center effects are confounded

mean ±SD (80% C.I.)

Affected studies had many sites with low enrollment

Studies with fewer sites (and more pts at each) were rarely affected

Dynamic Allocation reduced confounding slightly more effectively than permuted block

Randomization Metrics

How do we measure “badness” of a randomization sequence or system?

- Predictability
- Goal: an observer can guess no better than chance
⟶ Score based on Blackwell-Hodges guessing rule

- Easily calculated

- Goal: an observer can guess no better than chance
- Imbalance
Imbalance ⟶ reduced statistical power⟶ “Loss of Efficiency”

- Measure as effective loss in number of subjects

Use Blackwell-Hodges guessing rule

- Directly corresponds to game interpretation
- Investigator always guesses the most probable treatment assignment, based on past assignments
- “ bias factor F”
F ≣ abs(# Correct – Expected # Correct by chance alone)

- Measures potential for selection bias
- Modifications:
- Limits on knowledge of investigator (eg, can only know prior treatment allocation on own site)
- Score as percentage
e.g., Score ≣ abs(% Correct – 50%)

For treatment sequence “TCCC”

Initial guess ⟶ Expectation = ½

“T” ⟶ Imbalance =+1 ⟶ Guess C ⟶ Correct

“TC” ⟶ Imbalance=0 ⟶ Guess either ⟶ Expectation=½

“TCC” ⟶ Imbalance=-1 ⟶ Guess T ⟶ Wrong

“TCCC” ⟶ # Correct= ½ + 1+ ½ +0 =2

Score = #Correct - 2 = 2-2 = 0

For treatment sequence “TCCC”

“TCCC” ⟶ # Correct= ½ + 1+ ½ +0 =2

Complete Randomization ⇒ Pr{“TCCC”} = 1/16

Dynamic Allocation (p=0.15) ⇒ Pr{“TCCC”}= 0.5 *0.85 * 0.5 * 0.15 = 0.031875

Permuted Block (length≤4) ⇒ PR{“TCCC”} = 0

Strict Minimization ⇒ Pr{“TCCC”}=0

- Sequence “TCCT”
- # Correct= ½ + 1 + ½ + 1 = 3
- Score = 3 – 2 = 1
- Complete Randomization⇒ Pr{TCCT}= 1/16
- Strict Minimization ⇒ Pr{TCCT} = ½*1*½*1 = ¼
- Permuted Block⇒ Pr{TCCT} = 1/6
- (NB: 6 permutations of TTCC)
- Dynamic Allocation (2nd best prob.=0.15) ⇒ Pr{TCCT} = 0.5 * 0.85* 0.5 * 0.85 = 0.180625

Local PredictabilityONLY

Blackwell-Hodges

- Assesses potential selection bias ― Given known imbalance!¿¿ But which imbalance(s)??(Overall imbalance? Within strata? Within Factors?)
- Henceforth: only use imbalance within strata
- Proxy for center
- Assume observer only knows imbalance within “his center”
- Simple & unambiguous
M Requires some caution in interpretation

Inference in Covariate-Adaptive allocation

Elsa ValdésMárquez & Nick Fieller

EFSPI Adaptive Randomisation Meeting

Brussels, 7 December 2006

http://www.efspi.org/PDF/activities/international/adaptive-rando-docs/2ValdesMarquez.pdf

- Loss can be expressed as equivalent # Patients
- In a 100 patient study:Loss of Efficiency= 5
⇒ A perfectly designed study would require only 95

Designed Experiment (DOE):

⟶ Selectzand covariate values to minimize Ln

RCT ⟶ Select only z (No control of covariates)

X ≣ design matrix:

⟶n rows, 1 per pt

⟶K columns, 1 per covariate

z ≣ Treatment assignments

Dynamic Allocation

Sequentially assign Z to minimize

Randomization Performance Simulations

Plot B-H score

vs

Loss of Efficiency

Median

+

80% C.I. ⇒

10% lower& 10% higher

Both DA & PB are stratified.

Simulation: 48 subjects, 2 stratification factors, 6 strata, uneven sizes(DA) Dynamic Allocation (PB)Permuted Block (CR) Completely RandomDA( 2nd Best Probability ), PB( Allocation Ratio )Simulated subjects were randomized by all 3 methods

⟵Averages of Metrics

But for managing risk, need Worst Case

80% ⟶ Confidence Intervals

PB(1:1), DA(0)

PB(2:2), DA(0.15)

(Essentially Strict Minimization)

PB(8:8)

DA(0.5)CR

DA(0.5) ≣ CR PB⟶CR

PB(4:4)DA(0.33)

CR

DA(0.25)PB(3:3)

- 1,000 simulations per case
* 48 subjects each

* 6 Strata, 2 factor, Variety of proportions

DA(0.25)PB(3:3)

DA(0.25)PB(3:3)

Special Topics

Local PredictabilityONLY

Local PredictabilityONLY

DA(0) balanced only within strata Approximates PB(1:1)

DA(0) equal weighting Approximates PB(1:1)

DA(0) balanced on margins Intermediate properties

DA(0) balanced only overall Approximates CR (large N) NB: Predictability is limited to imbalance within a stratum

dafd

Weighting: (Strata, Margins, Overall)

DA(0) Equal Weighting (1,1,1) Strata Balance Dominates Approximates PB(1:1)

DA(0) Margin & Strata (1:9:0) Separates from PB(1:1)

DA(0) Unequal Weighting (1,6,20)

DA(0) Margin Balance (0,1,0)

DA(0) Overall Balance (0,0,1) Approx. CR

Local PredictabilityONLY

jjjjjj

Distance function ≣ Weighted Sum of Imbalances

- Relative Imbalance:
- Factor as Union of Strata ⇒
⇒ Strata Imbalances dominate Distance function

Over both sexes

Over both sexes

Over both sexes

- Stratified Randomization weights on strata, not margins or overall
- Imbalances within strata tend to dominate in DA

Males

Males

Males

Females

Females

Females

18-35 yo

18-35 yo

18-35 yo

35-65 yo

35-65 yo

35-65 yo

- Minimization weights on margins, not strata.
- DA can weight exclusively on margins

>65 yo

>65 yo

>65 yo

- If a Strata is balanced, the next assignment attempts to balance the margins.
- Since small groups are more likely to have imbalances which reduce efficiency, balancing strata 1stis appropriate

Pla

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Over all Ages:

Over all Ages:

Over all Ages:

Test

Test

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Over both sexes

- While Imbalances within strata tends to dominate in DA,if a Strata is balanced, the next assignment attempts to balance the margins
- Since small group imbalances tend to dominate, balancing tends to be sequential

Males

Females

18-35 yo

35-65 yo

- ⟵ This example:
- Balance within strata
- If balanced within the strata, balance by age group(since age groups tend to be smaller than sex groups)
- If balanced within age group, balance within sex group
- If balanced within sex group, balance overall
- However: cumulative imbalances may change this order

>65 yo

Pla

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Test

Test

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Over all Ages:

Test

Test

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dfd

Replacement Randomization

Patients discontinue

⟶ Imbalances

⟶ Reduced efficiency

“Tight” randomizations(PB with small blocks, DA with small 2nd best Prob.)

⟶ Lose more efficiency

“Loose” randomizations

(CR, PB with large blocks, DA with large 2nd best Prob.)

⟶ Lose less efficiency

⟶ Little or no change

25% DC

No DC

Dynamic Allocation: Can allocate new patients to restore balance

25% DC

DA Adj.

No DC

“Tight” randomizations(PB with small blocks, DA with small 2nd best Prob.)

⟶ Lose more efficiency

⟶ Benefit most

“Loose” randomizations

(CR, PB with large blocks, DA with large 2nd best Prob.)

⟶ Lose less efficiency

⟶ Little or no benefit

DA Adj.

25% DC

No DC

- High drop-out ⇒ PB, DA ⟶ CR
- Drop-out before becoming evaluable
- Constrained resources (small sample size, limited drug supply, ….)

- Crossover studies: Requires completers
- Evaluable Complete Sequence of Treatments

- Provisional Randomization / Randomize to ship
- Screening visit triggers:
- Randomize at screening
- If randomized treatment not on-site, ship blinded supplies

- Randomization visit:
- If patient eligible⇒ dispense assigned treatment
- If not eligible⇒store for next eligible patient

- Minimizes on-site drug supply

- Screening visit triggers:

Equipose ⇒ (less random is acceptable)

Small Study ⇒ Efficiency important ⟶ Lower 2nd Best Probability

Large Study ⇒ Are there small subgroups?All subgroups large ⟶ CR is acceptable

Small subgroups ⇒ Need more efficiency⟶ Smaller 2nd best Prob

- Smaller Studies
- Studies with small subgroups
- Early phase studies
- Interim Analyses
- Equipoise
- Strong Blinding
- Objective Endpoints
- Many Strata / Many centers
- Limited blinded supplies

- Large Studies
- Studies with large subgroups
- Late phase studies
- Strong Treatment preferences
- Weak Blinding
- Subjective Endpoints

⟶ ⟶ ⟶ Balanced

Unpredictable ⟵ ⟵⟵

Elsa ValdésMárquez & Nick Fieller. Inference in Covariate-Adaptive allocation. EFSPI Adaptive RandomisationMeeting, Brussels, 7 December 2006