Designs for Clinical Trials

Chapter 5 Reading Instructions

� 5.1: Introduction� 5.2: Parallel Group Designs (read)� 5.3: Cluster Randomized Designs (less important)� 5.4: Crossover Designs (read+copies)� 5.5: Titration Designs (read)� 5.6: Enrichment Designs (less important)� 5.7: Group Sequential Designs (read include 10.6)� 5.8: Placebo-Challenging Designs (less important)� 5.9: Blinded Reader Designs (less important)� 5.10: Discussion

Design issues

Design issues

Objective

Research question

Design, Variables

Control

Ethics

FeasibilityCost

Statistics

Variability

Confounding

Statistical optimality not enough!Use simulation models!

Parallel Group Designs

R

Test

Control 1

Control 2

ijiijY εµ +=

jni ,,1 K=kj ,,1 K=

( )2,0~ σε Nij�Easy to implement�Accepted�Easy to analyse�Good for acute conditions

Where does the varation go?

8 week blood pressure

Placebo

Drug X

εXβY +=

Anything we can explain Unexplained

Between and within subject variation

Baseline 8 weeks

Placebo

Drug XDBPmmHg

Female

Male

What can be done?

Stratify:

More observations per subject:

Run in period:

Randomize by baseline covariate and put the covariate in the model.

BaselineMore than 1 obsservation per treatment

Ensure complience, diagnosis

Parallell group design with baseline

R

Test

Control 1

Control 2

Baseline 8 weeks

Compare bloodpressure for three treatments, one test and two control.

Model: { } ijkkjiijkY εξτµ +++= =21 2,1=i3,2,1=j

( )2,0 iid σε Nijk

( )2,0 iid sk N σξjnk ,,1K=

Observation

Treatment

SubjectjτTreatment effect

Subject effectRandom error

Change from baseline

The variance of an 8 week value is

22 σσ +s

Change from baseline ( )jkjjkjkjkjk YYZ 2121 ετε +−=−=

The variance of change from baseline is 22σ

Usually 22222 2 ss σσσσσ +<⇒<

( ) { }( )ijkkjijk VarYVar εξτµ +++= =22 1

( ) ( ) ( )=+=+= ijkkijkk VarVarVar εξεξ

Baseline as covariate

R

Test

Control 1

Control 2

Baseline 8 weeks

ijijij xY εβτµ +++=

( )2,0 iid σε Nijk

Subject

TreatmentTreatment effect

jni ,,1K=

3,2,1=jjτ

Random error

Baseline value ix

Model:

Crossover studies

�All subject gets more that one treatment�Comparisons within subject

R

Was

h ou

t

A

AB

B

Period 1 Period 2

Sequence 1

Sequence 2

A B

B A

�Within subject comparison�Reduced sample size�Good for cronic conditions�Good for pharmaceutical studies

Model for a cross over study

021 =+γγ

021 =+ππ0=+ BA ττ

2,1 sequence ofeffect == iiγ( )2

)( ,0 iid sequence within ,,1subject ofeffect siki Nink σξ K==

2,1 period ofeffect == jjπ{ }BAtt , treatmentofeffect ∈=τ

( )20, iiderror random σε Nijk =

ijkrjtjkiiijkY ελδτπξγµ ++++++= )(

Obs=Period+sequence+subject+treament+carryover+error

2 by 2 Crossover design

[ ] =ijkYEAτπγ ++ 11 AB λτπγ +++ 21

Bτπγ ++ 12 BA γτπγ +++ 12

11µ 12µ22µ21µ

( ) ( )( )

( ) ( )( )

( )BABA

BBAABA

λλττ

λττππλττππ

µµµµ

+−−

=−−+−+−−+−

=−+−

21

2121

1221

21221211

Effect of treatment and carry over can not be separated!

Matrix formulation

( )T22211211 ,,, µµµµµ =( )TAτπγµβ 11=

βµ X=

−−−−−−

=

111111111111

1111

X

ijktjkiiijkY ετπξγµ +++++= )(

021 =+λλ021 =+ππ021 =+ττ

Model:

Sum to zero:

Matrix formulation

Matrix formulation

( )

=−

25.0000025.0000025.0000025.0

1XXT

( ) ( ) 2ˆˆ σβ XXTCov =

( ) YTT XXX 1ˆ −=β

( ) TYY TT Xβσ ˆˆ 2 −=

Parameter estimate:

( ) 2ˆ25.0ˆ στ =AVar

Estimates independent and

Alternatives to 2*2Compare A B

B A

A B

B A

B

Ato

Same model but with 3 periods and a carry over effect

2,1 sequence ofeffect == iiγ( )2

)( ,0 iid sequence within ,,1subject ofeffect siki Nink σξ K==

3,2,1 period ofeffect == jjπ{ }BAtt , treatmentofeffect ∈=τ

( )20, iiderror random σε Nijk =

021 =+γγ021 =+ππ

0=+ BA ττ

ijkrjtjkiiijkY ελδτπξγµ ++++++= )(

Parameters of the AAB, BBA design

11µ 12µ22µ21µ

13µ

23µ

ijktjkiiijkY ετπξγµ +++++= )( [ ] ijkijkYE µ=

Aτπγµµ +++= 1111

AB λτπγµµ ++++= 2112

BB λτπγµµ ++++= 3113

Bτπγµµ +++= 1221

BA λτπγµµ ++++= 2222

BA λτπγµµ ++++= 3223

021 =+λλ 0321 =++ πππ 021 =+ττ 0=+ BA λλ

Matrix again

−−−−−

−−−−−−

−

=

A

A

λτππγµ

β2

1

1

111111111011

010111111111

111011010111

X

( )

−−

=−

25.000000019.00006.000033.017.0000017.033.000006.00019.000000017.0

1XXT

Effect of treatment and carry over can be estimated independently!

Other 2 sequence 3 period designs

A B

B A

B

A

A A

B B

B

A

A B

B A

A

B

A A

B B

A

B

1

2

3

4

( )AVar τ̂

( )AVar λ̂

( )AACorr λτ ˆ,ˆ

1 2 3 4

0.19 0.75 0.25 N/A

0.25 1.00 1.00 N/A

0.0 0.87 0.50 N/A

Comparing the AB, BA and the ABB, BBA designs

A B

B A

B

A

A B

B A

( ) 2ˆ25.0ˆ στ =AVar ( ) 2ˆ19.0ˆ στ =AVarCan�t include carry over Carry over estimable2 treatments per subject 3 treatments per subject

Shorter duration Longer duration

Excercise: Find the best 2 treatment 4 period design

More than 2 treatments

Tool of the trade: Define the model

( ) 1−XXT

A B

B C

C

A

C A

A C

B

B

B A

C B

C

A

A B

B D

C

A

D

C

C A

D C

D

B

B

A

Investigate

A B

B C

C A

A C

B C

C B Watch out for drop outs!

Titration DesignsIncreasing dose panels (Phase I):

�SAD (Single Ascending Dose)�MAD (Multiple Ascending Dose)

Primary Objective:

�Establish Safety and Tolerability�Estimate Pharmaco Kinetic (PK) profile

Increasing dose panelse (Phase II):

Dose - response

Titration Designs (SAD, MAD)

�X on drug�Y on Placebo

Dose: Z1 mg

�X on drug�Y on Placebo

Dose: Z2 mg

�X on drug�Y on Placebo

Dose: Zk mg

Stop if any signs of safety issues

VERY careful with first group!

Titration DesignsWhich dose levels?

�Start dose based on exposure in animal models.�Stop dose based on toxdata from animal models.�Doses often equidistant on log scale.

Which subject?

�Healty volunteers�Young�Male

How many subjects?

�Rarely any formal power calculation.�Often 2 on placebo and 6-8 on drug.

Titration Designs

Not mandatory to have new subject for each group.

Gr. 1 Gr. 2 Gr. 3Gr. 1 Gr. 2 Gr. 4

X1 mg X2 mg X3 mg X4 mg X5 mg XY mg

12+2 12+212+212+212+2 12+2

�Slighty larger groups to have sufficiently many exposed.�Dose in fourth group depends on results so far.�Possible to estimate within subject variation.

Factorial designEvaluation of a fixed combination of drug A and drug B

A placeboB placebo

A activeB placebo

A placeboB active

A activeB active

The U.S. FDA�s policy (21 CFR 300.50) regarding the use of a fixed-dose combination

The agency requires:

Each component must make a contribution to the claimed effect ofthe combination.

Implication: At specific component doses, the combination must be superior to its components at the same respective doses

P

B AB

A

Factorial designUsually the fixed-dose of either drug under study has been approved for an indication for treating a disease.

Nonetheless, it is desirable to include placebo (P) to examine the sensitivity of either drug give alone at that fixed-dose (comparison of AB with P may be necessary in some situations).

Assume that the same efficacy variable is used for studying both drugs (using different endpoints can be considered and needs more thoughts).

Factorial design

Sample mean Yi ∼ N( µi , σ2/n ), i = A, B, AB n = sample size per treatment group (balanced design is assumed for simplicity).

H0: µAB ≤ µA or µAB ≤ µB

H1: µAB > µA and µAB > µB j=A, B

Min test and critical region:

BAjYYnT jAB

jAB , ; ˆ2: =

−=

σ

( ) CTT BABAAB >:: ,min

Group sequential designs

A large study is a a huge investment, $, ethics

�What if the drug doesn�t work or is much better than expected? �Could we take an early look at data and stop the study is it look good (or too bad)?

Repeated significance test

Let: ( )2,~ σµ jij NY Test: 210 : µµ =H

Test statistic:mk

Zmk 221

2

ˆˆ

σµµ −= ∑

=

=mk

iijj Y

mk 1

1µ̂

0HFor 1,1 −= Kk K If αCZk ≥ Stop, reject

otherwise Continue to group 1+k

If αCZk ≥ Stop, reject 0H

otherwise stop accept 0H

True type I error rate

Tests Critical value P(type I error)1 1.96 0.05 2 1.96 0.08 3 1.96 0.11 4 1.96 0.13 5 1.96 0.24

Repeat testing until H0 rejected

Pocock�s testSuppose we want to test the null hypothesis 5 times using the same critical value each time and keep the overall significance level at 5%

For 1,1 −= Kk K If ( )KCZ pk ,α≥ Stop, reject

otherwise Continue to group 1+k

If ( )KCZ pk ,α≥ Stop, reject 0H

otherwise stop accept 0H

Choose ( )KCp ,α Such that

α== )1 analysisany at Reject ( 0 KkHP K

After group K

0H

Pocock�s testTests Critical value P(type I error)1 1.960 0.05 2 2.178 0.05 3 2.289 0.05 4 2.361 0.05 5 2.413 0.05

All tests has the same nominal significance level

A group sequential test with 5 interrim tests has level

( )( ) 0158.0413.212' =−= φα

Pocock�s test

2.413

-2.413

kZ

k stage

0Reject H

0Reject H

0Accept H

O�Brian & Flemmings test

For 1,1 −= Kk K If ( ) kKKCZ pk /,α≥ Stop, reject

otherwise Continue to group 1+k

If ( )KCZ pk ,α≥ Stop, reject 0Hotherwise stop accept 0H

Choose ( )KCp ,α Such that

α== )1 analysisany at Reject ( 0 KkHP K

After group K :

:

Increasing nominal significance levels

0H

O�Brian & Flemmings test

Test (k) CB(K,αααα) CB(K,αααα)*Sqrt(K/k) αααα’ 1 2.04 4.56 0.000005 2 2.04 3.23 0.0013 3 2.04 2.63 0.0084 4 2.04 2.28 0.0225 5 2.04 2.04 0.0413

Critical values and nominal significance levels for a O�Brian Flemming test with 5 interrim tests.

Rather close to 5%

O�Brian & Flemmings test

-6

-4

-2

0

2

4

6

0 1 2 3 4 5 6 Stage K

Zk

0.000005

0.00130.0084 0.0225 0.0413

Comparing Pocock and O�Brian Flemming

O’Brian Flemming Pocock Test (k) CB(K,a)*Sqrt(K/k) αααα’ CP(K,a) αααα’ 1 4.56 0.00001 2.413 0.0158 2 3.23 0.0013 2.413 0.0158 3 2.63 0.0084 2.413 0.0158 4 2.28 0.0225 2.413 0.0158 5 2.04 0.0413 2.413 0.0158

Comparing Pocock and O�Brian Flemming

-6

-4

-2

0

2

4

6

0 1 2 3 4 5

Stage k

Zk

Group Sequential Designs

�Efficiency Gain (Decreasing marginal benefit)�Establish efficacy earlier�Detect safety problems earlier

�Smaller safety data base�Complex to run�Need to live up to stopping rules!

Pros:

Cons:

Selection of a design

The design of a clinical study is influenced by:

�Number of treatments to be compared�Characteristics of the treatment�Characteristics of the disease/condition�Study objectives�Inter and intra subject variability�Duration of the study�Drop out rates

Backup

( ) ( ) ( )⋅⋅⋅⋅ +=− 22212221 YVarYVarYYVar( ) ( )22

22

2221

11 σσσσ +++= ss nnBack up: