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Designs for Clinical Trials
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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
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Design issues
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Design issues
Objective
Research question
Design, Variables
Control
Ethics
FeasibilityCost
Statistics
Variability
Confounding
Statistical optimality not enough!Use simulation models!
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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
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Where does the varation go?
8 week blood pressure
Placebo
Drug X
εXβY +=
Anything we can explain Unexplained
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Between and within subject variation
Baseline 8 weeks
Placebo
Drug XDBPmmHg
Female
Male
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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
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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
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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
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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:
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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
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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
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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!
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Matrix formulation
( )T22211211 ,,, µµµµµ =( )TAτπγµβ 11=
βµ X=
−−−−−−
=
111111111111
1111
X
ijktjkiiijkY ετπξγµ +++++= )(
021 =+λλ021 =+ππ021 =+ττ
Model:
Sum to zero:
Matrix formulation
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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
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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 ++++++= )(
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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
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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!
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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
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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
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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!
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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
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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!
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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.
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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.
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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
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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).
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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
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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)?
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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
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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
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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
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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
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Pocock�s test
2.413
-2.413
kZ
k stage
0Reject H
0Reject H
0Accept H
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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
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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%
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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
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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
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Comparing Pocock and O�Brian Flemming
-6
-4
-2
0
2
4
6
0 1 2 3 4 5
Stage k
Zk
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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:
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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
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Backup
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22
2221
11 σσσσ +++= ss nnBack up: