Designs for Clinical Trials
Chapter 5 Reading Instructions
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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
Research question
Confounding
Statistics
Variability
Statistical optimality not enough!
Use simulation models!
Objective
Control
Design, Variables
Cost
Ethics
Feasibility
Parallel Group Designs
R
Test
Y ij   i   ij
Control 1
i  1,  , n j
Control 2
•Easy to implement
•Accepted
•Easy to analyse
•Good for acute conditions
j  1,  , k
 ij ~ N 0 , 
2

Where does the varation go?
Placebo
Drug X
8 week blood pressure
Y  Xβ  ε
Anything we can explain
Unexplained
Between and within subject
variation
Placebo
DBP
mmHg
Drug X
Female
Male
Baseline
8 weeks
What can be done?
Stratify: Randomize by baseline
covariate and put the
covariate in the model.
More observations per subject: Baseline
More than 1 obsservation
per treatment
Run in period:
Ensure complience, diagnosis
Parallell group design with
baseline
Test
Compare bloodpressure
for three treatments, one
test and two control.
Control 1
R
Control 2
Baseline
Observation i  1, 2
Model: Yijk    1i  2  j   k   ijk
Treatment effect
Subject effect
Random error

j
 k iid N 0 ,  s
2
 ijk iid N 0 , 

2
8 weeks

Treatment
j  1, 2 ,3
Subject
k  1,  , n j
Change from baseline
The variance of an 8 week value is
Var Y 2 jk   Var   1i  2  j   k   ijk 
 Var  k   ijk   Var  k   Var  ijk    s2  
Change from baseline
Z
jk
 Y1 jk  Y 2 jk   1 jk   j   2 jk 
The variance of change from baseline is 2
Usually 
2
  s  2
2
2
2

2
s
2
2
Baseline as covariate
Test
Control 1
R
Control 2
Baseline
Model:
8 weeks
Subject
Yij     j   x i   ij
Treatment effect

Baseline value
xi
Random error
Treatment
j
 ijk iid N 0 , 
2

i  1,  , n j
j  1, 2 ,3
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(C) Stephen Senn 2004
12
Crossover studies
•All subject gets more that one treatment
•Comparisons within subject
R
B
Period 1
B
Wash out
A
A
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
Obs=Period+sequence+subject+treament+carryover+error
Yijk     i   i ( k )   j   t   j  r   ijk
 i  effect of sequence
i  1, 2  1   2  0
 i ( k )  effect of subject k  1,  , n i within
 j  effect of period
 t  effect of treatment
1  2  0
j  1, 2
t  A , B 
 ijk  random error iid N 0, 
sequence
2

 A B  0

i iid N 0 , 
2
s

2 by 2 Crossover design
 
E Yijk 
1
2
1
2
 11  12
1 1  A
1   2  B  A
 22
 2 1 B
 2 1  A  
 21
  11   12     22
  21  
 1   2   A   B
  A    2   1   A   B   B  
 A  B 
1
2
 A   B 
Effect of treatment and carry
over can not be separated!
B
Matrix formulation
Model:
Yijk     i   i ( k )   j   t   ijk
1   2  0
Sum to zero:
1  2  0
1   2  0
Matrix formulation   X 
  
1
1
A
T
    11 ,  12 ,  21 ,  22 
T
1

1
X  
1

1
1
1
1
1
1
1
1
1
1 

1

 1

1 
Matrix formulation
Parameter estimate: ˆ  X T X  1 X T Y
2
T
T
ˆ  Y Y   ˆX T
 
T
2
Cov ˆ  X X ˆ
X X 
T
1
 0 . 25

0


 0

 0
0
0
0 . 25
0
0
0 . 25
0
0
0 

0

0 

0 . 25 
Estimates independent and
2
Var ˆ A   0 . 25 ˆ
Alternatives to 2*2
Compare
A
B
B
A
to
A
B
B
B
A
A
Same model but with 3 periods and a carry over effect
Yijk     i   i ( k )   j   t   j  r   ijk
 i ( k )  effect of subject k  1,  , n i within
 i  effect of sequence
 j  effect of period
i  1, 2
j  1, 2 ,3
sequence
1  2  0
1  2  0
 t  effect of treatment t  A , B 
2
 ijk  random error iid N 0,    A   B  0

i iid N 0 , 
2
s

Yijk
Parameters of the AAB, BBA
design
   
   
E Y   
i
i(k )
 11  12  13
 21
 22  23
j
t
ijk
ijk
ijk
 11     1   1   A
 12     1   2   B   A
 13     1   3   B   B
 21     2   1   B
 22     2   2   A   B
 23     2   3   A   B
1   2  0
1  2  3  0
1   2  0
A  B  0
Matrix again
1

1

1
X  
1
1

1
X X 
T
1
 0 . 17

0

 0
 
 0
 0

 0
1
1
0
1
1
0
1
1
1
1
1
1
1
1
0
1
1
0
1
1
1
1
1
1
0   
 

1
 1 
 1   1 
 
0   2 
 1  A 
 
1    A 
0
0
0
0
0 . 19
0
0
0 . 06
0
0 . 33
 0 . 17
0
0
 0 . 17
0 . 33
0
0 . 06
0
0
0 . 19
0
0
0
0
0 

0

0 

0 
0 

0 . 25 
Effect of treatment
and carry over can
be estimated
independently!
1
2
3
4
A
Other 2 sequence 3 period
designs
B
B
B
A
A
A
B
A
B
A
B
A
A
B
B
B
A
A
A
A
B
B
B
Var ˆ A 
 
Corr ˆ , ˆ 
Var ˆ A
A
A
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
A
B
B
B
A
B
A
A
2
Var ˆ A   0 . 25 ˆ
2
Var ˆ A   0 . 19 ˆ
Can’t include carry over
2 treatments per subject
Carry over estimable
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
Investigate X X 
T
1
A
B
A
B
C
A
B
C
D
B
C
B
C
A
B
D
A
C
C
A
C
A
B
C
A
D
B
A
C
A
C
B
D
C
B
A
B
C
B
A
C
C
B
C
B
A
Watch out for drop outs!
Titration Designs
Increasing 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)
Dose: Z1 mg
Dose: Z2 mg
Dose: Zk mg
•X on drug
•Y on Placebo
•X on drug
•Y on Placebo
•X on drug
•Y on Placebo
Stop if any signs of safety issues
VERY careful with first group!
Titration Designs
Which 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.
X1 mg
X2 mg
X3 mg
X4 mg
X5 mg
XY mg
Gr. 1
Gr. 2
Gr. 1
Gr. 2
Gr. 3
Gr. 4
12+2
12+2
12+2
12+2
12+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 design
Evaluation of a fixed combination of drug A and drug B
A placebo
B placebo
A active
B placebo
A placebo
B active
A active
B active
P
A
B
AB
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 of
the combination.
Implication: At specific component doses, the combination must be superior
to its components at the same respective doses
Factorial design
Usually 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
T AB : j 
n  Y AB  Y j

2 
ˆ
Min test and critical region:

 ; j  A, B


min T AB : A , T AB : B   C
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:

Y ij ~ N  j , 
2

ˆ 1  ˆ 2
Test statistic: Z mk 
For k  1,  K  1
Test:
2
If
2
mk
Z k  C
H 0 : 1   2
ˆ j 
1
mk
mk
Y
ij
i 1
Stop, reject H 0
otherwise Continue to group k  1
If
Z k  C
Stop, reject H 0
otherwise stop accept H 0
True type I error rate
Repeat testing until H0 rejected
Tests
C ritical valu e
P (typ e I erro r)
1
2
3
4
5
1.96
1.96
1.96
1.96
1.96
0.05
0.08
0.11
0.13
0.24
Pocock’s test
Suppose 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 k  1,  K  1
If
Z k  C p  , K
 Stop, reject
H0
otherwise Continue to group k  1
After group K
If Z k  C p  , K  Stop, reject H 0
otherwise stop accept H 0
Choose C p  , K  Such that
P ( Reject H 0 at any analysis
k  1 K )  
Pocock’s test
T e sts
C ritic al v a lu e
P (ty p e I e rro r)
1
2
3
4
5
1.960
2.178
2.289
2.361
2.413
0.05
0.05
0.05
0.05
0.05
All tests has the same nominal significance level
A group sequential test with 5 interrim tests has level
 '  2 1    2 . 413   0 . 0158
Pocock’s test
Zk
Reject H 0
2.413
Accept H 0
stage k
-2.413
Reject H 0
O’Brian & Flemmings test
Increasing nominal significance levels
For k  1,  K  1 : If
Z k  C p  , K  K / k Stop, reject H 0
otherwise Continue to group k  1
After group K :
If
Z k  C p  , K
 Stop, reject
otherwise stop accept H 0
Choose C p  , K  Such that
P ( Reject H 0 at any analysis
k  1 K )  
H0
O’Brian & Flemmings test
Critical values and nominal significance levels for
a O’Brian Flemming test with 5 interrim tests.
T e s t (k )
C B (K , )
C B (K , )*S q rt(K /k )
’
1
2
3
4
5
2 .0 4
2 .0 4
2 .0 4
2 .0 4
2 .0 4
4 .5 6
3 .2 3
2 .6 3
2 .2 8
2 .0 4
0 .0 0 0 0 0 5
0 .0 0 1 3
0 .0 0 8 4
0 .0 2 2 5
0 .0 4 1 3
Rather close to 5%
O’Brian & Flemmings test
Zk
6
4
2
0
0
1
2
3
4
5
0.0084
0.0225
0.0413
-2
-4
-6
0.0013
0.000005
6
Stage K
Comparing Pocock and O’Brian
Flemming
T e s t (k )
O ’B ria n F le m m in g
C B (K ,a )*S q rt(K /k )
’
Pocock
C P (K ,a )
’
1
2
3
4
5
4 .5 6
3 .2 3
2 .6 3
2 .2 8
2 .0 4
2 .4 1 3
2 .4 1 3
2 .4 1 3
2 .4 1 3
2 .4 1 3
0 .0 0 0 0 1
0 .0 0 1 3
0 .0 0 8 4
0 .0 2 2 5
0 .0 4 1 3
0 .0 1 5 8
0 .0 1 5 8
0 .0 1 5 8
0 .0 1 5 8
0 .0 1 5 8
Comparing Pocock and O’Brian
Flemming
6
4
Zk
2
0
0
1
2
3
-2
-4
-6
Stage k
4
5
Group Sequential Designs
Pros:
•Efficiency Gain (Decreasing marginal benefit)
•Establish efficacy earlier
•Detect safety problems earlier
Cons:
•Smaller safety data base
•Complex to run
•Need to live up to stopping rules!
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
Back up:
Var Y 21  Y 22    Var Y 21   Var Y 22  
1
1
2
2
2
2
 2  s     2  s   
n1
n2