Transcript Lecture 13

Common Designs for
Controlled Clinical Trials
A.
Parallel Group Trials
1. Simplest example - 2 groups, no stratification
2. Stratified design
3. Matched pairs
4. Factorial design
B.
Crossover Trials
Parallel Group Design
y ij = m + Ti + eij
m = overall mean
th
Ti = effect of i treatment
(i = A, B)
th
th
eij = error term for j patient in i group
eij ~ N(0, s 2 )
y ij ~ N (m + Ti , s 2 )
2
2
2
s = ss + se
Estimates of s and s for Selected
Response Variables
2
s
2
e
s 2s
s 2s
Diastolic BP (mmHg)
s 2e
Total
 s 2s + s 2e  s 2s + s 2e
58
36
94
0.62
Serum cholesterol
(mg/dl)
1200
400
1600
0.75
Viral load
(log10 copies/mL)
0.16
0.09
0.25
0.64
Overnight urine
excretion of Na+
(meq/L)
325
625
950
0.34
Carbohydrate intake
(% of calories)
110
208
318
0.35
A Poorly Designed “Crossover” Trial:
Why?
2n patients
Drug A
or
Low Dose
Drug B
or
Low Dose
Also referred to as “changeover” or “switchover”
...
Two-Period Crossover Trial
Period
1
Drug A
n
2n - Randomize Patients
2
W
A
S
H
Drug B
n
Drug B
O
U
T
Drug A
SNaP
100 mEq NaCI
Placebo
Group I (n=25)
WASH-OUT
Group II (n=23)
Placebo
100 mEq NaCI
Weeks
-8
0
4
6
Low Sodium Diet
Schematic design of study to determine effects of dietary sodium on
blood pressure in normotensive adults
Hypertension 1991; 17:I21-I26.
10
Crossover Trial
Advantages:
• Fewer patients required in most situations due to elimination of
between subject component of variability, therefore:
– recruitment may be easier
– can be more easily done in single center
– fewer patients exposed to experimental treatment
• Less data collection
Disadvantages:
• Interaction due to:
– differential carry-over effects of treatments
– treatment x period interaction
– differences between the two randomized groups (AB and BA)
at baseline
• Patients must be observed longer
• Losses to follow-up/missing data
• Naturally occurring changes in underlying disease state
Situations where crossover
design is most applicable:
1.
Rapid response and response is transitory
2.
Variability between patients is large
compared to variability within patients
3.
Steady state physiological condition; disease
or condition cannot be cured
4.
No residual or carry-over effects of treatment
expected
Survey of Randomized Trials in 2000
• 116 of 526 published trials were crossover
• Median sample size for crossover trials = 15
• 78% of crossover studies involved 2 treatments
• 70% of crossover studies reported a washout
period
• 17% reported a test for carry-over effect and
13% tested for period effects
Mills EJ et al. Trials 2009
General Model for 2-Period
Crossover Trial
y ijk = m + pk + Tu + lv + g ij + e ijk
m = overall mean
pk = effect of kth period (k = 1, 2)
Tu = effect of uth treatment (u = A, B)
(direct effect)
lv = residual effect of treatment given in first period
on second period response (v = A, B)
(l v = 0 for first period measurements,
i.e., when k = 1)
Hills and Armitage Br J Clin Pharmac 1979; 8:7-20.
Senn Stat Methods Med Res 1994;3:303-324.
g ij = random effect between subject of jth
patient in group i (I or II); same in each period
2
N(0,

ss )
e ijk = random within patient effect for kth period
2
N(0,

se )
Motivation for crossover
s 2s  s 2e
2-Period Crossover Trial
Fixed Effects
Period
1
2
I (AB)
m + p1 + TA
m + p2 + TB+l A
II (BA)
m + p1 + TB
m + p2 + TA+l B
Group
Group I (AB) Patient (i = 1)
Response - Patient j
Period 1: y1j1 = m + p 1 + TA + g 1j + e1j1
Period 2 : yij2 = m + p 2 + TB + l A + g 1j + e 1j2
Paired difference d1I = y1j1 - y1j2
= (TA - TB ) - l A + (p 1 - p 2 ) + (e1j1 - e 1j2 )
E(d I ) = (TA - TB ) - l A + (p 1 - p 2 )
Group II (BA) Patient (i = 2)
Response - Patient jI
Period 1: y 2j'1 = m + p 1 + TB + g 2j' + e 2j'1
Period 2 : y 2j'2 = m + p 2 + TA + lB + g 2j' + e 2j'2
Paired difference y 2j'1 - y 2j2
= (TB - TA ) - lB + (p 1 - p 2 )+ (e2j'1 - e 2j' 2 )
Consider difference in opposite direction and call it dll
dII = (TA - TB ) + lB + (p 2 - p 1)+ (e 2j' 2 - e 2j'1)
E(dII) = (TA - TB ) + lB + (p 2 - p 1)
dI + dII = 2(TA - TB) - (l A - lB )
d I - dII = 2(p 1 - p 2 ) - ( lA + l B )
d I + dII
2
estimates TA - TB if lA = l B
d I - dII
2
estimates p 1 - p 2 if l A = lB = 0
How Do We Convince
Ourselves That lA=lB
Consider the sum of the period 1
and period 2 responses.
Group I (AB) : E(SumI ) = 2 m + (p 1 + p 2 )
+ (TA + TB ) + l A
Group II (BA) : E(SumII) = 2 m + (p 1 + p 2 )
+ (TA + TB ) + lB
Note that SumI - SumII estimates
l A - lB
Test for Carryover Has Low Power
• It is a between, instead of within, patient test (even when
considering change from baseline, variance is 4 times larger)
• Although totals are used, direct information on carry over
only comes from 2nd period – effect is diluted with sum’s.
• If there is evidence of a carryover effect (p<0.10 or 0.15),
Grizzle proposed that the 1st period effects be used. This
has been shown to be a sequential testing procedure that is
not optimal. Some believe it is better not to carry out the
test at all.
• Better be sure in the design that there is no carryover effect!
Possible Reasons for Rejecting Hypothesis
of Equal Carry-over Effects
1. True carry-over effect for A or B, or both
2. Psychological carry-over effect
3. Difference between treatments depends on pretreatment level of response variable
4. Group I and Group II differ significantly by chance
A
Response
A
Group II
x
x
B
Group I
B
1
2
Period
1. Treatment effect
2. No period effect
3. No interaction
Response
A
x
A
x
B
B
Group II
Group I
1
2
Period
1. Treatment effect
2. Period effect
3. No interaction
Quantitative Interaction
A
Response
B
1
A
Group II
B
Group I
2
Period
Qualitative Interaction
A
B
Response
B
A
1
2
Period
Group I
Group II
Example from Senn
Statistical Issues in Drug Development
John Wiley & Sons, 1997
Beta-Agonist (A) vs. Placebo (B)
for Patients with Asthma
FEV1.0 (mL)
Period 1
Period 2
A-B
Difference
BA
2000
2300
300
AB
2300
2000
300
Sequence
300
Senn Example (cont.)
• Add 350 mL to BA
Sequence
BA
Period 1
2350
Period 2
2650
A-B
Difference
300
AB
2300
2000
300
300
Differences are recovered in spite
of difference between sequences
Senn Example (cont.)
• Add 350 mL to BA; and
• Add a secular trend causing FEV1.0 to be
100 mL HIGHER in 2nd period
Sequence
BA
Period 1
2350
Period 2
2750
A-B
Difference
400
AB
2300
2100
200
300
Difference in each sequence not recovered, but
average is okay – crossover still works!
Senn Example (cont.)
• Add 350 mL to BA; and
• Add a secular trend causing FEV1.0 to be 100
higher in 2nd period; and
• Add a carry-over effect of A (still completely
effective when B is given)
Sequence
BA
Period 1
2350
Period 2
2750
A-B
Difference
400
AB
2300
2400
-100
150
Crossover does not work!
Variance Estimates
d I + d II
2
d I - d II
Period Effect =
2
(1)
Carryover Effect = Sum I - Sum II
(3)
Treatment Effect =
(2)
Variance of (1) and (2)
d 
1  d
= s I + s II 
4  nI
nII 
2
s
2
2
= pooled variance
2
2
(nI - 1)s d I + (nII - 1)s d II
=
nI + nII - 2
d
Variance of (3)
= s
s
2
Sum
2
Sum I
nI
+ s
2
Sum II
nII
= pooled variance =
(nI - 1)s
+ (nII - 1)s
nI + nII - 2
2
Sum I
2
Sum II
Hypothesis Testing
1) Ho : no interaction
t(nI + nII - 2) =
SUMI - SUM II
1
 1
1  2
ˆ
sSUM
n + n ÷
÷
II 
 I
2) Ho : no treatment effect
dI + dII
2
t(nI + nII - 2) =
1
2


ˆ
1
1
sd  +
÷
2 
nII ÷
nI

3) Ho : no period effect
dI - dII
t(nI + nII - 2) =
2
1
1  2
sˆ d 

+
÷

÷
1
2 nI
nII 
Group I
Patient
Accession No.
Period 1
(A)
Period 2
(B)
A-B
Difference
Sum
1
3
4
6
7
9
11
13
16
18
19
21
22
24
25
27
28
8
14
8
9
11
3
6
0
13
10
7
13
8
7
9
10
2
5
10
0
7
6
5
0
0
12
2
5
13
10
7
0
6
2
3
4
8
2
5
-2
6
0
1
8
2
0
-2
0
9
4
0
13
24
8
16
17
8
6
0
25
12
12
26
18
14
9
16
4
Group II
Patient
Accession No.
Period 1
(B)
Period 2
(A)
A-B
Difference
Sum
2
12
11
-1
23
5
6
8
2
14
8
13
9
-4
22
10
8
8
0
16
12
8
9
1
17
14
4
8
4
12
15
8
14
6
22
17
2
4
2
6
20
8
`13
5
21
23
9
7
-2
16
26
7
10
3
17
29
7
6
-1
13
Group I (N = 17)
Mean
SD
SE
Period 1
(A)
Period 2
(B)
A-B
Difference
Sum
8.12
3.84
5.29
4.25
2.82
3.47
0.84
13.41
7.32
1.78
Group II (N = 12)
Mean
SD
SE
Period 1
(B)
Period 2
(A)
A-B
Difference
Sum
7.67
2.99
8.92
2.81
1.25
2.99
0.86
16.58
4.98
1.44
Ho : No interaction
1) Determine pooled variance of sum
s2SUM
(nI - 1)sI2 + (nII - 1)sII2
=
nI + nII - 2
2
16(7.32) + 11(4.98)
=
27
= 41.9
2
2) Calculate test statistic
t(nI + nII - 2) =
SUMI - SUMII
2
SUM
s
=
 1
1 

n + n ÷
÷
II 
 I
16 - 13.49
1
1
41.86
+
17 12
= 1.30
3) Compare with t-tables with 27df, p = 0.20
Ho : No treatment difference
same procedure
s
2
d
=
(nI - 1)s
2
Id
2
II d
+ (nII - 1)s
nI + nII - 2
2
16(3.47) + 11(2.99)
=
27
= 10.78
2
 dI + dII 

 2 ÷
÷


t(nI + nII - 2) =
s d  1
1 
+

÷

2 nI
nII ÷

=
2.82 + 1.24 

÷
2


= 3.28
10.78 1
1
+
2
17 12
p = 0.0028
Similarly for Ho : No Period Effect
dI - dII 

 2 ÷
÷


t(nI + nII - 2) =
s d  1
1 
+

÷

2 nI
nII ÷

=
2.82 - 1.24 

÷
2


= 1.27
10.78 1
1
+
2
17 12
p = 0.21
Two-Period Crossover Trial
Period
1
Drug A
n
2n - Randomize Patients
2
W
A
S
H
Drug B
n
Drug B
O
U
T
Drug A
Consider Period 1 Data
and Estimated SE of Treatment Difference
(Parallel Group Design)
2
2
(3.84)
(16)
+
(2.99)
(11)
2
sp =
27
= 12.38
SE = (12.38)
1
2
1
1
+
17 12
= 1.33
compared to 0.62 for crossover
Advantages of Baseline
Measurements
1.
Description of study participants at entry
2.
Comparability of treatment groups:
AB vs. BA
3.
More powerful test for treatment x period interaction
4.
Improved precision for estimating treatment
differences (e.g., analysis of covariance or change
from baseline when correlation >0.5.)
5.
Subgroup analysis
Advantage of a 2nd “Baseline” during
Washout Between Periods
• Differences between “baseline”
measurements for Group I and Group II may
provide support for unequal residual effects.
NOTE!
This comparison does not replace
comparison of sums of observation at the
end of periods for Group I and Group II that
was previously discussed.
Design for Estimating Direct and
Residual Treatment Effects
Square 1
Group
I
II
III
Period
1 2 3
A B C
B C A
C A B
Square 2
Group
I
II
III
Period
1 2 3
A C B
B A C
C B A
Recommendations
1.
2.
3.
Always measure initial baseline.
If washout is employed, measure B2.
Take multiple measures during each treatment
period, e.g., comparison of treatment A with
placebo (P).
Group
I
II
1
P
P
B1
Week of Study
Period 1
Period 2
2 3 4
5
6 7 8
A A A
P
P P P
P P P
P
A A A
B2