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
Epidemiology of Randomized Trials
• Among 519 trials published in December
2000
– 74% were parallel group with a median
enrollment of 80 participants total
– 22% were crossover with a median of 15
participants
Lancet 2005; 365:1159-1162.
Parallel Group Studies
Eligible Patients
Informed Consent
Yes, randomize
A
No
B
Variability of response =
=
 =
s +
between subject +  within subject

e
Parallel Group Studies
Stratified Randomization
Eligible Patients
Informed Consent
Yes
Stratum 1
A
B
No
Stratum 2 . . .
A
B
Variability of response = wi i
i = is +
Stratum i
A
ie
B
Immediate Versus Deferred
Treatment: Concorde Study
Randomization
If AIDS/ARC or CD4+
declines to
<500 cells/mm3
AZT
Placebo
Open-label
AZT
Open-label
AZT
Lancet 1994; 343:871-881.
Invasive versus Conservative Management
Strategy for Acute non-Q-wave Myocardial
Infarction (VANQUISH Trial)
Immediate vs Deferred Parallel Group Design
• Invasive: coronary angiography as the initial
diagnostic test
• Conservative: radionuclide ventriculography
followed by a symptom-limited treadmill test
with thallium scintigraphy
– Coronary angiography performed if:
• Recurrent post-infarction angina with ischemic ECG
changes
• ST segment depression during peak exercise
• Redistribution defects on scintigraphy
N Engl J Med 1998; 338:1785-1792
Parallel Group “Switchover” Trial: A Simple
Example of A Dynamic Treatment Regimen
(Treatment Individualized to Patient)
Eligible Patients
Informed Consent
Randomize
A
B
Use B
if A fails
Use A
if B fails
Didanosine (ddI) versus Zalcitabine (ddC) for
Patients with Advanced HIV
Patient Who Failed on AZT
OR
Intolerant to AZT
RANDOMIZATION
ddI
ddC
If fail/
intolerant
If fail/
intolerant
ddC
ddI
N Engl J Med 1994; 330:657-662
Sequential Randomization (Dynamic
Treatment Regimens)
OI
No.
Arms Treatment Groups
PCP
2
Daily vs. 3x weekly TMP/SMX (1:1)
PCP
2
2nd line treatment: dapsone vs. atovaquone
Sequential Randomization
Randomization (R)
Daily
3x week
R
A
For A vs. D,
R
D
A
D
daily/3x weekly is a baseline characteristic for defining subgroups
Stat Med 1996; 15:2445-2453.
Parallel Group Studies: 2 x 2 Factorial Design –
both A and B Controlled Experimentally
Eligible Patients
Informed Consent
Yes, randomize
A, B
A, no B
Variability of response =
No A, B
No A, no B
wi i
i
=
is +
No
ie
Two factors each at 2 levels = 2x2.
Fisher on Factorial Designs
“If the investigator…confines his
attention to a single factor, we may
infer either that he is the unfortunate
victim of a doctrinaire theory as to
how experimentation should proceed,
or the time, material, or equipment at
his disposal is too limited to allow him
to give attention to more than one
narrow aspect of his problem.”
The Design of Experiments (7th edition, 1st published 1935)
W.G. Cochran supposedly said
something like…
Factorial designs are useful when you
are interested in an interaction and
when there is unlikely to be an
interaction.
Parallel Group Study
Factorial Design
Another Representation
A
B
No A
No B
B
No B
Note: This is different than recording factor A (Y/N)
and/or possibly using it as a stratifying variable. The
analysis of the effect of B versus no B is similar but
inferences are different.
Example: 2x2 factorial study of pain from
peripheral neuropathy for individuals with HIV
Randomization
Placebo
+
Alt Points
Placebo
+
Acupuncture
Amitriptyline
+
Alt Points
JAMA 1998;280:1590-1595
Amitriptyline
+
Acupuncture
Example: Physician’s Health Study
2 x 2 Factorial Design
Factor 1: 2 levels, Factor 2: 2 levels
Factor 2
Factor 1
Placebo
Aspirin
Carotene
Placebo
Aspirin Main Effect:
No. Participants
No. CVD deaths
No. Fatal/non-fatal
myocardial infarctions
11,037
44
11,034
44
104
189
Factorial Design Considerations
1. Interest in multiple Rx – can be efficient
2. Interaction
3. Generalizability – treatments studied
experimentally under different conditions.
4. Safety, logistics – may not be feasible
5. Mechanism of action – if different, efficiency
increases
2 x 2 Factorial Design
Treatment B
Yes
No
Yes
n, XAB
n, XA
2n
No
n, XB
n, X
2n
Treatment A
2n
Main Effect of A :
Main Effect of B:
(XA - X) + (XAB - XB)
2
(XB - X) + (XAB - XA)
2
Interactions
A with B : (XA - X) - (XAB - XB)
B with A : (XB - X) - (XAB - XA)
2n
4n
2 x 2 Factorial
Variance of main effects,e.g., A
(XA - X) + (XAB - XB) 
Var 
=
2




1 4

4 
 n
2
  2
=

 n
Variance of AB interaction
Var [(XA - X) - (XAB
4
- XB)] = 
n
2
Quantitative Interaction
O Factor 2, Level 2
X Factor 2, Level 1
Response
O
X
1
2
Factor 1
Level 1-2 of Factor 2 is negative for Level 1 of Factor 1
Level 1-2 of Factor 2 is more negative for Level 2 of
Factor 1
Qualitative Interaction
O
X
X - Level 1 of Factor 2
Response
X
O
1
O - Level 2 of Factor 2
2
Factor 1
Level 1-2 of Factor 2 is negative for Level 1 of Factor 1
Level 1-2 of Factor 2 is positive for Level 2 of Factor 1
2 x 2 x 2 Factorial Design
A
–
B
–
C
–
No.
Patients
n
–
–
+
n
XC
–
+
–
n
XB
–
+
+
n
X BC
+
–
–
n
XA
+
–
+
n
X AC
+
+
–
n
X AB
+
+
+
n
XABC
Results
X
8n
Main effect of A:
1
[(X A - X) + (X AC - XC ) + (X AB - XB ) + (X ABC - X BC )]
4
AB Interaction:
1
{[(X A - X) - (X AB - X B )] + [(X AC - X C ) - (X ABC - XBC )]}
2
Two estimates of AB interaction, one in the
presence of C and one in the absence of C.
ABC Interaction:
{[(X A - X) - (X AB - X B )] - [(X AC - X C ) - (X ABC - X BC )]}
There are 3 equivalent interpretations of 3-way interaction,
like there are 2 equivalent interpretations for AB in 2x2 factorial.
One is that AB interaction depends on presence of C.
2 x 2 x 2 Factorial
Variance Estimates
Main effect:

2
2n
2
2
2 - way interaction:
2
4
=
n
2n
3 - way interaction:
8
16
=
n
2n
2
2
Women’s Antioxidant Cardiovascular
Study (WACS)
• 2x2x2 factorial double-blind study
– Vitamin C versus placebo
– Vitamin E versus placebo
– Beta-carotene versus placebo
• High risk women willing to forgo individual
supplements
• Primary endpoint: combined endpoint of
CVD morbidity and mortality
Arch Int Med 2007; 167:1610-1618.
How Do Interactions Arise?
• Both factors affect the endpoint through the
same biologic process
• Non-compliance resulting from more
complicated regimen and/or knowledge of
the intervention (unblinded trial)
• Scaling (e.g., additive on logarithmic or
arithmetic scale)
2 x 2 Factorial Design
Factor 1: 2 levels, Factor 2: 2 levels
Example: Physician’s Health Study
Factor 2
Factor 1
Placebo
Aspirin
Carotene
Placebo
Interaction unlikely.
2 x 2 Factorial Design
MRC/BHF Heart Protection Study
Simvastatin (F1)
No
Yes
Yes
Vitamin Supp. (F2)+
No
+
Vitamin E, Vitamin C, and beta carotene
Interaction unlikely.
2 x 2 Factorial Design
Pain from Peripheral Neuropathy
Acupuncture (F1)
No
Yes
Yes
Amitriptyline
(F2)
No
Pain reduction (F1 + F2) < pain reduction (F1) + pain reduction (F2)
Pain reduction (F1 + F2) > pain reduction (F1) > pain reduction (F2)
Quantitative interaction likely.
2 x 2 Factorial Design
HIV Infection
Microbicide (F1)
No
Yes
Yes
Behavioral Intervention
No
(F2)
Knowledge of microbicide could effect response to different behavioral
interventions, e.g., disinhibition on standard counseling arm if taking
microbicide but not on enhanced counseling arm
Quantitative interaction likely.
Women’s Health Initiative
Partial Factorial Design
Factor 1: Dietary modification (low fat)
vs.
Self-selected dietary behavior
N = 48,836 (2:3)
Women’s Health Initiative
Partial Factorial Design (cont.)
Factor 2: Postmenopausal hormone therapy
I. (Post-hysterectomy)
Estrogen
vs.
Placebo
N = 10,739 (1:1)
II. Intact uterus
Estrogen + Progestin
vs.
Placebo
N = 16,608 (1:1)
Women’s Health Initiative
Partial Factorial Design (cont.)
Factor 3: Calcium and vitamin D
supplementation
Ca+ + Vitamin D
vs.
Placebo
N = 36,282 (1:1)
N = 48,836 + 10,739 + 16,608 + 36,282 =
112,465
68,133 women = 60.6% of
total enrollments
Multiple Opportunistic Infection Prophylaxis Study
(MOPPS) Protocol: Sequential versus Simultaneous
(Factorial) Randomization
Sequential
Simultaneous
CD4+ Count
300
Candidiasis
200
PCP
100
CMV
50
MAC
2x2x2x3 Factorial
Treatments of Interest
OI
No.
Arms Treatment Groups
PCP
2
Daily vs. 3x weekly TMP/SMX (1:1)
Candidiasis
2
Fluconazole vs. Placebo (1:1)
MAC
3
Clarithromycin (C) vs.
Rifabutin (R) vs. C + R (1:1:1)
CMV
2
Ganciclovir vs. Placebo (2:1)
One randomization with 24 arms (factorial)
vs.
4 separate sequential randomizations
Sequential Randomization or Factorial
Randomization
Daily
Daily
3x week
or
R
G
For G vs. P,
G
G
P
R
P
3x
P
daily/3x weekly is a baseline characteristic for defining subgroups
Stat Med 1996; 15:2445-2453.
Concept of Interaction is Model
Dependent
A
no
yes
yes
10
20
no
30
40
B
no interaction - additive model
A
yes
no
yes
10
20
no
30
60
B
no interaction - multiplicative model
Possible Designs (Approaches)
for Comparing Two Experimental Treatments (A
and B) with a Control (C) Using a Parallel Design
1. A vs. C
then
B vs. C
2. A vs. B vs. C
3, A vs. B vs. C vs. AB
4. AB vs. C
Comparison of Power for Testing Main
Effects for 4 Designs
(Byar, Cancer Treatment Reports, 1985.)
Assumptions:  = .05 (1-sided)
pAB
= 0.1 (no interaction)
A
yes
no
yes
0.1
0.3
pA
= 0.3
no
0.3
0.5
pB
= 0.3
B
pAB
pAB(placebo)
= 0.2 (interaction)
A
yes
no
yes
0.2
0.3
no
0.3
0.5
B
= 0.5
Design 1
Two Separate Trials
Each with Two Treatments
Trial 1:
Trial 2:
A
Placebo
B
Placebo
No.
Patients
120
120
120
120
Total no. of patients = 480
• Independent assessments of A and B
• No information on interactions
Power
0.93
0.93
Design 2
One Trial with Three Treatment Groups
No.
Patients
A
120
B
120
Placebo
120
Power
0.93 (0.90)†
Total no. of patients = 360
• Comparisons of A and B with placebo are not independent
since they share the same control group
• No information on interactions
†
Dunnett’s procedure
Design 3
2 x 2 Factorial Design
Power
A
No.
Patients
60
B
60
0.96
AB
60
Placebo
60
Total no. of patients = 240
• Independent comparisons
• Information on interactions
Problem: considerable loss of power with interaction
Alternative Design 3
2 x 2 Factorial Design
No.
Patients
Power
A
90
B
90
i) 0.99
AB
90
ii) 0.92
Placebo
Total no. of patients = 360
90
Power for interaction
test = 0.18
Design 4
One Trial Comparing the Combination
of A and B with Placebo
No.
Patients
Power
AB
120
0.99*
Placebo
120
Total no. of patients = 240
• Not clear which treatment works
• Information on combined use only
* Power 0 = 0.93 if the failure rate for the combination of AB is 0.3, i.e.,
only one of the 2 treatments is effective
Another Approach – Same General
Idea
• Sample size for 40% versus 20% with
α = 0.05 (2-sided) and power = 0.90 is 110
per group in each main effect comparison
(55 per group for the 4 arms – 220 total).
• Determine sample size so that each
subgroup (e.g., A vs no A with and without
B) can be analyzed separately with power =
0.70.
• For that need 65 per group; 260 total.
Reporting of Factorial Studies
• State rationale for using factorial design
• Report number assigned to individual treatments
• Examine interaction for major efficacy and safety
outcomes
• Show data on outcomes for individual cells (as well
as margins) so others can assess possible
interaction (simply stating “no interaction” is not
sufficient)
JAMA 2003;289:2545-2553.
Summary
Factorial Designs
1.
Generally underutilized; should especially be considered by
groups conducting multiple studies
2.
Should be considered when multiple treatments (questions)
are of interest – can be efficient way to study two questions.
3.
If interest is in main effects and dilution due to interaction is
a possibility, sample size should be increased. Power
should be considered for each treatment vs no treatment
(e.g., A vs no A or B) and for each simple effect.
4.
If interest is in treatment interaction, sample size will have
to be substantially increased
5.
It is important to report “cell” summary statistics, e.g.,
summary statistics for each combination of factors.