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

Ti =effect of ith treatment

(i = A, B)

eij =error term for jth patient in ith group

eij ~ N(0,s2)

y ij ~N (m+Ti , s2)

s2= s s

2+ s e

2

Diastolic BP (mmHg) 58 36 94 0.62

Serum cholesterol 1200 400 1600 0.75(mg/dl)

Viral load 0.16 0.09 0.25 0.64(log10 copies/mL)

Overnight urine 325 625 950 0.34excretion of Na+

(meq/L)

Carbohydrate intake 110 208 318 0.35(% of calories)

Estimates of and for Selected Response Variables

2s

2e

2e

2s

2e

2s

2e

2s

Total

2s

A Poorly Designed “Crossover” Trial:

Why?

2n patients Drug A Drug B . . .

or or

Low Dose Low Dose

Also referred to as “changeover” or “switchover”

Two-Period Crossover Trial

2n - Randomize Patients

Drug A

Drug B

Drug B

Drug A

W

A

S

H

O

U

T

1 2

Period

n

n

Schematic design of study to determine effects of dietary sodium on blood pressure in normotensive adults

Low Sodium Diet

-8 0 4 6 10

Weeks

Group I (n=25)

Group II (n=23)

100 mEq NaCI

Placebo 100 mEq NaCI

Placebo

WASH-OUT

SNaP

Hypertension 1991; 17:I21-I26.

Crossover TrialAdvantages:• 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 + l v + g ij + eijk

m = overall mean

pk = effect of kth period (k = 1, 2)

Tu = effect of uth treatment (u = A, B)

(direct effect)

l v = 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.

2e2

s

g ij = random effect between subject of jth

patient in group i (I or II); same in each period

» N(0, s s2 )

eijk = random within patient effect for kth period

» N(0, s e2 )

Motivation for crossover

+ + T +

+ + T +

+ + T

+ + T

2-Period Crossover TrialFixed Effects

1 2

I (AB)

II (BA)

Group 1 A

1 B

2 B

2 A

A

B

Period

Group I (AB) Patient (i = 1)Response - Patient j

Period 1: y1j1 = m + p1 + TA + g1j + e1j1

Period 2: y ij2 = m + p2 + TB + l A + g1j + e1j2

Paired difference dI1 = y1j1 - y1j2

= (TA - TB ) - l A + (p1 - p2) + (e1j1 - e1j2)

E(d I) = (TA - TB) - l A + (p1 - p2)

Group II (BA) Patient (i = 2)Response - Patient jI

Period 1: y2j'1 = m + p1 + TB + g2j' + e2j'1

Period 2 : y2j'2 = m + p2 + TA + lB + g2j' + e2j'2

Paired difference y2j'1 - y2j2

= (TB - TA) - lB + (p1 - p2)+ (e2j'1 -e2j' 2)

Consider difference in opposite direction and call it dll

dII = (TA - TB) + lB + (p2 -p1)+ (e2j' 2 - e2j'1)

E(dII) = (TA - TB) + lB + (p2 -p1)

dI + dII = 2(TA - TB) - ( l A - lB)

dI - dII= 2(p1 -p2) - ( lA + l B)

dI + dII

2 estimates TA - TB if l A = l B

dI - dII

2 estimates p1 -p2 if l A = lB = 0

How Do We Convince Ourselves That AB

Consider the sum of the period 1and period 2 responses.

Group I (AB): E(SumI) = 2m + (p1+p2)

+ (TA + TB) + l A

Group II (BA): E(SumII) = 2m + (p1 +p2)

+ (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 Hypothesisof 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 pre-treatment level of response variable

4. Group I and Group II differ significantly by chance

ResponseGroup II

Group IB

A

B

A

1 2Period

x x

1. Treatment effect

2. No period effect

3. No interaction

1. Treatment effect

2. Period effect

3. No interaction

ResponseGroup II

Group I

A

A

B

B

1 2Period

x

x

Quantitative Interaction

Response

Group II

Group I

B

A

BA

1 2Period

Qualitative Interaction

Response

Group IIGroup I

B

AB

A

1 2Period

Example from SennStatistical Issues in Drug Development

John Wiley & Sons, 1997

BA 2000 2300 300

AB 2300 2000 300

300

Sequence Period 1 Period 2

A-B

Difference

FEV1.0 (mL)

Beta-Agonist (A) vs. Placebo (B)for Patients with Asthma

Senn Example (cont.)

BA 2350 2650 300

AB 2300 2000 300

300

Sequence Period 1 Period 2

A-B

Difference

• Add 350 mL to BA

Differences are recovered in spite

of difference between sequences

Senn Example (cont.)

BA 2350 2750 400

AB 2300 2100 200

300

Sequence Period 1 Period 2

A-B

Difference

• Add 350 mL to BA; and

• Add a secular trend causing FEV1.0 to be 100 mL HIGHER in 2nd period

Difference in each sequence not recovered, but average is okay – crossover still works!

Senn Example (cont.)

BA 2350 2750 400

AB 2300 2400 -100

150

Sequence Period 1 Period 2

A-B

Difference

• 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)

Crossover does not work!

Variance EstimatesTreatment Effect =

dI + dII

2

Period Effect =dI - dII

2

Carryover Effect = Sum I - Sum II

Variance of (1) and (2)

=14s

2d I

nI

+ s2

dII

nII

é

ë ê

ù

û ú

s2

d = pooled variance

= (nI - 1)s

2dI + (nII - 1)s

2dII

nI + nII - 2

Variance of (3)

= s2

Sum I

nI

+ s2

Sum II

nII

s2

Sum= pooled variance =

(nI - 1)s2

SumI + (nII - 1)s2

SumII

nI + nII - 2

(1)

(2)

(3)

Hypothesis Testing

1) Ho : no interaction

t(nI + nII - 2) =SUMI - SUMII

ˆ s SUM

1

nI

+ 1

nII

æ

è ç ç

ö

ø÷ ÷

12

2) Ho : no treatment effect

t(nI + nII - 2) =

dI + dII

2

ˆ s d

2

1

nI

+ 1

nII

æ

è ç ç

ö

ø÷ ÷

12

3) Ho : no period effect

t(nI + nII - 2) =

dI - dII

2

ˆ s d

2

1

nI

+ 1

nII

æ

è ç ç

ö

ø÷ ÷

12

Group I

1 8 5 3 133 14 10 4 244 8 0 8 86 9 7 2 167 11 6 5 179 3 5 -2 811 6 0 6 613 0 0 0 016 13 12 1 2518 10 2 8 1219 7 5 2 1221 13 13 0 2622 8 10 -2 1824 7 7 0 1425 9 0 9 927 10 6 4 1628 2 2 0 4

PatientAccession No.

Period 1(A)

Period 2(B)

A-BDifference Sum

Group II

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

PatientAccession No.

Period 1(B)

Period 2(A)

A-BDifference Sum

Group I (N = 17)

Mean 8.12 5.29 2.82 13.41SD 3.84 4.25 3.47 7.32SE 0.84 1.78

Period 1(A)

Period 2(B)

A-BDifference Sum

Group II (N = 12)

Mean 7.67 8.92 1.25 16.58

SD 2.99 2.81 2.99 4.98

SE 0.86 1.44

Period 1(B)

Period 2(A)

A-BDifference Sum

1) Determine pooled variance of sum

Ho : No interaction

sSUM2 =

(nI - 1)sI2 + (nII - 1)sII

2

nI + nII - 2

= 16(7.32)

2 + 11(4.98)

2

27

= 41.9

2) Calculate test statistic

3) Compare with t-tables with 27df, p = 0.20

t(nI + nII - 2) = SUMI - SUMII

sSUM2 1

nI

+ 1

nII

æ

è ç ç

ö

ø÷ ÷

= 16 - 13.49

41.861

17+

1

12

= 1.30

same procedure

Ho : No treatment difference

sd2 =

(nI - 1)sId

2 + (nII - 1)sII d

2

nI + nII - 2

= 16(3.47)

2 + 11(2.99)

2

27

= 10.78

t(nI + nII - 2) =

d I + d II2

æ

è ç ç

ö

ø÷ ÷

sd

2

1

nI

+ 1

nII

æ

è ç ç

ö

ø÷ ÷

=

2.82 + 1.24

2

æ

è ç

ö ø

÷

10.78

2

1

17+

1

12

= 3.28

p = 0.0028

Similarly for Ho : No Period Effect

t(nI + nII - 2) =

d I - d II2

æ

è ç ç

ö

ø÷ ÷

sd

2

1

nI

+ 1

nII

æ

è ç ç

ö

ø÷ ÷

=

2.82 - 1.24

2

æ

è ç

ö ø

÷

10.78

2

1

17+

1

12

=1.27

p = 0.21

Two-Period Crossover Trial

2n - Randomize Patients

Drug A

Drug B

Drug B

Drug A

W

A

S

H

O

U

T

1 2

Period

n

n

Consider Period 1 Dataand Estimated SE of Treatment Difference

(Parallel Group Design)

sp2 =

(3.84)2 (16) + (2.99)2 (11)

27

= 12.38

SE = (12.38)1

2 1

17+

1

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

Group 1 2 3

I A B C

II B C A

III C A B

PeriodSquare 1

Group 1 2 3

I A C B

II B A C

III C B A

PeriodSquare 2

Recommendations

1. Always measure initial baseline.

2. If washout is employed, measure B2.

3. Take multiple measures during each treatment period, e.g., comparison of treatment A with placebo (P).

Week of Study

Group 1 2 3 4 5 6 7 8

I P A A A P P P P

II P P P P P A A A

B1 B2

Period 1 Period 2