outline of stratification lectures
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Outline of Stratification Lectures. Definitions, examples and rationale (credibility) Implementation Fixed allocation (permuted blocks) Adaptive (minimization) Rationale - variance reduction. Stratification. - PowerPoint PPT PresentationTRANSCRIPT
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Outline of Stratification Lectures
• Definitions, examples and rationale (credibility)
• Implementation– Fixed allocation (permuted blocks)– Adaptive (minimization)
• Rationale - variance reduction
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Stratification
• A procedure in which factors known to be associated with the response (prognostic factors) are taken into account in the design (e.g., randomization)
• Pre-stratification refers to a stratified design; post-stratification refers to the analysis
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Pre- versus Post Stratification and Precision (Variance Reduction)
• As a general rule, the precision gained with pre- versus post-stratification is less than one might expect
• The gain in precision is greatest in small studies (where you need it the most) because the risk of chance imbalance is greater.
• Covariate adjustment for prognostic factors is usually carried out with regression (e.g., linear, logistic, or proportional hazards regression.
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Stratification Can Increase Precision
Simple versus stratified random sampling. Snedecor and Cochran note (p. 520): “If we
form strata so that a heterogeneous population is divided into parts each of which is fairly homogeneous, we may expect a gain in precision over simple random sampling”.
Ref. Snedecor and Cochran, Statistical Methods
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Stratification Can Increase Precision
Randomized block versus completely random design.
Snedecor and Cochran note (p. 299): “Knowledge (about predictors or response) can be used to increase the accuracy of experiments. If there are a treatments to be compared,…first arrange the experimental units in groups of a, often called replications. The rule is that units assigned to the same replication should be as similar in responsiveness as possible. Each treatment is then allocated by randomization to one unit in each replication…Replications are therefore usually compact areas of land…This experimental plan is called randomized blocks.”
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Pre-stratification Does Not Matter.
Peto et al note: “As long as good statistical methods,…,are used to analyze data from clinical trials, there is no need for randomization to be stratified by prognostic features.”
• Keep it simple so investigators are not discouraged from participating.
• Post-stratified analysis is needed with pre-stratification anyway.
• Improvement in sensitivity (precision) with pre-stratification compared to letting stratum sizes be determined by chance is small.
Peto R et al., Br. J Cancer, pp. 585-612,1976
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Stratified Design for Comparing Treatments
Stratum A B1
2
3
4
m1
m2
m3
m4
na nb
Treatment
m1A
m2A
m3A
m4A
m1B
m2B
m3B
m4B
• Typical situation:
m1 ≠ m2 m3 m4≠ ≠• Study is designed/powered based on na and nb
• Goal: miA = miB for all i.
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How much of a price does one pay with respect to precision by trusting randomization to achieve reasonable balance?
Consider the relative efficiency of a stratified design to an unstratified design:
Var (treatment contrast with stratification)
Var (treatment contrast with no stratification in design, but post-stratified analyses)
RE =
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Pooling Estimates
Estimates: E , E
Var (E ) = Var (E ) =
w E + w EPooled Estimate:
Best Pooled Est: w = , w =
w + w Variance Pooled Est:
w + w
1 1
1 2
2
121
22
1 1
1
2 2
21
21
22
(w + w )1 1
21
21
22
2
2
22
2
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Continuous response, equal variance - effect of chance imbalance
nA = total number randomly assigned to AnB = total number randomly assigned to Bg = fraction of those given A with prognostic factorh = fraction of those given B with prognostic factor
Treatment
Stratum
A BS1
S2
An g
Bn h
nA nB
(1-g)nA (1-h)nB
RE =A
n gB
n h(1-g) (1-h)An g
Bn h+
+ An g
Bn h+
An
Bn +1 -
-1
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RE obtained by noting:1) Var( y - y ) =
1A 1B 1 +n g A
1n h
+
( ) 2
2) Var(y - y ) = 2A 2B n
An
B( ) 2(1-g) (1-h)
3) Pooled variance is:
VarPooled(y - y ) = A B
wi2 Var (y - y )
iA iB2
wi 2
w1 =A
n gB
n h+An g
Bn h
An
Bn (1-g) (1-h)+w2 = A
n B
n (1-g) (1-h)
B1 1
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For Stratified Design, g = h
w =1 n
An B
+n
An
Bg
w =2 n
An
B+n An
B(1 - g)
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Assume A
n B
n =
RE = g(1-g) h(1-h)g + h
+g + h
21 - e.g., block randomization used
Consider the case of g = 2h:
0.10, 0.05 0.990.25, 0.1250.970.50, 0.25 0.930.75, 0.3750.86
g, h RE
-1
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Bernouli Response
Loss of efficiency =h2
n21 -
h = lack of balance2n = number in each stratum
Ref: Meier (Controlled Clinical Trials, 1981)
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This can be seen by noting:
1) Stratified design, for stratum 1
Var(p - p ) = A B
since n = n = nA1 B1
1 1
1nA 1
+ 1nB 1
p1q1( )
2n= p1 q1( () )
Note: q1 = 1- p1
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The ratio of these variances is proportional to:
2) No stratification in design; post-stratification in analysisVar(p - p ) =
A B11
1 n+h + 1
n-h p1q1( )
1/n+h + 1/n-h2/n = h2
n21 -
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n = 10
1 (11, 9) 0.99
2 (12, 8) 0.96
4 (14, 6) 0.86
5 (15, 5) 0.75
h RE(n , n )1A 1B
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Example:Brown et al. Clinical Trial of Tetanus Anti-toxin in Treatment of Tetanus. Lancet, 227-30;1960 (see also Meier, Cont Clin Trials, 1981; a slightly different approach is taken here).
Anti-Toxin (A)
Alive
Dead
21 9
20 29
No Anti-Toxin (B)
41 38
30
49
79
p = overall death rate == 0.620
4979
^
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= 20/41 = 0.488p̂A
= 29/38 = 0.763p̂B
p̂A
p̂B
- = -0.275
Var p̂A
p̂B
-( ) = +A
1n
B
1n ][ p̂ 1 p̂-( )
= 1 41
1 38
4979
49791 -[ ( ( ))]+
= 0.01195
SE 0.109
p̂A
p̂B
-( ) =
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Time from first symptoms to admission turned out to be an important prognostic factor; therefore, post-stratification was carried out.
A
Alive
Dead
10 4
18 26
B
28 30
A
Alive
Dead
11 5
2 3
B
13 8
< 72 Hours ≥ 72 Hours
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Stratum 1: < 72 hoursStratum 2: ≥ 72 hours
p̂1A
p̂1B
- = - 0.223
p̂1A= 0.643 p̂
1B= 0.866
p̂1= 0.759
SE( ) = 0.112
p̂1A
p̂1B
-
p̂2A
p̂2B
- = - 0.221
p̂2= 0.238
SE( ) = 0.191p̂2A
p̂2B
-
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Weighted diff. (ˆ p A - ̂ p B)wˆ
Let G = fraction of patients in Stratum 1
= 58/79 = 0.734(̂ p A - ̂ p B)w = ̂ G (̂ p 1A - ̂ p 1B) + (1 - G)(̂ p 2A - ̂ p 2B) = - 0.223
compared to - 0.275 unweighted
VAR(p - p ) = G VAR(p - p )ˆ A ˆ B w
ˆ 1A
ˆ 1B ̂
SE(p - pˆ A - ̂ B)w = .097+ (1-G ) VAR(̂ p 2A - ̂ p 2B) = 0.009382
2
ˆ
ˆ
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Gain in precision achieved with post-stratification
Var(post-stratification)Var(no stratification)
RE =
= 0.009380.01195
= 0.78 22% reduction
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How much gain in precision would be achieved if stratification was used in the
design?
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Force balance within stratum
Assume ̂ p ‘s don’t change ij
ˆ 1A ˆ 1B
ˆ 2A ˆ
2B
A
Alive
Dead
B
29 29
A
Alive
Dead
B
11 10
< 72 Hours ≥ 72 Hours
SE(p - p ) = 0.109instead of 0.112
SE(p - p ) = 0.186instead of 0.191
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Var(stratified design)Var(no stratification)
RE1
= (0.096)(0.109)
= 0.77 23% reduction
=
2
2
Var(stratified design)
Var(post-stratification)
RE2
= (0.096)(0.097)
= 0.98 2% reduction
=
2
2
SE stratified design = 0.096 (same weights are used)
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0.50 0.60 0.30 0.910.20 0.60 0.30 0.940.10 0.60 0.30 0.96
0.50 0.60 0.20 0.830.20 0.60 0.20 0.870.10 0.60 0.20 0.92
0.50 0.10 0.05 0.9910.20 0.10 0.05 0.9920.10 0.10 0.05 0.996
G REP1. P2.
Gp1.(1-p1.) + (1-G)p2.(1-p2.)[ Gp1. + (1-G)p2. ] [1 – Gp1. – (1-G)p2.]
RE =
If p1. = p2. Then RE = 1
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1) the distribution of the prognostic factor in the population;
2) the relative strength of the prognostic factor; and
3) the expected endpoint rate in the group studied.
The reduction in variance achieved with post-stratification depends on:
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Scott’s Survey of Trials Published in Lancet and N Eng J Med in 2001
StratificationPermuted block 43/150Minimization 6/150Other adaptive 3/150Other 19/150Unspecified 79/150
Scott et al. Cont Clin Trials 2002; 23:662-674
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Kahan’s Survey of 258 Trials Published in Four Major Medical
Journals in 2010Method of RandomizationSimple: 4Permuted blocks, no stratification: 40Permuted blocks, stratification: 85Minimization: 29Other: 4Unclear: 96
Kahan BC et al. BMJ 2012; 345:e5840
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Conclusions
1. Usually there is little loss of efficiency with post-stratification as compared to a stratified design.
2. Loss of efficiency results from large chance imbalances for important prognostic factors, which are more likely in small studies.
3. Stratified designs should be considered in small studies (n < 50) with important prognostic factors.
4. Strictly speaking, analysis should account for pre-stratification.
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Recommendation for Multi-Center Trials:Always Consider Stratification on Center
1. Clinic populations differ.
2. Treatment differs from clinic to clinic.
3. Each center represents a replicate of overall trial – can investigate treatment x clinic interactions.
4. In some trials (surgery), it may be better to stratify on surgeon within clinic.
5. If there are a very large number of clinical sites, small block size may have to be used and site combined into a priori defined larger strata (e.g., region or country) for analysis
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General Recommendations
• Large trials– Block randomization with stratification by center– Stratification on other factors not necessary (I am a lumper)– If needed, usually okay to carry out block
randomization within each stratum
• Small trials– Block randomization with stratification by center– If stratification on other factors is considered,
may have to use an adaptive approachThese are consistent with Freidman, Furberg and DeMets (see page 111)