1
PH 240A: Chapter 9
Steve ShiboskiUniversity of California Berkeley
(Slides by Nick Jewell)
2
CHD and Behavior Type, Stratified by Body Weight
CHD Event
Body Weight (lbs)
Yes No
150 Behavior Type
Type AType B
2210
253305
2.652 2.520
150+-160 Behavior Type
Type AType B
2110
235270
2.413 2.297
160+-170 Behavior Type
Type AType B
2921
297297
1.381 1.347
170+-180 Behavior Type
Type AType B
4719
248253
2.524 2.281
>180 Behavior Type
Type AType B
5919
378361
2.966 2.700
RO ˆ RR ˆ
≤
3
Pancreatic Cancer and Coffee Drinking, Stratified by Sex
Pancreatic Cancer
Sex Cases Controls
Females Coffee drinking
(cups/day)
10
14011
28056
2.545
Males Coffee drinking
(cups/day)
10
2079
27532
2.676
≥
RO ˆ
≥
4
2 x 2 Table Notation for ith Stratum
Disease Status
D not D
ExposureE ai bi ai+ bi
not E ci di ci+ di
ai+ci bi + di ni
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Testing of Independence Based on Stratified Tables
H0:OR1=OR2= . . . =ORI = 1HA1: at least one ORj differs from 1Do a test for each stratum
Multiple testing issue Low power
Add the test statistic from each stratum to give a test with I degrees of freedom Low power
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Variation of Stratum-Specific Odds Ratio
ORi
Stratum Indicator1 2 I
x
x
x
xx
x x
x
H0
Typical HA2
1
HA2:OR1=OR2= . . . =ORI 1≠
7
Cochran-Mantel-Haenszel test
In ith stratum, if D and E are independent E(ai) = Ai=(ai+bi)(ai+ci)/ni
Var(ai)=Vi
=(ai+bi)(ci+di)(ai+ci)(bi+di)/ni2(ni-1)
Look at deviations ai-Ai and their accumulation
∑
∑∑
=
==
⎟⎠
⎞⎜⎝
⎛−
= I
ii
I
ii
I
ii
CMH
V
Aa
1
2
112χ
8
CHD and Behavior Type, Stratified by Body Weight
CHD Event
Body Weight (lbs)
Yes No
150 Behavior Type
Type AType B
2210
253305
2.652 2.520
150+-160 Behavior Type
Type AType B
2110
235270
2.413 2.297
160+-170 Behavior Type
Type AType B
2921
297297
1.381 1.347
170+-180 Behavior Type
Type AType B
4719
248253
2.524 2.281
>180 Behavior Type
Type AType B
5919
378361
2.966 2.700
RO ˆ RR ˆ
≤
9
Cochran-Mantel-Haenszel Test Statistic Calculations for CHD Data
Body Weight
(lbs)
150 22 14.915 7.544
150+-160 21 14.806 7.301
160+-170 29 25.311 11.546
170+-180 47 34.339 14.581
>180 59 41.721 17.574
Totals 178 131.091
58.546
ia iA iV
≤
10
Cochran-Mantel-Haenszel Test Statistic Calculations for CHD Data
on)distributi reference (using10
6.37546.58
)091.131178(
2)1(
6
22
−≤
=−
=
p
CMH
9.39 statistic test Pooled 2 =
11
Woolf Estimator (and CI) for Common Odds Ratio
Average the log Odds Ratio estimates from the separate strata Weight the average with weight inversely
proportional to variability (high variance stratum estimates get low weight in the average and vice-versa)
( )( )
∑
∑
=
== I
ii
I
iii
W
w
ROwRO
1
1
ˆlogˆlog
))ˆr(log(av
1
i
iRO
w =
12
Woolf Estimator (and CI) for Common Odds Ratio
Sampling Distribution:
( ) ( )VORNRO W ),log(ˆlog ≈
( )( )( )
( )
( )
∑
∑
∑∑
∑
=
≅
=
=
i
i
ii
i
ii
W
w
w
ww
w
ROw
ROV
1
1
ˆlog(rav
ˆlogvar
2
2
2
2
Vassuming no interaction
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Mantel-Haenszel Estimator
Average the Odds Ratio estimates from each stratum
∑
∑
=
== I
ii
I
iii
MH
w
ROwRO
1
*
1
* ˆˆ
i
iii n
cbw =*
( )
( )∑
∑
=
== I
iiii
I
iiii
MH
ncb
ndaRO
1
1
/
/ˆ
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Woolf and Mantel-Haenszel Estimator Calculations for CHD Data
Woolf Method Mantel_Haenszel Method
Body Weight
(lbs) 150 0.949 6.807 11.373 4.288
150+-160 0.855 6.680 10.578 4.384
160+-170 0.316 11.477 13.374 9.685
170+-180 0.910 12.453 20.972 8.310
>180 1.070 13.606 26.070 8.791
Totals 51.023 82.367 35.458
)ˆlog( iRO iw iii nda / iii ncb /
≤
15
Summary of Findings for WCGS
Pooled results for Behavior Type test statistic is 39.9 OR estimate = 2.34 (1.79—3.11) Var of estimate of log(OR) = 0.01956
Adjusting Results for Body Weight Cochran-Mantel-Haenszel test statistic is 37.6 Woolf OR estimate = 2.26 (1.72—2.98) Mantel-Haenszel estimate = 2.32 (1.76—3.06) Var of estimates of log(OR) = 0.01960 (Woolf)
Little confounding seen from body weight
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Pancreatic Cancer and Coffee Drinking, Stratified by Sex
Pancreatic Cancer
Sex Cases Controls
Females Coffee drinking
(cups/day)
10
14011
28056
2.545
Males Coffee drinking
(cups/day)
10
2079
27532
2.676
≥
RO ˆ
≥
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Summary of Findings for Coffee Drinking and Pancreatic Cancer
Pooled results for Coffee Drinking test statistic is 16.6 OR estimate = 2.75 (1.66—4.55)
Adjusting Results for Sex Cochran-Mantel-Haenszel test statistic is
14.5 Woolf OR estimate = 2.51 (1.53—4.13) Mantel-Haenszel estimate = 2.60 (1.57—
4.32)Little confounding seen from sex
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Balance at Stratum Level
CHD Event
Body Weight (lbs) Yes No
150 Behavior Type
Type AType B
2210
253305
150+-160 Behavior Type
Type AType B
2110
235270
160+-170 Behavior Type
Type AType B
2921
297297
170+-180 Behavior Type
Type AType B
4719
248253
>180 Behavior Type
Type AType B
5919
378361
Pooled Table Occurrence of CHD
Yes No
Behavior Type Type A 178 1411 1589
Type B 79 1486 1565
257 2897 3154Var of estimate of log(OR) = 0.01956 (SD = 0.141)
Var of log(ORW) = 0.01960 (SD = 0.140)
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Balance at Stratum Level: Case-Control Study of Pancreatic Cancer
Pancreatic Cancer
Cases Controls
Coffee Drinking
(cups/day)
347 555 902
0 20 88 108
367 643 1010
1≥
0.255) (SD 065.0))ˆvar(log( ==RO
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Stratification by Hypothetical C
Pancreatic Cancer
Factor Cases Controls
C Coffee drinking
(cups/day)
10
31218
11318
2.76
Not C Coffee drinking
(cups/day)
10
352
44270
2.77
RO ˆ
0.308) (SD 095.0))ˆvar(log( ==WRO
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Increase in Variability Associated with Stratification
Loss of balance (on D) more common in case-control studies
Loss of precision not usually a problem with stratification on a single variable, but can quickly become an issue with stratification on several variables Think of how many strata are need with
5 (10) binary confounding variables
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Increase in Variability Associated with Stratification
Two remedies to loss of precision associated with excessive stratification Make modeling assumptions about
relationship of confounders to D • Regression models (logistic regression)
Maintain balance after stratification by design
• Matched designs
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Warning: Woolf’s Method
Woolf’s Method approximations fail when every stratum has small sample size (OK when some strata have large sample sizes) Mantel-Haenszel estimator still works
well in these situations (perhaps use exact CIs)
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