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

5

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

6

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

13

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

14

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

≥

17

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)