confounding, effect modification, and stratification

Confounding, Effect Confounding, Effect Modification, and StratificationModification, and Stratification

Adding a Third Dimension to Adding a Third Dimension to the the RxCRxC picture picture

Exposure Disease?

Mediator

Confounder

Effect modifier

1. Confounding1. Confounding

A confounding variable is associated with the exposure and it affects the outcome, but it is not an intermediate link in the chain of causation between exposure and outcome.

Alcohol Lung cancer

Smoking

Confounding: ExampleConfounding: Example

Confounding: exampleConfounding: example

Drinker

Non-drinker

100 200

Lung cancer No lung cancer

50 50

50 150

50% of cases are drinkers, but only 25% of controls are drinkers.

Therefore, it appears that drinking is strongly associated with lung cancer.

Confounding: exampleConfounding: example

Drinker

Non-drinker

Lung cancer No lung cancer

45 15

30 10

Drinker

Non-drinker

Lung cancer No lung cancer

5 35

20 140

Smoker

Non-smoker

Among smokers, 45/75=60% of lung

cancer cases drink and

15/25=60% of controls drink.

Among non-smokers 5/25=20% of lung cancer

cases drink and

35/175=20% of controls drink.

75

25

25

175

Confusion over Confusion over postmenopausal hormones: postmenopausal hormones:

Postmenopausal HRT Heart attacks (MI)?

High SES, high education, other confounders?

Mixture May Rival Estrogen in Preventing Heart Disease

August 15, 1996, Thursday    

“Widely prescribed hormone pills that combine estrogen and progestin appear to be just as effective as estrogen alone in preventing heart disease in women after menopause, a study has concluded.”

“Many women take hormones … to reduce the risk of heart disease and broken bones.”

“More than 30 studies have found that estrogen after menopause is good for the heart.”

Example: Nurse’s Health Study Example: Nurse’s Health Study

protective relative risks

Nurse’s Health StudyNurse’s Health Study

No apparent Confounding…No apparent Confounding…

e.g., the effect is the same among smokers and non-smokers, so the

association couldn’t be due to confounding by smokers (who may take less hormones and certainly get

more heart disease).

RCT: Women’s Health RCT: Women’s Health Initiative (2002)Initiative (2002)

On hormones

On placebo

Controlling for confounders in Controlling for confounders in medical studiesmedical studies

1. Confounders can be controlled for in the design phase of a study (randomization or restriction or matching).

2. Confounders can be controlled for in the analysis phase of a study (stratification or multivariate regression).

Analytical identification of Analytical identification of confounders through confounders through

stratification stratification

Mantel-Haenszel Procedure:Mantel-Haenszel Procedure:Non-regression technique used to Non-regression technique used to

identify confounders and to control identify confounders and to control for confounding in the for confounding in the statistical statistical

analysisanalysis phase rather than the phase rather than the designdesign phase of a study.phase of a study.

Stratification: “Series of 2x2 Stratification: “Series of 2x2 tables”tables”

Idea: Take a 2x2 table and break it into a series of smaller 2x2 tables (one table at each of J levels of the confounder yields J tables).

Example: in testing for an association between lung cancer and alcohol drinking (yes/no), separate smokers and non-smokers.

Stratification:“Series of 2xK Stratification:“Series of 2xK tables”tables”

Idea: Take a 2xK table and break it into a series of smaller 2xK tables (one table at each of J levels of the confounder yields J tables).

Example: In evaluating the association between lung cancer and being either a teetotaler, light drinker, moderate drinker, or heavy drinker (2x4 table), separate into smokers and non-smokers (two 2x4 tables).

Road MapRoad Map1. Test for Conditional Independence (Mantel-Haenszel, or “Cochran-Mantel-

Haenszel”, Test).Null hypothesis: when conditioned on the confounder, exposure and disease are

independent. Mathematically, (for dichotomous confounder): P(E&D/~C) = P(E/~C)*P(D/~C) and P(E&D/C)=P(E/C)*P(D/C)Example: once you condition on smoking, alcohol and lung cancer are

independent; M-H test comes out NS.

2. Test for homogeneity. Breslow-Day.Null hypothesis: the relationship (or lack of relationship) between exposure and

disease is the same in each stratum (homogeneity). Example: B-D test would come out significant if alcohol aggravated the risk of

cigarettes on lung cancer but did not increase lung cancer risk in non-smokers. Homogeneity does NOT require independence!!

3. If homogenous, for series of 2x2 tables, you can take a weighted average of OR’s or RR’s (which should be similar in each stratum !) from the strata to get an overall OR or RR that has been controlled for confounding by C.

Controlling for confounding by Controlling for confounding by stratificationstratification

Example: Gender Bias at Berkeley?(From: Sex Bias in Graduate Admissions: Data from Berkeley, Science 187: 398-403; 1975.)

Crude RR = (1276/1835)/(1486/2681) =1.25

(1.20 – 1.32)

Denied

Admitted

1835 2681

Female Male

1276 1486

559 1195

Program AProgram A

Stratum 1 = only those who applied to program A

Stratum-specific RR = .46 (.30-.70)

Denied

Admitted

108 825

Female Male

19 314

89 511

Program BProgram B

Stratum 2 = only those who applied to program B

Stratum-specific RR = 0.86 (.48-1.54)

Denied

Admitted

25 560

Female Male

8 208

17 352

Program CProgram C

Stratum 3 = only those who applied to program C

Stratum-specific RR = 1.05 (.94-1.16)

Denied

Admitted

593 325

Female Male

391 205

202 120

Program DProgram D

Stratum 4 = only those who applied to program D

Stratum-specific RR = 1.02 (.92-1.12)

Denied

Admitted

375 407

Female Male

248 265

127 142

Program EProgram E

Stratum 5 = only those who applied to program E

Stratum-specific RR = 0.96 (.87-1.05)

Denied

Admitted

393 191

Female Male

289 147

104 44

Program FProgram F

Stratum 6 = only those who applied to program F

Stratum-specific RR = 1.01 (.97-1.05)

Denied

Admitted

341 373

Female Male

321 347

20 26

SummarySummary

Crude RR = 1.25 (1.20 – 1.32)

Stratum specific RR’s:.46 (.30-.70)

0.86 (.48-1.54)1.05 (.94-1.16) 1.02 (.92-1.12)0.96 (.87-1.05)1.01 (.97-1.05)

Maentel-Haenszel Summary RR: .97

Cochran-Mantel-Haenszel Test is NS. Gender and denial of admissions are conditionally independent given program.

The apparent association (RR=1.25) was due to confounding.

Cochran-Mantel-Haenszel Test Cochran-Mantel-Haenszel Test of Conditional Independenceof Conditional Independence

The (Cochran)-Mantel-Haenszel statistic tests the null hypothesis that exposure and disease are independent when conditioned on the confounder.

CMH test of conditional CMH test of conditional independenceindependence

Exposed

Unexposed

 Disease No Disease

a b

c dStrata k

21

1

2

1 ~

)(

))](([

k

ik

k

k

ik

aVar

aEa

Nk

)1(

)(*)(*)(*)()(

)(*)()(

2

kk

kkkkkkkkk

k

kkkkk

NN

dbcadcbaaVar

N

cabaaE

CMH test of conditional CMH test of conditional independenceindependence

Exposed

Unexposed

 Disease No Disease

a b

c dStrata k

21

1

2

1 ~

)(

))](([

k

ik

k

k

ik

aVar

aEa

)1(

2*1*2*1)(

1*1)(

2

kk

kkkkk

k

kkk

NN

colcolrowrowaVar

N

colrowaE

Nk

E.g., for Berkeley…E.g., for Berkeley…

...]...)1)5118931419(()5118931419(

)511314(*)8919(*)51189(*)31419([(

...])....)352172088(

)178(*)2088((8()

)5118931419()8919(*)31419(

(19[(

2

2

Result is NS

SummarySummary

Crude RR = 1.25 (1.20 – 1.32)

Stratum specific RR’s:.46 (.30-.70)

0.86 (.48-1.54)1.05 (.94-1.16) 1.02 (.92-1.12)0.96 (.87-1.05)1.01 (.97-1.05)

Breslow-Day test rejects (p=.0023) because of the protective effect for women in program A. We will still

combine them—but that may obscure a potentially interesting pro-female bias in program A!

The Mantel-Haenszel The Mantel-Haenszel Summary Risk RatioSummary Risk Ratio

k

i i

iii

k

i i

iii

T

bacT

dca

1

1

)(

)(

Disease

Not Disease

Exposure Not Exposed

a c

b d

k strata

The Mantel-Haenszel The Mantel-Haenszel Summary Risk RatioSummary Risk Ratio

k

i i

iii

k

i i

iii

T

bacT

dca

1

1

)(

)(

Disease

Not Disease

Exposure Not Exposed

a c

b d

total unexposed

total exposed

Exposed &disease

Unexposed&disease

The Mantel-Haenszel The Mantel-Haenszel Summary Risk RatioSummary Risk Ratio

k

i i

iii

k

i i

iii

T

bacT

dca

1

1

)(

)(

Disease

Not Disease

Exposure Not Exposed

a c

b d

)(

)(

)/(

)/(

bac

dca

dcc

baaRR

E.g., for Berkeley…E.g., for Berkeley…

97.

714)341(347

584)393(147

782)375(265

918)593(205

585)25(208

933)108(314

714)373(321

584)191(289

782)407(248

918)325(391

585)560(8

933)825(19

Use computer to get confidence limits…

The Mantel-Haenszel The Mantel-Haenszel Summary Odds RatioSummary Odds Ratio

Exposed

Not Exposed

Case Control

a b

c d

k

i i

ii

k

i i

ii

T

cbT

da

1

1

Country

OR = 1.32 Spouse smokes

Spouse does not smoke

137 363

71 249US

Spouse smokes

Spouse does not smoke

19 38

5 16

Great

Britain

OR = 1.6

Spouse smokes

Spouse does not smoke

Lung Cancer Control

73 188

21 82Japan OR = 1.52

Source: Agresti. Introduction to Categorical Data Analysis. 2007. Chapter 3.

Example…

In SAS…In SAS…

 proc freq data=secondhand;

weight number; * specifies the size of each 2x2 cell;tables country*NoSpouse*NotCase/ cmh;run;

CMH test of conditional CMH test of conditional independence: independence: p=.0196p=.0196

  Summary Statistics for Spouse by Case Controlling for Country  Cochran-Mantel-Haenszel Statistics (Based on Table Scores) Statistic Alternative Hypothesis DF Value Probƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ 1 Nonzero Correlation 1 5.4497 0.0196

Significant CMH test means that there does appear to be an association between spousal smoking and

cancer, after controlling for country.

Breslow-Day test of Breslow-Day test of homogeneity: NShomogeneity: NS

Controlling for Country  Breslow-Day Test for Homogeneity of the Odds Ratios ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Chi-Square 0.2381 DF

Pr > ChiSq 0.8878  

Total Sample Size = 1262

NS means there’s no evidence that OR’s differ across strata (OK to combine them into summary

OR)

  Summary Statistics for Spouse by Case Controlling for Country  Estimates of the Common Relative Risk (Row1/Row2)  Type of Study Method Value ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ

Case-Control Mantel-Haenszel 1.3854 (Odds Ratio) Logit 1.3839  Cohort Mantel-Haenszel 1.2779 (Col1 Risk) Logit 1.2760  Cohort Mantel-Haenszel 0.9225 (Col2 Risk) Logit 0.9223  Type of Study Method 95% Confidence Limits ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Case-Control Mantel-Haenszel 1.0536 1.8217 (Odds Ratio) Logit 1.0521 1.8203 

MH OR and confidence limits…

Country

OR = 1.32 Spouse smokes

Spouse does not smoke

137 363

71 249US

Spouse smokes

Spouse does not smoke

19 38

5 16

Great

Britain

OR = 1.6

Spouse smokes

Spouse does not smoke

Lung Cancer Control

73 188

21 82Japan OR = 1.52

Example…

Source: Agresti. Introduction to Categorical Data Analysis. 2007. Chapter 3.

The Mantel-Haenszel The Mantel-Haenszel Summary Odds RatioSummary Odds Ratio

Exposed

Not Exposed

Case Control

a b

c d

k

i i

ii

k

i i

ii

T

cbT

da

1

1

Summary ORSummary OR

38.1

820363*71

785*38

36421*188

820137*249

7816*19

36482*73

Not Surprising!

MH OR assumptionsMH OR assumptions

OR or RR doesn’t vary across strata. (Homogeneity!)

If exposure/disease association does vary for different subgroups, then the summary OR or RR is not appropriate…

advantages and limitationsadvantages and limitationsadvantages…• Mantel-Haenszel summary statistic is easy to interpret

and calculate• Gives you a hands-on feel for the datadisadvantages…• Requires categorical confounders or continuous

confounders that have been divided into intervals • Cumbersome if more than a single confounder

To control for 1 and/or continuous confounders, a multivariate technique (such as logistic regression) is preferable.

2. Effect Modification2. Effect Modification

Effect modification occurs when the effect of an exposure is different among different subgroups.

Years of Life Lost Due to Obesity Years of Life Lost Due to Obesity ((JAMA.JAMA. Jan 8 2003;289:187-193) Jan 8 2003;289:187-193)

Data from US Life Tables and the National Health and Nutrition Examination Surveys (I, II, III).

ConclusionConclusion

Race and gender modify the effect of obesity on years-of-life-lost.

Among white women, stage of breast cancer at detection is associated with education.

However, no clear pattern among black women.

Colon cancer and obesity in pre- Colon cancer and obesity in pre- and post-menopausal womenand post-menopausal women

Obesity appears to be a risk

factor in pre-menopausal

womenBut appears to be

protective or unrelated in post-menopausal

women