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
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