Confounding, Effect Modification, and Stratification

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Confounding, Effect Modification, and
Stratification
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Adding a Third Dimension to the RxC picture
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1. 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.

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Examples of Confounding
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Confusion over postmenopausal hormones
?
Heart attacks (MI)
Postmenopausal HRT
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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.

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Example Nurses Health Study
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Nurses Health Study
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No apparent Confounding
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RCT Womens Health Initiative (2002)
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Controlling for confounders in medical 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).

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Analytical identification of confounders through
stratification
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Mantel-Haenszel ProcedureNon-regression
technique used to identify confounders and to
control for confounding in the statistical
analysis phase rather than the design phase of a
study.
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Stratification Series of 2x2 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.

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StratificationSeries of 2xK 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).

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Road Map

• 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(ED/C) P(E/C)P(D/C) and
  P(ED/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 ORs or RRs
  (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.

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From Agresti

• It is more informative to estimate the strength
  of association than simply to test a hypothesis
  about it.
• When the association seems stable across partial
  tables, we can estimate an assumed common value
  of the k true odds ratios.

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Controlling for confounding by stratification

• 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)
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Program A

• Stratum 1 only those who applied to program A

 
 
Stratum-specific RR .90 (.87-.94)
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Program B

• Stratum 2 only those who applied to program B

 
 
Stratum-specific RR .99 (.96-1.03)
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Program C

• Stratum 3 only those who applied to program C

 
 
Stratum-specific RR 1.08 (.91-1.30)
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Program D

• Stratum 4 only those who applied to program D

 
 
Stratum-specific RR 1.02 (.89-1.18)
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Program E

• Stratum 5 only those who applied to program E

 
 
Stratum-specific RR .88 (.67-1.17)
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Program F

• Stratum 6 only those who applied to program F

 
 
Stratum-specific RR 1.09 (.84-1.42)
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Summary

• Crude RR 1.25 (1.20 1.32)
• Stratum specific RRs
• .90 (.87-.94)
• .99 (.96-1.03)
• 1.08 (.91-1.30)
• 1.02 (.89-1.18)
• .88 (.67-1.17)
• 1.09 (.84-1.42)
• Maentel-Haenszel Summary RR .97
• Cochran-Mantel-Haenszel Test is NS. Gender and
  denial of admissions are conditionally
  independent given program.
• The apparent association (RR1.25) was due to
  confounding.

 
 
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Cochran-Mantel-Haenszel Test of Conditional
Independence

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

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CMH test of conditional independence
Strata k
Nk
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CMH test of conditional independence
Strata k
Nk
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E.g., for Berkeley
Result is NS
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Summary

• Crude RR 1.25 (1.20 1.32)
• Stratum specific RRs
• .90 (.87-.94)
• .99 (.96-1.03)
• 1.08 (.91-1.30)
• 1.02 (.89-1.18)
• .88 (.67-1.17)
• 1.09 (.84-1.42)
• Breslow-Day test is NS (ORs are similar across
  strata). Therefore, OK to combine them

 
 
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The Mantel-Haenszel Summary Risk Ratio
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The Mantel-Haenszel Summary Risk Ratio
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The Mantel-Haenszel Summary Risk Ratio
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E.g., for Berkeley
Use computer to get confidence limits
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The Mantel-Haenszel Summary Odds Ratio
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Example
Country
 
 
Source Agresti. Introduction to Categorical Data
Analysis. 2007. Chapter 3.
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In SAS
  proc freq datasecondhand weight number
specifies the size of each 2x2 cell tables
countryNoSpouseNotCase/ cmh run
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CMH test of conditional independence p.0196
Significant CMH test means that there does appear
to be an association between spousal smoking and
cancer, after controlling for country.
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Breslow-Day test of homogeneity NS
Controlling for Country  
Breslow-Day Test for
Homogeneity of the Odds Ratios

Chi-Square 0.2381
DF Pr gt ChiSq
0.8878     Total Sample Size 1262
NS means theres no evidence that ORs differ
across strata (OK to combine them into summary OR)
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MH OR and confidence limits
  Summary Statistics for Spouse by
Case Controlling for
Country   Estimates of the Common Relative
Risk (Row1/Row2)   Type of Study
Method Value

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
Case-Control
Mantel-Haenszel 1.0536 1.8217
(Odds Ratio) Logit 1.0521
1.8203  
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Example
Country
 
 
Source Agresti. Introduction to Categorical Data
Analysis. 2007. Chapter 3.
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The Mantel-Haenszel Summary Odds Ratio
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Summary OR
Not Surprising!
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MH OR assumptions

• OR or RR doesnt vary across strata.
  (Homogeneity!)
• If exposure/disease association does vary for
  different subgroups, then the summary OR or RR is
  not appropriate

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advantages and limitations

• advantages
• Mantel-Haenszel summary statistic is easy to
  interpret and calculate
• Gives you a hands-on feel for the data
• disadvantages
• 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.

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2. Effect Modification

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

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Years of Life Lost Due to Obesity (JAMA. Jan 8
2003289187-193)

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

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Conclusion

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

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Among white women, stage of breast cancer at
detection is associated with education.
However, no clear pattern among black women.
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Colon cancer and obesity in pre- and
post-menopausal women
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Hypothetical Example Effect Modification
OR 1.0 Conclusion Watching the World Series
doesnt affect anyones mood?
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Baseball-team preference
Rockies Fans
Red Sox Fans
Other/none
Should have highly significant Breslow-Day
statistic!