2014 Page 1Confounding: Methods to control or reduce confounding

• Methods used in study design to reduce confounding– Randomization– Restriction– Matching

• Methods used in study analysis to reduce confounding– Stratified analysis– Multivariate analysis

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• Basic goal of stratification is to evaluate the relationship between the predictor (“cause”) and outcome (“effect”) variable in strata homogenous with respect to potentially confounding variables

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Confounding:The use of stratification to reduce confounding• For example, to examine the relationship between

smoking and lung cancer while controlling for the potentially confounding effect of gender:– Create a 2x2 table (smoking vs. lung cancer) for

men and women separately– To control for multiple confounders simultaneously,

stratify by pairs (or triplets or higher) of confounding factors. For example, to control for gender and race/ethnicity determine the OR for smoking vs. lung cancer in multiple strata:

white women, blackwomen, Hispanic women, white men, black men, Hetics.panic men, 41

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• (From the earlier example): Goal: create a summary or “adjusted” estimate for the relationship between matches and lung cancer while adjusting for the two levels of smoking (the potential confounder)

• This process is analgous to the standardization of ratesearlier in the course—in those examples the purpose of adjustment was to remove the confounding effect of age on the relationship between populations (A vs. B etc.) and rates of disease or death.

• In the present example the goal is to remove the confounding effect of smoking on the relationship betweenmatches and lung cancer. 42

Confounding:Types of summary estimators to determine uniform effect over strata• Mantel-Haenszel– We will use this estimator in the present course– Resistant to the effects of small strata or cells with

a value of “0”– Computationally a piece of cake

• Directly pooled estimators (e.g. Woolf)– Sensitive to small strata and cells with value “0”– Computationally messy but doable

• Maximum likelihood– The most “appropriate” estimator– Resistant to the effects of small strata or cells with

a value of “0”– Computationallychallenging

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Confounding: smoking, matches, and lung cancer

• ORpooled = 8.84 (7.2, 10.9)

• ORsmokers = 1.0 (0.6, 1.5)

• ORnonsmokers = 1.0 (0.5, 2.0)

Pooled CancerNo

cancer

820180Cancer810

340660No cancer270

Matches No Matches Smokers Matches

No Matches Non-smoker Matches

No Matches

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90Cancer10

90

30No cancer70

630 44

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An aside: Terminology• Pooled = combined = collapsed = unadjusted• Adjusted = summary = weighted, etc.

– All of these reflect some adjustment process such as Mantel-Haenszel or Woolf or maximum likelihood estimation to weight the strata and develop confidence intervals about the estimate.

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Confounding:Notation used in Mantel- Haenszel estimators of relative risk

Case-control: RR = OR = ad / bc

Cohort: RR = Ie

I0

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a / (a + b)=

c/ (c + d)

• Notation for case-control or cohort studies with count data

Cases Controls Total

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a c b d a + b c + d

Exposed Nonexposed Total a + c b + d a + b + c + d = T

Confounding:Notation used in Mantel- Haenszel estimators of relative risk (cont.)• Notation for cohort studies with person-time

data

RR = Ie

I0

= a / PY1

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c / PY0

Cases ControlsExposedNonexposed

a c ------

PY1

PY0

Total a + c T

Confounding:Mantel-Haenszel estimators of relative risk for stratified dataCase-Control Study:

RRMH =∑(ad / T)

i

∑(bc / T)i

Cohort Study with Count Denominators:

RRMH =∑{a(c + d) / T}

i

∑{b(a + b) / T}ICohort Study with Person-years Denominators:RRMH =

∑{a(PY ) / T}0 i

∑{b(PY ) / T}1 i

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Confounding: smoking, matches, and lung cancer

• ORpooled = 8.84 (7.2, 10.9)

• ORsmokers = 1.0 (0.6, 1.5)

•

No Matches

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90 630 51

ORnonsmokers = 1.0 (0.5, 2.0)Pooled

Cancer

No cancer

Matches 820 340No Matches 180 660Smokers Cancer No cancerMatches 810 270No Matches 90 30Non-smoker Cancer No cancerMatches 10 70

Confounding:Mantel-Haenszel estimators of relative risk for stratified data (smoking, matches, lung cancer

RRMH = ∑(ad / T)i / ∑(bc / T)i

Numerator of MH estimator:

• For smokers: (ad/T)=(810*30)/1200=20.25;

• For nonsmokers: (ad/T)=(10*630)/800=7.88;

• Add these together: 20.25 + 7.88=28.13 (numerator)

Denominator of MH estimator:

• For smokers: (bc/T)=(270*90)/1200=20.25;

• For nonsmokers: (bc/T)=(90*70)/800=7.88;

• Add these together: 20.25 + 7.88=28.13•ORMH = 28.13 / 28.13 = 1.0 (as expected since both stratified OR’s were = 1.0)

•Be sure to try this on stratified data in which the two strata are not exactly equal to each other (but also not so different as to suggest that effect modification is present

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Confounding:Interpretation of ORMH• If ORMH (=1.0 in this example) “differs meaningfully”

from ORunadjusted (=8.8 in this example) then confounding is present

• What does “differs meaningfully” mean– This is a matter of judgment based on

biologic/clinical sense rather than on a statistical test– Even if they “differ” only slightly, generally the ORMH

rather than the ORcombined is reported as the summary effect estimate• But what is one disadvantage of reporting ORMH ?– Although there do exist statistical tests of confounding

they are not widely recommended (these tests evaluate 53

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Ho: OR = ORMH

unadjusted

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JC: test of homogeneity

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Hennekens, 1987, p305

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Review what the X^2 means in this context.

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• Confounding “pulls” the observed association away from the true association– It can either exaggerate/over-estimate the true association

(positive confounding)• Example

– RRcausal = 1.0

–RRobserved = 3.0

or

– It can hide/under-estimate the true association

(negative confounding)• Example

– RRcausal = 3.0

– RR = 1.0observed

Direction of Confounding Bias2014 Page 20

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Confounding:Summary of steps to evaluateconfoundingTable 12-10. Steps for the control of confounding and the evaluation of effect modification through stratified analysis1. Stratify by levels of the potential confounding factor.2. Compute stratum-specific unconfounded relative risk estimates.3. Evaluate similarity of the stratum-specific estimates by either eyeballing or

performing test of statistical significance. (More on this step later)4. If the effect is thought to be uniform, calculate a pooled unconfounded

summary If effect is not uniform (i.e. effect modification is present,estimate using RRMH. skip to step 6)

5. Perform hypothesis testing on the unconfounded estimate, using Mantel-Haenszel chi-square and compute confidence interval.

6. If effect is not thought to be uniform (i.e., if effect modification is present):a. Report stratum-specific estimates, results of hypothesis testing, and

confidence intervals for each estimate

b.If desired, calculate a summary unconfounded estimate using a standar6d6izedformula 2014 Page 80

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JC: test of homogeneity

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