C

1
E
?
C
D
2
DAGs also useful for
3
Confounding and Interaction Part II

• Methods to reduce confounding
• during study design
• Randomization
• Restriction
• Matching
• during study analysis
• Stratified analysis
• (Mathematical regression)
• Interaction
• What is it? How to detect it?
• Additive vs. multiplicative interaction
• Comparison with confounding
• Statistical testing for interaction
• Implementation in Stata

4
Confounding
Confounding occurs if there is a factor C that is
a Common Cause of both E and D
E
C
?
D

• C is part of a backdoor path to E and D

5
Confounding
E

• Adjusting/controlling for C blocks the backdoor
  path eliminates confounding

C
?
D
6
Methods to Prevent or Reduce Confounding

• By prohibiting at least one segment of the
  exposure- confounder - disease path, confounding
  is precluded

E
C
?
D

• C is part of a backdoor path to E and D

7
Confounding and Interaction Part II

• Methods to reduce confounding
• during study design
• Randomization
• Restriction
• Matching
• during study analysis
• Stratified analysis
• (Mathematical regression)

8
Randomization to Reduce Confounding

• Definition random assignment of subjects to
  exposure (e.g., treatment) categories
• All subjects ? Randomize
• Distribution of any variable is theoretically the
  same in the exposed group as the unexposed
• Theoretically, can be no association between
  exposure and any other variable
• Comes close to goal of exchangeability or
  counterfactual ideal (although still falls short)
• One of the most important inventions of the 20th
  Century!

Exposed (treatment)
Unexposed (no treatment)
9
Randomization to Prevent Confounding
Blocking the path confounder exposure explains
the exulted role of randomization in clinical
research
E
C
?
D
10
Randomization to Reduce Confounding

• All subjects ? Randomize
• Applicable only for ethically assignable
  exposures (ie, interventions, experiments)
• Not for naturally occurring exposures (e.g., air
  pollution)
• Special strength of randomization is its ability
  to control the effect of confounding variables
  about which the investigator is unaware
• Because distribution of any variable
  theoretically same across randomization groups
• Does not, however, always eliminate confounding!
• By chance alone, there can be imbalance
• Magnitude of bias contained in confidence
  interval
• Less of a problem in large studies
• Techniques exist to ensure balance of certain
  variables (e.g., blocked or stratified
  randomization)

Exposed (treatment)
Unexposed (no treatment)
11
But what if we cannot randomize?

• Methods to reduce confounding
• during study design
• Randomization
• Restriction
• Matching
• during study analysis
• Stratified analysis
• (Mathematical regression)

12
Restriction to Prevent Confounding

• AKA Specification
• Definition Restrict enrollment to only those
  subjects who have a specific value/range of the
  confounding variable

E
C
?
D

• e.g., when diet is a confounder, restrict to
  persons with a certain diet

13
Night lights and childhood myopia

• RQ Do night lights cause children to develop
  myopia?

Night Lights
Restrict to children with parents without myopia
Parental Myopia
?
Childs Myopia
14
Restriction to Prevent Confounding

• Particularly useful when confounder is
  quantitative in scale but difficult to measure

Behavioral factors (unmeasured)
Commercial sex

• e.g.
• RQ Does practice of commercial sex result in
  acquisition of HHV-8 infection?
• Issue Confounding by unmeasured behavioral
  factors operating through injection drug use

?
Injection drug use
HHV-8

• Problem degree of injection drug use is
  difficult to measure
• Solution restrict to subjects with no injection
  drug use, thereby precluding the need to measure
  degree of injection use
• Cannon et. al NEJM 2001
• Restricted to persons denying injection drug use

• e.g., Effect of HIV infection on pulmonary
  hypertension confounding by IDU (Hsue et al
  AIDS 2008)

15
Restriction to Reduce Confounding

• Advantages
• conceptually straightforward
• handles difficult to quantitate variables
• unlike matching, decisions can be made about
  individual subjects (include or not include)
  irrespective of other subjects
• can also be used in analysis phase

16
Restriction to Reduce Confounding

• Disadvantages
• may limit number of eligible subjects
• cost-inefficient to screen subjects, then not
  enroll
• residual confounding may persist if restriction
  categories not sufficiently narrow (e.g. 20 to
  30 years old restriction in Birth Order - Down
  syndrome question might be too broad)
• limits generalizability, but
• Validity before generalizabilty
• Including small numbers of persons in rare
  stratum of confounders (e.g., race) and then
  finding an effect for an exposure/treatment does
  not mean the effect is operative in that rare
  group
• Politics trumping science
• not possible to evaluate the relationship of
  interest at different levels of the restricted
  variable (i.e. cannot assess statistical
  interaction)
• Bottom Line
• Restriction not used as much as it should be

17

• Methods to reduce confounding
• during study design
• Randomization
• Restriction
• Matching
• during study analysis
• Stratified analysis
• (Mathematical regression)

18
Matching to Reduce Confounding

• Definition only unexposed/non-case subjects are
  enrolled who match those of the comparison group
  (either exposed or cases) in terms of the
  confounder in question
• Mechanics depends upon study design
• e.g. cohort study unexposed subjects are
  matched to exposed subjects according to their
  values for the potential confounder.
• e.g. matching on race
• One unexposedlatino enrolled for each
  exposedlatino
• One unexposedasian enrolled for each
  exposedasian
• e.g. case-control study non-diseased controls
  are matched to diseased cases
• e.g. matching on age
• One controlage 50 enrolled for each
  caseage 50
• One controlage 70 enrolled for each
  caseage 70
• can be in age ranges, e.g., /- 2.5 years
• Operationally, performed by individual matching
  (one-by-one) or frequency matching (e.g., select
  control group at the end to match distribution of
  confounding factor in case group)

19
Matching to Prevent Confounding

• Cross-sectional/cohort study

Uncommon in large cohort studies typically
because there is not just one exposure of
interest More common and can be valuable in
smaller studies with a single focused exposure

• Case-control study

More common use of matching Can be relevant for a
variety of exposures
20
Advantages of Matching

• 1. Useful in preventing confounding by factors
  which would be nearly impossible or statistically
  inefficient to manage in analysis phase
• e.g., neighborhood is a nominal variable with
  multiple values (complex nominal variable)
• e.g., Case-control study of the effect of a BCG
  vaccine in preventing TB (Int J Tub Lung Dis.
  2006)
• Cases newly diagnosed TB in Brazil
• Controls persons without TB
• Exposure receipt of a BCG vaccine
• Potential confounder neighborhood (village) of
  residence related to ambient TB incidence and
  practices regarding BCG vaccine
• Control sampling Relying upon random sampling
  without attention to neighborhood may result in
  (especially in a small study) choosing no
  controls from some of the neighborhoods seen in
  the case group (i.e., cases and controls lack
  overlap)
• Matching on neighborhood ensures overlap
• Even if all neighborhoods seen in the case group
  were represented in the control group, adjusting
  for neighborhood with analysis phase strategies
  is problematic

21
Neighborhood If you chose to stratify to manage
confounding, the number of strata is unwieldy
Crude
Stratified
Mission
Sunset
Richmond
Castro
Pacific Heights
Marina
Matching avoids this dilemma in the analysis phase
22
Advantages of Matching

• 2. Provides a way to ensure overlap between
  comparator groups (e.g., cases/controls) in the
  distribution of confounders other than complex
  nominal variables
• e.g., Case-control study of prostate cancer --
  potential confounding by age
• Cases will have many old individuals
• Random sampling of controls, especially in
  smaller studies, apt not to contain oldest
  individuals
• Matching age distribution of controls to age
  distribution of cases ensures complete overlap in
  age between cases and controls

cases
controls
23
Advantages of Matching

• 3. By ensuring a balanced number of cases and
  controls (in a case-control study) or
  exposed/unexposed (in a cohort study) within the
  various strata of the confounding variable,
  statistical precision may be increased

24
Smoking, Matches, and Lung Cancer
A. Random sample of controls Crude

OR crude 8.8
Stratified
Non-Smokers
Smokers
OR CF ORsmokers 1.0
OR CF- ORnon-smokers 1.0
Matching facilitates statistically efficient
stratification

• ORadj 1.0 (0.31 to 3.2)

B. Controls matched on smoking
Smokers
Non-Smokers
OR CF ORsmokers 1.0
OR CF- ORnon-smokers 1.0

• ORadj 1.0

(0.40 to 2.5)
Underappreciated benefit of matching Improved
precision
25
Advantages of Matching

• 4. People find it easy to understand, likely
  because it comes close to fulfilling
  exchangeability objective.
• So intuitive that it is often the first choice
  among the uninitiated (lets match on x, y, and
  z)
• This is both good and bad

26
Disadvantages of Matching

• 1. Finding appropriate matches may be difficult
  and expensive. Therefore, the gains in
  statistical efficiency can be offset by increases
  in overall costs.
• Exacerbated when matching gt 1 factors jointly
• 2. In a case-control study, factor used to match
  subjects cannot be itself evaluated as a risk
  factor for the disease. In general, matching
  decreases robustness of study to address
  secondary questions.
• 3. In a case-control study, must still perform
  either stratification or regression in the
  analysis phase.
• This is because matching artifactually induces
  cases and controls to look more similar regarding
  exposure
• If this extra step is forgotten (out of ignorance
  or the matching aspect simply gets lost over
  time) the crude OR is biased

27
More Disadvantages of Matching

• 4. Decisions are irrevocable
• if you happened to match on an intermediary
  factor, you have lost ability to evaluate role of
  exposure in question via that pathway
• study of effect of exercise on coronary artery
  disease. Matching on HDL cholesterol precludes
  ability to look assess total effect of exercise
• Inadvertently matching on a collider permanently
  induces bias
• 5. If potential confounding factor really isnt
  a confounder, statistical precision can be worse
  than no matching.
• Bottomline
• Matching very useful in certain situations but
  should not be done indiscriminately.
• Think carefully before you match and seek advice

28
Overmatching

• Often used term, poorly understood
• Two types of overmatching manifestations
• Overmatching resulting in precision losses
• In case-control studies, matching on factors
  which are truly not confounders will result in
  larger standard errors compared to not matching
• Especially bad for factors associated with
  exposure but not disease
• In case-control or cohort studies, matching on
  factors very strongly related to exposure results
  in collinearity
• Not unique to matching occurs with
  stratification or regression as well
• Overmatching resulting in bias
• Matching on intermediary factors
• Matching on colliders

29
Confounding and Interaction Part II

• Methods to reduce confounding
• during study design
• Randomization
• Restriction
• Matching
• during study analysis
• Stratified analysis
• (Mathematical regression)

30
Stratification to Reduce Confounding
Strategies in the analysis phase

• Goal evaluate the relationship between the
  exposure and outcome in strata homogeneous with
  respect to potentially confounding variables
• Each stratum is a mini-example of restriction!
• CF confounding factor

Crude
Stratified
CF Level I
CF Level 2
CF Level 3
31
Smoking, Matches, and Lung Cancer

Crude
OR crude
Stratified
Non-Smokers
Smokers
OR CF ORsmokers
OR CF- ORnon-smokers

• ORcrude 8.8
• Each stratum is unconfounded with respect to
  smoking
• ORsmokers 1.0
• ORnon-smoker 1.0

32
More than One Confounder
RQ Does Chlamydia pneumoniae infection cause
coronary artery disease (CAD)?
Chlamydia pneumoniae infection
Smoking


Age
?
CAD
33
Stratifying by Multiple Confounders
Crude

• Potential Confounders Age and Smoking
• To control for multiple confounders
  simultaneously, must construct mutually exclusive
  and exhaustive strata

34
Stratifying by Multiple Potential Confounders
Crude
Stratified
lt40 smokers
40-60 smokers
gt60 smokers
gt60 non-smokers
40-60 non-smokers
lt40 non-smokers
Each of these strata is unconfounded by age and
smoking
35
Adjusted Estimate from the Stratified Analyses

• After the stratum have been formed, what next?
• Process Summarize the unconfounded estimates
  from the two (or more) strata to form a single
  overall unconfounded adjusted estimate
• e.g., for matches-lung cancer example, summarize
  the odds ratios from the smoking stratum and
  non-smoking stratum into one odds ratio

36
Smoking, Matches, and Lung Cancer

Crude
OR crude
Stratified
Non-Smokers
Smokers
OR CF ORsmokers
OR CF- ORnon-smokers

• ORcrude 8.8
• ORsmokers 1.0
• ORnon-smoker 1.0
• ORadjusted 1.0

37
Smoking, Caffeine Use and Delayed Conception
RR risk ratio

Crude
RR crude 1.7
Stratified
No Caffeine Use
Heavy Caffeine Use
RRno caffeine use 2.4
RRcaffeine use 0.7
Stanton and Gray. AJE 1995
Is it appropriate to summarize these two
stratum-specific risk ratio estimates into a
single number?
38
Underlying Assumption Needed to Form a Summary of
the Unconfounded Stratum-Specific Estimates

• If the relationship between the exposure and the
  outcome varies meaningfully in a
  clinical/biologic sense and statistically across
  strata of a third variable
• it is not appropriate to create a single summary
  estimate of all of the strata
• i.e. When you summarize across strata, the
  assumption is that no statistical interaction
  is present

39
Statistical Interaction

• Definition
• when the magnitude of a measure of association
  (between exposure and disease) meaningfully
  differs according to the value of some third
  variable
• Synonyms
• Effect-measure modification
• Effect modification
• Heterogeneity of effect
• Heterogeneity of measure
• Nonuniformity of effect
• Effect variation
• Proper terminology
• e.g., Smoking, caffeine use, delayed conception
• Caffeine use modifies the effect of smoking on
  the risk for delayed conception.
• There is interaction between caffeine use and
  smoking in the risk for delayed conception.
• Caffeine is an effect modifier in the
  relationship between smoking and delayed
  conception.

40

RR 3.0
RR 3.0
Parallel lines means no interaction

RR 11.2
RR 3.0
Non-parallel lines means interaction
41


RR 2.4
RR 0.7
42
Interaction is everywhere

• Susceptibility to infectious diseases
• e.g.,
• exposure sexual activity
• disease HIV infection
• effect modifier chemokine receptor phenotype
• Susceptibility to non-infectious diseases
• e.g.,
• exposure smoking
• disease lung cancer
• effect modifier genetic susceptibility to smoke
• Susceptibility to drugs (efficacy and side
  effects)
• effect modifier genetic susceptibility to drug
• personalized medicine is an expression of
  interaction
• But in practice to date, difficult to document
• Genomics may change this

43
Smoking, Caffeine Use and Delayed Conception
Additive vs Multiplicative Interaction

Crude
RR crude 1.7 RD crude 0.07
Stratified
No Caffeine Use
Heavy Caffeine Use
RRno caffeine use 2.4 RDno caffeine use
0.12
RRcaffeine use 0.7 RDcaffeine use -0.06
Multiplicative interaction
Additive interaction
RD Risk Difference Risk exposed - Risk
Unexposed
44
Additive vs Multiplicative Interaction

• Assessment of whether interaction is present
  depends upon the measure of association
• ratio measure (multiplicative interaction) or
  difference measure (additive interaction)
• Hence, the term effect-measure modification
• Absence of multiplicative interaction implies
  presence of additive interaction (exception no
  association)

Additive interaction present
RR 3.0 RD 0.3
Multiplicative interaction absent
RR 3.0 RD 0.1
45
Additive vs Multiplicative Interaction

• Absence of additive interaction implies presence
  of multiplicative interaction

Multiplicative interaction present Additive
interaction absent
RR 1.7 RD 0.1
RR 3.0 RD 0.1
46
Additive vs Multiplicative Interaction

• Presence of multiplicative interaction may or may
  not be accompanied by additive interaction

RR 2.0 RD 0.1
No additive interaction
RR 3.0 RD 0.1
RR 3.0 RD 0.4
Additive interaction present
RR 2.0 RD 0.1
47
Additive vs Multiplicative Interaction

• Presence of additive interaction may or may not
  be accompanied by multiplicative interaction

RR 3.0 RD 0.4
Multiplicative interaction present
RR 2.0 RD 0.1
RR 3.0 RD 0.2
Multiplicative interaction absent
RR 3.0 RD 0.1
48
Additive vs Multiplicative Interaction

• Presence of qualitative multiplicative
  interaction is always accompanied by qualitative
  additive interaction

Multiplicative and additive interaction both
present
e.g., smoking, caffeine, delayed ocnception
49
Additive vs Multiplicative Scales

• Which do you want to use?
• Multiplicative measures (e.g., risk ratio)
• favored measure in etiologic research
• not dependent upon background incidence of
  disease
• Additive measures (e.g., risk difference)
• readily translated into impact of an exposure (or
  intervention) in terms of absolute number of
  outcomes prevented
• e.g. 1/risk difference no. needed to treat to
  prevent (or avert) one case of disease
• or no. of exposed persons one needs to take the
  exposure away from to avert one case of disease
• very dependent upon background incidence of
  disease
• gives public health impact of the exposure

50
Additive vs Multiplicative Scales

• Causally related but minor public health
  importance
• Risk ratio 2
• Risk difference 0.0001 - 0.00005 0.00005
• Need to eliminate exposure in 20,000 persons to
  avert one case of disease
• Causally related and major public health
  importance
• RR 2
• RD 0.2 - 0.1 0.1
• Need to eliminate exposure in 10 persons to avert
  one case of disease

51
Smoking, Family History and Cancer Additive vs
Multiplicative Interaction

Crude
Family History Present
Stratified
Family History Absent
Risk rationo family history 2.0 RDno family
history 0.05
Risk ratiofamily history 2.0 RDfamily history
0.20

• No multiplicative interaction but presence of
  additive interaction
• If etiology is goal, risk ratio is sufficient
• If goal is to define sub-groups of persons to
  target
• - Rather than ignoring, it is worth reporting
  that only 5 persons with a family history have
  to be prevented from smoking to avert one case
  of cancer

52
Confounding vs Interaction

• We discovered interaction by performing
  stratification as a means to evaluate for
  confounding
• This is where the similarities between
  confounding and interaction end!
• Confounding
• A backdoor path that an investigator hopes to
  prevent or rule out
• Interaction (Effect-measure modification)
• A more detailed description of the relationship
  between the exposure and disease
• A richer description of the biologic or
  behavioral system under study
• A finding to be reported, not a bias to be
  eliminated

53
Smoking, Caffeine Use and Delayed Conception
Crude

RR crude 1.7
Stratified
Heavy Caffeine Use
No Caffeine Use
RRno caffeine use 2.4
RRcaffeine use 0.7
RR adjusted 1.4 (95 CI 0.9 to 2.1) Is this
the best final answer? In etiologic research,
adjustment here is contraindicated. Instead,
report both stratum-specific risk ratios When
interaction is present, confounding becomes
irrelevant! (Exception sometimes in public
health research, the adjusted RR used to
understand net effect of the exposure across the
population)
54
Reciprocity of Interaction
Crude

RR crude 1.7
No Smoking
Stratified
Smoking
RRno caffeine use 2.3
RRcaffeine use 0.67
Caffeine use modifies the effect of smoking on
delayed conception or Smoking modifies the
effect of caffeine use on delayed conception
55
Chance as a cause of interaction? Are all
non-identical stratum-specific estimates
indicative of interaction?

Crude
OR crude 3.5
Stratified
Age gt 35
Age lt 35
ORage gt35 5.7
ORage lt35 3.4
Should we report interaction here?
56
Statistical Tests of Interaction Test of
Homogeneity (heterogeneity)

• Null hypothesis The individual stratum-specific
  estimates of the measure of association differ
  only by random variation (chance or sampling
  error)
• i.e., the strength of association is homogeneous
  across all strata
• i.e., there is no interaction
• Alternative there is heterogeneity (i.e. no
  homogeneity)
• If the test of homogeneity is significant
  (small p value), we reject the null in favor of
  the alternative hypothesis
• A variety of formal tests are available with the
  same general format, following a chi-square
  distribution
• where
• effecti stratum-specific measure of assoc.
• var(effecti) variance of stratum-specifc m.o.a.
• summary effect summary adjusted effect
• N no. of strata of third variable

57
Tests of Homogeneity with Stata

• 1. Determine crude measure of association
• e.g. for a cohort study
• command cs outcome-variable
  exposure-variable
• for smoking, caffeine, delayed conception
• -exposure variable smoking
• -outcome variable delayed
• -third variable caffeine
• command is cs delayed smoking
• 2. Determine stratum-specific estimates by
  levels of third variable
• command
• cs outcome-var exposure-var, by(third-variable)
• e.g. cs delayed smoking, by(caffeine)

58

• . cs delayed smoking
• smoking
• Exposed Unexposed
  Total
• ------------------------------------------------
  ---
• Cases 26 64
  90
• Noncases 133 601
  734
• ------------------------------------------------
  ---
• Total 159 665
  824
• Risk .163522 .0962406
  .1092233
• Point estimate 95
  Conf. Interval
• -------------------------------
  ---------------
• Risk difference .0672814
  .0055795 .1289833
• Risk ratio 1.699096
  1.114485 2.590369
• -----------------------------------------------
• chi2(1) 5.97
  Prgtchi2 0.0145
• . cs delayed smoking, by(caffeine)
• caffeine RR 95 Conf.
  Interval M-H Weight
• -------------------------------------------------
  -----------------

What does the p value mean?
59
Reporting or Ignoring Interaction

• When to report or ignore interaction is not clear
  cut.
• A clinical, statistical, and practical decision
• Clinical
• Is the magnitude of stratum-specific differences
  substantively (clinically) important?
• Is there prior evidence for the heterogeneity?
• Statistical
• There are inherent limitations in the power of
  the test of homogeneity
• Only relatively large effect sizes or large
  sample size can achieve p lt 0.05
• One approach is to report interaction for p lt
  0.10 if the magnitude of differences is
  clinically meaningful (threshold to report)
• However, meaning of p value is not different than
  other contexts
• Practical How complicated is the story?
• i.e., if it is not too complicated to report
  stratum-specific estimates, it is often more
  revealing to report potential interaction than to
  ignore it.

60
Report vs Ignore Effect-Measure
Modification?Some Guidelines
Is an art form requires consideration of both
clinical and statistical significance
61
Confounding and Interaction Part II

• Methods to reduce confounding
• during study design
• Randomization
• Restriction
• Matching
• during study analysis
• Stratified analysis
• (Mathematical regression)