Transcript No Slide Title

Confounding and Interaction: Part II

Methods to reduce confounding
– during study design:
» Randomization
» Restriction
» Matching
– during study analysis:
» Stratified analysis

Interaction
–
–
–
–
–
What is it? How to detect it?
Additive vs. multiplicative interaction
Comparison with confounding
Statistical testing for interaction
Implementation in Stata
Confounding
ANOTHER PATHWAY TO
GET TO THE DISEASE
(a mixing of effects)
Confounder
D
Methods to Prevent or Manage Confounding
By prohibiting at least
one “arm” of the
exposureconfounder - disease
structure,
confounding is
precluded
D
or
D
Randomization to Reduce
Confounding

Definition: random assignment of subjects to
exposure (e.g., treatment) categories
Exposed

All subjects  Randomize
(treatment)
Unexposed

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

One of the most important inventions of the
20th Century!
Randomization to Reduce Confounding
D
Explains the significant role
of randomization in clinical
research
Randomization prevents
confounding
Randomization to Reduce
Confounding
Exposed

All subjects  Randomize
Unexposed

Of course, applicable only for intervention
(experimental) studies

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
– Less of a problem in large studies
– Techniques exist to ensure balance of certain
variables
But what if we cannot randomize?
Restriction to Reduce Confounding

AKA Specification

Definition: Restrict enrollment to only those
subjects who have a specific value/range of the
confounding variable
– e.g., when age is confounder: include only
subjects of same narrow age range
Restriction to Reduce Confounding
D
e.g., restrict on age
Maternal
Age
Birth
Order
?
Down
Syndrome
Restriction to Prevent Confounding

Particularly useful when confounder is quantitative in scale but
difficult to measure

e.g.
– Research question: Is there an association between sexual
behavior and acquisition of HHV-8 infection?
– Issue: Is association confounded by injection drug use?
– 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
Commercial sex No.
No
311
Yes
160
% HHV-8-positive
9.6
18.8
Odds Ratio
1.0
2.2 (1.3 to 3.7)
Restriction to Reduce Confounding

Advantages:
– conceptually straightforward

Disadvantages:
– may limit number of eligible subjects
– inefficient to screen subjects, then not enroll
– “residual confounding” may persist if restriction
categories not sufficiently narrow (e.g. “20 to 30
years old” might be too broad)
– limits generalizability (but don’t worry too much
about this)
– not possible to evaluate the relationship of
interest at different levels of the restricted
variable (i.e. cannot assess interaction)
Matching to Reduce Confounding

A complex topic

Definition: only unexposed/non-case subjects are
chosen 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 unexposedblack enrolled for each exposedblack
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
Matching to Reduce Confounding
Cohort
design
D
or
Case-control
design
Also illustrates
a disadvantage
D
Advantages of Matching
1. Useful in preventing confounding by factors which would
be difficult to manage in any other way
– e.g. “neighborhood” is a nominal variable with
multiple values. (complex nominal variable)
– e.g. Cohort study of the effect of stop light cameras in
preventing motor vehicle accidents
» Exposed: persons going thru stop lights with
camera
» Unexposed: persons going thru stop lights
without camera
» Outcome: accidents
» Potential confounder: ambient driving practices in
the neighborhood
» Relying upon random sampling of unexposed
persons without attention to neighborhood may
result in (especially in a small study) choosing no
unexposed persons from some of the
neighborhoods seen in the exposed group
» Even if all neighborhoods seen in the exposed
group were represented in the unexposed group,
adjusting for neighborhood with “analysis phase”
strategies are problematic
If you chose to stratify to manage confounding,
the number of strata may be unwieldy
Crude
Accident No accident
Camera
No camera
Stratified
Mission
Sunset
Acc
No
Acc
Richmond
Acc
No
Acc
Camera
Camera
Camera
No camera
No camera
No camera
Marina
Pacific Heights
Acc
Acc
No
Acc
Acc
No
Acc
Acc
No
Acc
Castro
No
Acc
Camera
Camera
Camera
No camera
No camera
No camera
And strata for Western Addition, Russian Hill, Portero
Hill, Bayview, etc
Advantages of Matching
2. 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 is increased
Smoking, Matches, and Lung Cancer
A. Random sample of controls
Crude
Lung Ca No Lung Ca
Matches
820
340
No Matches
180
660
Stratified
Smokers
Lung Ca
Matches
No Matches
OR crude =
8.8
Non-Smokers
No
Lung Ca Lung CA
No
Lung CA
810
90
900
270
30
300
OR CF+ = ORsmokers = 1.0
Matches
No Matches
10
90
100
70
630
700
OR CF- = ORnon-smokers = 1.0
ORadj= 1.0 (0.69 to 1.45)
B. Controls matched on smoking
Smokers
Matches
No Matches
Lung Ca
No
Lung CA
810
90
900
810
90
900
OR CF+ = ORsmokers = 1.0
Non-Smokers
No
Lung Ca Lung CA
Matches
No Matches
10
90
100
10
90
100
OR CF- = ORnon-smokers = 1.0
ORadj= 1.0 (0.75 to 1.34)
Little known benefit of matching: Improved precision
Disadvantages of Matching
1. Finding appropriate matches may be difficult and
expensive and limit sample size (e.g., have to
throw out a case if cannot find a control).
Therefore, the gains in statistical efficiency can be
offset by losses in overall efficiency.
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. 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.
e.g. study of effect of sexual activity on
cervical cancer. Matching on HPV status
precludes ability to look at sexual activity
4. If potential confounding factor really isn’t a
confounder, statistical precision will be worse than
no matching.
Think carefully before you match and seek advice
Strategies in the analysis phase:
Stratification to Reduce Confounding

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!
Disease No Disease
Crude
Exposed
Unexposed
Stratified
CF Level I
Dis
CF Level 2
No
Dis
Dis
CF Level 3
No
Dis
Dis
Exp
Exp
Exp
Unexp
Unexp
Unexp

CF = confounding factor
No
Dis
Smoking, Matches, and Lung Cancer
Crude
Lung Ca No Lung Ca
Matches
820
340
No Matches
180
660
Smokers
Stratified
Matches
No Matches
Lung Ca
No
Lung CA
810
90
270
30
OR CF+ = ORsmokers

ORcrude
= 8.8

ORsmokers
= 1.0

ORnon-smoker= 1.0
OR crude
Non-Smokers
Matches
No Matches
Lung Ca
No
Lung CA
10
90
70
630
OR CF- = ORnon-smokers
Stratifying by Multiple
Confounders with More than 2 Levels
Crude
CAD
No CAD
Chlamydia
pneumoniae
infection
No Chlamydia
infection
Potential Confounders: Age and Smoking

To control for multiple confounders simultaneously,
must construct mutually exclusive and exhaustive
strata:
<40
Smokers
Non-smokers
40-60
>60
Stratifying by Multiple
Potential Confounders
Crude
CAD
No CAD
Chlamydia
No chlamydia
Stratified
<40 smokers
CAD
40-60 smokers
No
CAD
CAD
>60 smokers
No
CAD
CAD
Chlamydia
Chlamydia
Chlamydia
No
Chlamydia
No
Chlamydia
No
Chlamydia
<40 non-smokers
CAD
40-60 non-smokers
No
CAD
CAD
No
CAD
>60 non-smokers
No
CAD
CAD
Chlamydia
Chlamydia
Chlamydia
No
Chlamydia
No
Chlamydia
No
Chlamydia
No
CAD
Each of these strata is unconfounded by age and smoking
Summary Estimate from
the Stratified Analyses

After the stratum have been formed, what to do
next?

Goal: Create a single unconfounded (“adjusted”)
estimate for the relationship in question
– e.g., relationship between matches and lung
cancer after adjustment (controlling) for
smoking

Process: Summarize the unconfounded
estimates from the two (or more) strata to form a
single overall unconfounded “summary estimate”
– e.g., summarize the odds ratios from the
smoking stratum and non-smoking stratum into
one odds ratio
Smoking, Matches, and Lung Cancer
Crude
Lung Ca No Lung Ca
Matches
820
340
No Matches
180
660
Smokers
Stratified
Matches
No Matches
Lung Ca
No
Lung CA
810
90
270
30
OR crude
Non-Smokers
Matches
No Matches
OR CF+ = ORsmokers

ORcrude
= 8.8 (7.2, 10.9)

ORsmokers
= 1.0 (0.6, 1.5)

ORnon-smoker= 1.0 (0.5, 2.0)

ORadjusted
Lung Ca
No
Lung CA
10
90
70
630
OR CF- = ORnon-smokers
= 1.0 (0.69 to 1.45)
Smoking, Caffeine Use
and Delayed Conception
Crude
Delayed
Smoking
26
No Smoking
64
Stratified
Heavy
Caffeine Use
Not Delayed
133
RR crude = 1.7
601
No Caffeine
Use
Not
Delayed Delayed
Not
Delayed Delayed
Smoking
No Smoking
11
17
RRcaffeine use = 0.7
72
73
Smoking
No Smoking
15
47
61
528
RRno caffeine use = 2.4
Is it appropriate to summarize these
two stratum-specific estimates?
Underlying Assumption When Forming
a Summary of the Unconfounded
Stratum-Specific Estimates

If the relationship between the exposure and
the outcome varies meaningfully in a
clinical/biologic sense across strata of a third
variable, then it is often not appropriate to
create a single summary estimate of all of the
strata

i.e. the assumption is that no “interaction” is
present
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 modification
– Effect-measure modification
– Heterogeneity of effect

Proper terminology
– e.g. Smoking, caffeine use, and 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.
No Multiplicative Interaction
Third Variable Present
Risk of Disease
10
Third Variable Absent
1
0.45
0.1
0.15
0.15
RR = 3.0
RR = 3.0
0.05
0.01
Unexposed
Exposed
Multiplicative Interaction
Third Variable Present
Risk of Disease
10
Third Variable Absent
1
0.1
0.9
0.15
0.08
0.05
0.01
Unexposed
Exposed
RR = 11.2
RR = 3.0
Qualitative Interaction
10
Third Variable Present
Risk of Disease
Third Variable Absent
1
RR = 2.5
0.18
0.1
0.2
0.13
RR = 0.72
0.08
0.01
Unexposed
Exposed
Interaction is likely 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

But in practice to date, difficult to document
– Genomics may change this
Smoking, Caffeine Use
and Delayed Conception:
Additive vs Multiplicative Interaction
Crude
Delayed
Smoking
26
No Smoking
64
Not Delayed
RR crude = 1.7
133
601
RD crude =
0.07
Heavy
Caffeine Use
Stratified
No Caffeine
Use
Not
Delayed Delayed
Not
Delayed Delayed
Smoking
No Smoking
11
17
RRcaffeine use = 0.7
RDcaffeine use = -0.06
72
73
Smoking
No Smoking
15
47
61
528
Multiplicative
interaction
RRno caffeine use = 2.4
Additive
interaction
RDno caffeine use = 0.12
RD =
Risk Difference = Risk exposed - Risk Unexposed
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 typically
implies presence of additive interaction
1
Risk of Disease

RR = 3.0 RD = 0.3
0.45
0.1
0.15
0.15
0.05
RR = 3.0 RD = 0.1
0.01
Unexposed
Exposed
Additive
interaction
present
Multiplicative
interaction
absent
Additive vs Multiplicative Interaction
Absence of additive interaction typically implies
presence of multiplicative interaction
Multiplicative
interaction
present
1
Risk of Disease

RR = 1.7 RD = 0.1
0.1
0.25
0.15
0.15
0.05
RR = 3.0 RD = 0.1
0.01
Unexposed
Exposed
Additive
interaction
absent
Additive vs Multiplicative Interaction
Presence of multiplicative interaction may or may
not be accompanied by additive interaction
Risk of Disease
1
RR = 2.0 RD = 0.1
0.1
0.2
0.15
No additive
interaction
0.1
0.05
RR = 3.0 RD = 0.1
0.01
Unexposed
1
Risk of Disease

Exposed
RR = 3.0 RD = 0.4
0.6
0.2
0.1
0.2
0.1
RR = 2.0 RD = 0.1
0.01
Unexposed
Exposed
Additive
interaction
present
Additive vs Multiplicative Interaction
Presence of additive interaction may or may not
be accompanied by multiplicative interaction
Risk of Disease
1
RR = 3.0 RD = 0.4
0.6
0.2
0.1
0.2
Multiplicative
interaction
present
0.1
RR = 2.0 RD = 0.1
0.01
Unexposed
1
Risk of Disease

Exposed
RR = 3.0 RD = 0.2
0.3
0.1
0.15
0.1
0.05
RR = 3.0 RD = 0.1
0.01
Unexposed
Exposed
Multiplicative
interaction
absent
Additive vs Multiplicative Interaction
Presence of qualitative multiplicative interaction
is always accompanied by qualitative additive
interaction
Qualitative Interaction
1
Third Variable Present
Third Variable Absent
Risk of Disease

0.18
0.2
0.13
0.1
0.08
0.01
Unexposed
Exposed
Multiplicative
and additive
interaction
both present
Additive vs Multiplicative Scales

Additive measures (e.g., risk difference):
– readily translated into impact of an exposure (or
intervention) in terms of 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
– gives “public health impact” of the exposure

Multiplicative measures (e.g., risk ratio)
– favored measure when looking for causal association
(etiologic research)
Additive vs Multiplicative Scales

Causally related but minor public health
importance
Disease
No Disease
Exposed
10
99990
Unexposed
5
99995
– 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
Disease
No Disease
Exposed
20
80
Unexposed
10
90
– RR = 2
– RD = 0.2 - 0.1 = 0.1
– Need to eliminate exposure in 10 persons to
avert one case of disease
Smoking, Family History
and Cancer:
Additive vs Multiplicative Interaction
Crude
Cancer
Smoking
50
No Smoking
25
Stratified
Smoking
No Smoking
Family
History
Present
Cancer
No
Cancer
40
20
60
80
No Cancer
150
175
Family
History
Absent
Smoking
No Smoking
Cancer
No
Cancer
10
5
90
95
Risk ratiofamily history = 2.0
Risk rationo family history = 2.0
RDfamily history = 0.20
RDno family history = 0.05
• No multiplicative interaction but presence of
additive interaction
• If etiology is goal, risk ratio’s may be 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
Confounding vs Interaction

Confounding
– An extraneous or nuisance pathway that an
investigator hopes to prevent or rule out

Interaction
– 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
Smoking, Caffeine Use
and Delayed Conception
Crude
Delayed
Smoking
26
No Smoking
64
Stratified
Heavy
Caffeine Use
Not Delayed
133
RR crude = 1.7
601
No Caffeine
Use
Not
Delayed Delayed
Smoking
No Smoking
11
17
RRcaffeine use = 0.7
72
73
Not
Delayed Delayed
Smoking
No Smoking
15
47
RRno caffeine use = 2.4
RR adjusted = 1.4 (95% CI= 0.9 to 2.1)
Here, adjustment is contraindicated
When interaction is present,
confoundng becomes irrelevant!
61
528
Chance as a Cause of Interaction?
Are all non-identical stratum-specific
estimates indicative of interaction?
Crude
Down’s
Spermicide Use
4
No Spermicide
12
Stratified
Age < 35
No Down’s
109
1145
Age > 35
No
Down’s Down’s
Spermicide
No Spermicide
3
9
ORage <35 = 3.4
104
1059
OR crude = 3.5
Spermicide
No Spermicide
Down’s
No
Down’s
1
3
5
86
ORage >35 = 5.7
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

A variety of formal tests are available with the same
general format, following a chi-square distribution:
chi  squareN 1

(effecti  summary effect) 2

var(effecti )
i
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
Interpreting Tests of Homogeneity

If the test of homogeneity is “significant”, this is
evidence that there is heterogeneity (i.e. no
homogeneity)
– i.e., interaction may be present

The choice of a significance level (e.g. p <
0.05) for reporting interaction is not clear cut.
– There are inherent limitations in the power
of the test of homogeneity
» p < 0.05 may be too conservative
– One approach is to report interaction for p <
0.20 if the magnitude of differences is high
enough
» 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.
» However, meaning of p value is not
different than other contexts
» Not a purely statistical decision
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)

. cs delayed smoking



| smoking
|
Exposed
|
Unexposed
|
Total
-----------------+------------------------+----------

Cases |
26
64
|
90

Noncases |
133
601
|
734

-----------------+------------------------+----------

Total |
159
665
|
824

|

Risk |

|

|------------------------+----------------------
|
.163522
.0962406
Point estimate
|
.1092233
|
[95% Conf. Interval]

Risk difference |
.0672814
|
.0055795
.1289833

Risk ratio |
1.699096
|
1.114485
2.590369




– +----------------------------------------------chi2(1) =
5.97 Pr>chi2 = 0.0145
. cs delayed smoking, by(caffeine)
caffeine |
RR
[95% Conf. Interval]
M-H Weight
-----------------+-------------------------------------------------

no caffeine |
2.414614
1.42165
4.10112
5.486943

heavy caffeine |
.70163
.3493615
1.409099
8.156069

-----------------+-------------------------------------------------

Crude |
1.699096
1.114485
2.590369

M-H combined |
1.390557
.9246598
2.091201

-----------------+-------------------------------------------------

Test of homogeneity (M-H)
chi2(1) =
What does the p value mean?
7.866
Pr>chi2 = 0.0050
Report vs Ignore Interaction?
Some Guidelines
Is an art form: requires consideration of both
clinical and statistical significance
Relative Risks for a
Given Exposure and
Disease
Potential Effect Modifier
P value for
heterogeneity
Present
Absent
2.3
2.6
0.45
Report or
Ignore
Interaction
Ignore
2.3
2.6
0.001
Ignore
2.0
20.0
0.001
Report
2.0
20.0
0.20
Report
2.0
20.0
0.40
Ignore
3.0
4.5
0.30
Ignore
3.0
4.5
0.001
+/-
0.5
3.0
0.001
Report
0.5
3.0
0.20
Report
0.5
3.0
0.30
+/-
When Assessing the Association
Between an Exposure and a Disease,
What are the Possible Effects of a
Third Variable?
No Effect
C
I
+
_
Confounding:
ANOTHER
PATHWAY TO
GET TO THE
DISEASE
Intermediary
ON CAUSAL
Variable:
PATHWAY
D
EM
Effect
Modifier
(Interaction):
MODIFIES
THE EFFECT
OF THE
EXPOSURE