Upload
leonard-nichols
View
222
Download
2
Tags:
Embed Size (px)
Citation preview
3 Causal Models Part II:Counterfactual Theory and Traditional Approaches to
Confounding (Bias?)
Confounding, Identifiability, Collapsibility and Causal inference
Thursday reception at lunch time at SACEMA
Review Yesterday
Causes – definition Sufficient causes model
– Component causes– Attributes– Causal complements
Lessons– Disease causation is poorly understood– Diseases don’t have induction periods– Strength of effects determined by prevalence of complements– Only need to prevent one component to prevent disease
This Morning
Counterfactual model– Susceptibility types– Potential outcomes
Confounding under the counterfactual susceptibility model of causation
Stratification– Identifying confounders– Standardization versus pooling
What is Confounding?
Give me the definition you were taught or describe how you understand it
What is an “adjusted” measure of effect?
Is red wine cardio-protective?
In an adjusted model to remove confounding of the E-D relationship, is it reasonable to remove variables that are not statistically significant
and include those that are?
Counterfactual Theory
Potential Outcomes,
Susceptibility types
Poor Clare
Doctor prescribes antibiotics
3 days later she is cured
Did the antibiotic cure her?
Cinema d’Counterfactual
The counterfactual model:The counterfactual ideal
Disease experience, given exposedHypothetical disease experience, if unexposed
TheCounterfactual
Ideal
Counterfactual theory
Only one can actually be observed– The other is “counterfactual” in that it is counter to
what is actually observed
Ask, what would have happened had things been different, all other things being equal?– Leads to the causal contrast
Exposure must be changeable to have effect– We will come back to this
The counterfactual model:The counterfactual ideal
Disease experience, given exposed
Substitute disease experience of truly unexposed
Approximation to
The Counterfactual
Ideal
Take home message 1:We’re often interested in what happens
to index (exposed). Reference (unexposed) are useful only insofar as
they tell us about index group.
Must Specify a Causal Contrast
Events are not causes themselves– Only causes as part of a causal contrast
What is the effect of oral contraceptives on risk of death?– The question, as defined, has no meaning
Compared to condoms, increased risk– Through stroke and heart attack
Compared to no contraceptive, maybe decreased risk
– Some places childbirth may be a greater risk
Take home message 2:“Effects” of exposures only have meaning when defined in contrast
to an alternative
If ethics were not a concern, how would you design an RCT of smoking and lung cancer?
Think about dose, duration
What about gender and cancer?
What about obesity and MI?
Effects Must be Amenable to Action
To have an effect, must be changeable– What is effect of sex on heart disease?– How would you change sex?
Defining the action helps define the causal contrast well– What is the effect of obesity on death?– How would you change obesity?
Each has a different effect, some good, some bad
To remind us, use A for Action, not E
Think of the action, inclusion criteria, the placebo, etc.
Take Home Message 3:For etiologic observational studies,
think of RCT you would do first. Develop your observational study
with the RCT in mind.
To identify a causal effect in an individual
Need three things:– Outcome, actions compared, person whose
2+ counterfactual outcomes comparedCall the counterfactual outcomes:
– Ya=1 vs Ya=0, read: Y that would occur if A=aNote counterfactuals different from:
– Y|A=1 (or just Y), read: Y given A=1Effect can be precisely defined as:
– Ya=1 ≠Ya=0
All examples, assume each person represents 1,000,000 people exactly the same as them so no random error problem
Assume infinite population with no information or selection bias, a
dichotomous A and Y
A (E) Ya=1 Ya=0 Y
Person A 1 1 0 1
Person B 1 1 1 1
Person C 1 0 1 0
Person D 1 0 0 0
Person E 0 0 0 0
Person F 0 1 0 0
Person G 0 1 1 1
Person H 0 0 1 1
Effect : [Pr(Ya=1=1) - Pr(Ya=0=1)] =
Assume each person represents 100,000 people
A (E) Ya=1 Ya=0 Y
Person A 1 1 0 1
Person B 1 1 1 1
Person C 1 0 1 0
Person D 1 0 0 0
Person E 0 0 0 0
Person F 0 1 0 0
Person G 0 1 1 1
Person H 0 0 1 1
Effect : [Pr(Ya=1=1) - Pr(Ya=0=1)] =
Assume each person represents 100,000 people
[4/8 – 4/8] = 0
Association : [Pr(Y=1|A=1) - Pr(Y=1|A=0)] =
A (E) Ya=1 Ya=0 Y
Person A 1 1 0 1
Person B 1 1 1 1
Person C 1 0 1 0
Person D 1 0 0 0
Person E 0 0 0 0
Person F 0 1 0 0
Person G 0 1 1 1
Person H 0 0 1 1
Effect : [Pr(Ya=1=1) - Pr(Ya=0=1)] =
Assume each person represents 100,000 people
Association : [Pr(Y=1|A=1) - Pr(Y=1|A=0)] =
[4/8 – 4/8] = 0
[2/4 – 2/4] = 0
The counterfactual modelSusceptibility types
Envision 4 responses to exposure, relative to unexposed– Type 1 - Doomed– Type 2 - E causal– Type 3 - E preventive– Type 4 - Immune
Susceptibility Type
Exposed Outcome
(Ya=1)
Unexposed Outcome
(Ya=0)
Type 1 – doomed
Type 2 – E causal
Type 3 – E preventive
Type 4 – immune
CST: Counterfactual susceptibility type
1 1
1 0
0 1
0 0
The counterfactual model
The index condition, relative to the reference condition, affects only susceptibility types 2 and 3– Types 2 get the disease, but would not get disease
had they had the reference condition– Types 3 do not get the disease, but would have got
the disease had they had the reference condition
Individual Susceptibility under the CST model
Individual Susceptibility
Risk Difference Ya=1 – Ya=0
Risk Ratio Ya=1 / Ya=0
Type 1
Type 2
Type 3
Type 4
1 – 1 = 0 1 / 1 = 1
1 – 0 = 1 1 / 0 = undef
0 – 1 = -1 0 / 1 = 0
0 – 0 = 0 0 / 0 = undef
Can type 2 and 3 co-exist?
Are there exposures that can both prevent and causes disease? – Vaccination and polio– Exercise and heart attack– Seat belts and death in a motor vehicle accident– Heart transplant and mortality
So what does RD = 0 or RR=1 mean?– Could mean no effect– Could be balance of causal/preventive mechanisms– We call no effect “sharp null” but it is not identifiable
Take home message 4:If exposures can be causal and
preventive, estimates of effect only tell us about the balance of causal and
preventive effects
Average causal effects
Individual effects rarely identifiable because we don’t have both conditions– But average causal effects may be
identifiable in populationsAn average causal effect of treatment A
on outcome Y occurs when:– Pr(Ya=1 = 1) ≠ Pr(Ya=0 = 1) – Or more generally, E(Ya=1) ≠ E(Ya=0)
Note makes no reference to relative vs. absolute
Effects vs. Associations
Effects measures– RD: Pr(Ya=1 = 1) - Pr(Ya=0 = 1) – RR: Pr(Ya=1 = 1) / Pr(Ya=0 = 1) – OR: Pr(Ya=1 = 1)/Pr(Ya=1 = 0)/
Pr(Ya=0 = 1)/Pr(Ya=0 = 0)
Associational measures– RD: Pr(Y = 1|A=1) - Pr(Y = 1|A=0) – RR: Pr(Y = 1|A=1) / Pr(Y = 1|A=0) – OR: Pr(Y = 1|A=1) / Pr(Y = 0|A=1) /
Pr(Y = 1|A=0) / Pr(Y = 0|A=0)
Traditional Approaches to Confounding and Confounders
Extend the CST model of causation to populations
Susceptibility Type
Index Outcome
Reference Outcome
Proportion in
Index Pop
Proportion in
Reference Pop
Type 1 – doomed
1 1 p1 q1
Type 2 – index causal
1 0 p2 q2
Type 3 – index preventive
0 1 p3 q3
Type 4 – immune
0 0 p4 q4
1 1
What is the risk of disease in exposed?
Susceptibility Type
Index Outcome
Reference Outcome
Proportion in
Index Pop
Proportion in
Reference Pop
Type 1 – doomed
1 1 p1 q1
Type 2 – index causal
1 0 p2 q2
Type 3 – index preventive
0 1 p3 q3
Type 4 – immune
0 0 p4 q4
1 1
Observed risk in exposed is p1 + p2, but we cannot tell how many of each
What would the risk of disease be in the exposed had they been unexposed?
Susceptibility Type
Index Outcome
Reference Outcome
Proportion in
Index Pop
Proportion in
Reference Pop
Type 1 – doomed
1 1 p1 q1
Type 2 – index causal
1 0 p2 q2
Type 3 – index preventive
0 1 p3 q3
Type 4 – immune
0 0 p4 q4
1 1
Counterfactual risk is the risk the
exposed would have had had they
been exposed: p1+p3
When can reference group stand in for the exposed had they been unexposed?
Susceptibility Type
Index Outcome
Reference Outcome
Proportion in
Index Pop
Proportion in
Reference Pop
Type 1 – doomed
1 1 p1 q1
Type 2 – index causal
1 0 p2 q2
Type 3 – index preventive
0 1 p3 q3
Type 4 – immune
0 0 p4 q4
1 1
To have a valid comparison, we require the disease experience of reference
group be able to stand in for the counterfactual risk. This
is partial exchangeability
Exchangeability
Full exchangeability means the two groups can stand in for each other– Risk exposed had = risk unexposed would
have had if they were exposedPr(Ya=1=1|A=1) = Pr(Ya=1=1|A=0)
– Risk unexposed had = risk exposed would have had if they were unexposed
Pr(Ya=0=1|A=1) = Pr(Ya=0=1|A=0)
ObservedCounterfactual
Exchangeability
Partial exchangeability means the E- can stand in for what would have happened to the E+ had they been unexposed– Risk unexposed had = risk exposed would
have had if they were unexposedPr(Ya=0=1|A=1) = Pr(Ya=0=1|A=0)
ObservedCounterfactual
Take Home Message 5:The unexposed have to be able to
stand in for the exposed had they been unexposed. Not vice versa.
Partial exchangeability
Two possible definitions of no confounding (1)
Definition One — the risk of disease due to background causes is equal in the index and reference populations
So p1 = q1 under this definition.
The risk difference [(p1 + p2) - (q1 + q3)] equals
(p2 - q3), assuming partial exchangeability.
p1 = q1
p1
p2 – q3
But effect should be based only on exposed
Two possible definitions of no confounding (2)
Definition Two -- the risk of disease in the reference population equals the risk the index population would have had, if they had been unexposed
So p1 + p3 = q1 + qunder this definition.
The risk difference [(p1 + p2) - (q1 + q3)]
equals (p2 - p3 ), assuming partial
exchangeability.
p1 + p3 = q1+ q3
p1 + p3
p2 – p3
NOTE that RD related to balance of p2 and p3
We choose the second definition
First forces inclusion of effect of absence of exposure in reference group
Second measures effect of exposure only in index group – Holds under randomization– However, it is counterfactual
If exposure is never preventive, they are same
We choose the second definition
A measure of association is unconfounded if:– Experience of the reference group = the
disease occurrence the index population would have had, had they been unexposed
Risk difference tells about balance of causal/preventive action in index– Effect, not an estimate
To put it mathematically
Suppose we have two populations A and B We want to observe: IAE+ - IAE-
We observe: IAE+ - IBE-
If we add IAE- - IAE- to this we get: (IAE+ - IAE-) + (IAE- - IBE-) (IAE+ - IAE-) is the causal RD (IAE- - IBE-) is a bias factor (i.e. confounding)
Bias is difference between counterfactual unexposed experience of exposed and experience of truly unexposed
Causal RD vs. Observed
Susceptibility Type
Index Frequency
Reference Frequency
Type 1 – doomed
10 10
Type 2 – index causal
5 10
Type 3 – index preventive
10 5
Type 4 – immune
75 75
Causal RD?– p2 – p3
– 5/100 – 10/100 = -5/100
Observed RD?– (p1+p2) – (q1+q3)
– 15/100 – 15/100 = 0 Confounding?
– Does (p1+p3) = (q1+q3) ?
– 20/100 ≠ 15/100, Yes Causal = Observed?
– No100 100
Causal RD vs. Observed
Causal RD?– p2 – p3
– 5/100 – 5/100 = 0
Observed RD?– (p1+p2) – (q1+q3)
– 15/100 – 15/100 = 0 Confounding?
– Does (p1+p3) = (q1+q3) ?
– 15/100 = 15/100, No Causal = Observed?
– Yes100 100
Susceptibility Type
Index Frequency
Reference Frequency
Type 1 – doomed
10 5
Type 2 – index causal
5 10
Type 3 – index preventive
5 10
Type 4 – immune
80 75
Take Home Message 6:Lack of confounding doesn’t mean
perfect balance of CST types which we would expect under randomization
Take Home Message 7:If there is no confounding, the causal risk difference (i.e. the true effect) is
the observed effect
Assuming no other bias and random error
Getting the observed contrast close to the counterfactual ideal
Design– Randomization– Creating similar
populations Matching Restriction
Analysis– Stratification based
methods Stratification, Mantel-
Haenszel, Regression
– Standardization based methods
Standardization, G-estimation, IPTW, Marginal structural models
Confounders
Note we have defined confounding with no reference to imbalances in covariates– Separate confounder from confounding– Confounder is a factor that explains discrepancy
between observed risk in reference and desired counterfactual risk
Must be imbalanced in index/reference groups, a cause of disease and not on causal pathway– Use data as guide only
Non-identifiability and collapsibility:Identifying confounding in practice
Because we can’t identify individuals’ CST types, can’t use comparability definition in practice – Call this “ the non-identifiability problem” – Except thoughtfully
Instead a traditional approach uses the collapsibility criterion – If crude measure equals adjusted for potential
confounder, no confounding by that variable
What adjusted measure of effect?
Take Home Message 8: Confounding is when the
unexposed can’t stand in for the exposed had they been unexposed.
Confounders are variables that explain confounding.
Stratified Analysis: Introduction
One method for control of extraneous variables in the analysis– Analysis of disease-exposure association within
categories of confounder / modifier prevents external influence of that variable
Advantages/disadvantages– Straight-forward, few statistical assumptions– Data become thin with many categories/ variables
Candidate variables– Confounders, Modifiers, Matched factors
Stratified Analysis 1 VariableStratify then ask:
Are measures of effect within each stratum heterogeneous? – Yes = Interaction, stratified analysis?– No = No Interaction, assess confounding
Does summary measure of effect across strata equal crude? – Yes = No confounding, collapse– No = Confounding, use summary measure
Note, this is about change in estimate of effect, nothing about p-values
Example (1-1) (CST balance within strata)
All agesMen Women
cases 150 70non-cases 850 930total 1000 1000risk 0.15 0.07risk ratio 2.1
Old Young All agesMen Women Men Women
cases 90 60 60 10non-cases 243 607 607 323total 333 667 667 333risk 0.27 0.09 0.09 0.03risk ratio 3.0 3.0
Example (1-2) (CST balance within strata)
Take home message 9:In practice, confounding USUALLY presents as – within levels of the
confounder, uneven distribution of the exposure and different risk of outcome
among unexposed
But be careful, as this can be misleadingas this is NECESSARY but not SUFFICIENT
Example (1-3) (CST balance within strata)All ages
Men Womencases 150 70non-cases 850 930total 1000 1000risk 0.15 0.07risk ratio 2.1
type1 30 35type2 120 40type3 20 35type4 830 890total 1000 1000confounding -0.020
Does p1 + p3 = q1 + q3?
(30+20)/1000 <>(35+35)/1000
Example (1-4) (CST balance within strata)Old Young All ages
Men Women Men Womencases 90 60 60 10non-cases 243 607 607 323total 333 667 667 333risk 0.27 0.09 0.09 0.03risk ratio 3.0 3.0
type1 20 30 10 5type2 70 30 50 10type3 10 30 10 5type4 233 577 597 313total 333 667 667 333confounding 0.000 0.000
Does p1 + p3 = q1 + q3
within strata?
Example (2-1) (choice of effect measure) E+ E-
cases 100 50 non-cases 900 950
total 1000 1000 risk 0.1 0.05 RR 2
odds 0.111 0.053 OR 2.11
males females E+ E- E+ E-
cases 90 45 cases 10 5 non-cases 110 155 non-cases 790 795
total 200 200 total 800 800 risk 0.45 0.225 0.0125 0.00625 RR 2 2
MHRR 2 odds 0.818 0.290 0.013 0.006 OR 2.82 2.01
MHOR 2.68
Collapsible? Does crude = adjusted?Collapsible? Does crude = adjusted?
Outcome needs to be rare in all levels of the
exposure/confounder
Take Home Message 10:The odds ratio is not strictly
collapsible. Change in estimate of effect after adjustment can be just an artifact of the data. Outcome must be
rare in ALL strata.
But this can go wrong
Counfounding?
Does p1+p3=q1+q3?
Is the exposure distribution different across strata?
Is the risk in the unexposed different?
Take Home Message 11:Statistical Criteria Are Not Sufficient
to Determine What to Keep in a Model to Observe Causal Effects
Pooled adjusted estimate
Assumes uniform RR/RD across strata – Precision enhancing
Pooled estimates are weighted averages of effects in strata– Pooled estimate are between stratum estimates– Weights measure information in strata (inverse
variance) but can be computed differently
Ex: Mantel-Haenzel, Logistic/Cox Reg– So long as there are no interaction terms– Regression models are analogous to stratification
Review of weighting
Pooling means we average the stratum specific estimates to get one estimate– Thus the pooled estimate must be between the two
stratum specific estimates
We can choose the weights however we like– Different weighting schemes have different
properties and logics
wwR
wwR
0
1
RR weighted
Example: MH Pooling
95% CI:
1.4, 2.79.1
2422290
*39215691520
*4
24222132
*1001569
49*320
ˆR MH1
0
NbNNaN
R
3.12181/396
1810/420ˆ RCrudeR
p for heterogeneity 0.09
Crude C1 C0
E+ E- E+ E- E+ E-
D+ 420 396 D+ 320 4 D+ 100 392
D- 1390 1785 D- 1200 45 D- 190 1740
Total 1810 2181 Total 1520 49 Total 290 2132
RR 1.3 RR 2.6 RR 1.9
Example: MH Pooling
9.1
2422290
*39215691520
*4
24222132
*1001569
49*320
*
*
ˆR MH0
1
10
0
10
10
1
10
NbNNaN
NNN
Nc
NNN
NNN
Na
NNN
R
Weight is N1*N0/N which weights towards the strata with highest total N and most evenly distributed exposure distribution
Crude C1 C0
E+ E- E+ E- E+ E-D+ 420 396 D+ 320 4 D+ 100 392D- 1390 1785 D- 1200 45 D- 190 1740Total 1810 2181 Total 1520 49 Total 290 2132RR 1.3 RR 2.6 RR 1.9
Mantel Haenszel Weights
The weight, (N1*N0 )/ N is at its minimum if N1=1 so N0 = (N-1). Weight is then (N-1)/N which is about 1
The weight, (N1*N0 )/ N is at its maximum if N1= N0 = N/2. Weight is then (N/2)2/N which is N/4
So a larger sample size will increase the weight, as will an even distribution of exposed an unexposed subjects
Summary estimates: Mantel-Haenszel
A pooled summary estimate: – Weighted average of estimates of effect from each
stratum– Weight is highest for stratum with most information
(subjects)
Precision optimizing Calculation depends on design
MH estimates for 3 designs
g g
gg
g g
gg
MH
n
cb
n
da
OR design control-Case
Exposed
(Index)
Unexposed
(reference)
Casesag bg
Controlscg dg
MH estimates for 3 designs
g g
gg
g g
gg
MH
n
nb
n
na
RR1
0
design Risk
Exposed
(Index)
Unexposed
(reference)
Casesag bg
Undiseasedcg dg
Total n1g n0g
MH estimates for 3 designs
g g
gg
g g
gg
MH
L
Lb
L
La
IRR1
0
design Rate
Exposed
(Index)
Unexposed
(reference)
Casesag bg
Person-timeL1g L0g
Summary estimates:Standardized RR (SMR)
Standardize the risk or rate– Weighted average of risk or rate in strata, using the
index group’s experience as the weight
Choose index group because:– Want reference group to reflect the rate we would
have seen in the exposed had they been unexposed
No assumption of homogeneity across strata
Example: Standardization
95% CI:
0.9, 6.4
3.12181/396
1810/420ˆ RCrudeR
4.2
2132392
*290494
*1520
100320
*
ˆ1
oNN
b
ORSM
1.7, 2.7
0.7, 2.50.9, 6.4
1.4, 2.7
Crude C1 C0
E+ E- E+ E- E+ E-D+ 420 396 D+ 320 4 D+ 100 392D- 1390 1785 D- 1200 45 D- 190 1740Total 1810 2181 Total 1520 49 Total 290 2132RR 1.3 RR 2.6 RR 1.9
Example: Standardization
When we standardize, we can use whatever distribution we want. If we use the distribution of the exposed group, we call this an SMR.
Crude C1 C0
E+ E- E+ E- E+ E-D+ 420 396 D+ 320 4 D+ 100 392D- 1390 1785 D- 1200 45 D- 190 1740Total 1810 2181 Total 1520 49 Total 290 2132RR 1.3 RR 2.6 RR 1.9
4.2
2132392
*290494
*1520
290100
*2901520320
*1520
**
*
*
*
ˆ
01
01
11
01
11
Nb
N
a
Nb
N
Na
N
WNb
N
WNa
N
RSM
Example: Standardization
0.2
2132392
*2422494
*1569
290100
*24221520320
*1569
*
*
*
*
ˆ
0
1
0
1
Nb
N
Na
N
WNb
W
WNa
W
RSM
We could also ask what would happen if everyone was both exposed and unexposed: corresponds to PO model
Crude C1 C0
E+ E- E+ E- E+ E-D+ 420 396 D+ 320 4 D+ 100 392D- 1390 1785 D- 1200 45 D- 190 1740Total 1810 2181 Total 1520 49 Total 290 2132RR 1.3 RR 2.6 RR 1.9
MH estimates for 3 designs
g g
gg
gg
d
bc
a
SMR design control-Case
Exposed
(Index)
Unexposed
(reference)
Casesag bg
Controlscg dg
MH estimates for 3 designs
g g
gg
gg
n
bn
a
SMR
01
design Risk
Exposed
(Index)
Unexposed
(reference)
Casesag bg
Undiseasedcg dg
Total n1g n0g
MH estimates for 3 designs
g g
gg
gg
L
bL
a
SMR
01
design Rate
Exposed
(Index)
Unexposed
(reference)
Casesag bg
Person-timeL1g L0g
Practical summary
Use the RRc to measure the direction and magnitude of confounding:
cRR = SMR*RRc
RRc = cRR/SMR
Use pooled estimates to maximize precision when effects are homogeneous within strata.
Use the SMR as an unconfounded summary estimate when effects are heterogeneous
1.3 1.31
Practical summary
Use the RRc to measure the direction and magnitude of confounding:
cRR = SMR*RRc
RRc = cRR/SMR
Use pooled estimates to maximize precision when effects are homogeneous within strata.
Use the SMR as an unconfounded summary estimate when effects are heterogeneous
1.3 2.10.6
Take Home Message 12:Mantel-Haenszel is only appropriate when no interaction. Standardization can be used with interaction but isn’t
precision optimizing.
Conclusion
Counterfactual model Causal contrast is between disease experience
of exposed and counterfactual experience they would have had had they been unexposed
Use unexposed group to stand in for counterfactual ideal
Confounding occurs when the unexposed can’t stand in for exposed had they been unexposed