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Statistical Rethinking
06: Good & Bad Controls
2022
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G P
U
C
grandparent
education
parent
education
child
education
unobserved
confound
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G P
C
P is a mediator
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G P
U
C
P is a collider
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G P
U
C
Can estimate total
effect of G on C
Cannot estimate
direct effect
G P
U
C
C
i
∼ Normal(μ
i
, σ)
μ
i
= α + β
G
G
i
C
i
∼ Normal(μ
i
, σ)
μ
i
= α + β
G
G
i
+ β
P
P
i
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N <- 200 # num grandparent-parent-child triads
b_GP <- 1 # direct effect of G on P
b_GC <- 0 # direct effect of G on C
b_PC <- 1 # direct effect of P on C
b_U <- 2 #direct effect of U on P and C
set.seed(1)
U <- 2*rbern( N , 0.5 ) - 1
G <- rnorm( N )
P <- rnorm( N , b_GP*G + b_U*U )
C <- rnorm( N , b_PC*P + b_GC*G + b_U*U )
d <- data.frame( C=C , P=P , G=G , U=U )
m6.11 <- quap(
alist(
C ~ dnorm( mu , sigma ),
mu <- a + b_PC*P + b_GC*G,
a ~ dnorm( 0 , 1 ),
c(b_PC,b_GC) ~ dnorm( 0 , 1 ),
sigma ~ dexp( 1 )
), data=d )
Page 180
b_GC
b_PC
-1.0 -0.5 0.0 0.5 1.0 1.5
Value
True values
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Stratify by parent centile
(collider)
Two ways for parents to
attain their education: from
G or from U
b_GC
b_PC
-1.0 -0.5 0.0 0.5 1.0 1.5
Value
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From Theory to Estimate
Our job is to
(1) Clearly state assumptions
(2) Deduce implications
(3) Test implications
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Avoid Being Clever At All Costs
Being clever is neither reliable nor
transparent
Now what?
Given a causal model, can use logic to
derive implications
Others can use same logic to verify/
challenge your work
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X Z Y
The Pipe
X Z Y
The Fork
X Z Y
The Collider
X and Y associated
unless stratify by Z
X and Y associated
unless stratify by Z
X and Y not associated
unless stratify by Z
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A B C
X Z Y
F G
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A B C
X Z Y
F G
Forks
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A B C
X Z Y
F G
Forks
Pipes
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A B C
X Z Y
F G
Forks
Pipes
Colliders
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Thousands of Years Ago
0
50
100
150
0
50
100
Effective Population Size (thousands)
Effective Population Size (thousands)
Thousands of Years Ago
0
100
200
300
400
500
0
50
100
Africa
Andes
Central Asia
Europe
Near-East & Caucasus
Southeast &
East Asia
Siberia
South Asia
Region
10 10
2. Cumulative Bayesian skyline plots of Y chromosome and mtDNA diversity by world regions. The red dashed lines highlight the horizons
d 50 kya. Individual plots for each region are presented in Supplemental Figure S4A.
Y chromosome MtDNA
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DAG Thinking
In an experiment, we cut causes of
the treatment
We randomize (hopefully)
So how does causal inference
without randomization ever work?
Is there a statistical procedure that
mimics randomization?
X Y
U
Without randomization
X Y
U
With randomization
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DAG Thinking
Is there a statistical procedure
that mimics randomization? X Y
U
Without randomization
X Y
U
With randomization
P(Y|do(X)) = P(Y|?)
do(X) means intervene on X
Can analyze causal model to
find answer (if it exists)
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Example: Simple Confound
X Y
U
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Example: Simple Confound
X Y
U Non-causal path
X <– U –> Y
Close the fork!
Condition on U
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Example: Simple Confound
X Y
U Non-causal path
X <– U –> Y
Close the fork!
Condition on U
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Example: Simple Confound
X Y
U Non-causal path
X <– U –> Y
Close the fork!
Condition on U
P(Y|do(X)) =
∑
U
P(Y|X, U)P(U) = E
U
P(Y|X, U)
“The distribution of Y, stratified by X and U,
averaged over the distribution of U.”
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The causal effect of X on Y is not (in
general) the coefficient relating X to Y
It is the distribution of Y when we change
X, averaged over the distributions of the
control variables (here U)
P(Y|do(X)) =
∑
U
P(Y|X, U)P(U) = E
U
P(Y|X, U)
“The distribution of Y, stratified by X and U,
averaged over the distribution of U.”
X Y
U
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Marginal Effects Example
B G
C
cheetahs
baboons gazelle
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Marginal Effects Example
B G
C
cheetahs present
B G
C
cheetahs absent
Causal effect of baboons depends upon distribution of cheetahs
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do-calculus
For DAGs, rules for finding
P(Y|do(X)) known as do-calculus
do-calculus says what is possible
to say before picking functions
Additional assumptions yield
additional implications
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do-calculus
do-calculus is worst case:
additional assumptions often
allow stronger inference
do-calculus is best case:
if inference possible by do-
calculus, does not depend on
special assumptions
Judea Pearl, father of
do-calculus, in 1966
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Backdoor Criterion
Very useful implication of do-calculus is
the Backdoor Criterion
Backdoor Criterion is a shortcut to
applying rules of do-calculus
Also inspires strategies for research
design that yield valid estimates
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Backdoor Criterion
Backdoor Criterion: Rule to find a set of variables
to stratify (condition) by to yield P(Y|do(X))
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Backdoor Criterion
Backdoor Criterion: Rule to find a set of variables
to stratify (condition) by to yield P(Y|do(X))
(1) Identify all paths connection the treatment
(X) to the outcome (Y)
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Backdoor Criterion
Backdoor Criterion: Rule to find a set of variables
to stratify (condition) by to yield P(Y|do(X))
(1) Identify all paths connection the treatment
(X) to the outcome (Y)
(2) Paths with arrows entering X are backdoor
paths (non-causal paths)
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Backdoor Criterion
Backdoor Criterion: Rule to find a set of variables
to stratify (condition) by to yield P(Y|do(X))
(1) Identify all paths connection the treatment
(X) to the outcome (Y)
(2) Paths with arrows entering X are backdoor
paths (non-causal paths)
(3) Find adjustment set that closes/blocks all
backdoor paths
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(1) Identify all paths connection the
treatment (X) to the outcome (Y)
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(2) Paths with arrows entering X are
backdoor paths (non-causal paths)
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(3) Find a set of control variables that
close/block all backdoor paths
Block the pipe: X ⫫ U | Z
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(3) Find a set of control variables that
close/block all backdoor paths
P(Y|do(X)) =
∑
U
P(Y|X, Z)P(Z)
μ
i
= α + β
X
X
i
+ β
Z
Z
i
Y
i
∼ Normal(μ
i
, σ)
Block the pipe: X ⫫ U | Z
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X Y
List all the paths connecting X and Y.
Which need to be closed to estimate
effect of X on Y?
C
Z
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X Y
List all the paths connecting X and Y.
Which need to be closed to estimate
effect of X on Y?
C
Z
X Y
C
Z
X Y
C
Z
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X Y
C
Z
X Y
C
Z
X Y
C
Z
Adjustment set: nothing!
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X Y
Z B
List all the paths connecting X and Y.
Which need to be closed to estimate
effect of X on Y?
A
C
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No content
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X Y
Z B
A
C
P(Y|do(X))
X Y
Z B
A
C
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X Y
Z B
A
C
P(Y|do(X))
X Y
Z B
A
C
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X Y
Z B
A
C
X Y
Z B
A
C
X Y
Z B
A
C
X Y
Z B
A
C
X Y
Z B
A
C
X Y
Z B
A
C
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X Y
Z B
A
C
X Y
Z B
A
C
X Y
Z B
A
C
X Y
Z B
A
C
X Y
Z B
A
C
X Y
Z B
A
C
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X Y
Z B
A
C
X Y
Z B
A
C
X Y
Z B
A
C
X Y
Z B
A
C
X Y
Z B
A
C
X Y
Z B
A
C
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X Y
Z B
A
C
X Y
Z B
A
C
X Y
Z B
A
C
X Y
Z B
A
C
X Y
Z B
A
C
X Y
Z B
A
C
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X Y
Z B
A
C
X Y
Z B
A
C
X Y
Z B
A
C
X Y
Z B
A
C
X Y
Z B
A
C
X Y
Z B
A
C
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X Y
Z B
A
C
X Y
Z B
A
C
X Y
Z B
A
C
Adjustment set: C, Z, and either A or B
(B is better choice)
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www.dagitty.net
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Backdoor Criterion
Backdoor Criterion: Rule to find adjustment set to
yield P(Y|do(X))
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Backdoor Criterion
Backdoor Criterion: Rule to find adjustment set to
yield P(Y|do(X))
Beware non-causal paths that you open while closing
other paths!
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Backdoor Criterion
Backdoor Criterion: Rule to find adjustment set to
yield P(Y|do(X))
Beware non-causal paths that you open while closing
other paths!
More than backdoors:
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Backdoor Criterion
Backdoor Criterion: Rule to find adjustment set to
yield P(Y|do(X))
Beware non-causal paths that you open while closing
other paths!
More than backdoors:
Also solutions with simultaneous equations
(instrumental variables e.g.)
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Backdoor Criterion
Backdoor Criterion: Rule to find adjustment set to
yield P(Y|do(X))
Beware non-causal paths that you open while closing
other paths!
More than backdoors:
Also solutions with simultaneous equations
(instrumental variables e.g.)
Full Luxury Bayes: use all variables, but in separate
sub-models instead of single regression
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PAUSE
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http://www.blackswanman.com/
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Good & Bad Controls
“Control” variable: Variable introduced to an
analysis so that a causal estimate is possible
Common wrong heuristics for choosing
control variables
Anything in the spreadsheet YOLO!
Any variables not highly collinear
Any pre-treatment measurement (baseline)
CONTROL
ALL THE
THINGS
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X
Cinelli, Forney, Pearl 2021 A Crash Course in Good and Bad Controls
Y
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X Y
u v
Z
Cinelli, Forney, Pearl 2021 A Crash Course in Good and Bad Controls
unobserved
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X Y
u v
Z
Cinelli, Forney, Pearl 2021 A Crash Course in Good and Bad Controls
Health
person 1
Health
person 2
Hobbies
person 1
Hobbies
person 2
Friends
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X Y
u v
Z
(1) List the paths
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X Y
u v
Z
(1) List the paths
X → Y
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X Y
u v
Z
(1) List the paths
X → Y
X ← u → Z ← v → Y
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X Y
u v
Z
(1) List the paths
X → Y
X ← u → Z ← v → Y
frontdoor & open
backdoor & closed
(2) Find backdoors
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X Y
u v
Z
(1) List the paths
X → Y
X ← u → Z ← v → Y
frontdoor & open
backdoor & closed
(2) Find backdoors
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X Y
u v
Z
(1) List the paths
X → Y
X ← u → Z ← v → Y
frontdoor & open
backdoor & closed
(2) Find backdoors (3) Close backdoors
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X Y
u v
Z
What happens if you stratify by
Z?
Opens the backdoor path
Z could be a pre-treatment
variable
Not safe to always control pre-
treatment measurements
Health
person 1
Health
person 2
Hobbies
person 1
Hobbies
person 2
Friends
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X Y
Z
u
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X Y
Z
u
Win
lottery
Lifespan
Happiness
Contextual
confounds
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X Y
Z
u
X → Z → Y
X → Z ← u → Y
No backdoor, no need
to control for Z
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X Y
Z
u
f <- function(n=100,bXZ=1,bZY=1) {
X <- rnorm(n)
u <- rnorm(n)
Z <- rnorm(n, bXZ*X + u)
Y <- rnorm(n, bZY*Z + u )
bX <- coef( lm(Y ~ X) )['X']
bXZ <- coef( lm(Y ~ X + Z) )['X']
return( c(bX,bXZ) )
}
sim <- mcreplicate( 1e4 , f() , mc.cores=8 )
dens( sim[1,] , lwd=3 , xlab="posterior mean" )
dens( sim[2,] , lwd=3 , col=2 , add=TRUE )
1 1
1 1
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X Y
Z
u
-1.0 0.0 0.5 1.0 1.5 2.0
0.0 1.0 2.0
posterior mean
Density
f <- function(n=100,bXZ=1,bZY=1) {
X <- rnorm(n)
u <- rnorm(n)
Z <- rnorm(n, bXZ*X + u)
Y <- rnorm(n, bZY*Z + u )
bX <- coef( lm(Y ~ X) )['X']
bXZ <- coef( lm(Y ~ X + Z) )['X']
return( c(bX,bXZ) )
}
sim <- mcreplicate( 1e4 , f() , mc.cores=8 )
dens( sim[1,] , lwd=3 , xlab="posterior mean" )
dens( sim[2,] , lwd=3 , col=2 , add=TRUE )
Y ~ X
correct
Y ~ X + Z
wrong
1 1
1 1
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X Y
Z
u
Y ~ X
correct
Y ~ X + Z
wrong
Change bZY to zero
f <- function(n=100,bXZ=1,bZY=1) {
X <- rnorm(n)
u <- rnorm(n)
Z <- rnorm(n, bXZ*X + u)
Y <- rnorm(n, bZY*Z + u )
bX <- coef( lm(Y ~ X) )['X']
bXZ <- coef( lm(Y ~ X + Z) )['X']
return( c(bX,bXZ) )
}
sim <- mcreplicate( 1e4 , f(bZY=0) , mc.cores=8 )
dens( sim[1,] , lwd=3 , xlab="posterior mean" )
dens( sim[2,] , lwd=3 , col=2 , add=TRUE )
-1.0 -0.5 0.0 0.5
0.0 1.0 2.0
posterior mean
Density
1 0
1 1
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X Y
Z
u
X → Z → Y
X → Z ← u → Y
No backdoor, no need
to control for Z
Controlling for Z biases
treatment estimate X
Controlling for Z opens biasing
path through u
Can estimate effect of X; Cannot
estimate mediation effect Z
Win
lottery
Lifespan
Happiness
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Post-treatment bias is common
IMENTS 761
-
s;
g
s,
d
6;
-
s
o
t
-
h.
-
-
e
-
t
m
TABLE 1 Posttreatment Conditioning
in Experimental Studies
Category Prevalence
Engages in posttreatment conditioning 46.7%
Controls for/interacts with a
posttreatment variable
21.3%
Drops cases based on posttreatment
criteria
14.7%
Both types of posttreatment conditioning
present
10.7%
No conditioning on posttreatment variables 52.0%
Insufficient information to code 1.3%
Note: The sample consists of 2012–14 articles in the American Po-
litical Science Review, the American Journal of Political Science, and
the Journal of Politics including a survey, field, laboratory, or lab-
in-the-field experiment (n = 75).
avoid posttreatment bias. In many cases, the usefulness
Montgomery et al 2018 How Conditioning on Posttreatment Variables Can Ruin Your Experiment
Regression with confounds
Regression with post-
treatment variables
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X Y
Z
Do not touch the collider!
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X Y
Z u
Colliders not always so obvious
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X Y
Z u
education
values
income
family
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X Y
Z
“Case-control bias”
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X Y
Z
“Case-control bias”
Education Occupation
Income
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X Y Z
“Case-control bias”
f <- function(n=100,bXY=1,bYZ=1) {
X <- rnorm(n)
Y <- rnorm(n, bXY*X )
Z <- rnorm(n, bYZ*Y )
bX <- coef( lm(Y ~ X) )['X']
bXZ <- coef( lm(Y ~ X + Z) )['X']
return( c(bX,bXZ) )
}
sim <- mcreplicate( 1e4 , f() , mc.cores=8 )
dens( sim[1,] , lwd=3 , xlab="posterior mean" )
dens( sim[2,] , lwd=3 , col=2 , add=TRUE )
0.0 0.5 1.0 1.5
0 1 2 3 4 5
posterior mean
Density
Y ~ X
correct
Y ~ X + Z
wrong
1 1
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X Y
Z
“Precision parasite”
No backdoors
But still not good to
condition on Z
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X Y
Z
“Precision parasite”
f <- function(n=100,bZX=1,bXY=1) {
Z <- rnorm(n)
X <- rnorm(n, bZX*Z )
Y <- rnorm(n, bXY*X )
bX <- coef( lm(Y ~ X) )['X']
bXZ <- coef( lm(Y ~ X + Z) )['X']
return( c(bX,bXZ) )
}
sim <- mcreplicate( 1e4 , f(n=50) , mc.cores=8 )
dens( sim[1,] , lwd=3 , xlab="posterior mean" )
dens( sim[2,] , lwd=3 , col=2 , add=TRUE )
0.6 0.8 1.0 1.2 1.4
0 1 2 3 4
posterior mean
Density
Y ~ X
correct
Y ~ X + Z
wrong
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X Y
Z
u
“Bias amplification”
X and Y confounded by u
Something truly awful happens
when we add Z
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f <- function(n=100,bZX=1,bXY=1) {
Z <- rnorm(n)
u <- rnorm(n)
X <- rnorm(n, bZX*Z + u )
Y <- rnorm(n, bXY*X + u )
bX <- coef( lm(Y ~ X) )['X']
bXZ <- coef( lm(Y ~ X + Z) )['X']
return( c(bX,bXZ) )
}
sim <- mcreplicate( 1e4 , f(bXY=0) , mc.cores=8 )
dens( sim[1,] , lwd=3 , xlab="posterior mean" )
dens( sim[2,] , lwd=3 , col=2 , add=TRUE )
X Y
Z
u
-0.5 0.0 0.5 1.0
0 1 2 3 4 5
posterior mean
Density
Y ~ X
biased
Y ~ X + Z
more bias
true value
is zero
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X Y
Z
u
-0.5 0.0 0.5 1.0
0 1 2 3 4 5
posterior mean
Density
Y ~ X
biased
Y ~ X + Z
more bias
true value
is zero
WHY?
Covariation X & Y requires
variation in their causes
Within each level of Z, less
variation in X
Confound u relatively more
important within each Z
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-5 0 5 10
-2 0 2 4
X
Y
X Y
Z
u
0
+ + +
n <- 1000
Z <- rbern(n)
u <- rnorm(n)
X <- rnorm(n, 7*Z + u )
Y <- rnorm(n, 0*X + u )
Z = 0 Z = 1
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Good & Bad Controls
“Control” variable: Variable
introduced to an analysis so that a
causal estimate is possible
Heuristics fail — adding control
variables can be worse than omitting
Make assumptions explicit
MODEL
ALL THE
THINGS
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PAUSE
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Table 2 Fallacy
Not all coefficients are causal
effects
Statistical model designed to
identify X –> Y will not also
identify effects of control
variables
Table 2 is dangerous
Westreich & Greenland 2013 The Table 2 Fallacy
724 THE AMERICAN
EC
TABLE 2-ESTIMATED PROBIT MODELS
FOR THE USE OF A SCREEN
Finals
Preliminaries blind blind
(1) (2) (3)
(Proportion female),_ 2.744 3.120 0.490
(3.265) (3.271) (1.163)
[0.006] [0.004] [0.011]
(Proportion of orchestra -26.46 -28.13 -9.467
personnel with <6 (7.314) (8.459) (2.787)
years tenure),- 1 [-0.058] [-0.039] [-0.207]
"Big Five" orchestra 0.367
(0.452)
[0.001]
pseudo R2 0.178 0.193 0.050
Number of observations 294 294 434
Notes: The dependent variable is 1 if the orchestra adopts a
screen, 0 otherwise. Huber standard errors (with orchestra
random effects) are in parentheses. All specifications in-
clude a constant. Changes in probabilities are in brackets.
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A
X Y
S
Westreich & Greenland 2013 The Table 2 Fallacy
Stroke
HIV
Smoking
Age
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No content
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Use Backdoor Criterion
A
X Y
S
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Use Backdoor Criterion
A
X Y
S X Y
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Use Backdoor Criterion
A
X Y
S X Y
X Y
S
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Use Backdoor Criterion
A
X Y
S X Y
X Y
S
A
X Y
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Use Backdoor Criterion
A
X Y
S X Y
X Y
S
A
X Y
A
X Y
S
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Use Backdoor Criterion
A
X Y
S X Y
X Y
S
A
X Y
A
X Y
S
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Y
i
∼ Normal(μ
i
, σ)
μ
i
= α + β
X
X
i
+ β
S
S
i
+ β
A
A
i
A
X Y
S
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A
X Y
S
Confounded by A
and S
Unconditional
X
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Coefficient for X:
Effect of X on Y
(still must
marginalize!)
A
X Y
S
Confounded by A
and S
A
X Y
S
Unconditional Conditional on A and S
X
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A
X Y
S
Effect of S
confounded by A
Unconditional
S
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Coefficient for S:
Direct effect of S on Y
A
X Y
S
Effect of S
confounded by A
Unconditional Conditional on A and X
A
X Y
S
S
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A
X Y
S
Total causal effect
of A on Y flows
through all paths
Unconditional
A
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Coefficient for A:
Direct effect of A on Y
A
X Y
S
Total causal effect
of A on Y flows
through all paths
Unconditional Conditional on X and S
A
X Y
S
A
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A
X Y
S
Stroke
HIV
Smoking
Age
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A
X Y
S
Stroke
HIV
Smoking
Age
u
unobserved
confound
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Table 2 Fallacy
Not all coefficients created equal
So do not present them as equal
Options:
Do not present control coefficients
Give explicit interpretation of each
No causal model, no interpretation
A
X Y
S
u
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Imagine Confounding
Often we cannot credibly adjust for all
confounding
Do not give up!
Biased estimate can be better than no
estimate
Sensitivity analysis: draw the
implications of what you don’t know
Find natural experiment or design one
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Course Schedule
Week 1 Bayesian inference Chapters 1, 2, 3
Week 2 Linear models & Causal Inference Chapter 4
Week 3 Causes, Confounds & Colliders Chapters 5 & 6
Week 4 Overfitting / MCMC Chapters 7, 8, 9
Week 5 Generalized Linear Models Chapters 10, 11
Week 6 Integers & Other Monsters Chapters 11 & 12
Week 7 Multilevel models I Chapter 13
Week 8 Multilevel models II Chapter 14
Week 9 Measurement & Missingness Chapter 15
Week 10 Generalized Linear Madness Chapter 16
https://github.com/rmcelreath/stat_rethinking_2022
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