Presentación de PowerPoint3. Adjusted causal effect
Treated Untreated
Association Causation
versus versus
c: control
- An outcome Y with two possible manifestations:
Yi(c): outcome Y in the unit ui when ui is allocated to c
Yi(t): outcome Y in the unit ui when ui is allocated to t
Example:
DBP of patient 7 if he receives the control c: Y7(c)
DBP of patient 7 if he receives the treatment t: Y7 (t)
Definition of Potential Outcomes
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The effect of the cause ‘t’ relative to the control ‘c’
on the outcome Y in the individual ui
is defined as: t causes the effect yi(t) – yi(c)
In the example: The effect of the cause ‘new treatment’
relative to the control on the outcome DBP
in patient 7 is defined as: the new treatment causes the effect y7(t) - y7(c)
Definition of Causal Effect in a unit i
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(i) We define the effect of a cause and not the reverse.
Just answer the question prospectively: ¿what is the effect of this cause?
Avoid “competence” with causes of the same effect. Example: Toxin versus germs.
Implications of the definition
We’re talking about allocation:
“ Outcome when ‘t’ was assigned“ This definition only considers manipulables aspects It doesn’t apply to attributes Z such as gender
Remember: i) Attributes can help to predict or to characterize the response ii) The gender isn’t assignable, but a photo and a name (masculine o
feminine) can be assigned to a curriculum iii) Smoking isn’t assignable, but the “advice not to smoke” is iv) Age isn’t assignable, but a molecule that stops the aging is
(ii) Allocation
- emulates the ‘practice’ decision procedure
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(iv) Potential response
Unit Baseline Z Potential response Causal effect Y(t)-Z Y(c)-Z [Y(t)-Z]-[Y(c)-Z]
You 18 -13 -3 -10
Unit Potential response Causal effect Y(t) Y(c) Y(t)-Y(c)
You 5 15 -10
Example: Y is the pain in a scale from 0 to 20
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Unit Potential response Causal effect
Y(t) Y(c) Y(t)-Y(c) 1 2 3 4 5 6 7 8
14 0 1 2 3 1 10 9
13 6 4 5 6 6 8 8
1 -6 -3 -3 -3 -5 2 1
Mean -2
Unit Alloca-
Potential response
Causal effect
Y(t) Y(c) Y(t)-Y(c) 1 2 3 4 5 6 7 8
1 1 1 1 0 0 0 0
14 0 1 2 3 2 10 9
13 6 4 5 6 6 8 8
? ? ? ? ? ? ? ?
Mean 4.25 7 ?
If patients choose their treatment, no rationale to compare 4.25 to 7!!
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A non observed potential response implies
the fundamental problem of causal inference:
Yi(t), Yi(c) can't be observed at once and in the same conditions:
Y(t) can be observed in some units
Y(c) can be observed in others.
Or: Yi(t) can be observed in a specific conditions
Yi(c) can be observed in others
Thus, one is observed, but not its complementary.
In summary,
The causal effect in the unit can’t be observed
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Taking into account the entire population removes the problem because Y(t) can be observed in some units and Y(c) in others.
t causes the (average) effect E[ Yi(t) ] - E[ Yi(c) ]
2.- Causal effect on the population
Association Causation
versus versus
E (Yi|t ) – E (Yi|c) ≠ E[ Yi(t) ] - E[ Yi(c) ]
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what cases apply the mean to?
Unit Potential response Causal effect Y(t) Y(c) Y(t)-Y(c)
1 2 3 4 5 6 7 8
14 0 1 2 3 1 10 9
13 6 4 5 6 6 8 8
1 -6 -3 -3 -3 -5 2 1
Mean -2
Mean 1.4 9.67 -8.23 ≠ -2
Without constant effect, clinician should allocate best treatment to any case
Changing ‘best’ by ‘worst’ at any unit
But, how to know…?
Mean -2
In all the units, the same effect is observed
(It’s exact because there is no within-case variability in this example.)
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An assumption for the effect is necessary: Simplest assumption: constant and additive effect t
Advantages: 1) The average effect is pertinent for each unit [ the same dosage can be use for all patients ]
2) Implies ‘parametric’ assumptions: homoscedasticity, same distribution,… [ If not, how to know if there is an unit-treatment interaction? ]
Causal effect on the population. Constant effect
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Different allocations involve different estimations.
Random allocation: no bias and known variance estimation.
Unit Alloca-
Potential response
Causal effect
Y(t) Y(c) Y(t) - Y(c) 1 2 3 4 5 6 7 8
1 0 0 0 0 0 1 1
11 ? ? ? ? ? 6 7
? ? ? ? ? ? ? ?
The assumption of NON-INTERFERENCE or the effect independence to other allocations is also needed
You take I take
t t
c t
t c
c c
You=1 Y1(t,t)=0 Y1(c,t)=100 Y1(t,c)=0 Y1(c,c)=100 I=2 Y2(t,t)=0 Y2(c,t)=50 Y2(t,c)=75 Y2(c,c)=100
In your case, the effect is independent of my allocation
If I take: Y1(t,t) - Y1(c,t)= 0-100 = -100
If I don’t take: Y1(t,c) - Y1(c,c)= 0-100 = -100
In my case, the effect depends on your allocation
If you take: Y2(t,t) - Y2(t,c)= 0 - 75 = -75
If you don’t take: Y2(c,t) - Y2(c,c)= 50-100 = -50
¿Is necessary another assumption about the effect?
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The effect could be constant under some conditions/attributes Z:
t causes the adjusted (average) effect E[Y(t)|z] - E[Y(c)|z]
Example: increase in the adjusted DBP, when the treatment C is replaced by the treatment T in a patient with fixed age and center.
Conditioning, adjusting by Z…
2) Reduces outcome/response variability → higher efficiency, accuracy and power (lower estimator SE)
3) Controls possible confounders
3.- Adjusted causal effect
Options for the adjustment. Adapted from Kleimbaum et al. Epidemiologic Research, 1982
Option Stage* Name Advantages Inconvenients
Restriction Design Eligibility criteria Complete control
Inexpensive Easy to design
analysis
Without assumptions Direct
(matching)
Allows several exposure variables
A lot of assumptions Model choice difficult Predictors choice difficult Interpretation Software parameterizationAnalysis Covariance
Regression
Generalizability is not lost Logistics are sophisticated Analysis Optimal pair
*Only an adjustment specified in the protocol allows a confirmatory interpretation. Adjustment in the analysis phase doesn’t allow to correct design flaws.
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Directed: arrows have origin and end.
Acyclic: a variable cannot cause it self
Graphs: Diagrams
Z: Baseline BP X: Treatment BP Y: Outcome BP
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Associational words: Z2 and Y are associated
CONFOUNDING
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Example: Genetics and high lipids (invented data)
Assumption: High Lipids (Lip) cause bad events (AVC) such stroke, infarct,.. Gen causes High Lipids (Lip)
Temporal order: Gen, Lip, AVC
Hypothesis: gen (X) causes bad events (Y) independently of High Lipids (Z)
Without adjusting by lipids (Z) we see overall effects of X on Y Adjusting, we estimate direct effect, accounting by lipids For example: can we eliminate all the gen effects by controlling lipids?
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Z: Smoking Z2 : Yellow fingers Y: Lung cancer
Causal words: Z2 has no effect on Y
Associational words: Z2 and Y are NOT associated conditional on Z
Conditional independence
Z+ Z-
80 20
40 40
Z+ Z-
40 40
20 80
Z+ Z-
120 60
60 120
Y+ Z2+ Z2 Y- Z2+ Z2- Z2+ Z2-
Z+ Z-
80 20
40 40
Z+ Z-
40 40
20 80
Z+ Z-
120 60
60 120
Z+ Y+ Y- Z- Y+ Y- Y+ Y-
Z2+ Z2-
80 40
40 20
Z2+ Z2-
20 40
40 80
Z2+ Z2-
100 80
80 100
Conditional Independence
CONFOUNDING
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Mean (n) Treated Control ORZX = 10·10 / 10·10 = 1
Z+ 110 (10) 120 (10) 115 (20)
Z- 130 (10) 140 (10) 135 (20)
Total 120 (20) 130 (20) 125 (40), SD=10
adjusted effect (-10) =
= raw effects (-10)
Mean (n) Exposed Control ORZZ2 = 2·2 / 18·18 = 1/81
Z+ 110 ( 2 ) 110 (18) 110 (20)
Z- 130 (18) 130 ( 2 ) 130 (20)
Total 128 (20) 112 (20) 125 (40), SD=10
adjusted effect (0) ≠
≠ raw effects (+6)
Z1
Z2 10 (100) All
You have to be able to draw an imaginary data set with collinearity.
For both, a continuous and a dichotomous response
Colineality: the best friend of statisticians
Z1 Y+ Y- Z2 Y+ Y- Y+ Y-
X+ X+ 100 X+
Z Z2 Y
Causal words: Z has no effect on Z2
Associational words: Z and Z2 are independent
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Causal words: Z has no effect on Z2
Associational words: Z and Z2 are associated conditional on Y
Selection BIAS
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6
7
8
9
Z+ Z-
64 40
16 40
Z+ Z-
40 16
40 64
Z+ Z-
104 56
56 104
Z+ Y+ Y- Z- Y+ Y- Y+ Y-
Z2+ Z2-
64 40
16 40
Z2+ Z2-
40 16
40 64
Z2+ Z2-
104 56
56 104
Y+ Z2+ Z2 Y- Z2+ Z2- Z2+ Z2-
Z+ Z-
64 40
40 16
Z+ Z-
16 40
40 64
Z+ Z-
80 80
80 80
Conditioning on common effects
Z and Z2 are associated conditional on Y SELECTION BIAS
Z has NO effect on Z2
Z has effect on Y
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Another example of Selection bias: Genetics and high lipids (invented data)
Previous model New model
Selection bias by an intermediate variable: What would we have seen about gen and lipids Z if, in our source data, we only account for people with AVC+ (as in an hospital service) ?
AVC+ Lip+ Lip- AVC- Lip+ Lip-
Gen+ 80 45 Gen+ 10 45
Gen- 45 10 Gen- 45 80 OR=0,4 CI95%=0,18 to 0,86 OR=0,4 CI95%=0,18 to 0,86
Overall (previous in time to the occurrence of AVC), Gen is independent to Lipids
All Lip+ Lip-
Gen+ 90 90
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Z+ Z-
Z+ Z-
Z+ Z-
Z2+ Z2-
Z2+ Z2-
Z2+ Z2-
Z+ Z-
Z+ Z-
Z+ Z-
Conditioning on common effects: YOUR OWN DATA
YOUR OWN DATA: Propose some variables and conditions were selection bias may be devisable.
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Over-adjustment
Independent interpretation:
effect on Y of one X variable at fixed level of the remaining Z ones.
It should be possible and make sense!
Example: risk of Down sd. depending on the age of mother and father
Note the higher relationship between both ages:
Older is one of them, older is expected to be the other.
If the age of one is fixed, the age variability of the other decreases.
Can we act on the mother’s age without modifying the father’s one?
Example: SBP and DBP. Another time, a great relationship.
Can we change one without modifying the other?
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3
1
3
1
1
2
3
6
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5.- IP weighting
Assume nobody is treated
Assume everyone is treated
Y0 Y1
Nobody treated
Everyone treated
Y0 Y1
Reconstructed data
Observed data
1/½=2
1/½=2
kiW Y0 Y1
Inverse Probability Weighting W=1/f[X|Z]
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EJERCICIO
El protocolo de intervención de una neoplasia aconseja que los casos de Nivel I (N=I), que suponen el 50%, deben seguir una pauta (X) de Cirugía (X=C), mientras que los de Nivel II (N=II) deben seguir Quimioterapia (X=Q).
En nuestro estudio, en ambos niveles, una cuarta parte de los 144 casos observados no siguen las recomendaciones y son finalmente tratados con la otra opción.
Ambos tratamientos tienen la misma eficacia, de forma que en el Nivel I siempre fallecen 1/3 (Y=1), pero en el nivel II, 2/3 –independientemente del tratamiento recibido.
a. Reconstruya el árbol de probabilidades b. Calcule el OR observado entre X (C/Q) e Y (0/1) c. Utilice el IPW para estimar el efecto de cambiar C por Q
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ITT: (12+34)/(12+34+9663+2385) versus (74 )/(74 +1154 ): -0.0025
PP: (12 )/(12 +9663 ) versus (74 )/(74 +1154 ): -0.0052
AT: (12 )/(12 +9663 ) versus (74+34)/(74+34+1154+2385): -0.0065
Allocated treatment
Treatment received
9663 12
2385 34
1154 74
ACE (Averaged Causal Effect = ITT) = -0.0025
CACE (Compliers Averaged Causal Effect) PN = (2385+34)/(9663+12+2385+34)
NACE (Non compliers Averaged Causal Effect) PC = 1-PN
ACE = PN ·NACE + PC·CACE -> -.0025= .2·NACE + .8·CACE
Si NACE = 0 -> CACE= -.0025/.8 = -.0031
Allocated treatment
C C N N ? ?
9663 12
2385 34
1154 74
(Instrumental variable)
The propensity to be treated as an instrumental variable
If Z is the vector of confounders, the usual LM/GLM adjustment first subtracts from the outcome Y, its part explained by Z before estimating X-effect.
Alternatively, obtain the propensity (probability) to be treated as a function of Z X = f(Z) (e.g.: logistic regression)
and estimate X-effect on cases with similar propensity through:
- matching - stratification, - modeling, with the adjustment of Y = g(X, f(Z))
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1) Stratification
2) Matching
Otherwise:
The propensity to be treated (PS) as an instrumental variable
The idea is that given f(Z), the allocation is ignorant
Simulation studies have shown either that:
PS has a better behavior than
PS has a behavior as bad as
The process is more explicit
It prevents "overconfidence" in the adjustment process
This allows a sophisticated model at X = f (Z) and a subsequent simple analysis
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Yp /Z
Z V
Unknown variables V may also influence the outcome [V should also include variables measured without enough accuracy -such stress,…]
They must be included in the definition:
t causes the adjusted (average) effect E[Y(1)|z,v] - E[Y(0)|z,v] [to rule out that they can not be an alternative explanation]
However, as these V variables aren’t observable, we need further assumptions.
7.- Unknown variables V
A premise/assumption/postulate P allows testing a causal hypothesis H if…
given P: statistic independence ↔ no causal effect
More formally
given P: E[Y(1)|Z] = E[Y(0)|Z] ↔ E[Y(1)|Z,V] = E[Y(0)|Z,V] (estimable effect) (defined effect)
That is,
the conclusion based only on (observed) Z variables
to include also (unobservable) V variables ?
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X
Y/Z
Model sufficiency There are no V Deterministic sciences
No confusion There are some V but they are independent to X Epidemiology
Random allocation V are balanced between groups Clinical trials
Ignorant allocation Their overall influence is balanced Social sciences
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W
V
Z
/Z
Yp
Y/Z
X
Then, it is irrelevant to condition or not, and so:
Given Pms: E[Y(1)|Z] = E[Y(0)|Z] ↔ E[Y(1)|Z,V] = E[Y(0)|Z,V]
In our example, If we assume that there aren’t more variables, other than the controlled Z (age, center, ...), that affect the DBP of a patient,
the observed difference estimation will also estimate the real causal effect.
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B) Randomization:
Pr: V ⊥ X|Z → The unobservable covariates are independents of treatment.
It is the only a premise that can be “established” through the design: If the treatment is randomly assigned, all ‘free’ variables have the same distribution in both treatment groups:
X ⊥ Vo in consequence: Yp|Z ⊥ X X ⊥ Vc Y |Z ⊥ X
and given Pr: E[Y(1)|Z] = E[Y(0)|Z] ↔ E[Y(1)|Z,V] = E[Y(0)|Z,V]
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C) Absence of confounders:
Pnc: Vc ⊥X|Z → the unobservable covariates related to the outcome are independent of the treatment.
Thus: Yp|Z ⊥ X
Pnc requires to believe that the unobserved variables: either they are independent of the outcome (Vo ⊥ Yp|Z ) or they are independent of the treatment (Vc ⊥ X|Z )
Given Pnc: E[Y(1)|Z] = E[Y(0)|Z] ↔ E[Y(1)|Z,V] = E[Y(0)|Z,V]
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D) Ignorant allocation of the treatment:
Piat: Yp|Z ⊥ X → the potential response is independent of the treatment.
In our example, if the baseline DBP can be considered as the result of several factors or previous variables, some of them could be unbalanced in the groups and would favor to one or another group, but their global influence, is balanced.
Given Piat: E[Y(1)|Z] = E[Y(0)|Z] ↔ E[Y(1)|Z,V] = E[Y(0)|Z,V]
Note that, it’s necessary to allocate X “independently” (ignorant)
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Confounded treatment allocation: → allocation probability depends on the potential response (selection bias)
No confounded treatment allocation : → allocation probability doesn’t depend on the potential response values
No ignorable treatment allocation : → allocation probability depends on the unobserved potential response values
Ignorable treatment allocation : → allocation probability only depends (at most) on the observed potential response
values
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Slide Number 24
Slide Number 25
Conditioning on common effects
Over-adjustment
Slide Number 57
A) Model sufficiency:
Rubin terminology