Simpson’s Paradox

This notebook follows the famous example from Causality (Pearl, 2009).

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aGrUM

interactive online version

In [1]:
from IPython.display import display, Math, Latex

import pyAgrum as gum
import pyAgrum.lib.notebook as gnb
import pyAgrum.causal as csl
import pyAgrum.causal.notebook as cslnb

In a statistical study about a drug, we try to evaluate the latter’s efficiency among a population of men and women. Let’s note: - \(Drug\) : drug taking - \(Patient\) : cured patient - \(Gender\) : patient’s gender

The model from the observed date is as follow :

In [2]:
m1 = gum.fastBN("Gender{F|M}->Drug{Without|With}->Patient{Sick|Healed}<-Gender")

m1.cpt("Gender")[:]=[0.5,0.5]
m1.cpt("Drug")[:]=[[0.25,0.75],  #Gender=F
                   [0.75,0.25]]  #Gender=M

m1.cpt("Patient")[{'Drug':'Without','Gender':'F'}]=[0.2,0.8] #No Drug, Male -> healed in 0.8 of cases
m1.cpt("Patient")[{'Drug':'Without','Gender':'M'}]=[0.6,0.4] #No Drug, Female -> healed in 0.4 of cases
m1.cpt("Patient")[{'Drug':'With','Gender':'F'}]=[0.3,0.7] #Drug, Male -> healed 0.7 of cases
m1.cpt("Patient")[{'Drug':'With','Gender':'M'}]=[0.8,0.2] #Drug, Female -> healed in 0.2 of cases
gnb.flow.row(m1,m1.cpt("Gender"),m1.cpt("Drug"),m1.cpt("Patient"))

G Patient Patient Gender Gender Gender->Patient Drug Drug Gender->Drug Drug->Patient
Gender
F
M
0.50000.5000
Drug
Gender
Without
With
F
0.25000.7500
M
0.75000.2500
Patient
Gender
Drug
Sick
Healed
F
Without
0.20000.8000
With
0.30000.7000
M
Without
0.60000.4000
With
0.80000.2000
In [3]:
def getCuredObservedProba(m1,evs):
    evs0=dict(evs)
    evs1=dict(evs)
    evs0["Drug"]='Without'
    evs1["Drug"]='With'

    return gum.Potential().add(m1["Drug"]).fillWith([
            gum.getPosterior(m1,target="Patient",evs=evs0)[1],
            gum.getPosterior(m1,target="Patient",evs=evs1)[1]
        ])

gnb.sideBySide(getCuredObservedProba(m1,{}),
               getCuredObservedProba(m1,{'Gender':'F'}),
               getCuredObservedProba(m1,{'Gender':'M'}),
               captions=[r"$P(Patient = Healed \mid Drug )$<br/>Taking $Drug$ is observed as efficient to cure",
                         r"$P(Patient = Healed \mid Gender=F,Drug)$<br/>except if the $gender$ of the patient is female",
                         r"$P(Patient = Healed \mid Gender=M,Drug)$<br/>... or male."])

Drug
Without
With
0.50000.5750

$P(Patient = Healed \mid Drug )$
Taking $Drug$ is observed as efficient to cure
Drug
Without
With
0.80000.7000

$P(Patient = Healed \mid Gender=F,Drug)$
except if the $gender$ of the patient is female
Drug
Without
With
0.40000.2000

$P(Patient = Healed \mid Gender=M,Drug)$
... or male.

Those results form a paradox called Simpson paradox :

\[P(C\mid \neg{D}) = 0.5 < P(C\mid D) = 0.575\]
\[P(C\mid \neg{D},G = Male) = 0.8 > P(C\mid D,G = Male) = 0.7\]
\[P(C\mid \neg{D},G = Female) = 0.4 > P(C\mid D,G = Female) = 0.2\]

Actuallay, giving a drug is not an observation in our model but rather an intervention. What if we use intervention instead of observation ?

How to compute causal impacts on the patient’s health ?

We propose this causal model.

In [4]:
d1 = csl.CausalModel(m1)
cslnb.showCausalModel(d1)
../_images/notebooks_62-Causality_SimpsonParadox_9_0.svg

Computing \(P (Patient = Healed \mid \text{do}(Drug = Without))\)

In [5]:
cslnb.showCausalImpact(d1, "Patient", doing="Drug",values={"Drug" : "Without"})
G Gender Gender Drug Drug Gender->Drug Patient Patient Gender->Patient Drug->Patient
Causal Model
$$\begin{equation*}P( Patient \mid \text{do}(Drug)) = \sum_{Gender}{P\left(Patient\mid Drug,Gender\right) \cdot P\left(Gender\right)}\end{equation*}$$
Explanation : backdoor ['Gender'] found.
Patient
Sick
Healed
0.40000.6000

Impact

We have, \(P (Patient = Healed \mid \hookrightarrow Drug = without) = 0.6\)

Computing \(P (Patient = Healed \mid \text{do}(Drug = With))\)

In [6]:
d1 = csl.CausalModel(m1)
cslnb.showCausalImpact(d1, "Patient", "Drug",values={"Drug" : "With"})

G Gender Gender Drug Drug Gender->Drug Patient Patient Gender->Patient Drug->Patient
Causal Model
$$\begin{equation*}P( Patient \mid \text{do}(Drug)) = \sum_{Gender}{P\left(Patient\mid Drug,Gender\right) \cdot P\left(Gender\right)}\end{equation*}$$
Explanation : backdoor ['Gender'] found.
Patient
Sick
Healed
0.55000.4500

Impact

And then : $P(Patient = Healed \mid `:nbsphinx-math:text{do}`(Drug = With)) = 0.45 $

Therefore : $P(Patient = Healed:nbsphinx-math:mid `:nbsphinx-math:text{do}`(Drug = Without)) = 0.6 > P(Patient = Healed:nbsphinx-math:mid `:nbsphinx-math:text{do}`(Drug = With)) = 0.45 $

Which means that taking this drug would not enhance the patient’s healing process, and it is better not to prescribe this drug for treatment.

Simpson paradox solved by interventions

So to summarize, the paradox appears when wrongly dealing with observations on \(Drug\) :

In [7]:
gnb.sideBySide(getCuredObservedProba(m1,{}),
               getCuredObservedProba(m1,{'Gender':'F'}),
               getCuredObservedProba(m1,{'Gender':'M'}),
               captions=[r"$P(Patient = Healed \mid Drug )$<br/>Taking $Drug$ is observed as efficient to cure",
                         r"$P(Patient = Healed \mid Gender=F,Drug)$<br/>except if the $gender$ of the patient is female",
                         r"$P(Patient = Healed \mid Gender=M,Drug)$<br/>... or male."])

Drug
Without
With
0.50000.5750

$P(Patient = Healed \mid Drug )$
Taking $Drug$ is observed as efficient to cure
Drug
Without
With
0.80000.7000

$P(Patient = Healed \mid Gender=F,Drug)$
except if the $gender$ of the patient is female
Drug
Without
With
0.40000.2000

$P(Patient = Healed \mid Gender=M,Drug)$
... or male.

… and disappears when dealing with intervention on \(Drug\) :

In [8]:
gnb.sideBySide(csl.causalImpact(d1,on="Patient",doing="Drug",values={"Patient":"Healed"})[1],
               csl.causalImpact(d1,on="Patient",doing="Drug",knowing={"Gender"},values={"Patient":"Healed","Gender":"F"})[1],
               csl.causalImpact(d1,on="Patient",doing="Drug",knowing={"Gender"},values={"Patient":"Healed","Gender":"M"})[1],
               captions=[r"$P(Patient = 1 \mid \text{do}(Drug) )$<br/>Effectively $Drug$ taking is not efficient to cure",
                         r"$P(Patient = 1 \mid \text{do}(Drug), gender=F )$<br/>, the $gender$ of the patient being female",
                         r"$P(Patient = 1 \mid \text{do}(Drug), gender=M )$<br/>, ... or male."])
Drug
Without
With
0.60000.4500

$P(Patient = 1 \mid \text{do}(Drug) )$
Effectively $Drug$ taking is not efficient to cure
Drug
Without
With
0.80000.7000

$P(Patient = 1 \mid \text{do}(Drug), gender=F )$
, the $gender$ of the patient being female
Drug
Without
With
0.40000.2000

$P(Patient = 1 \mid \text{do}(Drug), gender=M )$
, ... or male.