Walking Example (p135)

Authors: Aymen Merrouche and Pierre-Henri Wuillemin.

This notebook follows the example from “The Book Of Why” (Pearl, 2018) chapter 4 page135

Confounding

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

In 1998 a study unveiled a correlation between physical exercise and longevity among nonsmoking retired men. Of course what we want to know is whether men who exercise more live longer, suggesting a causal relationship. Study measurements are to be found at the end of this notebook.

corresponding causal diagram

The corresponding causal diagram is the following:

## We create the causal diagram
we = gum.fastBN("Walking{casual|normal|intense}->Mortality{dead|alive}")

## We fill the CPTs
we.cpt("Walking")[:] = [151 / 707, 379 / 707, 177 / 707]
we.cpt("Mortality")[{"Walking": "casual"}] = [0.43, 0.57]
we.cpt("Mortality")[{"Walking": "intense"}] = [0.215, 0.785]
we.cpt("Mortality")[{"Walking": "normal"}] = [0.277, 0.723]

gnb.sideBySide(
  we,
  we.cpt("Walking") * we.cpt("Mortality"),
  we.cpt("Walking"),
  we.cpt("Mortality"),
  captions=["the BN", "the joint distribution", "the marginal for $Walking$", "the CPT for $Mortality$"],
)

the BN

Walking Mortality casual normal intense dead 0.0918 0.1485 0.0538 alive 0.1217 0.3876 0.1965 the joint distribution

Walking casual normal intense 0.2136 0.5361 0.2504 the marginal for $Walking$

Mortality Walking dead alive casual 0.4300 0.5700 normal 0.2770 0.7230 intense 0.2150 0.7850 the CPT for $Mortality$

The study showed that after 12 years, 43% of casual walkers died while only 21,5% of intense walkers died.

Causal effect of walking on mortality in this model:

weModele = csl.CausalModel(we)
cslnb.showCausalImpact(weModele, "Mortality", doing="Walking", values={})

Causal Model

$$\begin{equation*}P( Mortality \mid \text{do}(Walking)) = P\left(Mortality\mid Walking\right)\end{equation*}$$

Explanation : Do-calculus computations

	Mortality
Walking	dead	alive
casual	0.4300	0.5700
normal	0.2770	0.7230
intense	0.2150	0.7850

Impact

Before jumping to any conclusions, we should consider the presence of possible confounders. We need to ask the following question: what characterizes intense walkers from casual walkers?
Without abandoning the idea of a possible cause-and-effect relationship between walking and mortality, we introduce a third variable, a “confounder”, a common cause of the two variables that could explain the correlation that exists between them. Our aim is to distinguish between the causal effect of walking on mortality (if there is a cause and effect relationship) the bias induced by this third variable. For this purpose, we need to adjust for it.

weModele1 = csl.CausalModel(we, [("confounder", ["Walking", "Mortality"])], True)
gnb.show(weModele1)

cslnb.showCausalImpact(weModele1, "Mortality", "Walking", values={"Walking": "intense"})

Causal Model

Hedge Error: G={'Walking', 'Mortality'}, G[S]={'Mortality'}
Impossible

No result
Impact

Introducing age as a confounder:

We want to measure the causal effect of walking on mortality, the introduction of a confounding bias occurs when a third variable called “confounding variable” influences both walking and mortality.
An obvious confounder is age, younger subjects exercise more and have more time to live! (there are other confounders)

Let’s use fictitious data:

wea = gum.fastBN("Age{cat1|cat2|cat3}->Walking{casual|normal|intense}->Mortality{dead|alive}<-Age{cat1|cat2|cat3}")

gnb.sideBySide(
  wea,
  wea.cpt("Age"),
  wea.cpt("Walking"),
  wea.cpt("Mortality"),
  captions=["the BN", "the marginal for $Age$", "the CPT for $Walking$", "the CPT for $Mortality$"],
)

the BN

Age cat1 cat2 cat3 0.4098 0.3557 0.2345 the marginal for $Age$

Walking Age casual normal intense cat1 0.3935 0.3978 0.2086 cat2 0.5396 0.3272 0.1332 cat3 0.2616 0.4733 0.2651 the CPT for $Walking$

Mortality Age Walking dead alive cat1 casual 0.3069 0.6931 normal 0.4508 0.5492 intense 0.3586 0.6414 cat2 casual 0.3457 0.6543 normal 0.5842 0.4158 intense 0.6148 0.3852 cat3 casual 0.4517 0.5483 normal 0.7280 0.2720 intense 0.4739 0.5261 the CPT for $Mortality$

Causal effect of walking on mortality with age as a confounder:

weModele2 = csl.CausalModel(wea)
cslnb.showCausalImpact(weModele2, "Mortality", "Walking", values={})

Causal Model

$$\begin{equation*}P( Mortality \mid \text{do}(Walking)) = \sum_{Age}{P\left(Mortality\mid Age,Walking\right) \cdot P\left(Age\right)}\end{equation*}$$

Explanation : backdoor [‘Age’] found.

	Mortality
Walking	dead	alive
casual	0.3546	0.6454
normal	0.5633	0.4367
intense	0.4768	0.5232

Impact

We adjusted for Age using the back-door criterion (Age blocks all back-door paths from Walking to Mortality, setting Walking= “intense” or conditioning on Walking=“intense” has the same effect on Mortality)

Conclusion:

After adjusting for age, we obtain that 40.5% (43% unadjusted) of casual walkers died, whereas only 23.8% (21,5% unadjusted) of intense walkers died. The correlation induced by Age between the two variables is negligible.
Even after adjusting for all plausible confounders, after getting rid of all confounding bias, Walking is still associated to Mortality. Unless we missed any other confounders, in which case the remaining uncertainty is proportional to the correlation induced by these hidden variables, we can say that intentional walking prolongs life among the studied population.

In an observational study, adjusting for confounding factors is systematic in order to measure the causal effect of a treatment on an outcome.

Study measurements both unadjusted and age-adjusted: