The Practical Uses of Causal Diagrams

Michael Joffe Imperial College London

The use of DAGs in epidemiology • the theory of Directed Acyclic Graphs (DAGs) has

developed formal rules for controlling confounding, equivalent to algebraic formulations in their rigour, but simpler to use and less error-prone

• the resulting graphical theory is found to conform to traditional “rules of thumb” – but is a better guide in difficult conditions

A typical DAG in epidemiology

from Hernán, MA, Hernández-Díaz S, Robins, JM. A structural approach to selection bias. Epidemiology 2004; 15(5): 615-25

L is parental socioeconomic status U is attraction towards physical activity (unmeasured) C is “risk” of becoming a firefighter E is being physically active (“exposure”) D is heart disease (“outcome”)

The use of DAGs in epidemiology • the theory of Directed Acyclic Graphs (DAGs) has

developed formal rules for controlling confounding, equivalent to algebraic formulations in their rigour, but simpler to use and less error-prone

• the resulting graphical theory is found to conform to traditional “rules of thumb” – but is a better guide in difficult conditions

The use of DAGs in epidemiology • the theory of Directed Acyclic Graphs (DAGs) has

developed formal rules for controlling confounding, equivalent to algebraic formulations in their rigour, but simpler to use and less error-prone

• the resulting graphical theory is found to conform to traditional “rules of thumb” – but is a better guide in difficult conditions

• the focus is on a single link: effect of E on D • arrows mean causation: a variable alters the magnitude,

probability and/or severity of the next variable • it can readily cope with multi-causation

– the representation of effect modification is still problematic

Pearl: causal & statistical languages

associational concept: can be defined as a joint

distribution of observed variables

• correlation • regression • risk ratio • dependence • likelihood • conditionalization • “controlling for”

causal concept: • influence • effect • confounding • explanation • intervention • randomization • instrumental variables • attribution • “holding constant”

Four ways of explaining a robust statistical association

X

X

X

X

Y

Y

Y

Y

C

C

causation

reverse causation

common ancestor(confounding)

common descendant(Berkson bias)

The SIR model of infections

β ν

• this type of “compartmental model” is widely used in infectious disease epidemiology

• it can only be used where the population can be divided unambiguously into categories: a flow chart

• another example is Ross’ classic equation for malaria: N = p.m.i.a.b.s.f

where N is new infections/month; p is population; m is proportion infected; i is proportion infectious among the infected; a is av. no. of mosquitoes/person; b is proportion of uninfected mosquitoes, s is proportion of mosquitoes that survive; f is proportion of infected mosqitoes that feed on humans

• this only applies to a uni-causal situation (mosquitoes) • what is the equivalent in typical multi-causal situations?

Modelling the whole system I

Transport-related health problems

Respiratory morbidity

& mortality

Cardiovascular morbidity &

mortality

Impaired mental health

Fatal and non-fatal injuries

Osteo-porosis

etc

Air pollution

Physical activity Access

Community severance Noise

Collisions: number, severity

Traffic speed

Traffic volume

Distribution of vehicle emissions

Safe walking &

cycling

Modelling the whole system II • “web of causation”; “upstream” influences • causal diagrams are constructed based on substantive

knowledge of the topic area plus evidence • chains of causation, not just one link; and multiple chains

– assumption of independence

• multidisciplinary • individual & group levels are combined

– as is routine in infectious disease epidemiology

• organised by economic/policy sector • health determines the content of the diagram – “driven

by the bottom line”

Conditional independence

• Z = genotype of parents • X, Y = genotypes of 2 children• If we know the genotype of the

parents, then the children’s genotypes are conditionally independent

Z

X Y

diagram adapted from Best, Richardson & Jackson

X and Y are conditionally independent given Z if, knowing Z, discovering Y tells you nothing more about X

P(X | Y, Z) = P(X | Z)

Conditional independence provides mathematical basis for splitting up large system into smaller components

D

EB

CA

F

CA

C

B

D

E

D

E

F

diagram adapted from Best, Richardson & Jackson

Functions of diagrams: scientific • the aim is a diagram that describes causal relations

“out there” in the world, not a mental map • a framework for analysis, e.g. statistical modelling • to make assumptions and hypotheses explicit for

discussion, and for planning data collection and analysis

• to place hypotheses in the public domain prior to testing – a conjecture that is open to refutation

• to identify evidence gaps • to generate a research agenda

Empirical aspects • default: “all arrows” (saturated model) is conservative –

omission is a stronger statement than inclusion • corollary: deletion following statistical analysis is the

strong step – uses model selection methods, e.g. AIC • quantification of the links that remain • transmissibility: X → Y and Y → Z does not necessarily

imply that X → Y → Z, e.g. in the case of a threshold • a diagram is not like a single study, it’s more like a

synthesis, => the issue of generalisability • a single diagram can be used to integrate multiple

datasets • suitable both for qualitative and quantitative analysis

Diagrams and evidence • a conjectural diagram can be formed from substantive

knowledge of a subject, as a basis for analysis • diagrams evolve from conjectural to well-supported,

as evidence is accumulated • it is crucial to specify the status of any particular

diagram – an analysis of the assumptions and judgements that have been made, the degree of uncertainty and the strength of evidence for the structure of the diagram and for each of the links (including those thought to be absent)

Causes of the causes of health

Underlying causes e.g. socioeconomic factors

Determinants (risk factors)

Health status (diseases etc)

Transport-related health problems

Respiratory morbidity

& mortality

Cardiovascular morbidity &

mortality

Impaired mental health

Fatal and non-fatal injuries

Osteo-porosis

etc

Air pollution

Physical activity Access

Community severance Noise

Collisions: number, severity

Traffic speed

Traffic volume

Distribution of vehicle emissions

Safe walking &

cycling

Altering the causes of the causes

Policy options alterable causes

Changes in alterable risk factors

Changes in health status

Health impact of transport policies

resp. morbidity &

mortality

cardiovascular morbidity &

mortality

impaired mental health

fatal and non-fatal injuries

osteo-porosis

etc

air pollution

physical activity

access

community severance

noise

collisions: number, severity

Speed control policies

Traffic reduction policies

Emissions control policies

Promotion of active transport

“Change” models: advantages • Parsimony: the immense complexity of the

pathways can be greatly reduced by focusing on changes, especially in the absence of effect modification;

• Philosophy: causality is more readily grasped when something is altered, e.g. a particular road layout rather than “roads” as a necessary condition of “road deaths”;

• Pragmatism: changes in the determinants of health determinants link naturally to policy options (cf Wanless: “natural experiments”).

Effect of the coal ban, Dublin, 1990 • before-after comparison of pollution

concentration, adjusted for weather etc • 72 months before and after the ban • also controls for influenza and age structure • all-Ireland controls for secular changes

Speed control and health gain

resp. morbidity &

mortality

cardiovascular morbidity &

mortality

impaired mental health

fatal and non-fatal injuries

osteo-porosis

etc

air pollution

physical activity

access

community severance

noise

collisions: number, severity

Lower speed limits

Better enforcement

Traffic calming

Public education

Speed

Emissions control as a technical fix

resp. morbidity &

mortality

cardiovascular morbidity &

mortality

impaired mental health

fatal and non-fatal injuries

osteo-porosis

etc

air pollution

physical activity

access

community severance

noise

collisions: number, severity

Emissions control policies

Car dependence

reduction of active transport

traffic growth

community severance

unpleasantness & inconvenience of

non-car travel

increased car ownership

reduction of public transport

congestion

increased prosperity

vicious circle of decline

car dependence affecting e.g.

shopping

pro-bus policies

labor productivity

nutritional intake

healthstatus

labor productivity

nutritional intake

healthstatus

exposure to infectious agents contaminants e.g. aflatoxins chemicals e.g. pesticideswar, natural catastrophe, etc

micronutrient contentinfant feeding practices

land quantity & fertilityclimate & weatherpests, e.g. fungi, ratsinputs, e.g. irrigation,

chemicalstools & technology

The SIR model of infections

β ν

The SIR model of infections

The basic reproductive number R0 is given by:

where β is the contact rate (infectivity), and ν is the recovery rate (= 1/D where D is duration)

β ν

The SIR model of infections

The basic reproductive number R0 is given by:

where β is the contact rate (infectivity), and ν is the recovery rate (= 1/D where D is duration)

β ν

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