Graphical approaches to causality

Graphical approaches to causality

Holger Steinmetz University of Paderborn/Germany

Contents  Introduction to graphical approaches to causality ( DAGs)  Data implications of causal models: Path tracing rules and d-separation  The concepts of open vs. closed paths and path blocking

 Collider bias and endogeneous selection  Graph theoretical approaches to endogeneity and instrumental variables  Introduction to the free browser-based DAGitty software  Equivalent models

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DAGs (directed acyclic graphs)  DAGs represent the presumed causal model of the researcher  Arrows – Unidirectional – bidirectional (confounder)  Nodes: – Parents vs. children – Ancestors and descentents  Encodes and communicates all causal assumptions: – Weak assumptions: Effects – Strong assumptions: Non-existing effects  Most important: Models make statements about the implied data (i.e., how the data should look like if the model was true  Testability 3

DAGs (directed acyclic graphs)  Contrasts to structural equation modeling – Bidirectional arrows (confounder vs. „unanalyzed correlation“) – No depiction of error terms – Can be non-parametric (i.e., no assumptions about distributions and functional form of the effects)

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Model subtypes  Every complex model can be decomposed to three basic subtypes

Chain („mediator model“)

Fork („confounder model“, „common cause model)

Inverted fork („common effect model“; „regression model“) „Collider“

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Connection between model and reality (data)  In order to

a)

test the model and

b) to estimate parameters, we have to know how the model connects to the observed world  „Correlation implies not causality “ BUT „Causality implies correlation“

 More specific: A model with its structure implies a certain pattern of correlations among the model variables („model-implied correlations“).

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Model-Implied Correlations  Example: We suppose a chain (i.e., full mediation)

 This model implies … – A correlation between X and M of Kor(X, M) = a – A correlation between X and Y to the amount of Kor(X,Y) = ab

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Model-Implied Correlations  Second example: Fork (Common Cause Model)

Kor(Y1,Y2) = ab

 Third example: Inverted fork (Common Effect Model)

Kor(X,Y) = a + cb

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Wright’s Path Tracing Rules  For the just depicted 3 model elements the identification of the implied correlations was simple.  But how’s about this one:

 Of which paths does e.g. – Kor(X1,Y1) – Kor(M1,M2)

 Wright’s path tracing rules 9

Wright’s Path Tracing Rules  The correlation between two variables follows from the sum of relevant paths which connect these two variables.  Path: A sequence of arrows linking two variables

Example:  A possible path from X1 to Y1 is ae.  Which further paths can you identify that connect X1 with Y1? 10

Wright’s Path Tracing Rules  Rule No. 1: „No variable can be passed twice within one path“. Example Kor(F,A): adecf is no legitimate path to determine, because C is passed twice. But this does not mean that a variable cannot be passed twice – it must not be in the same path.

E

c

f

F

C

e

d D

a A

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Wright’s Path Tracing Rules  Rule No. 2: „Not first forward then backward“. If one walks along an arrow first foward, it is forbidden to walk along another arrow backwards. It is permitted the other way round. To determine Kor(B,C), the path „ba“ is legitimate – „dc“ is not.

a

C

c D

A b

B

d

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Wright’s Path Tracing Rules  Rule No. 3: “Only one bivariate arrow per path” For the determination of Kor(D,F), “agc” is a legitimate path – “afec” is not because this would require that the path contains two bidirectional arrows. C

c

F

e B

g f

A

b

a

E D

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Wright’s Path Tracing Rules  Examples: X1

c

a Z1

d b

X2

f

X1

Y

c

e X3

Kor(X1,Y)=

#1: No variable can be crossed twice. #2: Not „first foward“ then backward #3: Only one bivariate arrow per path

b1

e

Z2 b2

b3

b4

b

X2

Y1

Kor(Z1, X1)= Kor(X1, X2)= Kor(X1, Y2)=

Y2

M

d

Y

X2

g

f X1

a

c X3

Kor(X1, Y)= Kor(X3, M)=

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Open and closed paths  The path tracing rules define open and closed paths  Open path: Connection between two variables that creates a correlation between both, e.g.,

 Closed path: Connection between two variables that imply a zero correlation due to a collider

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Path blocking and path opening  Open paths can be blocked/closed by controlling / adjusting variables located in the path  Both variables become conditionally independent (Kor(X,Y | M) = 0)

 Closed paths (due to a collider) will be opened by controlling the collider

 Korr(X,Y | M) != 0 16

Path blocking and path opening

 How many paths connect A and F?  How can each of the paths be blocked?

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Path blocking and path opening

1) 2) 3) 4)

Is the path between A and D open or closed? What happens if you control for C? What happens if you control for B? What happens if you control for C and B?

A closed path will be opened if you control / adjust for a collider or its descendants

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Collider bias  Controlling for a collider introduces bias  Controlling for D – strongly biases the effect of B – and: biases the effect of A Simulation: Population with each arrow = .5 (N = 10,000) Estimate Std. Error t value Pr(>|t|) (Intercept) 0.003473 0.010074 0.345 0.73 A 0.511914 0.011143 45.940