2 Rosenstein example

For Lia’s project, she is interested in testing the relationships among affective instability (AI), identity disturbance (ID), and impulsivity (IMP) in a cross-sectional undergraduate sample. The primary hypothesis is that identity disturbance mediates the relationship between affective instability and impulsivity.

g1 <- dagitty('dag {
    Aff_Ins [pos="2,1"] 
    Ident [pos="1,0"]
    Impul [pos="4,0"]

    Aff_Ins -> Ident
    Ident -> Impul
}')
plot(g1)

We might also consider whether the mediation through ID is not complete:

g2 <- dagitty('dag {
    Aff_Ins [pos="2,1"] 
    Ident [pos="1,0"]
    Impul [pos="4,0"]

    Aff_Ins -> Ident
    Ident -> Impul
    Aff_Ins -> Impul
}')
plot(g2)

An alternative model would be to change the mediator and independent variable. That is, what if AI mediates the relationship between ID and IMP?

g3 <- dagitty('dag {
    Aff_Ins [pos="1,0"] 
    Ident [pos="2,1"]
    Impul [pos="4,0"]

    Ident -> Aff_Ins
    Aff_Ins -> Impul
}')
plot(g3)

What are the testable implications for causal mediation?

2.1 d-Separation

First, are AI and IMP d-separated if we do not condition on anything? That is, do we expect their marginal association to be zero?

dseparated(g1, "Impul", "Aff_Ins")

## [1] FALSE

No, under the DAG specified, we expect:

\[ \begin{align*} \textrm{IMP} {\thinspace\not\mkern-7mu{\perp\!\!\perp}\thinspace}\textrm{AI} \\ \textrm{IMP} {\thinspace\perp\!\!\perp\thinspace}\textrm{AI} \: | \: \textrm{ID} \end{align*} \]

What if we condition the AI-Impul relationship on identity disturbance? In this situation, affective instability and impulsivity are d-separated because conditioning on the mediator blocks the relationship between them. Thus, if there is a causal relationship between affective instability and impulsivity via identity disturbance, a model in which we regress impulsivity on affective instability and identity disturbance should reveal a zero conditional association between affective instability and impulsivity.

dseparated(g1, "Impul", "Aff_Ins", "Ident")

## [1] TRUE

What about the partial mediation case in which some of the causal relationship between AI and IMP is direct?

First, what is the expected marginal association of AI and IMP (empty conditioning set)?

dseparated(g2, "Impul", "Aff_Ins")

## [1] FALSE

We do not expect AI and IMP to be marginally independent.

What if we condition on the mediator, as before?

dseparated(g2, "Impul", "Aff_Ins", "Ident")

## [1] FALSE

Unfortunately, in this case, there is no testable prediction of conditional independence based on the mediator.

\[ \begin{align*} \textrm{IMP} {\thinspace\not\mkern-7mu{\perp\!\!\perp}\thinspace}\textrm{AI} \\ \textrm{IMP} {\thinspace\not\mkern-7mu{\perp\!\!\perp}\thinspace}\textrm{AI} \: | \: \textrm{ID} \end{align*} \]

2.2 Implied conditional independencies

What are the implied conditional independencies of these alternative graphs?

For the first, the only implied CI is the one mentioned above.

impliedConditionalIndependencies(g1, "missing.edge")

## Aff_Ins _||_ Impul | Ident

For the second DAG, partial mediation, there are no CIs. Remember that this is a saturated model. As such, changing the direction of the paths will yield equivalent DAGs. A model with no conditional indepencies between two variables means that we cannot test hypotheses about causal directionality.

impliedConditionalIndependencies(g2, "missing.edge")

How many equivalent models can we fit to the saturated partial mediation case?

eqdags <- equivalentDAGs( g2 )
dummy <- lapply(eqdags, plot)

Here we have 6 equivalent models:

length(eqdags)

## [1] 6

What about in the full mediation case?

eqdags <- equivalentDAGs( g1 )
dummy <- lapply(eqdags, plot)

#plot(g1)

Here, there are only 3 equivalent models, though they are quite conceptually distinct from a causality standpoint. The first is our candidate, the second is a fork arrangement in which identity disturbance is a common cause. The third reverses the direction of the mediated effects.

Are there covariate adjustment sets that we could use to render the IV and DV conditionally independent?

adjustmentSets(g1, exposure='Aff_Ins', outcome = 'Impul', type="all")

##  {}

Nope…

The general point is that causality is hard to corroborate in small models.

Here’s an example of when this can help:

grViz("
digraph g {
  # a 'graph' statement
  graph [overlap = true, fontsize = 12, rankdir=LR]

  # nodes for observed and latent
  node [shape = circle, fontname = Arial]
  x y a b z i

{rank=same; a i z b}

  edge [dir=none, fontname = Arial]
  a:w -> z:w
  a:w -> i:w
  a:w -> b:w

  edge[dir=forward, fontname=Arial]
  a -> x
  z -> x
  i -> x

  x -> y
  z -> y
  b -> y

}")

The paths lacking arrows are part of a partially directed acyclic graph. This is a variant of DAGs in which the direction of some edges is unknown. This is different than a standard DAG where the double-headed arrow \(\leftarrow \! \rightarrow\) denotes an unknown (latent) variable, \(U\), that causes association.

 gadj <- dagitty("pdag { x[e] y[o] a -- {i z b}; {a z i} -> x -> y <- {z b} }")
#plot(gadj)
adjustmentSets( gadj )

##  { b, z }
##  { a, z }

In this scenario, conditioning the \(x-y\) relationship on b and z, or a and z, would identify the unique causal relationship between x and y.

impliedConditionalIndependencies(gadj, "missing.edge")

## a _||_ y | b, x, z
## b _||_ i | a
## b _||_ x | a
## b _||_ z | a
## i _||_ y | b, x, z
## i _||_ y | a, x, z
## i _||_ z | a

DAG Demo in SEM

PSY 597 Week 13

Michael Hallquist

16 Nov 2017

1 DAG example from Venners video

1.1 Feel free to extend the DAG

2 Rosenstein example

2.1 d-Separation

2.2 Implied conditional independencies