meta.transport {causaleffect}

Derive a transport formula for a causal effect between a target domain and multiple source domains

Description

This function returns an expression for the transport formula of a causal effect between a target domain and multiple source domains. The formula is returned for the interventional distribution of the set of variables (y) given the intervention on the set of variables (x). The multiple source domains are given as a list of selection diagrams (D). If the effect is non-transportable, an error is thrown describing the graphical structure that witnesses non-transportability. The vertices of any diagram in (D) that correspond to selection variables must have a description parameter of a single character "S" (shorthand for "selection").

Usage

meta.transport(y, x, D, expr = TRUE, simp = TRUE, 
  steps = FALSE, primes = FALSE, stop_on_nonid = TRUE)

Arguments

y

A character vector of variables of interest given the intervention.

x

A character vector of the variables that are acted upon.

D

A list of igraph objects describing the selection diagrams in the internal syntax.

expr

A logical value. If TRUE, a string is returned describing the expression in LaTeX syntax. Else, a list structure is returned which can be manually parsed by the function get.expression

simp

A logical value. If TRUE, a simplification procedure is applied to the resulting probability object. d-separation and the rules of do-calculus are applied repeatedly to simplify the expression.

steps

A logical value. If TRUE, returns a list where the first element corresponds to the expression of the transport formula and the second to the a list describing intermediary steps taken by the algorithm.

primes

A logical value. If TRUE, prime symbols are appended to summation variables to make them distinct from their other instantiations.

stop_on_nonid

A logical value. If TRUE, an error is produced when a non-identifiable effect is discovered. Otherwise recursion continues normally.

Value

If steps = FALSE, A character string or an object of class probability that describes the transport formula. Otherwise, a list as described in the arguments.

Author(s)

Santtu Tikka

References

Bareinboim E., Pearl J. 2013b Meta-Transportability of Causal Effects: A Formal Approach. Proceedings of the 16th International Conference on Artificial Intelligence and Statistics, 135–143.

See Also

parse.graphml, get.expression, transport, generalize

Examples

library(igraph)

# Selection diagram corresponding to the first source domain.
# We set simplify = FALSE to allow multiple edges.
d1 <- graph.formula(X -+ Z, W_1 -+ W_2, W_2 -+ Z, 
  W_3 -+ Z, X -+ W_3, W_2 -+ X, Z -+ Y, # Observed edges
  S_1 -+ X, S_2 -+ W_2, S_3 -+ W_3, S_4 -+ Y, # Edges related to selection variables
  X -+ W_3, W_3 -+ X, X -+ W_2, W_2 -+ X, X -+ W_1, 
  W_1 -+ X, W_1 -+ Z, Z -+ W_1, simplify = FALSE)

# Here the bidirected edges are set to be unobserved in the selection diagram d1.
# This is denoted by giving them a description attribute with the value "U".
# The first 7 edges are observed and the next 4 are related to the selection variables.
# The rest of the edges are unobserved.
d1 <- set.edge.attribute(d1, "description", 12:19, "U")

# The variables "S_1", "S_2", "S_3" and "S_4" are selection variables.
# This is denoted by giving them a description attribute with the value "S".
d1 <- set.vertex.attribute(d1, "description", 7:10, "S")

# Selection diagram corresponding to the second 
# source domain is constructed in a similar fashion.
d2 <- graph.formula(X -+ Z, W_1 -+ W_2, W_2 -+ Z, W_3 -+ Z, 
 X -+ W_3, W_2 -+ X, Z -+ Y, # Observed edges
 S_1 -+ X, S_2 -+ W_2, S_3 -+ W_1, 
 S_4 -+ Y, S_5 -+ Z, # Edges related to selection variables
 X -+ W_3, W_3 -+ X, X -+ W_2, W_2 -+ X, X -+ W_1, 
 W_1 -+ X, W_1 -+ Z, Z -+ W_1, simplify = FALSE)
d2 <- set.edge.attribute(d2, "description", 13:20, "U")
d2 <- set.vertex.attribute(d2, "description", 7:11, "S")

# We combine the diagrams as a list.
d.comb <- list(d1, d2)
meta.transport(y = "Y", x = "X", D = d.comb)