generalize {causaleffect}

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

Description

This function returns an expression for the transport formula of a causal effect between a target domain and multiple source domains with limited experiments. The formula is returned for the interventional distribution of the set of variables (y) given the intervention on the set of variables (x). Available experiments are depicted by a list (Z) where the first element describes the elements available at the target and the rest at the sources. The multiple domains are given as a list (D) where the first element is the underlying causal diagram without selection variables, and the rest correspond to the selection diagrams. 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

generalize(y, x, Z, D, expr = TRUE, simp = FALSE, 
  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.

Z

A list of character vectors describing the available interventions at each domain.

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. 2014 Transportability from Multiple Environments with Limited Experiments: Completeness Results. Proceedings of the 27th Annual Conference on Neural Information Processing Systems, 280–288.

See Also

aux.effect, causal.effect, get.expression, meta.transport, parse.graphml, recover, transport

Examples

library(igraph)

# Selection diagram corresponding to the target domain (no selection variables).
# We set simplify = FALSE to allow multiple edges.
d1 <-  graph.formula(Z_1 -+ X, Z_2 -+ X, X -+ Z_3, Z_3 -+ W,
 Z_3 -+ U, U -+ Y, W -+ U, Z_1 -+ Z_3, # Observed edges
 Z_1 -+ Z_2, Z_2 -+ Z_1, Z_1 -+ X, X -+ Z_1,
 Z_2 -+ Z_3, Z_3 -+ Z_2, Z_2 -+ U, U -+ Z_2,
 W -+ Y, Y -+ W, 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 8 edges are observed and the next 10 are unobserved.
d1 <- set.edge.attribute(d1, "description", 9:18, "U")

# We can use the causal diagram d1 to create selection diagrams 
# for two source domains, a and b.
d1a <- union(d1, graph.formula(S_1 -+ Z_2, S_2 -+ Z_3, S_3 -+ W))

# The variables "S_1", "S_2", and "S_3" are selection variables.
# This is denoted by giving them a description attribute with the value "S".
# The graph already has 7 vertices, so the last three depict the new ones.
d1a <- set.vertex.attribute(d1a, "description", 8:10, "S")

# Selection diagram corresponding to the second 
# source domain is constructed in a similar fashion.
d1b <- union(d1, graph.formula(S_1 -+ Z_1, S_2 -+ W, S_3 -+ U))
d1b <- set.vertex.attribute(d1b, "description", 8:10, "S")

# We combine the diagrams as a list.
d.comb <- list(d1, d1a, d1b)

# We still need the available experiments at each domain.
z <- list(c("Z_1"), c("Z_2"), c("Z_1"))
# This denotes that the variable "Z_1" is available for intervention 
# in both the target domain, and the second source domain. 
# The variable "Z_2" is available for intervention in the first source domain.

generalize(y = "Y", x = "X", Z = z, D = d.comb)