Adjust parameter matrix to allow confounding.
Arguments
- model
A
causal_model. A model object generated bymake_model.- confound
A
listof statements indicating pairs of nodes whose types are jointly distributed (e.g. list("A <-> B", "C <-> D")).
Details
Confounding between X and Y arises when the nodal types for X and Y are not independently distributed. In the X -> Y graph, for instance, there are 2 nodal types for X and 4 for Y. There are thus 8 joint nodal types:
| | t^X | | |
|-----|----|--------------------|--------------------|-----------|
| | | 0 | 1 | Sum |
|-----|----|--------------------|--------------------|-----------|
| t^Y | 00 | Pr(t^X=0 & t^Y=00) | Pr(t^X=1 & t^Y=00) | Pr(t^Y=00)|
| | 10 | . | . | . |
| | 01 | . | . | . |
| | 11 | . | . | . |
|-----|----|--------------------|--------------------|-----------|
| |Sum | Pr(t^X=0) | Pr(t^X=1) | 1 |
This table has 8 interior elements and so an unconstrained joint distribution would have 7 degrees of freedom. A no confounding assumption means that Pr(t^X | t^Y) = Pr(t^X), or Pr(t^X, t^Y) = Pr(t^X)Pr(t^Y). In this case there would be 3 degrees of freedom for Y and 1 for X, totaling 4 rather than 7.
set_confound lets you relax this assumption by increasing the
number of parameters characterizing the joint distribution. Using the fact
that P(A,B) = P(A)P(B|A) new parameters are introduced to capture P(B|A=a)
rather than simply P(B). For instance here two parameters
(and one degree of freedom) govern the distribution of types X and four
parameters (with 3 degrees of freedom) govern the types for Y given the
type of X for a total of 1+3+3 = 7 degrees of freedom.
See also
Other set:
set_prior_distribution(),
set_restrictions()
Examples
make_model('X -> Y; X <-> Y') |>
inspect("parameters")
#>
#> parameters
#> Model parameters with associated probabilities:
#>
#> X.0 X.1 Y.00_X.0 Y.10_X.0 Y.01_X.0 Y.11_X.0 Y.00_X.1 Y.10_X.1
#> 0.50 0.50 0.25 0.25 0.25 0.25 0.25 0.25
#> Y.01_X.1 Y.11_X.1
#> 0.25 0.25
make_model('X -> M -> Y; X <-> Y') |>
inspect("parameters")
#>
#> parameters
#> Model parameters with associated probabilities:
#>
#> X.0 X.1 M.00 M.10 M.01 M.11 Y.00_X.0 Y.10_X.0
#> 0.50 0.50 0.25 0.25 0.25 0.25 0.25 0.25
#> Y.01_X.0 Y.11_X.0 Y.00_X.1 Y.10_X.1 Y.01_X.1 Y.11_X.1
#> 0.25 0.25 0.25 0.25 0.25 0.25
model <- make_model('X -> M -> Y; X <-> Y; M <-> Y')
inspect(model, "parameters_df")
#>
#> parameters_df
#> Mapping of model parameters to nodal types:
#>
#> param_names: name of parameter
#> node: name of endogeneous node associated
#> with the parameter
#> gen: partial causal ordering of the
#> parameter's node
#> param_set: parameter groupings forming a simplex
#> given: if model has confounding gives
#> conditioning nodal type
#> param_value: parameter values
#> priors: hyperparameters of the prior
#> Dirichlet distribution
#>
#>
#> snippet (use grab() to access full 38 x 8 object):
#>
#> param_names node gen param_set nodal_type given param_value priors
#> 1 X.0 X 1 X 0 0.50 1
#> 2 X.1 X 1 X 1 0.50 1
#> 3 M.00 M 2 M 00 0.25 1
#> 4 M.10 M 2 M 10 0.25 1
#> 5 M.01 M 2 M 01 0.25 1
#> 6 M.11 M 2 M 11 0.25 1
#> 7 Y.00_M.00_X.0 Y 3 Y.M.00.X.0 00 M.00, X.0 0.25 1
#> 8 Y.10_M.00_X.0 Y 3 Y.M.00.X.0 10 M.00, X.0 0.25 1
#> 9 Y.01_M.00_X.0 Y 3 Y.M.00.X.0 01 M.00, X.0 0.25 1
#> 10 Y.11_M.00_X.0 Y 3 Y.M.00.X.0 11 M.00, X.0 0.25 1
# Example where set_confound is implemented after restrictions
make_model("A -> B -> C") |>
set_restrictions(increasing("A", "B")) |>
set_confound("B <-> C") |>
inspect("parameters")
#>
#> parameters
#> Model parameters with associated probabilities:
#>
#> A.0 A.1 B.00 B.10 B.11 C.00_B.00 C.10_B.00 C.01_B.00
#> 0.5000000 0.5000000 0.3333333 0.3333333 0.3333333 0.2500000 0.2500000 0.2500000
#> C.11_B.00 C.00_B.10 C.10_B.10 C.01_B.10 C.11_B.10 C.00_B.11 C.10_B.11 C.01_B.11
#> 0.2500000 0.2500000 0.2500000 0.2500000 0.2500000 0.2500000 0.2500000 0.2500000
#> C.11_B.11
#> 0.2500000
# Example where two parents are confounded
make_model('A -> B <- C; A <-> C') |>
set_parameters(node = "C", c(0.05, .95, .95, 0.05)) |>
make_data(n = 50) |>
cor()
#> Warning: Possible ambiguity: use additional arguments or check behavior in parameters_df.
#> A C B
#> A 1.00000000 -0.95431352 -0.06189845
#> C -0.95431352 1.00000000 0.02432316
#> B -0.06189845 0.02432316 1.00000000
# Example with two confounds, added sequentially
model <- make_model('A -> B -> C') |>
set_confound(list("A <-> B", "B <-> C"))
inspect(model, "statement")
#>
#> Causal statement:
#> A -> B; B -> C; B <-> A; C <-> B
# plot(model)