Deterministic Nodes: Interactions, tresholds and mathematical transformations

Achaz von Hardenberg

In standard Bayesian regression, variables are typically modeled as Stochastic Nodes: $$Y_i \sim \text{Normal}(\alpha + \beta X_i, \sigma)$$ This means $Y$ is correlated with $X$, but has its own independent “noise”.

However, in many causal models, some variables are logical consequences of others. For example: Interactions: Risk = Exposure * Toxicity Thresholds: IsAdult = (Age > 2) Transformations: LogSize = log(Size)

If you treat IsAdult as a stochastic node, the model might learn the correlation, but it treats the relationship as “soft”. Crucially, if Age is missing (NA), the model might impute Age = 5 but IsAdult = 0. This violates the laws of logic.

because solves this with Deterministic Nodes. These are variables defined by a hard equation (<-), ensuring they always stay perfectly synchronized with their parents, even during imputation and counterfactual interventions.

Setup

library(because)
set.seed(123)

Example 1: Interactions (A * B)

Standard R formulas (y ~ A * B) handle interactions automatically on complete data. However, if values of A or B are missing, and unless you use the standard procedure of eliminating rows with missing data (with a cosequent loss of power), imputation introduces a trap. If you follow the common workflow of “impute first, then model”, the imputation step usually assumes simple linear relationships and ignores the interaction. This often biases the interaction estimate towards zero.

With because, you just write the formula naturally. The model imputes missing values and estimates the interaction simultaneously, ensuring the interaction structure is preserved in the imputed data.

N <- 100

# Predictors
Rain <- rnorm(N)
Temp <- rnorm(N)

# Interaction: Growth depends on Rain * Temp
Growth <- 0.5 * (Rain * Temp) + rnorm(N, sd = 0.1)

# remove some random values in Rain and Temp
Rain[sample(1:N, 10)] <- NA
Temp[sample(1:N, 15)] <- NA


data <- data.frame(Growth = Growth, Rain = Rain, Temp = Temp)

# Note the formula: Growth ~ Rain * Temp
fit_int <- because(
    equations = list(
        Growth ~ Rain * Temp
    ),
    data = data,
    n.iter = 1000,
    quiet = TRUE
)
# The model creates a deterministic node for the interaction

# Notice the parameter name "beta_Growth_Rain_x_Temp"
summary(fit_int)

Example 2: Logic and Thresholds (Age > 2)

You can model “tipping points” or categorical definitions. For example, a marmot (Marmota marmota) is only “Mature” if it is older than 2 years.

# Predictors
Age <- runif(N, 0, 10)

# Deterministic threshold: 1 if Age > 2, else 0
IsMature <- as.numeric(Age > 2)

# Outcome: Mating success depends on Maturity
Mating <- 2 * IsMature + rnorm(N, sd = 0.5)

data_logic <- data.frame(Mating = Mating, Age = Age)

# Use I() to wrap the logic
fit_logic <- because(
    equations = list(
        Mating ~ I(Age > 3)
    ),
    data = data_logic,
    n.iter = 1000,
    quiet = TRUE
)

# The model recovers the effect of the threshold
# Parameter: "beta_Mating_Age_gt_2" where "gt" means "greater than"
summary(fit_logic)

Example 3: Multi-Class Definitions

What if you have multiple life stages? * Newborn: Age = 0 * Juvenile: Age 1 * Subadult: Age 2-4 * Adult: Age >= 5

You can define this using a sum of logical conditions:

# Define 4-level class: 1=Newborn, 2=Juv, 3=Sub, 4=Adult
# IMPORTANT: The class must be derived from Age so the model understands the structure.
Age <- round(runif(N, 0, 10), 0)

AgeClass <- 1 * (Age == 0) +
    2 * (Age > 0 & Age < 2) +
    3 * (Age >= 2 & Age < 5) +
    4 * (Age >= 5)

# Outcome: Social Rank increases with Life Stage
Rank <- 1.5 * AgeClass + rnorm(N, sd = 0.5)

data_multi <- data.frame(Rank = Rank, Age = Age, AgeClass = AgeClass)

# because() recovers the slope (~1.5) and confirms the causal path Age -> AgeClass -> Rank
fit_multi <- because(
    equations = list(
        # 1. Link AgeClass to Rank (Causal: Class -> Rank)
        Rank ~ AgeClass
    ),
    data = data_multi,
    n.iter = 1000,
    quiet = TRUE
)

# Check effect of AgeClass on Rank
summary(fit_multi)

Example 4: Mathematical transformations (log(A))

You can also use mathematical transformations directly in the formula.

Mass <- runif(N, 1, 100) # Ensure positive for log
Metabolism <- 0.75 * log(Mass) + rnorm(N, sd = 0.1)

data_math <- data.frame(Metabolism = Metabolism, Mass = Mass)

fit_math <- because(
    equations = list(
        Metabolism ~ log(Mass)
    ),
    data = data_math,
    n.iter = 1000,
    quiet = TRUE
)

# Parameter: "beta_Metabolism_log_Mass"
summary(fit_math)

When do I need I()?

You might notice we used I() for logic and thresholds but not for mathematical transformations. The rule is:

1. Standard Functions: You interpret things like log(x), sqrt(x), exp(x) directly as transformations. No I() needed.
   • y ~ log(x) (OK)
2. Interactions: Use * or :.
   • y ~ A * B (OK)
3. Complex Operators: If you use operators like >, <, +, - inside a term, R gets confused. Use I() to wrap them.
   • y ~ A + B (Means “A and B are predictors”)
   • y ~ I(A + B) (Means “The SUM of A and B is the predictor”)
   • y ~ I(A > 5) (Means “True/False is the predictor”)

Why “Deterministic” Matters (The Imputation Advantage)

You might ask: “Why not just create a column IsMature before loading the data?”

If your data is complete (no NAs), that works fine. But if you have missing data, pre-calculation is dangerous.

Scenario: The “Broken Logic” of Standard Imputation

Imagine you have a missing Age value. You want to impute it.

1. Standard Approach (e.g., MICE):
   • The imputer sees two separate columns: Age (NA) and IsMature (NA).
   • It guesses Age = 6 based on body size.
   • It independently guesses IsMature = 0 based on hormone levels.
   • Result: A marmot that is 6 years old but “Immature”. Impossible!
2. The because Approach:
   • IsMature is not a variable you impute; it is a formula.
   • In MCMC Step 1, the model guesses Age = 1. It automatically calculates IsMature = 0.
   • In MCMC Step 2, the model guesses Age = 6. It automatically calculates IsMature = 1.
   • Result: Every single sample in your posterior distribution is logically consistent. The “physics” of your model are preserved.

D-Separation with Deterministic Nodes

How does because treat interactions and deterministic nodes in d-separation tests?

1. Standard Interactions (A * B)

If you use a standard interaction in your formula:

equations = list(Y ~ A * B)

because follows standard DAG theory (Shipley/Pearl). The interaction describes the functional form, not the topology.

• The DAG contains edges: A -> Y and B -> Y.
• There is no separate “interaction node” (A_x_B).
• D-separation tests involving Y will condition on the parents: A and B.
• You do not need to condition on the interaction term itself. Once A and B are known, A*B provides no new causal information (it is collinear).

2. Explicit Deterministic Nodes (Compound ~ I(A * B))

If you explicitly define the interaction as a separate variable (a deterministic node):

equations = list(
    Compound ~ I(A * B),
    Y ~ Compound
)

because treats Compound as a distinct node in the DAG.

• The DAG contains edges: A -> Compound -> Y and B -> Compound -> Y.
• This allows you to test for Mediation. You can test if Y is independent of A conditional on Compound ($Y \perp A \mid Compound$).

Use Case: The “Blocking” Test

This “Explicit Node” approach is powerful for testing hypotheses about mechanisms. In Example 3 above, we defined AgeClass explicitly:

AgeClass ~ I(... Age ...)
Rank ~ AgeClass

Because AgeClass is a distinct node, because will test if Age is d-separated from Rank given AgeClass.

• Hypothesis: “Age only affects Rank through the Life Stage classification.”
• Test: I(Rank, Age | AgeClass)
• If this test passes, it confirms that AgeClass effectively blocks the information from Age, validating your structural assumption.

fit_multi_dsep <- because(
    equations = list(
        AgeClass ~ I(
            1 * (Age < 1) +
                2 * (Age >= 1 & Age < 2) +
                3 * (Age >= 2 & Age < 5) +
                4 * (Age >= 5)
        ),
        Rank ~ AgeClass
    ),
    data = data_multi,
    n.iter = 1000,
    dsep = TRUE,
    quiet = TRUE
)

summary(fit_multi_dsep)