Total Effect

The Total Effect of a node on the Target Node is the change in the mean of the Target Node that accompanies a small change in the mean of that node, with all other nodes in the network free to respond. It is BayesiaLab’s standard measure of how strongly a driver and the Target Node are related.

Its counterpart, the Direct Effect, holds the other nodes fixed and thereby isolates the direct relationship between a node and the Target Node.

Definition

TE_X = \frac{\delta_Y}{\delta_X}

where $X$ is the Driver Node and $Y$ is the Target Node. $\delta_X$ is a small change in the mean value of $X$ , and $\delta_Y$ is the inferred change in the mean value of $Y$ . The Total Effect is the ratio of the two.

How BayesiaLab Computes the Total Effect

BayesiaLab does not estimate an equation. It obtains the Total Effect by inference with the network:

The mean value of the Driver Node is moved across its range of values, using Minimum Cross-Entropy (MinXEnt) to set the corresponding soft evidence.

For each value of the Driver Node, the network infers the posterior mean of the Target Node.

The resulting x-y pairs form the Total Effect Curve of the Driver Node. Target Mean Analysis plots these curves.

The Total Effect is the derivative of the curve at the a priori mean of the Driver Node, i.e., at Delta Mean = 0.

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In this plot, Delta Mean = 0 is marked with a black dashed vertical line, indicating the mean value of all Driver Nodes. For the Driver Node $\mathit{Intensity}$ , the derivative at its mean value is shown as a cyan dashed tangent. The Total Effect of $\mathit{Intensity}$ is the slope of that tangent.

The plotted curves are not based on equations. No functions or parameters were estimated. The curves are purely the result of inference performed with the given network.

Because the Total Effect is a single number, it summarizes a curve well only where the curve is close to linear around the mean. In the plot above, the Total Effect describes the three near-linear drivers well, but not $\mathit{Intensity}$ , whose curve bends sharply. When in doubt, look at the curves with Target Mean Analysis before relying on the single-number summary.

Standardized Total Effect

The Standardized Total Effect is the Total Effect multiplied by the ratio of the standard deviation of the Driver Node to the standard deviation of the Target Node.

STE_X = \frac{\delta_Y}{\delta_X}\,\frac{\sigma_X}{\sigma_Y}

Why Standardize

Standardizing the effect of $X$ on $Y$ answers “how important is $X$ relative to typical variation,” not “what happens if I change $X$ by one unit.”
A unit change in $X$ often depends on an arbitrary scale. A unit effect answers a technically correct question, but not necessarily a meaningful one. Standardization removes this dependence on how the variable happened to be measured.
When $X$ is standardized, its effect on $Y$ is expressed in terms of typical variability, not raw units, e.g., “a one-standard-deviation increase in $X$ goes with a 0.4-standard-deviation increase in $Y$ .”
This makes it possible to compare the effect of $X$ with the effect of another node $Z$ , even if they are measured in completely different units, and to see which drivers matter more in practice, not just statistically.
Without standardization, larger effects may simply reflect larger units, not stronger influence.

In BayesiaLab’s reports, positive effects are shown in blue and negative effects in red.

Association or Cause

The Total Effect takes into account every open path between the Driver Node and the Target Node: direct and indirect, causal and non-causal. With a network learned from observational data and no adjustment, the Total Effect therefore describes an association: how the Target Node is expected to change when the Driver Node is observed to change. That is the right quantity for prediction, but it is not, in general, the causal effect.

The Total Effect becomes a causal estimate once the non-causal paths are blocked, for instance by placing the nodes in Intervention Mode (Graph Surgery) or by fixing the appropriate confounders (Likelihood Matching). It then measures the effect through all causal paths, direct and mediated. Chapter 10 of the e-book, Causal Effect Identification and Estimation, explains both adjustment methods. To isolate the direct path alone, use the Direct Effect.