Total Effect

The Total Effect (TE) is estimated as the derivative of the Target Node with respect to a Driver Node under study.

$\displaystyle TE(X,Y) = \frac{\delta_Y}{\delta_X}$

where $X$ is the node of interest and $Y$ is the Target Node.

The Total Effect represents the change in the mean of the Target Node associated with — and not necessarily caused by — a small modification of the mean of a Driver Node. The Total Effect is the ratio of these two values.

Standardized Total Effect

The Standardized Total Effect (STE) represents the Total Effect multiplied by the ratio of the standard deviation of a Driver Node and the standard deviation of the Target Node.

$\displaystyle STE(X,Y) = \frac{\delta_Y}{\delta_X}\,\frac{\sigma_X}{\sigma_Y}$

Motivation for Calculating the Standardized Total Effect

• Standardizing the Total 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. Therefore, 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 $X$ leads to a 0.4-standard-deviation increase in $Y$.”

• This makes it possible to compare:

  • the effect of $X$ versus the effect of another variable $Z$, even if they are measured in completely different units
  • which predictors matter more in practice, not just statistically

• Without standardization, larger coefficients may simply reflect larger units, not stronger influence.