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In a linear structural equation model, each variable is a linear combination of its graphical parents and an exogenous error term. Edge coefficients quantify direct effects.

Take all components to be centered and let their marginal variances be $V_A,V_C,V_E$. A linear structural equation model iswhere $C_i,E_{i1},E_{i2}$ are mutually independent and the $E_{ij}$ are identically distributed. The causal directed acyclic graph has arrows

For monozygotic twins, set $A_{i1}=A_{i2}=G_i$ with $\mathrm{Var}(G_i)=V_A$; the genetic cause is completely shared. For dizygotic twins, one explicit construction iswhere $G_i,G_{i1},G_{i2}$ are independent with variance $V_A/2$. Then both additive genetic terms have variance $V_A$, while their covariance is $V_A/2$ and their correlation coefficient is $1/2$. In a fuller graph, the shared $G_i$ points to both genetic components and the unique $G_{ij}$ points only to its own component.

In a standardized linear structural equation model, attach to every directed edge its path coefficient. Wright path tracing rule says that a covariance is the sum, over admissible unblocked paths between the two variables, of the product of the coefficients along each path; an admissible path does not pass through a collider and does not enter and later leave the same variable in a way that reverses direction twice. The total causal effect of $X$ on $Y$ is the sum of the products of edge coefficients over all directed paths from $X$ to $Y$.

Sensitivity analysis for unmeasured confounding asks how an estimated causal effect changes under explicit departures from the assumed conditional exchangeability. It varies parameters describing the omitted variable's assignment imbalance and outcome effect, rather than assuming that measured covariate balance proves the omission harmless.

For a transparent additive linear structural equation model, take a pretreatment variable $U$ andAssume intervening on $A$ leaves $U$ unchanged, so the constant treatment effect in this model is $\tau$. Subtracting the two conditional outcome means givesThus the adjusted effect for a hypothesized parameter pair is $\tau =D_{\mathrm{obs}}-\gamma {\Delta }_U$. An observed difference five would fall to three when $\gamma =2$ and ${\Delta }_U=1$; explaining it entirely would require $\gamma {\Delta }_U=5$.

The product is a bias term derived for this particular additive model. Other outcome scales and interactions need different formulas. Plausible ranges can be compared with measured pretreatment predictors or external evidence, but the observed treatment-outcome table alone cannot determine the omitted parameters. A sensitivity result quantifies a model-dependent robustness claim; it does not establish that unmeasured confounding is absent.

Wright path tracing rule computes covariances in a linear structural equation model by adding contributions along the admissible paths of its graph. For an acyclic system $X=BX+\epsilon$ with centered disturbances and disturbance covariance matrix $\Omega$,Acyclicity makes $B$ nilpotent in a topological ordering, so $(I-B{)}^{-1}=I+B+\cdots +B^{p-1}$. Every matrix entry sums products of coefficients along directed paths. Substituting this finite expansion into the covariance formula gives the path-tracing construction. With uncorrelated disturbances, contributions join two directed paths at a common disturbance; correlated disturbances add the corresponding cross-disturbance terms.

For a worked example, let $X={\epsilon }_X$, $M=aX+{\epsilon }_M$ and $Y=bM+cX+{\epsilon }_Y$, with mutually uncorrelated centered disturbances. Substitution gives $Y=(ab+c)X+b{\epsilon }_M+{\epsilon }_Y$, henceThe direct path contributes $c$ and the mediated path contributes $ab$, each times the source variance. For standardized variables this becomes a sum of coefficient products for a correlation coefficient. Reading those coefficients as causal effects requires the structural intervention interpretation; an arbitrary covariance-fitting graph does not supply it by itself.