Residualized psychological resilience score ($R_i=\hat{m}(S_i)-D_i$)

An exposure-conditioned adaptation measure. With distress $D_i$, exposure $S_i$ and predicted distress $\hat{m}(S_i)$, the displayed convention makes higher scores mean less distress than predicted. The result depends on exposure measurement, reference population and model. A regression residual can include measurement error and omitted causes rather than a distinct protective capacity.

For ordinary least squares $D=X\hat{\beta }+e$, the normal equation gives $X^{T}e=0$. Thus $R=-e$ is also orthogonal to the fitted predictors; with an intercept its sample mean is zero. A zero in-sample linear association between exposure and this score can therefore be an algebraic consequence of its construction, not evidence that exposure has no effect. Independent validation is needed before assigning substantive meaning to the regression residual.