Table 2.

Equations corresponding to treatment effect heterogeneity assessment methods

Risk modeling A multivariate regression model f that predicts the risk of an outcome y based on the predictors x1…, xp is identified or developed: $risk(x_1,\dots ,x_p)=E\{y|x_1,\dots x_p\}=f(\alpha +{\beta }_1{x}_1+\dots {\beta }_p{x}_p)(1)$ The expected outcome of a patient with measured predictors x1, …, xp receiving treatment T (where T = 1, when patient is treated and 0 otherwise) based on the linear predictor lp(x1, …xp) = a + β1x1 + …βpxp from a previously derived risk model can be described as: $E\{y|x_1,\dots ,x_p,T\}=f(lp+{\gamma }_{0}T+\gamma Tlp)(2)$ When the assumption of constant relative treatment effect across the entire risk distribution is made (risk magnification), equation (2) takes the form: $E\{y|x_1,\dots ,x_p,T\}=f(lp+{\gamma }_{0}T)(3)$ Treatment effect modeling The expected outcome of a patient with measured predictors x1, …, xp receiving treatment T can be derived from a model containing predictor main effects and potential treatment interaction terms: $E\{y|x_1,\dots ,x_p,T\}=f(\alpha +{\beta }_1{x}_1+\cdots +{\beta }_p{x}_p+{\gamma }_{0}T+{\gamma }_{1}T{x}_1+\cdots +{\gamma }_{p}T{x}_p)(4)$ Optimal treatment regime A treatment regime T(x1, …, xp) is a binary treatment assignment rule based on measured predictors. The optimal treatment regime maximizes the overall expected outcome across the entire target population: $T_{optimal}=argma{x}_{T}E\left\{E\{y|x_1,\dots x_p,T(x_1,\dots ,x_p)\}\right\}(5)$