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# Table 2 Equations corresponding to treatment effect heterogeneity assessment methods

From: Predictive approaches to heterogeneous treatment effects: a scoping review

Risk modelingA multivariate regression model f that predicts the risk of an outcome y based on the predictors x_{1}…, x_{p} is identified or developed:\( risk\left({x}_1,\dots, {x}_p\right)=E\left\{y|{x}_1,\dots {x}_p\right\}=f\left(\alpha +{\beta}_1{x}_1+\dots {\beta}_p{x}_p\right)\kern0.3em (1) \) The expected outcome of a patient with measured predictors x_{1}, …, x_{p} receiving treatment T (where T = 1, when patient is treated and 0 otherwise) based on the linear predictor lp(x_{1}, …x_{p}) = a + β_{1}x_{1} + …β_{p}x_{p} from a previously derived risk model can be described as:\( E\left\{y|{x}_1,\dots, {x}_p,T\right\}=f\left( lp+{\gamma}_0T+\gamma Tlp\right)\kern0.3em (2) \) When the assumption of constant relative treatment effect across the entire risk distribution is made (risk magnification), equation (2) takes the form: \( E\left\{y|{x}_1,\dots, {x}_p,T\right\}=f\left( lp+{\gamma}_0T\right)\kern0.28em (3) \) Treatment effect modelingThe expected outcome of a patient with measured predictors x_{1}, …, x_{p} receiving treatment T can be derived from a model containing predictor main effects and potential treatment interaction terms:\( E\left\{y|{x}_1,\dots, {x}_p,T\right\}=f\left(\alpha +{\beta}_1{x}_1+\cdots +{\beta}_p{x}_p+{\gamma}_0T+{\gamma}_1T{x}_1+\cdots +{\gamma}_pT{x}_p\right)\kern0.28em (4) \) Optimal treatment regimeA treatment regime T(x_{1}, …, x_{p}) is a binary treatment assignment rule based on measured predictors. The optimal treatment regime maximizes the overall expected outcome across the entire target population:\( {T}_{optimal}= argma{x}_T\kern0.28em E\left\{E\left\{y|{x}_1,\dots {x}_p,T\left({x}_1,\dots, {x}_p\right)\right\}\right\}\kern0.28em (5) \) |