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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 modeling
A multivariate regression model f that predicts the risk of an outcome y based on the predictors x1…, xp is identified or developed:
\( risk\left({x}_1,\dots, {x}_p\right)=E\left\{y|{x}_1,\dots {x}_p\right\}=f\left(\alpha +{\beta}_1{x}_1+\dots {\beta}_p{x}_p\right)\kern0.3em (1) \)
The expected outcome of a patient with measured predictors 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:
\( E\left\{y|{x}_1,\dots, {x}_p,T\right\}=f\left( lp+{\gamma}_0T+\gamma Tlp\right)\kern0.3em (2) \)
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 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:
\( E\left\{y|{x}_1,\dots, {x}_p,T\right\}=f\left(\alpha +{\beta}_1{x}_1+\cdots +{\beta}_p{x}_p+{\gamma}_0T+{\gamma}_1T{x}_1+\cdots +{\gamma}_pT{x}_p\right)\kern0.28em (4) \)
Optimal treatment 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:
\( {T}_{optimal}= argma{x}_T\kern0.28em E\left\{E\left\{y|{x}_1,\dots {x}_p,T\left({x}_1,\dots, {x}_p\right)\right\}\right\}\kern0.28em (5) \)