Chapter 04
Semiparametric Regression
Regression methods for relating composite outcomes to treatment and other covariates.
Chapter overview · Full exposition to follow
4.1 Proportional Win-Fractions Model
The proportional win-fractions (PW) model connects a covariate-specific win ratio to differences in covariates. It extends the two-sample comparison to regression while retaining the specified outcome hierarchy.
\[ \mathrm{WR}(t; Z_i, Z_j) = \exp\{\beta^{\mathsf T}(Z_i-Z_j)\} \]
The right-hand side does not depend on time. This proportionality assumption gives the coefficients their interpretation and motivates model checking.
4.2 Estimation and Diagnostics
The slides develop estimation and inference for the PW model, followed by cumulative score-residual methods for assessing model adequacy. Stratification allows separate baseline structures while retaining a common regression effect.
The HF-ACTION application illustrates preparation of covariates, fitting through WR::pwreg(), joint hypothesis tests, and residual plots. These steps are available together in the chapter’s original R code.
Specify the target before adjustment.
A conditional regression effect and a marginal treatment effect answer different questions. Chapter 5 returns to covariate adjustment for marginal estimands.
4.3 Variable Selection and Prediction
The final part of the chapter introduces regularized PW regression, using an elastic net penalty for variable selection and prediction. Cross-validation, variable importance, and assessment of concordance connect model fitting to predictive performance.
The accompanying WRNet material provides the software context. The full exposition of this topic will be developed as the book chapters take shape.