Proportional-hazards time interaction (${\beta }_k(t)={\beta }_k+{\gamma }_{k}g(t)$)

A time interaction extends a constant-coefficient Cox proportional-hazards model by a covariate term $z_{k}g(t)$ and tests whether its coefficient is zero. The time function is evaluated at the current event/risk-set time, not each person's eventual outcome time. Scaled Schoenfeld residuals provide related graphical and score-test diagnostics for coefficient trends.