Z-residuals for Standard Cox Models (Internal Worker) — Zresidual_coxph

Internal function to compute randomized Z-residuals for a standard Cox proportional hazards model (without frailty).

Usage

Zresidual_coxph_survival(fit_coxph, newdata, n.rep = 1, ...)

Arguments

fit_coxph: A fitted survival::coxph model object.
newdata: Optional data frame on which to compute the residuals. If NULL, the original model frame is used.
n.rep: Integer. Number of randomized residual replicates to generate. Default is 1.
...: Additional arguments (currently unused).

Value

A matrix containing the Z-residuals ($N \times nrep$) with diagnostic attributes: Survival.Prob, linear.pred, covariates, censored.status, object.model.frame, and type = "survival".

Details

The Z-residual for an observation $i$ is calculated as $$Z_i = -\Phi^{-1}(\hat{S}_i(t_i, \text{rand}))$$ where $\Phi^{-1}$ is the inverse standard normal CDF, and $\hat{S}_i(t_i, \text{rand})$ is the predicted survival probability at time $t_i$.

For uncensored observations ($t_i$ is an event time), $\hat{S}_i(t_i, \text{rand}) = \hat{S}_i(t_i)$. For censored observations, $\hat{S}_i(t_i, \text{rand}) = \hat{S}_i(t_i) \cdot U$, where $U \sim \text{Unif}(0, 1)$.

The predicted survival is calculated as $$\hat{S}_i(t_i) = \exp(-\exp(\mathbf{x}_i \mathbf{\hat{\beta}}) \hat{H}_0(t_i))$$.