Proportional hazards model for interval-censored event times

Internal fitter behind survivalSub() when the outcome is interval-censored (Surv(L, R, type = "interval2")). coxph cannot fit such data, so the marginal model is a proportional hazards model \(H(t \mid x) = H_0(t) e^{x\beta}\) with a parametric baseline, fit by maximum likelihood. Each subject contributes \(S(L) - S(R)\) for \(L < T \le R\), \(S(L)\) when right-censored (\(R = \infty\)), \(1 - S(R)\) when left-censored, and \(h(T) S(T)\) for an exactly observed \(T\). Two baselines:

"spline" (default)

Royston–Parmar: \(\log H_0(t)\) is a natural cubic spline in \(\log t\) with df degrees of freedom (df - 1 internal knots at quantiles of the log interval endpoints, boundary knots at their extremes). df = 1 is a Weibull model.

"piecewise"

a piecewise-constant hazard on df pieces cut at quantiles of the interval endpoints.

The spline is the default because interval-censored data say little about the shape of the hazard within a visit interval, and that shape is exactly what imputeEventTime() draws \(T\) from: the piecewise-constant hazard is flat there, which in simulations biased the longitudinal sub-model fit by fitIntervalBJM() when the true hazard was increasing, while the smooth spline carries the trend across neighbouring intervals into each one.

The baseline cumulative hazard is then tabulated on a fine grid in the same hazard/time layout as basehaz, so the prediction code can use it exactly as it uses the Breslow estimate of a right-censored fit (see survival_accessors).

Usage

icphFit(
  formula,
  data,
  event_time,
  baseline = c("spline", "piecewise"),
  df = 3,
  lower_tail = c("spline", "weibull")
)

Arguments

formula

Surv(L, R, type = "interval2") ~ covariates. A strata() term gives each stratum its own baseline (spline knots or cut points from that stratum's interval endpoints), with the regression coefficients shared, as in a stratified Cox model.

data

One row per subject.

event_time

Name of the event-time column the longitudinal sub-model uses.

baseline

"spline" or "piecewise".

df

Spline degrees of freedom, or number of pieces.

lower_tail

Spline only: how \(\log H_0\) is extrapolated before the earliest interval endpoint (the lower boundary knot), where it never enters the likelihood. "spline" (default) continues the spline's own linear tail; "weibull" uses the slope of the nested Weibull fit (df = 1) instead. In simulations where a third to a half of the events fell before the first visit, "weibull" made fitIntervalBJM()'s estimates less variable but biased them when the true early hazard was not Weibull-shaped (Gompertz), so it is kept only for sensitivity analysis: the shape there is not identified by interval-censored data, and every choice is an assumption.

Value

An object of class "icph.BJM".