TemporalHazard: Temporal Parametric Hazard Modeling

Native R implementation of the multiphase parametric hazard model of Blackstone, Naftel, and Turner (1986), with a focus on behavioral parity, transparent numerics, and reproducible validation against the original ‘C’/‘SAS’ HAZARD program. The package fits time-varying hazards as an additive sum of parametric phases — early, constant, and late risk streams — each carrying its own covariate effects.

The multiphase model

The total cumulative hazard decomposes additively across $J$ phases:

$$H(t \mid \mathbf{x}) = \sum_{j=1}^{J} \mu_j(\mathbf{x}) \, \Phi_j(t)$$

where $\mu_j(\mathbf{x}) = \exp(\alpha_j + \mathbf{x}_j^\top \beta_j)$ is the phase-specific log-linear scale (intercept plus covariate effects) and $\Phi_j(t)$ is the temporal shape contributed by phase $j$, with its own parameters depending on the phase type (see the phase vocabulary below). The instantaneous hazard and survival follow directly:

$$h(t \mid \mathbf{x}) = \sum_{j=1}^{J} \mu_j(\mathbf{x}) \, \varphi_j(t), \qquad S(t \mid \mathbf{x}) = \exp\!\bigl(-H(t \mid \mathbf{x})\bigr)$$

with $\varphi_j = d\Phi_j/dt$. Each phase is specified with hzr_phase() and the model is fit by maximum likelihood with hazard().

Phase vocabulary

Phase type	$\Phi_j(t)$	Domain	Use
`"cdf"`	$G(t)$	$[0, 1]$	Early risk that resolves over time
`"hazard"`	$-\log(1 - G(t))$	$[0, \infty)$	Late or aging risk that accumulates
`"g3"`	$G_3(t)$	$[0, \infty)$	Unbounded late risk (original C/SAS late phase)
`"constant"`	$t$	$[0, \infty)$	Flat background rate (no shape parameters)

Here $G(t)$ is the generalized temporal decomposition CDF computed by hzr_decompos(), and $G_3(t)$ is the unbounded late-phase intensity from hzr_decompos_g3(). See vignette("mf-mathematical-foundations") for the full derivation.

SAS/C HAZARD bridge

The classic three-phase HAZARD model maps directly onto hzr_phase() calls:

HAZARD phase	Role	R equivalent
G1 (early)	Early resolving risk	`hzr_phase("cdf", ...)`
G2 (constant)	Flat background rate	`hzr_phase("constant")`
G3 (late)	Rising late risk	`hzr_phase("g3", ...)`

TemporalHazard generalizes the fixed three-phase structure to $N$ phases of any type. hzr_argument_mapping() gives the full parameter translation table between the SAS/C parameterization and the R arguments.

Main entry points

Model fitting: hazard() — build and fit single- or multiphase models; hzr_phase() — specify one phase; hzr_stepwise() — forward, backward, or bidirectional covariate selection.
Prediction: predict() on a fitted hazard object — survival, cumulative hazard, and per-phase decomposed hazard; summary() — coefficient tables with Wald inference.
Parametric family: hzr_decompos() — the early-phase (G1) decomposition $G(t)$, $g(t)$, $h(t)$; hzr_decompos_g3() — the late-phase (G3) intensity; hzr_phase_cumhaz() and hzr_phase_hazard() — per-phase $\Phi(t)$ and $\varphi(t)$.
Diagnostics: hzr_kaplan(), hzr_nelson() — nonparametric references; hzr_gof() — goodness of fit; hzr_calibrate(), hzr_deciles() — calibration; hzr_bootstrap() — resampling CIs; hzr_competing_risks() — cumulative incidence.

Vignettes

vignette("getting-started"): First fit, end to end.
vignette("mf-mathematical-foundations"): The decomposition family and multiphase model, with derivations.
vignette("fitting-hazard-models"): Single-phase through multiphase fitting.
vignette("prediction-visualization"): Prediction types and decomposed-hazard plots.
vignette("inference-diagnostics"): Bootstrap CIs and diagnostics.
vignette("sas-to-r-migration"): Translating SAS/C HAZARD code.

References

Blackstone EH, Naftel DC, Turner ME Jr. The decomposition of time-varying hazard into phases, each incorporating a separate stream of concomitant information. J Am Stat Assoc. 1986;81(395):615–624. doi:10.1080/01621459.1986.10478314

Rajeswaran J, Blackstone EH, Ehrlinger J, Li L, Ishwaran H, Parides MK. Probability of atrial fibrillation after ablation: Using a parametric nonlinear temporal decomposition mixed effects model. Stat Methods Med Res. 2018;27(1):126–141. doi:10.1177/0962280215623583

Author

Maintainer: John Ehrlinger john.ehrlinger@gmail.com [copyright holder]

Authors:

John Ehrlinger john.ehrlinger@gmail.com [copyright holder]

Phase type	\(\Phi_j(t)\)	Domain	Use
`"cdf"`	\(G(t)\)	\([0, 1]\)	Early risk that resolves over time
`"hazard"`	\(-\log(1 - G(t))\)	\([0, \infty)\)	Late or aging risk that accumulates
`"g3"`	\(G_3(t)\)	\([0, \infty)\)	Unbounded late risk (original C/SAS late phase)
`"constant"`	\(t\)	\([0, \infty)\)	Flat background rate (no shape parameters)