ExponentialPH

Parametric proportional hazards with constant baseline hazard

from _api_doc_utils import *

1 Where it fits

Group: Survival / event-time models

ExponentialPH is the smallest fully parametric proportional-hazards model in the module. It assumes

\[ h_i(t \mid x_i) = \lambda_0 \exp(x_i'\beta), \]

so the baseline hazard is constant over time. Because the full hazard is identified, the class can expose the whole prediction stack: log hazard, hazard, cumulative hazard, and survival.

2 Likelihood and predictions

The model has constant baseline hazard

\[ h(t\mid x)=\lambda\exp(x'\beta), \qquad H(t\mid x)=\lambda t\exp(x'\beta), \qquad \lambda=\exp(\alpha). \]

For observed time \(t_i\) and event indicator \(d_i\), the implemented right-censored log-likelihood is

\[ \ell(\alpha,\beta) = \sum_i \left[ d_i(\alpha+x_i'\beta) -\exp(\alpha+x_i'\beta)t_i \right]. \]

Newton iterations use the analytic score and Hessian, subtract \(10^{-8}\) from the log-likelihood Hessian diagonal before solving, clamp each proposed parameter step to \([-2,2]\), and backtrack until log-likelihood does not decrease. The initial log baseline hazard is \(\log\{\sum_i d_i/\sum_i t_i\}\) and slopes start at zero. Convergence requires either \(\|\nabla\ell\|_\infty<\tau\) or an accepted likelihood change below \(\tau(1+|\ell|)\), where \(\tau\) is tolerance.

Prediction returns survival \(\exp\{-H(t\mid x)\}\) by default. The linear prediction is the full log hazard \(\alpha+x'\beta\), not only a relative-risk index.

3 Inference

The returned covariance is the inverse observed information

\[ \widehat V = \{-\nabla^2\ell(\hat\alpha,\hat\beta)\}^{-1}. \]

It is model-based under independent observations, correct exponential proportional hazards, and noninformative right censoring. There is no robust, clustered, bootstrap, or Wald interface. The covariance includes the log baseline hazard first and slope coefficients afterward, although the summary does not separately report standard errors. Reaching max_iterations raises ValueError before covariance or fitted state is stored. A successful summary exposes converged, iterations, termination_reason, and objective, where objective is \(-\ell(\hat\alpha,\hat\beta)\).

4 Performance and numerical behavior

Each score evaluation is \(O(np)\); forming the dense Hessian is \(O(np^2)\) and solving it is up to \(O(p^3)\) per Newton iteration. The implementation stores the dense design but no risk sets. Exponential evaluations are not clipped, so extreme covariates or steps can overflow. The class supports only positive finite stop times, binary events, no delayed entry, no ties issue beyond ordinary parametric likelihood, and no observation weights.

5 Python API

Constructor: cm.ExponentialPH

Use fit(x, time, event). The layered prediction surface is predict_lin(x) for log hazard, predict_hazard(x), predict_cumulative_hazard(x, time), and predict_survival(x, time). The default predict(x, time) returns survival probabilities. Times must be positive and finite, events must be binary, and x must be finite.

print(inspect.signature(cm.ExponentialPH))
()
cls = cm.ExponentialPH
display(HTML(html_table(["Public method"], public_methods(cls))))
Public method
fit(self, /, x, time, event, max_iterations=100, tolerance=1e-08)
predict(self, /, x, time)
predict_cumulative_hazard(self, /, x, time)
predict_hazard(self, /, x)
predict_lin(self, /, x)
predict_log_hazard(self, /, x)
predict_survival(self, /, x, time)
summary(self, /)
survival(self, /, x, time)

6 Minimal example

rng=np.random.default_rng(31)
x=rng.normal(size=(250,2)); rate=0.04*np.exp(x@np.array([0.5,-0.3])); t_event=rng.exponential(1.0/rate); c=rng.exponential(30,size=250)
time=np.minimum(t_event,c); event=(t_event<=c).astype(float)
model=cm.ExponentialPH(); model.fit(x,time,event)
fit=model.summary(); print({key: fit[key] for key in ['converged', 'iterations', 'termination_reason', 'objective']})
print(model.predict_lin(x[:3]))
print(model.predict_hazard(x[:3]))
print(model.predict_cumulative_hazard(x[:3], time[:3]))
print(model.predict(x[:3], time[:3]))
{'converged': True, 'iterations': 4, 'termination_reason': 'Relative objective tolerance reached', 'objective': 484.03927327282383}
[-3.63150107 -2.69355206 -2.87532579]
[0.02647641 0.06764025 0.05639776]
[0.14886781 0.2138672  1.47682181]
[0.86168302 0.80745561 0.22836231]

7 summary() contract

The table below is generated by fitting the live class in this repository and then inspecting summary(). Shapes are shown because most values are plain NumPy arrays or scalars.

rng=np.random.default_rng(131); x=rng.normal(size=(120,2)); rate=0.05*np.exp(x@np.array([0.4,-0.2])); te=rng.exponential(1.0/rate); c=rng.exponential(20,size=120); time=np.minimum(te,c); event=(te<=c).astype(float)
model=cm.ExponentialPH(); model.fit(x,time,event)
summary = model.summary()
display(HTML(html_table(["summary() key", "shape"], summary_shape_rows(summary))))
summary() key shape
log_baseline_hazard ()
baseline_hazard ()
coef (2,)
hazard_ratio (2,)
vcov (3, 3)
log_likelihood ()
converged ()
iterations ()
termination_reason ()
objective ()