Alternatives to Cox: Accelerated Failure Time and Other Models

TL;DR

When Cox proportional hazards doesn't fit—because curves cross, effects change over time, or hazard ratios aren't interpretable—consider alternatives. Accelerated failure time (AFT) models express effects as time ratios ("treatment doubles survival time"). Parametric models (Weibull, log-normal) are efficient but require correct distributional assumptions. Restricted mean survival time (RMST) needs no assumptions but requires choosing a time horizon. This guide covers when and how to use each.

Why Look Beyond Cox?

Cox Regression Requires

1. Proportional hazards: HR constant over time
2. Interpretable hazard ratios: HR as the effect measure makes sense

When These Fail

• Curves cross: HR positive early, negative late (or vice versa)
• Effect wears off: Treatment helps initially, fades over time
• Time ratio is more intuitive: "Lives twice as long" vs "hazard halved"
• Prediction is goal: Cox doesn't easily predict survival times

Accelerated Failure Time (AFT) Models

The Concept

AFT models express covariate effects as speeding up or slowing down time:

$$\log(T) = \beta_0 + \beta_1 X_1 + ... + \sigma \epsilon$$

Interpretation: Covariates multiply survival time.

If $\beta_1 = 0.5$ (log scale), then $e^{0.5} = 1.65$:

• Treatment extends survival time by 65%
• Someone who would survive 100 days survives 165 days

Time Ratio vs. Hazard Ratio

These are related but not interchangeable. HR ≠ 1/Time Ratio except for specific distributions.

Common AFT Distributions

Code: AFT Models

from lifelines import WeibullAFTFitter, LogNormalAFTFitter, LogLogisticAFTFitter


def fit_aft_models(data, time_col, event_col, covariates):
    """
    Fit multiple AFT models and compare.
    """
    model_data = data[[time_col, event_col] + covariates]

    results = {}

    # Weibull AFT
    weibull = WeibullAFTFitter()
    weibull.fit(model_data, duration_col=time_col, event_col=event_col)
    results['weibull'] = {
        'model': weibull,
        'aic': weibull.AIC_,
        'time_ratios': np.exp(weibull.params_['mu_'])
    }

    # Log-normal AFT
    lognormal = LogNormalAFTFitter()
    lognormal.fit(model_data, duration_col=time_col, event_col=event_col)
    results['lognormal'] = {
        'model': lognormal,
        'aic': lognormal.AIC_,
        'time_ratios': np.exp(lognormal.params_['mu_'])
    }

    # Log-logistic AFT
    loglogistic = LogLogisticAFTFitter()
    loglogistic.fit(model_data, duration_col=time_col, event_col=event_col)
    results['loglogistic'] = {
        'model': loglogistic,
        'aic': loglogistic.AIC_,
        'time_ratios': np.exp(loglogistic.params_['alpha_'])
    }

    # Compare AIC
    comparison = pd.DataFrame({
        'Model': ['Weibull', 'Log-Normal', 'Log-Logistic'],
        'AIC': [results['weibull']['aic'],
                results['lognormal']['aic'],
                results['loglogistic']['aic']]
    }).sort_values('AIC')

    results['comparison'] = comparison

    return results

library(survival)
library(flexsurv)

# Weibull AFT
weibull_aft <- survreg(Surv(time, event) ~ treatment + age,
                        data = data, dist = "weibull")

# Log-normal AFT
lognormal_aft <- survreg(Surv(time, event) ~ treatment + age,
                          data = data, dist = "lognormal")

# Compare AIC
AIC(weibull_aft, lognormal_aft)

# Time ratio (acceleration factor)
exp(coef(weibull_aft))

Parametric Survival Models

The Approach

Specify a distribution for survival times, estimate parameters.

$$S(t) = f(\theta, t)$$

Where f is a known distributional form (Weibull, exponential, etc.) with parameters θ.

Advantages Over Cox

1. Efficiency: More precise estimates if distribution is correct
2. Prediction: Can predict survival times and probabilities
3. Extrapolation: Can extrapolate beyond observed data (with caution)
4. Interpretability: Parameters often have intuitive meaning

Disadvantages

1. Misspecification: Wrong distribution → biased results
2. Rigidity: Less flexible than non-parametric methods
3. Requires choice: Must select a distribution

Choosing a Distribution

Method 1: Visual comparison to Kaplan-Meier

from lifelines import KaplanMeierFitter, WeibullFitter

# Fit KM
kmf = KaplanMeierFitter()
kmf.fit(data['time'], data['event'])

# Fit parametric
wf = WeibullFitter()
wf.fit(data['time'], data['event'])

# Plot both
ax = kmf.plot_survival_function(label='Kaplan-Meier')
wf.plot_survival_function(ax=ax, label='Weibull')

Method 2: AIC/BIC comparison

Fit multiple distributions, compare information criteria.

Method 3: Hazard shape

• Constant hazard → Exponential
• Monotonic (increasing or decreasing) → Weibull
• Peak then decline → Log-logistic or log-normal

Restricted Mean Survival Time (RMST)

The Concept

RMST is the area under the survival curve up to a time horizon τ:

$$\text{RMST}(\tau) = \int_0^{\tau} S(t) dt$$

Interpretation: Expected survival time restricted to τ.

Advantages

1. No proportional hazards assumption
2. Works when curves cross
3. Interpretable: "X extra days of survival"
4. Non-parametric: No distributional assumptions

Disadvantages

1. Requires choosing τ: Results depend on time horizon
2. Less efficient: Doesn't use full distributional information
3. Not a full model: Doesn't model the whole survival curve

Code: RMST

from lifelines import restricted_mean_survival_time, KaplanMeierFitter


def compare_rmst(data, time_col, event_col, group_col, tau):
    """
    Compare RMST between groups.
    """
    groups = sorted(data[group_col].unique())
    results = {}

    for group in groups:
        mask = data[group_col] == group
        kmf = KaplanMeierFitter()
        kmf.fit(data.loc[mask, time_col], data.loc[mask, event_col])

        rmst = restricted_mean_survival_time(kmf, t=tau)
        results[group] = rmst

    diff = results[groups[1]] - results[groups[0]]

    return {
        'rmst_by_group': results,
        'difference': diff,
        'interpretation': f"Group {groups[1]} survives {diff:.1f} days longer on average (up to day {tau})"
    }

library(survRM2)

# Compare RMST between groups
rmst_result <- rmst2(
    data$time,
    data$event,
    data$treatment,
    tau = 180  # Time horizon
)

print(rmst_result)

Piecewise and Time-Varying Models

When Effects Change Over Time

If treatment helps early but not late (or vice versa), a single HR misleads.

Piecewise approach: Fit different HRs for different time periods.

from lifelines import CoxPHFitter
import pandas as pd

def piecewise_cox(data, time_col, event_col, covariate, cutpoints):
    """
    Fit piecewise Cox model with different HRs by period.
    """
    results = []
    prev_cut = 0

    for cut in cutpoints + [float('inf')]:
        # Filter to observations at risk in this period
        period_data = data[data[time_col] > prev_cut].copy()

        # Censor at end of period
        period_data['_time'] = np.minimum(period_data[time_col] - prev_cut, cut - prev_cut)
        period_data['_event'] = np.where(
            period_data[time_col] <= cut,
            period_data[event_col],
            0
        )

        if period_data['_event'].sum() > 5:  # Need events
            cph = CoxPHFitter()
            cph.fit(period_data[['_time', '_event', covariate]],
                    duration_col='_time', event_col='_event')

            hr = np.exp(cph.params_[covariate])
            results.append({
                'period': f'{prev_cut}-{cut}' if cut < float('inf') else f'{prev_cut}+',
                'HR': hr,
                'p_value': cph.summary['p'][covariate]
            })

        prev_cut = cut

    return pd.DataFrame(results)

Time-Varying Coefficients in Cox

Allow HR to change smoothly over time:

# Include interaction with log(time)
cph = CoxPHFitter()
cph.fit(data, duration_col='time', event_col='event',
        formula='treatment + treatment:np.log(time)')

Decision Framework: Which Model?

START: Survival data analysis
       ↓
QUESTION: Does proportional hazards hold?
├── Yes → Cox regression (standard)
└── No or Unsure → Continue
       ↓
QUESTION: What's more interpretable for your question?
├── Time ratios ("lives X times longer") → AFT model
├── Hazard still meaningful → Consider below
└── Neither → RMST
       ↓
QUESTION: Is parametric distribution appropriate?
├── Yes, and you know which → Parametric AFT/PH
├── Unsure → Compare distributions via AIC
└── No → Non-parametric alternatives
       ↓
QUESTION: Do effects change over time?
├── Yes, distinct periods → Piecewise model
├── Yes, gradual change → Time-varying coefficient
└── No → Single model
       ↓
QUESTION: What's your primary goal?
├── Prediction → Parametric models
├── Causal inference → Appropriate model + sensitivity
└── Description → RMST or KM comparison

Comparison Summary

Related Methods

• Time-to-Event and Retention Analysis (Pillar) - Full survival framework
• Cox Proportional Hazards - Standard approach
• Log-Rank Test - Non-parametric comparison
• Hazard Ratio Interpretation - Understanding HRs

Key Takeaway

Cox regression is the default for survival analysis, but it's not the only option. When proportional hazards fails, consider AFT models (intuitive time ratios), parametric models (efficient if correctly specified), RMST (no assumptions, interpretable as "days gained"), or piecewise models (when effects change over time). Choose based on what makes your results interpretable and your assumptions defensible.