I consider using the lifelines package to fit a Cox-Proportional-Hazards-Model. I read that lifelines uses a nonparametric approach to fit the baseline hazard, which results in different baseline_hazards for some time points (see code example below). For my application, I need an
exponential distribution leading to a baseline hazard h0(t) = lambda which is constant across time.
So my question is: is it (in the meantime) possible to run a Cox-Proportional-Hazards-Model with an exponential distribution for the baseline hazard in lifelines or another Python package?
Example code:
from lifelines import CoxPHFitter
import pandas as pd
df = pd.DataFrame({'duration': [4, 6, 5, 5, 4, 6],
'event': [0, 0, 0, 1, 1, 1],
'cat': [0, 1, 0, 1, 0, 1]})
cph = CoxPHFitter()
cph.fit(df, duration_col='duration', event_col='event', show_progress=True)
cph.baseline_hazard_
gives
baseline hazard
T
4.0 0.160573
5.0 0.278119
6.0 0.658032
đź‘‹lifelines author here.
So, this model is not natively in lifelines, but you can easily implement it yourself (and maybe something I'll do for a future release). This idea relies on the intersection of proportional hazard models and AFT (accelerated failure time) models. In the cox-ph model with exponential hazard (i.e. constant baseline hazard), the hazard looks like:
h(t|x) = lambda_0(t) * exp(beta * x) = lambda_0 * exp(beta * x)
In the AFT specification for an exponential distribution, the hazard looks like:
h(t|x) = exp(-beta * x - beta_0) = exp(-beta * x) * exp(-beta_0) = exp(-beta * x) * lambda_0
Note the negative sign difference!
So instead of doing a CoxPH, we can do an Exponential AFT fit (and flip the signs if we want the same interpretation as the CoxPH). We can use the custom regession model syntax to do this:
from lifelines.fitters import ParametricRegressionFitter
from autograd import numpy as np
class ExponentialAFTFitter(ParametricRegressionFitter):
# this is necessary, and should always be a non-empty list of strings.
_fitted_parameter_names = ['lambda_']
def _cumulative_hazard(self, params, T, Xs):
# params is a dictionary that maps unknown parameters to a numpy vector.
# Xs is a dictionary that maps unknown parameters to a numpy 2d array
lambda_ = np.exp(np.dot(Xs['lambda_'], params['lambda_']))
return T / lambda_
Testing this,
from lifelines.datasets import load_rossi
from lifelines import CoxPHFitter
rossi = load_rossi()
rossi['intercept'] = 1
regressors = {'lambda_': rossi.columns}
eaf = ExponentialAFTFitter().fit(rossi, "week", "arrest", regressors=regressors)
eaf.print_summary()
"""
<lifelines.ExponentialAFTFitter: fitted with 432 observations, 318 censored>
event col = 'arrest'
number of subjects = 432
number of events = 114
log-likelihood = -686.37
time fit was run = 2019-06-27 15:13:18 UTC
---
coef exp(coef) se(coef) z p -log2(p) lower 0.95 upper 0.95
lambda_ fin 0.37 1.44 0.19 1.92 0.06 4.18 -0.01 0.74
age 0.06 1.06 0.02 2.55 0.01 6.52 0.01 0.10
race -0.30 0.74 0.31 -0.99 0.32 1.63 -0.91 0.30
wexp 0.15 1.16 0.21 0.69 0.49 1.03 -0.27 0.56
mar 0.43 1.53 0.38 1.12 0.26 1.93 -0.32 1.17
paro 0.08 1.09 0.20 0.42 0.67 0.57 -0.30 0.47
prio -0.09 0.92 0.03 -3.03 <0.005 8.65 -0.14 -0.03
_intercept 4.05 57.44 0.59 6.91 <0.005 37.61 2.90 5.20
_fixed _intercept 0.00 1.00 0.00 nan nan nan 0.00 0.00
---
"""
CoxPHFitter().fit(load_rossi(), 'week', 'arrest').print_summary()
"""
<lifelines.CoxPHFitter: fitted with 432 observations, 318 censored>
duration col = 'week'
event col = 'arrest'
number of subjects = 432
number of events = 114
partial log-likelihood = -658.75
time fit was run = 2019-06-27 15:17:41 UTC
---
coef exp(coef) se(coef) z p -log2(p) lower 0.95 upper 0.95
fin -0.38 0.68 0.19 -1.98 0.05 4.40 -0.75 -0.00
age -0.06 0.94 0.02 -2.61 0.01 6.79 -0.10 -0.01
race 0.31 1.37 0.31 1.02 0.31 1.70 -0.29 0.92
wexp -0.15 0.86 0.21 -0.71 0.48 1.06 -0.57 0.27
mar -0.43 0.65 0.38 -1.14 0.26 1.97 -1.18 0.31
paro -0.08 0.92 0.20 -0.43 0.66 0.59 -0.47 0.30
prio 0.09 1.10 0.03 3.19 <0.005 9.48 0.04 0.15
---
Concordance = 0.64
Log-likelihood ratio test = 33.27 on 7 df, -log2(p)=15.37
"""
Notice the sign change! So if you want the constant baseline hazard in the model, it's exp(-4.05).