Bayesian Parametric Survival Analysis#
import warnings
import arviz as az
import numpy as np
import pymc as pm
import pytensor.tensor as pt
import scipy as sp
import seaborn as sns
from matplotlib import pyplot as plt
from matplotlib.ticker import StrMethodFormatter
from statsmodels import datasets
print(f"Running on PyMC v{pm.__version__}")
Running on PyMC v5.16.2
%config InlineBackend.figure_format = 'retina'
az.style.use("arvizdarkgrid")
warnings.filterwarnings("ignore")
Survival analysis studies the distribution of the time between when a subject comes under observation and when that subject experiences an event of interest. One of the fundamental challenges of survival analysis (which also makes is mathematically interesting) is that, in general, not every subject will experience the event of interest before we conduct our analysis. In more concrete terms, if we are studying the time between cancer treatment and death (as we will in this post), we will often want to analyze our data before every subject has died. This phenomenon is called censoring and is fundamental to survival analysis.
This post illustrates a parametric approach to Bayesian survival analysis in PyMC. Parametric models of survival are simpler to both implement and understand than semiparametric models; statistically, they are also more powerful than non or semiparametric methods when they are correctly specified. For an example of a semiparametric Cox proportional hazards model, you can read this blogpost, but be aware that the post used and old version of PyMC and that Implementing a semiparametric model in PyMC involved some fairly complex numpy code and nonobvious probability theory equivalences.
We will analyze the mastectomy data from R
’s HSAUR
package.
sns.set()
blue, green, red, purple, gold, teal = sns.color_palette(n_colors=6)
pct_formatter = StrMethodFormatter("{x:.1%}")
df = datasets.get_rdataset("mastectomy", "HSAUR", cache=True).data.assign(
metastized=lambda df: 1.0 * (df.metastized == "yes"), event=lambda df: 1.0 * df.event
)
df.head()
time  event  metastized  

0  23  1.0  0.0 
1  47  1.0  0.0 
2  69  1.0  0.0 
3  70  0.0  0.0 
4  100  0.0  0.0 
The column time
represents the survival time for a breast cancer patient after a mastectomy, measured in months. The column event
indicates whether or not the observation is censored. If event
is one, the patient’s death was observed during the study; if event
is zero, the patient lived past the end of the study and their survival time is censored. The column metastized
indicates whether the cancer had metastized prior to the mastectomy. In this post, we will use Bayesian parametric survival regression to quantify the difference in survival times for patients whose cancer had and had not metastized.
Accelerated failure time models#
Accelerated failure time models are the most common type of parametric survival regression models. The fundamental quantity of survival analysis is the survival function; if \(T\) is the random variable representing the time to the event in question, the survival function is \(S(t) = P(T > t)\). Accelerated failure time models incorporate covariates \(\mathbf{x}\) into the survival function as
where \(S_0(t)\) is a fixed baseline survival function. These models are called “accelerated failure time” because, when \(\beta^{\top} \mathbf{x} > 0\), \(\exp\left(\beta^{\top} \mathbf{x}\right) \cdot t > t\), so the effect of the covariates is to accelerate the effective passage of time for the individual in question. The following plot illustrates this phenomenon using an exponential survival function.
S0 = sp.stats.expon.sf
fig, ax = plt.subplots(figsize=(8, 6))
t = np.linspace(0, 10, 100)
ax.plot(t, S0(5 * t), label=r"$\beta^{\top} \mathbf{x} = \log\ 5$")
ax.plot(t, S0(2 * t), label=r"$\beta^{\top} \mathbf{x} = \log\ 2$")
ax.plot(t, S0(t), label=r"$\beta^{\top} \mathbf{x} = 0$ ($S_0$)")
ax.plot(t, S0(0.5 * t), label=r"$\beta^{\top} \mathbf{x} = \log\ 2$")
ax.plot(t, S0(0.2 * t), label=r"$\beta^{\top} \mathbf{x} = \log\ 5$")
ax.set_xlim(0, 10)
ax.set_xlabel(r"$t$")
ax.yaxis.set_major_formatter(pct_formatter)
ax.set_ylim(0.025, 1)
ax.set_ylabel(r"Survival probability, $S(t\ \ \beta, \mathbf{x})$")
ax.legend(loc=1)
ax.set_title("Accelerated failure times");
Accelerated failure time models are equivalent to loglinear models for \(T\),
A choice of distribution for the error term \(\varepsilon\) determines baseline survival function, \(S_0\), of the accelerated failure time model. The following table shows the correspondence between the distribution of \(\varepsilon\) and \(S_0\) for several common accelerated failure time models.
Loglinear error distribution (\(\varepsilon\)) 
Baseline survival function (\(S_0\)) 

Extreme value (Gumbel) 

Accelerated failure time models are conventionally named after their baseline survival function, \(S_0\). The rest of this post will show how to implement Weibull and loglogistic survival regression models in PyMC using the mastectomy data.
Weibull survival regression#
In this example, the covariates are \(\mathbf{x}_i = \left(1\ x^{\textrm{met}}_i\right)^{\top}\), where
We construct the matrix of covariates \(\mathbf{X}\).
The likelihood of the data is specified in two parts, one for uncensored samples, and one for censored samples. Since \(Y = \eta + \varepsilon\), and \(\varepsilon \sim \textrm{Gumbel}(0, s)\), \(Y \sim \textrm{Gumbel}(\eta, s)\). For the uncensored survival times, the likelihood is implemented as
with weibull_model:
censored = pm.Data("censored", df.event.values == 0.0)
We transform the observed times to the log scale and standardize them.
We place independent, vague normal prior distributions on the regression coefficients,
with weibull_model:
beta = pm.Normal("beta", mu=0.0, sigma=5.0, shape=2)
The covariates, \(\mathbf{x}\), affect value of \(Y = \log T\) through \(\eta = \beta^{\top} \mathbf{x}\).
with weibull_model:
eta = beta.dot(predictors.T)
For Weibull regression, we use
with weibull_model:
s = pm.HalfNormal("s", 5.0)
with weibull_model:
events = pm.Gumbel("events", eta[~censored], s, observed=y_obs)
For censored observations, we only know that their true survival time exceeded the total time that they were under observation. This probability is given by the survival function of the Gumbel distribution,
This survival function is implemented below.
We now specify the likelihood for the censored observations.
with weibull_model:
censored_like = pm.Potential("censored_like", gumbel_sf(y_cens, eta[censored], s))
We now sample from the model.
SEED = 845199 # from random.org, for reproducibility
SAMPLE_KWARGS = {"chains": 4, "tune": 1000, "random_seed": [SEED + i for i in range(4)]}
with weibull_model:
weibull_trace = pm.sample(**SAMPLE_KWARGS)
Autoassigning NUTS sampler...
Initializing NUTS using jitter+adapt_diag...
Multiprocess sampling (4 chains in 4 jobs)
NUTS: [beta, s]
Sampling 4 chains for 1_000 tune and 1_000 draw iterations (4_000 + 4_000 draws total) took 1 seconds.
The energy plot and Bayesian fraction of missing information give no cause for concern about poor mixing in NUTS.
az.plot_energy(weibull_trace, fill_color=("C0", "C1"));
The \(\hat{R}\) statistics also indicate convergence.
<xarray.DataArray 'beta' ()> Size: 8B array(1.00442271)
Below we plot posterior distributions of the parameters.
az.plot_forest(weibull_trace, figsize=(10, 4));
These are somewhat interesting (especially the fact that the posterior of \(\beta_1\) is fairly wellseparated from zero), but the posterior predictive survival curves will be much more interpretable.
The advantage of using Data
variables is that we can now change their values to perform posterior predictive sampling. For posterior prediction, we set \(X\) to have two rows, one for a subject whose cancer had not metastized and one for a subject whose cancer had metastized. Since we want to predict actual survival times, none of the posterior predictive rows are censored.
with weibull_model:
pp_weibull_trace = pm.sample_posterior_predictive(weibull_trace)
Sampling: [events]
The posterior predictive survival times show that, on average, patients whose cancer had not metastized survived longer than those whose cancer had metastized.
t_plot = np.linspace(0, 230, 100)
weibull_pp_surv = np.greater_equal.outer(
np.exp(
y.mean()
+ y.std() * az.extract(pp_weibull_trace.posterior_predictive["events"])["events"].values
),
t_plot,
)
weibull_pp_surv_mean = weibull_pp_surv.mean(axis=1)
fig, ax = plt.subplots(figsize=(8, 6))
ax.plot(t_plot, weibull_pp_surv_mean[0], c=blue, label="Not metastized")
ax.plot(t_plot, weibull_pp_surv_mean[1], c=red, label="Metastized")
ax.set_xlim(0, 230)
ax.set_xlabel("Weeks since mastectomy")
ax.set_ylim(top=1)
ax.yaxis.set_major_formatter(pct_formatter)
ax.set_ylabel("Survival probability")
ax.legend(loc=1)
ax.set_title("Weibull survival regression model");
Loglogistic survival regression#
Other accelerated failure time models can be specified in a modular way by changing the prior distribution on \(\varepsilon\). A loglogistic model corresponds to a logistic prior on \(\varepsilon\). Most of the model specification is the same as for the Weibull model above.
with pm.Model() as log_logistic_model:
predictors = pm.Data("predictors", X)
censored = pm.Data("censored", df.event.values == 0.0)
y_obs = pm.Data("y_obs", y_std[df.event.values == 1.0])
y_cens = pm.Data("y_cens", y_std[df.event.values == 0.0])
beta = pm.Normal("beta", 0.0, 5.0, shape=2)
eta = beta.dot(predictors.T)
s = pm.HalfNormal("s", 5.0)
We use the prior \(\varepsilon \sim \textrm{Logistic}(0, s)\). The survival function of the logistic distribution is
so we get the likelihood
def logistic_sf(y, mu, s):
return 1.0  pm.math.sigmoid((y  mu) / s)
with log_logistic_model:
events = pm.Logistic("events", eta[~censored], s, observed=y_obs)
censored_like = pm.Potential("censored_like", logistic_sf(y_cens, eta[censored], s))
We now sample from the loglogistic model.
with log_logistic_model:
log_logistic_trace = pm.sample(**SAMPLE_KWARGS)
Autoassigning NUTS sampler...
Initializing NUTS using jitter+adapt_diag...
Multiprocess sampling (4 chains in 4 jobs)
NUTS: [beta, s]
Sampling 4 chains for 1_000 tune and 1_000 draw iterations (4_000 + 4_000 draws total) took 2 seconds.
All of the sampling diagnostics look good for this model.
az.plot_energy(log_logistic_trace, fill_color=("C0", "C1"));
<xarray.DataArray 'beta' ()> Size: 8B array(1.00301488)
Again, we calculate the posterior expected survival functions for this model.
with log_logistic_model:
pm.set_data(
{"predictors": X_pp, "censored": cens_pp, "y_obs": np.zeros(2), "y_cens": np.zeros(0)}
)
pp_log_logistic_trace = pm.sample_posterior_predictive(log_logistic_trace)
Sampling: [events]
log_logistic_pp_surv = np.greater_equal.outer(
np.exp(
y.mean()
+ y.std()
* az.extract(pp_log_logistic_trace.posterior_predictive["events"])["events"].values
),
t_plot,
)
log_logistic_pp_surv_mean = log_logistic_pp_surv.mean(axis=1)
fig, ax = plt.subplots(figsize=(8, 6))
ax.plot(t_plot, weibull_pp_surv_mean[0], c=blue, label="Weibull, not metastized")
ax.plot(t_plot, weibull_pp_surv_mean[1], c=red, label="Weibull, metastized")
ax.plot(t_plot, log_logistic_pp_surv_mean[0], "", c=blue, label="Loglogistic, not metastized")
ax.plot(t_plot, log_logistic_pp_surv_mean[1], "", c=red, label="Loglogistic, metastized")
ax.set_xlim(0, 230)
ax.set_xlabel("Weeks since mastectomy")
ax.set_ylim(top=1)
ax.yaxis.set_major_formatter(pct_formatter)
ax.set_ylabel("Survival probability")
ax.legend(loc=1)
ax.set_title("Weibull and loglogistic\nsurvival regression models");
This post has been a short introduction to implementing parametric survival regression models in PyMC with a fairly simple data set. The modular nature of probabilistic programming with PyMC should make it straightforward to generalize these techniques to more complex and interesting data set.