Skip to content

Latest commit

 

History

History
193 lines (144 loc) · 6.02 KB

weibull_aft.myst.md

File metadata and controls

193 lines (144 loc) · 6.02 KB
jupytext kernelspec
text_representation
extension format_name format_version
.md
myst
0.13
display_name language name
pymc-examples
python
pymc-examples

(weibull_aft)=

Reparameterizing the Weibull Accelerated Failure Time Model

:::{post} January 17, 2023 :tags: censored, survival analysis, weibull :category: intermediate, how-to :author: Junpeng Lao, George Ho, Chris Fonnesbeck :::

import arviz as az
import numpy as np
import pymc as pm
import pytensor.tensor as pt
import statsmodels.api as sm

print(f"Running on PyMC v{pm.__version__}")

%config InlineBackend.figure_format = 'retina'
RANDOM_SEED = 8927
np.random.seed(RANDOM_SEED)
az.style.use("arviz-darkgrid")

Dataset

The {ref}previous example notebook on Bayesian parametric survival analysis <bayes_param_survival_pymc3> introduced two different accelerated failure time (AFT) models: Weibull and log-linear. In this notebook, we present three different parameterizations of the Weibull AFT model.

The data set we'll use is the flchain R data set, which comes from a medical study investigating the effect of serum free light chain (FLC) on lifespan. Read the full documentation of the data by running:

print(sm.datasets.get_rdataset(package='survival', dataname='flchain').__doc__).

# Fetch and clean data
data = (
    sm.datasets.get_rdataset(package="survival", dataname="flchain")
    .data.sample(500)  # Limit ourselves to 500 observations
    .reset_index(drop=True)
)

y = data.futime.values
censored = ~data["death"].values.astype(bool)

y[:5]

censored[:5]

Using pm.Potential

We have an unique problem when modelling censored data. Strictly speaking, we don't have any data for censored values: we only know the number of values that were censored. How can we include this information in our model?

One way do this is by making use of pm.Potential. The PyMC2 docs explain its usage very well. Essentially, declaring pm.Potential('x', logp) will add logp to the log-likelihood of the model.

+++

Parameterization 1

This parameterization is an intuitive, straightforward parameterization of the Weibull survival function. This is probably the first parameterization to come to one's mind.

def weibull_lccdf(x, alpha, beta):
    """Log complementary cdf of Weibull distribution."""
    return -((x / beta) ** alpha)

with pm.Model() as model_1:
    alpha_sd = 10.0

    mu = pm.Normal("mu", mu=0, sigma=100)
    alpha_raw = pm.Normal("a0", mu=0, sigma=0.1)
    alpha = pm.Deterministic("alpha", pt.exp(alpha_sd * alpha_raw))
    beta = pm.Deterministic("beta", pt.exp(mu / alpha))

    y_obs = pm.Weibull("y_obs", alpha=alpha, beta=beta, observed=y[~censored])
    y_cens = pm.Potential("y_cens", weibull_lccdf(y[censored], alpha, beta))

with model_1:
    # Change init to avoid divergences
    data_1 = pm.sample(target_accept=0.9, init="adapt_diag")

az.plot_trace(data_1, var_names=["alpha", "beta"])

az.summary(data_1, var_names=["alpha", "beta"], round_to=2)

Parameterization 2

Note that, confusingly, alpha is now called r, and alpha denotes a prior; we maintain this notation to stay faithful to the original implementation in Stan. In this parameterization, we still model the same parameters alpha (now r) and beta.

For more information, see this Stan example model and the corresponding documentation.

with pm.Model() as model_2:
    alpha = pm.Normal("alpha", mu=0, sigma=10)
    r = pm.Gamma("r", alpha=1, beta=0.001, initval=0.25)
    beta = pm.Deterministic("beta", pt.exp(-alpha / r))

    y_obs = pm.Weibull("y_obs", alpha=r, beta=beta, observed=y[~censored])
    y_cens = pm.Potential("y_cens", weibull_lccdf(y[censored], r, beta))

with model_2:
    # Increase target_accept to avoid divergences
    data_2 = pm.sample(target_accept=0.9)

az.plot_trace(data_2, var_names=["r", "beta"])

az.summary(data_2, var_names=["r", "beta"], round_to=2)

Parameterization 3

In this parameterization, we model the log-linear error distribution with a Gumbel distribution instead of modelling the survival function directly. For more information, see this blog post.

logtime = np.log(y)


def gumbel_sf(y, mu, sigma):
    """Gumbel survival function."""
    return 1.0 - pt.exp(-pt.exp(-(y - mu) / sigma))

with pm.Model() as model_3:
    s = pm.HalfNormal("s", tau=5.0)
    gamma = pm.Normal("gamma", mu=0, sigma=5)

    y_obs = pm.Gumbel("y_obs", mu=gamma, beta=s, observed=logtime[~censored])
    y_cens = pm.Potential("y_cens", gumbel_sf(y=logtime[censored], mu=gamma, sigma=s))

with model_3:
    # Change init to avoid divergences
    data_3 = pm.sample(init="adapt_diag")

az.plot_trace(data_3)

az.summary(data_3, round_to=2)

Authors

  • Originally collated by Junpeng Lao on Apr 21, 2018. See original code here.
  • Authored and ported to Jupyter notebook by George Ho on Jul 15, 2018.
  • Updated for compatibility with PyMC v5 by Chris Fonnesbeck on Jan 16, 2023.
%load_ext watermark
%watermark -n -u -v -iv -w

:::{include} ../page_footer.md :::