http://i.imgur.com/EOowdSD.png

Time-lagged conversion rates and cure models

Suppose in our population we have a subpopulation that will never experience the event of interest. Or, for some subjects the event will occur so far in the future that it’s essentially at time infinity. The survival function should not asymptically approach zero, but some positive value. Models that describe this are sometimes called cure models or time-lagged conversion models.

There’s a serious fault in using parametric models for these types of problems that non-parametric models don’t have. The most common parametric models like Weibull, Log-Normal, etc. all have strictly increasing cumulative hazard functions, which means the corresponding survival function will always converge to 0.

Let’s look at an example of this problem. Below I generated some data that has individuals who will not experience the event, not matter how long we wait.

[1]:
%matplotlib inline
%config InlineBackend.figure_format = 'retina'

from matplotlib import pyplot as plt
import autograd.numpy as np
from autograd.scipy.special import expit, logit
import pandas as pd
plt.style.use('seaborn-colorblind')

[2]:
N = 200
U = np.random.rand(N)
T = -(logit(-np.log(U) / 0.5) - np.random.exponential(2, N) - 6.00) / 0.50

E = ~np.isnan(T)
T[np.isnan(T)] = 50
[3]:
from lifelines import KaplanMeierFitter
kmf = KaplanMeierFitter().fit(T, E)
kmf.plot(figsize=(8,4))
plt.ylim(0, 1);
../_images/jupyter_notebooks_Modelling_time-lagged_conversion_rates_3_0.png

It should be clear that there is an asymptote at around 0.6. The non-parametric model will always show this. If this is true, then the cumulative hazard function should have a horizontal asymptote as well. Let’s use the Nelson-Aalen model to see this.

[4]:
from lifelines import NelsonAalenFitter

naf = NelsonAalenFitter().fit(T, E)
naf.plot(figsize=(8,4))
[4]:
<matplotlib.axes._subplots.AxesSubplot at 0x11b8b9b50>
../_images/jupyter_notebooks_Modelling_time-lagged_conversion_rates_5_1.png

However, when we try a parametric model, we will see that it won’t extrapolate very well. Let’s use the flexible two parameter LogLogisticFitter model.

[5]:
from lifelines import LogLogisticFitter

fig, ax = plt.subplots(nrows=2, ncols=2, figsize=(10, 6))

t = np.linspace(0, 40)
llf = LogLogisticFitter().fit(T, E, timeline=t)


llf.plot_survival_function(ax=ax[0][0])
kmf.plot(ax=ax[0][0])

llf.plot_cumulative_hazard(ax=ax[0][1])
naf.plot(ax=ax[0][1])

t = np.linspace(0, 100)
llf = LogLogisticFitter().fit(T, E, timeline=t)

llf.plot_survival_function(ax=ax[1][0])
kmf.plot(ax=ax[1][0])

llf.plot_cumulative_hazard(ax=ax[1][1])
naf.plot(ax=ax[1][1])
[5]:
<matplotlib.axes._subplots.AxesSubplot at 0x108136890>
../_images/jupyter_notebooks_Modelling_time-lagged_conversion_rates_7_1.png

The LogLogistic model does a quite terrible job of capturing the not only the asymptotes, but also the intermediate values as well. If we extended the survival function out further, we would see that it eventually nears 0.

Custom parametric models to handle asymptotes

Focusing on modeling the cumulative hazard function, what we would like is a function that increases up to a limit and then tapers off to an asymptote. We can think long and hard about these (I did), but there’s a family of functions that have this property that we are very familiar with: cumulative distribution functions! By their nature, they will asympotically approach 1. And, they are readily available in the SciPy and autograd libraries. So our new model of the cumulative hazard function is:

\[H(t; c, \theta) = c F(t; \theta)\]

where \(c\) is the (unknown) horizontal asymptote, and \(\theta\) is a vector of (unknown) parameters for the CDF, \(F\).

We can create a custom cumulative hazard model using ParametricUnivariateFitter (for a tutorial on how to create custom models, see this here). Let’s choose the Normal CDF for our model.

Remember we must use the imports from autograd for this, i.e. from autograd.scipy.stats import norm.

[6]:
from autograd.scipy.stats import norm
from lifelines.fitters import ParametricUnivariateFitter


class UpperAsymptoteFitter(ParametricUnivariateFitter):

    _fitted_parameter_names = ["c_", "mu_", "sigma_"]

    _bounds = ((0, None), (None, None), (0, None))

    def _cumulative_hazard(self, params, times):
        c, mu, sigma = params
        return c * norm.cdf((times - mu) / sigma, loc=0, scale=1)
[7]:
uaf = UpperAsymptoteFitter().fit(T, E)
uaf.print_summary(3)
uaf.plot(figsize=(8,4))
model lifelines.UpperAsymptoteFitter
number of observations 200
number of events observed 78
log-likelihood -357.204
hypothesis c_ != 1, mu_ != 0, sigma_ != 1
coef se(coef) coef lower 95% coef upper 95% z p -log2(p)
c_ 0.494 0.056 0.383 0.604 -8.973 <0.0005 61.592
mu_ 16.233 0.490 15.273 17.193 33.150 <0.0005 798.089
sigma_ 4.260 0.343 3.587 4.933 9.498 <0.0005 68.659
[7]:
<matplotlib.axes._subplots.AxesSubplot at 0x11e29a490>
../_images/jupyter_notebooks_Modelling_time-lagged_conversion_rates_10_2.png

We get a lovely asympotical cumulative hazard. The summary table suggests that the best value of \(c\) is 0.586. This can be translated into the survival function asymptote by \(\exp(-0.586) \approx 0.56\).

Let’s compare this fit to the non-parametric models.

[8]:
fig, ax = plt.subplots(nrows=2, ncols=2, figsize=(10, 6))

t = np.linspace(0, 40)
uaf = UpperAsymptoteFitter().fit(T, E, timeline=t)

uaf.plot_survival_function(ax=ax[0][0])
kmf.plot(ax=ax[0][0])

uaf.plot_cumulative_hazard(ax=ax[0][1])
naf.plot(ax=ax[0][1])

t = np.linspace(0, 100)
uaf = UpperAsymptoteFitter().fit(T, E, timeline=t)
uaf.plot_survival_function(ax=ax[1][0])
kmf.survival_function_.plot(ax=ax[1][0])

uaf.plot_cumulative_hazard(ax=ax[1][1])
naf.plot(ax=ax[1][1])
[8]:
<matplotlib.axes._subplots.AxesSubplot at 0x108307090>
../_images/jupyter_notebooks_Modelling_time-lagged_conversion_rates_12_1.png

I wasn’t expect this good of a fit. But there it is. This was some artificial data, but let’s try this technique on some real life data.

[9]:
from lifelines.datasets import load_leukemia, load_kidney_transplant

T, E = load_leukemia()['t'], load_leukemia()['status']
uaf.fit(T, E)
ax = uaf.plot_survival_function(figsize=(8,4))
uaf.print_summary()

kmf.fit(T, E).plot(ax=ax, ci_show=False)
print("---")
print("Estimated lower bound: {:.2f} ({:.2f}, {:.2f})".format(
        np.exp(-uaf.summary.loc['c_', 'coef']),
        np.exp(-uaf.summary.loc['c_', 'coef upper 95%']),
        np.exp(-uaf.summary.loc['c_', 'coef lower 95%']),
    )
)
model lifelines.UpperAsymptoteFitter
number of observations 42
number of events observed 30
log-likelihood -118.60
hypothesis c_ != 1, mu_ != 0, sigma_ != 1
coef se(coef) coef lower 95% coef upper 95% z p -log2(p)
c_ 1.63 0.36 0.94 2.33 1.78 0.07 3.75
mu_ 13.44 1.73 10.06 16.82 7.79 <0.005 47.07
sigma_ 7.03 1.07 4.94 9.12 5.65 <0.005 25.91
---
Estimated lower bound: 0.20 (0.10, 0.39)
../_images/jupyter_notebooks_Modelling_time-lagged_conversion_rates_14_2.png

So we might expect that about 20% will not have the even in this population (but make note of the large CI bounds too!)

[10]:
# Another, less obvious, dataset.

T, E = load_kidney_transplant()['time'], load_kidney_transplant()['death']
uaf.fit(T, E)
ax = uaf.plot_survival_function(figsize=(8,4))
uaf.print_summary()

kmf.fit(T, E).plot(ax=ax)
print("---")
print("Estimated lower bound: {:.2f} ({:.2f}, {:.2f})".format(
        np.exp(-uaf.summary.loc['c_', 'coef']),
        np.exp(-uaf.summary.loc['c_', 'coef upper 95%']),
        np.exp(-uaf.summary.loc['c_', 'coef lower 95%']),
    )
)
model lifelines.UpperAsymptoteFitter
number of observations 863
number of events observed 140
log-likelihood -1458.88
hypothesis c_ != 1, mu_ != 0, sigma_ != 1
coef se(coef) coef lower 95% coef upper 95% z p -log2(p)
c_ 0.29 0.03 0.24 0.35 -24.37 <0.005 433.49
mu_ 1139.88 101.60 940.75 1339.01 11.22 <0.005 94.62
sigma_ 872.59 66.31 742.62 1002.56 13.14 <0.005 128.67
---
Estimated lower bound: 0.75 (0.70, 0.79)
../_images/jupyter_notebooks_Modelling_time-lagged_conversion_rates_16_2.png

Using alternative functional forms

An even simplier model might look like \(c \left(1 - \frac{1}{\lambda t + 1}\right)\), however this model cannot handle any “inflection points” like our artificial dataset has above. However, it works well for this Lung dataset.

With all cure models, one important feature is the ability to extrapolate to unforseen times.

[11]:
from autograd.scipy.stats import norm
from lifelines.fitters import ParametricUnivariateFitter

class SimpleUpperAsymptoteFitter(ParametricUnivariateFitter):

    _fitted_parameter_names = ["c_", "lambda_"]

    _bounds = ((0, None), (0, None))

    def _cumulative_hazard(self, params, times):
        c, lambda_ = params
        return c * (1 -  1 /(lambda_ * times + 1))
[12]:
# Another, less obvious, dataset.

saf = SimpleUpperAsymptoteFitter().fit(T, E, timeline=np.arange(1, 10000))
ax = saf.plot_survival_function(figsize=(8,4))
saf.print_summary(4)

kmf.fit(T, E).plot(ax=ax)
print("---")
print("Estimated lower bound: {:.2f} ({:.2f}, {:.2f})".format(
        np.exp(-saf.summary.loc['c_', 'coef']),
        np.exp(-saf.summary.loc['c_', 'coef upper 95%']),
        np.exp(-saf.summary.loc['c_', 'coef lower 95%']),
    )
)
model lifelines.SimpleUpperAsymptoteFitter
number of observations 863
number of events observed 140
log-likelihood -1392.1610
hypothesis c_ != 1, lambda_ != 1
coef se(coef) coef lower 95% coef upper 95% z p -log2(p)
c_ 0.4252 0.0717 0.2847 0.5658 -8.0151 <5e-05 49.6912
lambda_ 0.0006 0.0002 0.0003 0.0009 -5982.3551 <5e-05 inf
---
Estimated lower bound: 0.65 (0.57, 0.75)
../_images/jupyter_notebooks_Modelling_time-lagged_conversion_rates_19_2.png