def survival_difference_at_fixed_point_in_time_test(
point_in_time,
durations_A,
durations_B,
event_observed_A=None,
event_observed_B=None,
**kwargs) -> StatisticalResult:
"""
Often analysts want to compare the survival-ness of groups at specific times, rather than comparing the entire survival curves against each other.
For example, analysts may be interested in 5-year survival. Statistically comparing the naive Kaplan-Meier points at a specific time
actually has reduced power (see [1]). By transforming the Kaplan-Meier curve, we can recover more power. This function uses
the log(-log) transformation.
Parameters
----------
point_in_time: float,
the point in time to analyze the survival curves at.
durations_A: iterable
a (n,) list-like of event durations (birth to death,...) for the first population.
durations_B: iterable
a (n,) list-like of event durations (birth to death,...) for the second population.
event_observed_A: iterable, optional
a (n,) list-like of censorship flags, (1 if observed, 0 if not), for the first population.
Default assumes all observed.
event_observed_B: iterable, optional
a (n,) list-like of censorship flags, (1 if observed, 0 if not), for the second population.
Default assumes all observed.
kwargs:
add keywords and meta-data to the experiment summary
Returns
-------
StatisticalResult
a StatisticalResult object with properties ``p_value``, ``summary``, ``test_statistic``, ``print_summary``
Examples
--------
.. code:: python
T1 = [1, 4, 10, 12, 12, 3, 5.4]
E1 = [1, 0, 1, 0, 1, 1, 1]
T2 = [4, 5, 7, 11, 14, 20, 8, 8]
E2 = [1, 1, 1, 1, 1, 1, 1, 1]
from lifelines.statistics import survival_difference_at_fixed_point_in_time_test
results = survival_difference_at_fixed_point_in_time_test(12, T1, T2, event_observed_A=E1, event_observed_B=E2)
results.print_summary()
print(results.p_value) # 0.893
print(results.test_statistic) # 0.017
Notes
-----
Other transformations are possible, but Klein et al. [1] showed that the log(-log(c)) transform has the most desirable
statistical properties.
References
-----------
[1] Klein, J. P., Logan, B. , Harhoff, M. and Andersen, P. K. (2007), Analyzing survival curves at a fixed point in time. Statist. Med., 26: 4505-4519. doi:10.1002/sim.2864
"""
kmfA = KaplanMeierFitter().fit(durations_A,
event_observed=event_observed_A)
kmfB = KaplanMeierFitter().fit(durations_B,
event_observed=event_observed_B)
sA_t = kmfA.predict(point_in_time)
sB_t = kmfB.predict(point_in_time)
# this is doing a prediction/interpolation between the kmf's index.
sigma_sqA = interpolate_at_times_and_return_pandas(kmfA._cumulative_sq_,
point_in_time)
sigma_sqB = interpolate_at_times_and_return_pandas(kmfB._cumulative_sq_,
point_in_time)
log = np.log
clog = lambda s: log(-log(s))
X = (clog(sA_t) - clog(sB_t))**2 / (sigma_sqA / log(sA_t)**2 +
sigma_sqB / log(sB_t)**2)
p_value = _chisq_test_p_value(X, 1)
return StatisticalResult(
p_value,
X,
null_distribution="chi squared",
degrees_of_freedom=1,
point_in_time=point_in_time,
test_name="survival_difference_at_fixed_point_in_time_test",
**kwargs)
def survival_difference_at_fixed_point_in_time_test(point_in_time, fitterA, fitterB, **result_kwargs) -> StatisticalResult:
"""
Often analysts want to compare the survival-ness of groups at specific times, rather than comparing the entire survival curves against each other.
For example, analysts may be interested in 5-year survival. Statistically comparing the naive Kaplan-Meier points at a specific time
actually has reduced power (see [1]). By transforming the survival function, we can recover more power. This function uses
the log(-log(·)) transformation.
Parameters
----------
point_in_time: float,
the point in time to analyze the survival curves at.
fitterA:
A lifelines univariate model fitted to the data. This can be a ``KaplanMeierFitter``, ``WeibullFitter``, etc.
fitterB:
the second lifelines model to compare against.
result_kwargs:
add keywords and meta-data to the experiment summary
Returns
-------
StatisticalResult
a StatisticalResult object with properties ``p_value``, ``summary``, ``test_statistic``, ``print_summary``
Examples
--------
.. code:: python
T1 = [1, 4, 10, 12, 12, 3, 5.4]
E1 = [1, 0, 1, 0, 1, 1, 1]
kmf1 = KaplanMeierFitter().fit(T1, E1)
T2 = [4, 5, 7, 11, 14, 20, 8, 8]
E2 = [1, 1, 1, 1, 1, 1, 1, 1]
kmf2 = KaplanMeierFitter().fit(T2, E2)
from lifelines.statistics import survival_difference_at_fixed_point_in_time_test
results = survival_difference_at_fixed_point_in_time_test(12.0, kmf1, kmf2)
results.print_summary()
print(results.p_value) # 0.77
print(results.test_statistic) # 0.09
Notes
-----
1. Other transformations are possible, but Klein et al. [1] showed that the log(-log(·)) transform has the most desirable
statistical properties.
2. The API of this function changed in v0.25.3. This new API allows for right, left and interval censoring models to be tested.
References
-----------
[1] Klein, J. P., Logan, B. , Harhoff, M. and Andersen, P. K. (2007), Analyzing survival curves at a fixed point in time. Statist. Med., 26: 4505-4519. doi:10.1002/sim.2864
"""
if type(fitterB) != type(fitterA):
warnings.warn(
"This test compares survival functions, but your fitters are estimating the survival functions differently. This means that this test is also testing the different ways to estimate the survival function and will be unreliable.",
UserWarning,
)
log = np.log
clog = lambda s: log(-log(s))
sA_t = fitterA.predict(point_in_time)
sB_t = fitterB.predict(point_in_time)
from lifelines.fitters import NonParametricUnivariateFitter, ParametricUnivariateFitter
if isinstance(fitterA, NonParametricUnivariateFitter):
sigma_sqA = interpolate_at_times_and_return_pandas(fitterA._cumulative_sq_, point_in_time)
elif isinstance(fitterA, ParametricUnivariateFitter):
sigma_sqA = fitterA._compute_variance_of_transform(fitterA._survival_function, [point_in_time]).squeeze()
if isinstance(fitterB, NonParametricUnivariateFitter):
sigma_sqB = interpolate_at_times_and_return_pandas(fitterB._cumulative_sq_, point_in_time)
elif isinstance(fitterB, ParametricUnivariateFitter):
sigma_sqB = fitterB._compute_variance_of_transform(fitterB._survival_function, [point_in_time]).squeeze()
X = (clog(sA_t) - clog(sB_t)) ** 2 / (sigma_sqA / log(sA_t) ** 2 + sigma_sqB / log(sB_t) ** 2)
p_value = _chisq_test_p_value(X, 1)
return StatisticalResult(
p_value,
X,
null_distribution="chi squared",
degrees_of_freedom=1,
point_in_time=point_in_time,
test_name="survival_difference_at_fixed_point_in_time_test",
fitterA=fitterA,
fitterB=fitterB,
**result_kwargs
)