delicatessen.estimating_equations.survival.ee_exponential_model

ee_exponential_model(theta, t, delta)

Estimating equation for an exponential model. Let \(T_i\) indicate the time of the event and \(C_i\) indicate the time to right censoring. Therefore, the observable data consists of \(t_i = \min(T_i, C_i)\) and \(\Delta_i = I(t_i = T_i)\). The estimating equation is

\[\sum_{i=1}^n \left\{ \frac{\Delta_i}{\lambda} - t_i \right\} = 0\]

Here, \(\theta\) is a single parameter that corresponds to the scale parameter for the exponential distribution. The hazard from the exponential model is parameterized as the following

\[h(t) = \lambda\]
Parameters

  • theta (ndarray, list, vector) – Theta in the case of the exponential model consists of a single value. Furthermore, the parameter will be non-negative. Therefore, an initial value like the [1, ] should be provided.

  • t (ndarray, list, vector) – 1-dimensional vector of n observed times.

  • delta (ndarray, list, vector) – 1-dimensional vector of n event indicators, where 1 indicates an event and 0 indicates right censoring.

Returns

Returns a 1-by-n NumPy array evaluated for the input theta

Return type

array

Examples

Construction of a estimating equation(s) with ee_exponential_model should be done similar to the following

>>> import numpy as np
>>> import pandas as pd
>>> from delicatessen import MEstimator
>>> from delicatessen.estimating_equations import ee_exponential_model

Some generic survival data to estimate an exponential survival model

>>> n = 100
>>> data = pd.DataFrame()
>>> data['C'] = np.random.weibull(a=1, size=n)
>>> data['C'] = np.where(data['C'] > 5, 5, data['C'])
>>> data['T'] = 0.8*np.random.weibull(a=1.0, size=n)
>>> data['delta'] = np.where(data['T'] < data['C'], 1, 0)
>>> data['t'] = np.where(data['delta'] == 1, data['T'], data['C'])

Defining psi, or the stacked estimating equations

>>> def psi(theta):
>>>         return ee_exponential_model(theta=theta,
>>>                                     t=data['t'], delta=data['delta'])

Calling the M-estimator

>>> estr = MEstimator(stacked_equations=psi, init=[1.])
>>> estr.estimate(solver='lm')

Inspecting the parameter estimates, variance, and confidence intervals

>>> estr.theta
>>> estr.variance
>>> estr.confidence_intervals()

Inspecting parameter the specific parameter estimates

>>> estr.theta[0]     # lambda (scale)

References

Collett D. (2015). Modelling survival data in medical research. CRC press.