Parametric Estimation of the Immunes Proportion based on BCH Model and Exponential Distribution using Left Censored Data

MATERIALS AND METHODS

Suppose that T is a random variable with probability density function f (t; θ), θ to be estimated and t₁, t₂, þ, t_n is a random sample of size n. We are interesting in the likelihood function using the left censored data, because it gives us the possibility to compute the Maximum Likelihood Estimates (MLE) in order to fit a model for censored data. In order to analyze such data let α_i and c_i are indicators for the left censoring and cured, respectively where for the ith patient:

If α_i = 1, then c_i = 1 but if α_i = 0, then c_i is not observed and it can be either one or zero, assuming that censoring is independent of failure times.

In parametric maximum likelihood method the cumulative distribution function F (.) and the probability density function f (.) for the entire population are known. Thus, given α_i and c_i (i.e., the complete data are available), then the joint probability density function can be written as:

(4)

Consequently, the complete log likelihood function is:

(5)

where, f_u (t_i) and S_u (t_i) are the p.d.f and the survival function for the uncured patients, respectively.

This study considers the exponential distribution for S_u (t_i) and f_u (t_i) such that:

A datum t_i is said to be left-censored if the event occurs at a time before a left bound but it is unknown when it happens, for example, when the date of starting a cancer clinical trial is assigned but for a cancer patient we don’t know when the patient has been died. However, in case of left censoring the survival function of the uncured patients becomes S_u (t) = 1-e^-λτ.

Therefore, the log-likelihood function becomes:

(6)

The solutions of:

and:

are the desired estimates of θ and λ, where,

(7)

(8)

Solving Eq. 7 implies:

(9)

While Eq. 8 can be solved numerically since no explicit solution can be found.

As the cure status c_i is not fully observed, the Expectation Maximization (EM) algorithm will employ.

Before implementing the EM algorithm, let’s define g_i as the expected value of the ith patient to be uncured conditional on the current estimates of α_i and the survival function of uncured patients, S_u (t_i) (Peng and Dear, 2000):

(10)

For censored individuals α_i = 0 and hence the equation giving g_i can be re-written as follows:

For simplicity, let p_i to be the probability of cured patients such that p_i = E (1-c_t) = 1-g_i:

THE EM ALGORITHM

EM algorithm is an iterative optimization method which alternates between performing an Expectation (E) step which computes the expectation of the log-likelihood function using the current estimate for the latent variables and Maximization (M) step which computes parameters maximizing the expected log-likelihood found on the E step. These parameter-estimates are then used to determine the distribution of the latent variables in the next E step. The EM algorithm (and its faster variant ordered subset expectation maximization) is widely used in many different fields, especially in data clustering in machine learning, computer vision and medicine (Liu et al., 2006b; Safarinejadian et al., 2009).

However, suppose that the data vector is in the form of (t_i, α_i, c_i). For i = 1…n, the observed data is the lifetime (t_i) and censoring status (α_i = 1) for i = 1…n and also the cure status (c_i = 1), i = 1…m while the unobserved data is the cure status (c_i) for i = (m+1)…n, ∀m<n.

In the presence of unobserved data (c_i), only a function of the complete-data vector is observed. However, in the E-Step we find the expected value of the log likelihood function given by Eq. 6 as follows:

are the sufficient statistics for the parameters vector (λ, θ)^T.

It follows that the log-likelihood based on complete data is linear in complete data sufficient statistics and then the E-step requires the computation of :

and

Let:

(11)

(12)

(13)

For the M-Step we can use the complete data maximum likelihood estimates of (λ, θ) given by Eq. 8 and 9 and then substituting the expectations derived in the E-step for the complete data sufficient statistics, such that on grounds of the sufficient statistics, the maximum likelihood equation of θ implies:

(14)

While Eq. 8 could be re-rewritten as follows:

(15)

Thus, the E-step involves evaluating the sufficient statistics given by Eq. 11, 12 and 13 and also p_i using some initial values for the parameters (θ^o, λ^o) followed by M-step involves substituting these values in Eq. 14 and solving Eq. 15 numerically with respect to λ. The convergence t^th iteration is our desired estimates of θ and λ and eventually the desired cure fraction is exp (-θ^t+1).

CONCLUSION

We investigated the maximum likelihood estimation methodology for cure rate estimation based on the bounded cumulative hazard model when the exponential distribution can be used to represent the survival function of the uncured patients. A novel development of the EM algorithm was used to obtain maximum likelihood estimates when the data set has some left censoring observations.