Likelihood for Censored Data

Definition

Likelihood for right-censored data:

$L = \prod_{i = 1}^{n} [f (t_{i})]^{δ_{i}} [S (t_{i})]^{1 - δ_{i}}$

where $δ_{i}$ is the event indicator:

For left-truncated data with entry time $L_{i}$ :

$L = \prod_{i = 1}^{n} \frac{f ( t _{i} )}{S ( L _{i} )}$

NOTE

Subjects must survive until $L_{i}$ toenter the study. The likelihood uses conditional probabilities.

For subject entering at $L_{i}$ with event/censoring at $t_{i}$ :

$L_{i} = [\frac{f ( t _{i} )}{S ( L _{i} )}]^{δ_{i}} [\frac{S ( t _{i} )}{S ( L _{i} )}]^{1 - δ_{i}}$

Scenario: 5 patients in a clinical trial. 3 die at times 2, 5, 8; 2 are censored at times 3, 6.

Construction:

$L = f (2) \cdot f (5) \cdot f (8) \cdot S (3) \cdot S (6)$

For exponential model with hazard $λ$ :

$L = λ e^{- λ \cdot 2} \cdot λ e^{- λ \cdot 5} \cdot λ e^{- λ \cdot 8} \cdot e^{- λ \cdot 3} \cdot e^{- λ \cdot 6} = λ^{3} e^{- λ \cdot 24}$

Log-likelihood: $ln L = 3 ln λ - 24 λ$

MLE: $\frac{d}{d λ} ln L = \frac{3}{λ} - 24 = 0 \Rightarrow \hat{λ} = \frac{3}{24} = 0.125$

Scenario: Nursing home study. Subject enters at age 70, dies at age 85.

The subject must survive until age 70 to be observed. Contribution to likelihood:

$L_{i} = \frac{f ( 85 )}{S ( 70 )}$

This is the conditional density given survival until entry.