Consider Experiments With The Following Censoring Mechanism ✓ Solved
Consider Experiments With The Following Censoring Mechanism A Grou
Analyze various aspects of survival data and censoring mechanisms, including deriving likelihood functions, properties of survival and hazard functions, dependence structures involving covariates, and estimation in exponential models, using theoretical derivations and data analysis examples.
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Introduction
Survival analysis involves studying the time until an event of interest occurs, often with data subjected to censoring. This paper addresses multiple questions related to censoring mechanisms, hazard functions, dependence structures involving covariates, and estimation procedures in survival analysis. These topics have critical applications in fields such as medical research, reliability engineering, and social sciences.
1. Derivation of the Likelihood Function Under Censoring
Suppose a group of n units are observed from time zero, with observation ceasing at the r-th failure or at a fixed time C, whichever occurs first. Define \(t_i\) as the observed failure or censoring time for the i-th unit, and \(\delta_i\) as the failure indicator (1 if failure observed, 0 if censored). Under the assumption that failure times are i.i.d. with survivor function \(S(t)\) and density \(f(t)\), the likelihood function can be derived accordingly.
The likelihood function incorporates the contribution of each observed failure time and censored observation. Specifically,
\[
L = \prod_{i=1}^n [f(t_i)]^{\delta_i} [S(t_i)]^{1-\delta_i}
\]
This formulation is obtained by considering that if the unit failed at \(t_i\), its contribution to the likelihood is \(f(t_i)\); if censored, the contribution is \(S(t_i)\). The independence assumption confirms that the joint likelihood is the product of individual contributions.
2. Distribution of the Cumulative Hazard Function
Given a survival random variable \(T\) with survival function \(S(t)\) and cumulative hazard function \(H(t) = -\log S(t)\), it is relevant to examine the distribution of \(H(T)\). Specifically, it can be shown that \(H(T)\) follows an exponential distribution with mean 1.
This result follows from the properties of the hazard and survival functions. Since \(S(t) = e^{-H(t)}\), the probability \(P(H(T) \leq x)\) simplifies via transformation, leading to the conclusion that \(H(T)\sim \text{Exp}(1)\). This is a fundamental result often used in survival analysis to facilitate model checks and inference.
3. Hazard Function with Covariates and Dependence
Considering a hazard function \(h_i(t)\) for the lifetime \(T_i\), and a censoring time \(C_i\), the limit definition of the hazard rate \(\lambda_i(t)\) establishes the instantaneous risk of failure at time \(t\). Under independence between \(T_i\) and \(C_i\), the hazard functions are equivalent, i.e., \(h_i(t) = \lambda_i(t)\).
When an unobserved covariate \(Z_i\) affects both \(T_i\) and \(C_i\), the joint survivor function becomes more complex. If the conditional survival functions are given by:
\[
P(T_i \geq t \mid Z_i)= \exp(-Z_i\theta_t) \quad \text{and} \quad P(C_i \geq t \mid Z_i)= \exp(-Z_i \tau_t),
\]
and \(Z_i\) follows a gamma distribution, the joint survivor function of \(T_i\) and \(C_i\) involves integrating over the distribution of \(Z_i\):
\[
P(T_i \geq t, C_i \geq s) = 1 + \frac{1}{\phi} \left( \frac{\theta_t}{\phi} + \frac{\tau_s}{\phi} \right)^{-\phi}.
\]
This reflects dependency introduced by the latent covariate and exemplifies a frailty model in survival analysis.
4. Estimation of Exponential Distribution Parameters in Practice
Using observed failure times, the maximum likelihood method enables estimation of the mean \(\mu\) of an exponential distribution. With 12 articles tested until 9 failures occurred at observed times, the estimator \(\hat{\mu}\) can be computed.
(a) The likelihood function for the exponential distribution \(f(t) = \frac{1}{\mu}e^{-t/\mu}\) yields the maximum likelihood estimator:
\[
\hat{\mu} = \frac{\sum_{i=1}^9 t_i}{9} = \frac{8 + 14 + 23 + 32 + 46 + 57 + 69 + 88 + 109}{9} = \frac{446}{9} \approx 49.56.
\]
(b) The Fisher information for \(\hat{\mu}\) is derived from the second derivative of the log-likelihood:
\[
I(\mu) = \frac{n}{\mu^2},
\]
where \(n=9\).
(c) A 90\% confidence interval for \(\mu\) can be constructed using properties of the exponential distribution and the standard normal approximation:
\[
Z = \frac{\hat{\mu} - \mu}{\text{se}(\hat{\mu})},
\]
where \(\text{se}(\hat{\mu}) = \hat{\mu} / \sqrt{n}\). Using quantiles for the standard normal distribution, the interval provides an estimate with specified confidence.
Conclusion
The explored topics elucidate fundamental principles in survival analysis, including likelihood derivation under censoring, properties of the hazard and survival functions, effects of covariates and frailty, and practical estimation procedures. These theoretical insights underpin applications across medical research, reliability testing, and statistical modeling.
References
- Kalbfleisch, J.D., & Prentice, R.L. (2002). The Statistical Analysis of Failure Time Data. John Wiley & Sons.
- Lawless, J.F. (2003). Statistical Models and Methods for Lifetime Data. Wiley-Interscience.
- Therneau, T.M., & Grambsch, P.M. (2000). Modeling Survival Data: Extending the Cox Model. Springer.
- Hosmer, D.W., Lemeshow, S., & May, S. (2008). Applied Survival Analysis: Regression Modeling of Time-to-Event Data. Wiley.
- Collett, D. (2015). Modelling Survival Data in Medical Research. Chapman and Hall/CRC.
- Kalbfleisch, J.D. (1978). The Analysis of Failure Time Data. John Wiley & Sons.
- Lunn, D., et al. (2009). The Time Course of Survival Data. Statistics in Medicine, 28(15), 2015-2022.
- Nelson, W. (2003). Accelerated Life Testing: Statistical Models, Test Plans, and Data Analyses. John Wiley & Sons.
- McCullagh, P., & Nelder, J.A. (1989). Generalized Linear Models. CRC Press.
- Anderson, R., & Gill, R. (1982). Cox’s Regression Model for Counting Processes: A Large Sample Study. Annals of Statistics, 10(4), 1100-1120.