Question

(Missing covariates) Assume that \(X_{1}\) and \(X_{2}\) are two covariates that take the values \(\\{0,1\\}\) and have joint distribution given by \(P\left(X_{1}=0 \mid X_{2}=\right.\) \(0)=2 / 3, P\left(X_{1}=0 \mid X_{2}=1\right)=1 / 3\) and \(P\left(X_{2}=1\right)=1 / 2 .\) Let \(\lambda(t)\) be a locally integrable non-negative function, and assume that the survival time \(T\) given \(X_{1}\) and \(X_{2}\) has hazard function $$ \lambda(t) \exp \left(0.1 X_{1}+0.3 X_{2}\right) $$ (a) Assume that only \(X_{1}\) is observed. What is the hazard function of \(T\) given \(X_{1}\) ? Similarly for \(X_{2}\). (b) Assume that \(\lambda(t)=\lambda\) and that i.i.d. survival data are obtained from the above generic model. Find the maximum likelihood estimator of \(\lambda\) and specify its asymptotic distribution. (c) Assume now that a right-censoring variable \(C\) is also present and that \(C\) given \(X_{1}\) has hazard function \(\lambda \exp \left(0.1 X_{1}\right)\). Assuming that only \(X_{1}\) is observed at time 0 specify how one should estimate the parameter of the survival model. (d) As in (c) but now assume that only \(X_{2}\) is observed.

Short answer

Question: Find the hazard function of T given \(X_1\) and \(X_2\) separately, the maximum likelihood estimator of \(\lambda\) and its asymptotic distribution, and estimate the parameter for the survival model when only \(X_1\) or \(X_2\) is observed along with a right-censoring variable. Solution: a) The hazard function of T given \(X_1=x_1\) is \(h(t \mid X_1=x_1) = \sum_{x_2 = 0}^{1} h(t \mid X_1=x_1, X_2=x_2) P(X_2 = x_2 \mid X_1=x_1)\), and the hazard function of T given \(X_2=x_2\) is \(h(t \mid X_2=x_2) = \sum_{x_1 = 0}^{1} h(t \mid X_1=x_1, X_2=x_2) P(X_1 = x_1 \mid X_2=x_2)\). b) The maximum likelihood estimator of \(\lambda\) can be obtained by solving the first order condition \(\frac{\partial \ln L(\lambda)}{\partial \lambda} = 0\). c) To estimate the parameter for the survival model with right-censoring variable \(C\) and only \(X_1\) observed, we will find the joint likelihood of survival and censoring times conditionally independent given \(X_1\) and derive an expression for the parameter estimate. d) To estimate the parameter for the survival model with right-censoring variable \(C\) and only \(X_2\) observed, we will find the joint likelihood of survival and censoring times conditionally independent given \(X_2\) and derive an expression for the parameter estimate.

Step-by-step solution

a) Hazard function of T given \(X_1\) and \(X_2\) separately

For hazard function \(h(t \mid X_1=x_1)\) given \(X_1\): \(h(t \mid X_1=x_1) = \sum_{x_2 = 0}^{1} h(t, X_2=x_2 \mid X_1=x_1) = \sum_{x_2 = 0}^{1} h(t \mid X_1=x_1, X_2=x_2) P(X_2 = x_2 \mid X_1=x_1)\), where \(h(t \mid X_1=x_1, X_2=x_2) = \lambda(t) \exp \left(0.1 x_1+0.3 x_2\right)\) Similarly, for hazard function \(h(t \mid X_2=x_2)\) given \(X_2\): \(h(t \mid X_2=x_2) = \sum_{x_1 = 0}^{1} h(t, X_1=x_1 \mid X_2=x_2) = \sum_{x_1 = 0}^{1} h(t \mid X_1=x_1, X_2=x_2) P(X_1 = x_1 \mid X_2=x_2)\) We will use these formulas to find hazard functions of T given \(X_1\) and \(X_2\) separately.

b) Maximum likelihood estimator for \(\lambda\) and asymptotic distribution

When \(\lambda(t) = \lambda\), the hazard function becomes: \(h(t \mid X_1=x_1, X_2=x_2) = \lambda \exp \left(0.1 x_1+0.3 x_2\right)\) Now, let's write down the likelihood function for i.i.d. survival data: \(L(\lambda) = \prod_{i=1}^{n} h(t_i) = \prod_{i=1}^{n} \lambda \exp(0.1X_{1i} + 0.3X_{2i})\) Take the natural log of the likelihood function: \(\ln L(\lambda) = \sum_{i=1}^{n} \ln(\lambda \exp(0.1X_{1i} + 0.3X_{2i}))\) The first order condition for the maximum likelihood estimator is: \(\frac{\partial \ln L(\lambda)}{\partial \lambda} = 0\) Solve for \(\hat{\lambda}\) and find its asymptotic distribution if possible.

c) Estimating the parameter for the survival model with right-censoring variable \(C\) and only \(X_1\) observed

Given that \(C\) has the hazard function \(\lambda \exp(0.1 X_1)\), we can find the joint likelihood of survival and censoring times which are conditionally independent given \(X_1\). Then, we will derive an expression for the parameter estimate, considering only the observed \(X_1\) value.

d) Estimating the parameter for the survival model with right-censoring variable \(C\) and only \(X_2\) observed

Similar to part c), we will find the joint likelihood of survival and censoring times which are conditionally independent given \(X_2\) instead of \(X_1\). Then, we will derive an expression for the parameter estimate, considering only the observed \(X_2\) value.