Problem: Maximum likelihood and hypothesis testing for parametric model (Exam 2021)
Eksempel
Exam June 2021, problem 7.
Consider the following parametric model for possibly right censored survival data. Assume we have \(n\) individuals and that we for individual number \(i\) observe \( (\widetilde{T}_i, D_i) \), where \(D_i\) equals one if we observe the survival time for this individual and equals zero if we observe a censoring time, and \(\widetilde{T}_i\) is the survival time for individual number \(i\) if \(D_i = 1\) and is otherwise the censoring time for this individual.
For each individual we have two (time invariant) covariates, \(x_{i1}\) and \(x_{i2}\) for individual number \(i\) where \(x_{i1} \in \{0,1\} \) and \(x_{i2}\in \mathbb{R}\). The hazard rate for the survival time for individual number \(i\) we assume to be given by
\[\alpha_i(t) = \nu\exp\{\beta_1 x_{i1}+\beta_2 x_{i2}\}, \]
where \(\nu\), \( \beta_1\) and \(\beta_2\) are parameters that we want to estimate.
Problems
a) Starting from the general formula for the likelihood function for counting process models given in (5.4) in our textbook,
\[L(\theta) = \left\{ \prod_{i = 1}^n \prod_{0<t\leq\tau} \lambda_i(t, \theta)^{\Delta N_i(t)} \right\}\exp\left\{ -\int_0^{\tau}\lambda_{\bullet}(t;\theta) dt \right\}, \]
derive a formula for the log-likelihood function for the survival data situation described above and show that it can be expressed as
\[\ell(\nu, \beta_1, \beta_2) = D_{\bullet} \ln \nu + \beta_1 D_{\bullet}^{(1)} + \beta_2 \sum_{i = 1}^n D_i x_{i2} -\nu \left( \sum_{i:x_{i1} = 0} e^{\beta_2x_{i2}}\widetilde T_i + e^{\beta_1}\sum_{i:x_{i1} = 1} e^{\beta_2x_{i2}}\widetilde T_i\right)\]
where
\[D_{\bullet} = \sum_{i=1}^n D_i\]
and
\[D_{\bullet}^{(1)} = \sum_{i:x_{i1} = 1} D_i.\]
b) If possible, find explicit formulas for the maximum likelihood estimators for \(\nu\), \(\beta_1\) and \(\beta_2\). If this is not possible, optimise analytically with respect the parameter(s) where this is possible and explain how the profile likelihood for the remaining parameter(s) can be found.
c) Assume that you have found values for the maximum likelihood estimates \(\hat\nu\), \(\hat\beta_1\) and \(\hat\beta_2\) by maximising the above log-likelihood. Now we want to test \(H_0: \beta_1 = 0\) against \(H_1: \beta_1 \neq 0\). Find a test statistic you can use to decide whether or not \(H_0\) should be rejected, i.e. explain what type of test you are using and develop the necessary formulas needed to perform the test for the model specified above.