Solution: Score functions and observed information matrix

Assuming that the counting processes \(N_i(t)\), for \(i = 1,2,\ldots,n\), have intensity processes of the form \(\lambda_i(t) = Y_i(t)\alpha_0(t)\exp(\beta^T x_i)\), where the \(Y_i(t)\) are at risk indicators and the \(x_i = (x_{i1},...,x_{ip})^T\) are fixed covariates, and with \(r(\beta ,x_i) = \exp(\beta^T x_i)\) we get that the partial likelihood in \((4.7)\) becomes: \[L(\beta) = \prod_{T_j} \frac{\exp(\beta^T x_{i_j})}{\sum_{l \in R_j}\exp(\beta^T x_l)},\]

where \(i_j\) is the index of the individual with event time \(T_j\) and \(R_j = \{l: T_l \geq T_j\}\) are the indices of individuals still at risk right before \(T_j\). Note that \(\beta^T x_l = \beta_1 x_{l1} + \ldots + \beta_k x_{lk} + \ldots + \beta_p x_{lp}\), so that \(\frac{\partial \beta^T x_l}{\partial \beta_k} = x_{lk}\).

To simplify some later calculations we denote \(\theta_l := \exp(\beta^T x_l)\), and have \(\frac{\partial \theta_l}{\partial \beta_k} = \frac{\partial \exp(\beta^T x_l)}{\partial \beta_k} = \exp(\beta^T x_l)\cdot\frac{\partial \beta^T x_l}{\partial \beta_k} = \theta_l x_{lk}\).

We will now derive an expression for the score vector, \(U(\beta) = \frac{\partial \log L(\beta)}{\partial \beta}\), for the partial likelihood. The \(k\)th element of this vector, with \(k \in \{1,2,\ldots,p\}\), is:

\begin{align*} U(\beta)_k &= \frac{\partial \log L(\beta)}{\partial \beta_k} \\ &= \frac{\partial}{\partial \beta_k} \log \left[ \prod_{T_j} \frac{\exp(\beta^T x_{i_j})}{\sum_{l \in R_j}\exp(\beta^T x_l)} \right] \\ &= \sum_{T_j} \left\{\frac{\partial}{\partial \beta_k}\beta^T x_{i_j} - \frac{\partial}{\partial \beta_k}\log \left[ \sum_{l \in R_j}\exp(\beta^T x_l)\right] \right\} \\ &= \sum_{T_j} \left\{x_{i_j k} - \frac{\partial}{\partial \beta_k}\log \left[ \sum_{l \in R_j}\theta_l\right] \right\} \\ &= \sum_{T_j} \left\{x_{i_j k} - \frac{\frac{\partial}{\partial \beta_k}\left[ \sum_{l \in R_j}\theta_l\right]}{\sum_{l \in R_j}\theta_l} \right\} \\ &= \sum_{T_j} \left\{x_{i_j k} - \frac{\sum_{l \in R_j}\frac{\partial}{\partial \beta_k}\theta_l}{\sum_{l \in R_j}\theta_l} \right\} \\ &= \sum_{T_j} \left\{x_{i_j k} - \frac{\sum_{l \in R_j}\theta_l x_{lk}}{\sum_{l \in R_j}\theta_l } \right\}.\end{align*}

The first term can be simplified very slightly since we sum over all event times, so that we get

\[\underline{\underline{U(\beta)_k = \sum_{i=1}^n x_{i k} - \sum_{T_j} \left\{\frac{\sum_{l \in R_j}\theta_l x_{lk}}{\sum_{l \in R_j}\theta_l } \right\}}}.\]

The observed information matrix \(I(\beta)\) is defined as minus the hessian of \(\log L(\beta)\), meaning it is a symmetric \(p \times p\)-matrix whose entry in the \(k\)th row and \(m\)th column is

\begin{align*} I(\beta)_{km} &= -\frac{\partial \log L(\beta)}{\partial \beta_k \partial \beta_m}\\ &= -\frac{\partial U(\beta)_k}{\partial \beta_m} \\ &= \frac{\partial}{\partial \beta_m} \sum_{T_j} \left\{\frac{\sum_{l \in R_j}\theta_l x_{lk}}{\sum_{l \in R_j}\theta_l } \right\} \\ &= \sum_{T_j} \left\{\frac{\partial}{\partial \beta_m} \frac{\sum_{l \in R_j}\theta_l x_{lk}}{\sum_{l \in R_j}\theta_l } \right\} \\ &= \sum_{T_j} \left\{\frac{\left[\frac{\partial}{\partial \beta_m} \sum_{l \in R_j}\theta_l x_{lk}\right]\cdot \left[\sum_{l \in R_j}\theta_l \right] - \left[ \sum_{l \in R_j}\theta_l x_{lk}\right]\cdot\left[\frac{\partial}{\partial \beta_m}\sum_{l \in R_j}\theta_l \right]}{\left[\sum_{l \in R_j}\theta_l \right]^2} \right\} \\ &= \sum_{T_j} \left\{\frac{\left[\frac{\partial}{\partial \beta_m} \sum_{l \in R_j}\theta_l x_{lk}\right]}{\left[\sum_{l \in R_j}\theta_l \right]} - \frac{\left[ \sum_{l \in R_j}\theta_l x_{lk}\right]\cdot\left[\frac{\partial}{\partial \beta_m}\sum_{l \in R_j}\theta_l \right]}{\left[\sum_{l \in R_j}\theta_l \right]^2} \right\} \\ &= \sum_{T_j} \left\{\frac{\left[\sum_{l \in R_j}\theta_l x_{lk} x_{lm}\right]}{\left[\sum_{l \in R_j}\theta_l \right]} - \frac{\left[ \sum_{l \in R_j}\theta_l x_{lk}\right]\cdot\left[\sum_{l \in R_j}\theta_l x_{lm}\right]}{\left[\sum_{l \in R_j}\theta_l \right]^2} \right\} .\end{align*}