The PHREG Procedure

Newton-Raphson Method

Let $L({\bbeta })$ be one of the likelihood functions described in the previous subsections. Let $l({\bbeta })=\textrm{log} L({\bbeta })$. Finding $\bbeta $ such that $L({\bbeta })$ is maximized is equivalent to finding the solution $\hat{\bbeta }$ to the likelihood equations

\[  \frac{\partial l(\bbeta ) }{\partial \bbeta } = 0  \]

With $\hat{\bbeta }^{0} = \Strong{0}$ as the initial solution, the iterative scheme is expressed as

\[  \hat{\bbeta }^{j+1}=\hat{\bbeta }^{j}-\left[ \frac{ \partial ^{2} l ( \hat{\bbeta }^{j} ) }{ \partial {\bbeta }^{2}} \right]^{-1} \frac{ \partial l ( \hat{\bbeta }^{j}) }{ \partial {\bbeta } }  \]

The term after the minus sign is the Newton-Raphson step. If the likelihood function evaluated at $\hat{\bbeta }^{j+1}$ is less than that evaluated at $\hat{\bbeta }^{j}$, then $\hat{\bbeta }^{j+1}$ is recomputed using half the step size. The iterative scheme continues until convergence is obtained—that is, until $\hat{\bbeta }_{j+1}$ is sufficiently close to $\hat{\bbeta }_ j$. Then the maximum likelihood estimate of $\bbeta $ is $\hat{\bbeta }=\hat{\bbeta }_{j+1}$.

The model-based variance estimate of $\hat{\bbeta }$ is obtained by inverting the information matrix $\mc {I}(\hat{\bbeta })$

\[  \hat{\mb {V}}_ m ( \hat{\bbeta })= \mc {I}^{-1}(\hat{\bbeta }) = -\left[ \frac{\partial ^{2} l ( \hat{\bbeta }) }{ \partial \bbeta ^2 } \right] ^{-1}  \]