The HPLMIXED Procedure

Iteration History

The Iteration History table describes the optimization of the restricted log likelihood or log likelihood. The function to be minimized (the objective function) is $-2l$ for ML and $-2l_ R$ for REML; the column name of the objective function in the Iteration History table is -2 Log Like for ML and -2 Res Log Like for REML. The minimization is performed by using a ridge-stabilized Newton-Raphson algorithm, and the rows of this table describe the iterations that this algorithm takes in order to minimize the objective function.

The Evaluations column of the Iteration History table tells how many times the objective function is evaluated during each iteration.

The Criterion column of the Iteration History table is, by default, a relative Hessian convergence quantity given by

\[  \frac{ \mb {g} _{k}\mb {H} ^{-1}_{k}\mb {g} _{k} }{ |{f_ k}| }  \]

where $ f_ k $ is the value of the objective function at iteration k, $\mb {g} _ k$ is the gradient (first derivative) of $f_ k$, and $\mb {H} _ k$ is the Hessian (second derivative) of $f_ k$. If $\mb {H} _ k$ is singular, then PROC HPLMIXED uses the following relative quantity:

\[  \frac{ \mb {g} _{k}\mb {g} _{k} }{ |f_ k| }  \]

To prevent division by $|f_ k|$, specify the ABSGCONV option in the PROC HPLMIXED statement. To use a relative function or gradient criterion, specify the FCONV or GCONV option, respectively.

The Hessian criterion is considered superior to function and gradient criteria because it measures orthogonality rather than lack of progress (Bates et al., 1987). Provided that the initial estimate is feasible and the maximum number of iterations is not exceeded, the Newton-Raphson algorithm is considered to have converged when the criterion is less than the tolerance specified with the FCONV or GCONV option in the PROC HPLMIXED statement. The default tolerance is 1E–8. If convergence is not achieved, PROC HPLMIXED displays the estimates of the parameters at the last iteration.

A convergence criterion that is missing indicates that a boundary constraint has been dropped; it is usually not a cause for concern.