Checking convergence of REML analyses

WOMBAT employs iterative schemes to locate the maximum of the (log) likelihood function. The program stops when a certain maximum number of iterates have been carried out or when it considers changes between iterates to be sufficiently small.

The default convergence criteria employed are fairly stringent. Hence, WOMBAT may stop after the maximum number of iterates allowed with the warning message that full convergence has not been achieved. The question then is whether this is really the case or whether this is overzealous and that – for practical purposes – the analysis has converged after all.

Alternatively, WOMBAT may not have reached the maximum number of iterations but progress (i.e. increase in log likelihood) is small and we want to decide whether to stop the run or to continue.

WOMBAT reports a number of statistics aimed at helping you to make this decision. Places to look are:

• The file Iterates written out during the analysis
• The file SumEstimates.out generated when WOMBAT stops.

Characteristics to look at (when using the AI algorithm) are:

1. The change in log likelihood between iterates:
   Inspect not only the last change (that's what WOMBAT uses), but the changes for the last few iterates. Are they consistently small, i.e. less than 0.01 or 0.001 ?
2. The norm of the vector of first derivatives (gradients):
   At the maximum of the likelihood function, we expect this to be close to zero. Note though that in some cases this value might not become close to zero for good reason, in particular with parameters at the boundary of the parameter space.
3. The so-called Newton decrement:
   This is a measure of the (estimated) proximity of the current point to the maximum log likelihood. It is calculated using both the first and second derivatives of the likelihood function. If this is close to zero, we are likely to be close to convergence. Alternatively, the likelihood surface may be very flat and little is to be gained by further iterations.

If all this is inconclusive, a continuation run – specifying a only few iterates and using another maximisation algorithm than the one used previously – is highly recommended !