WOMBAT – A program for Mixed Model Analyses by Restricted Maximum Likelihood

• Q: When I run WOMBAT I sometimes get a message to the screen, saying either small, negative or invalid pivot.
What does it mean and what should or can I do about it?
• A: All the calculations in WOMBAT assume that the coefficient matrix in the mixed model equations for the model fitted is positive definite. In essence, the coefficient matrix is factored sequentially and the pivot' is the current diagonal element, adjusting for (or absorbing) all rows and columns already processed. If this quantity is close to zero, we suspect trouble and this can cause an analysis to fail - Hence the warning message. There are several reasons why this may occur and different actions to take.
1. If your model of analysis contains several, cross-classified fixed effects, we cannot estimate all these effects. WOMBAT automatically accounts for this by setting the solution for the first levels of all fixed effects other than the first to zero. However, there may be additional dependencies among your fixed effects, for example due to some confounding or if an effect is, in fact, nested within the levels of another effect fitted. While WOMBAT does try to find some of these additional dependencies, the mechanism used is not all that reliable and may fail due to numerical inaccuracies, especially for larger data sets.
Hence, if you get such warning or error message right from the start of the analysis, unidentified dependencies among the fixed effects fitted are most likely the cause. The remedy is to identify these (I know, that's easier said than done) and specify them in the parameter file, using the ZEROUT directive.
2. In other cases, this warning message will start to appear several iterates into the analysis, most commonly for multivariate or random regression analysis. Usually this is due to a covariance matrix to be estimated which is close to the bounds of the parameter space, i.e. has one or more eigenvalues close to zero. WOMBAT will attempt some damage control and if you are lucky, the analysis will converge anyway, though it may need a few extra iterates. If you ran an analysis where you expect high (absolute value) correlations among traits (or RR coefficients), specifying —-logdia as a run time option may be beneficial. In other cases the analysis fails to converge and you may need to change the maximization algorithm used (try run options —-emalg or —-simplex`) or your model and parameterization. For instance, if a covariance matrix pertains to several strongly correlated variables, estimating it at slightly reduced rather than full rank may help convergence and make the warning messages disappear.
3. In rare cases, invalid pivots reported may be negative (and not close to zero) and WOMBAT usually (but not always) simply stops. As a rule that is indicative of a serious problem with the data or model. For instance, this can occur if you think you have single records per trait and animal but, in reality, have duplicated records or animal codes. If no permanent environmental effects of the individual have been fitted, this can cause serious deviations of the coefficient matrix from its assumed structure and thus wreak havouc. Last but not least, this may be due to a bug in the program - if you have ruled out all other possible causes, please file a bug report.