FO vs FOCE vs LAPLACIAN

-----Original Message----- From: Nick Holford [mailto:n.holford@auckland.ac.nz] Sent: Monday, July 21, 2003 11:46 PM To: nmusers@globomaxnm.com Subject: Re: [NMusers] Re: FO vs FOCE vs LAPLACIAN Ken, I am primarily interested in avoiding local minima (so that I can test model building hypotheses) and obtaining minimally biased and imprecise parameter estimates. I agree with you that success or failure of $COV probably does not help diagnose a local minimum problem. I have no evidence to support this. But what about bias and imprecision? [Ken Kowalski] Actually, what I said was that a successful $COV does not guarantee that one has converged to a global minimum. However, a failure of $COV should raise a concern that we might not have converged to a global minimum. That is, conditions may be ripe for converging to a local minimum. I have encountered many instances (as I'm sure you have as well) where for two hierarchical models, the bigger model converges to a higher OFV than the smaller model (ie., delta-OFV is negative). In these instances the bigger model has clearly converged to a local minimum and often the $COV will fail for the bigger model. However, I have also seen instances where the $COV runs successfully for the bigger model but inspection of the output indicates large SEs for one or more parameters (as well as high pairwise correlations of the estimates) and this is the diagnostic information that leads one to be concerned that the bigger model is over-parameterized, leading to an ill-conditioning of the R-matrix (e.g., nearly singular) and instability in the estimation. The bigger model is unstable in the sense that if I change the starting values, I may converge to a different set of final estimates. When the likelihood surface for the bigger model is very flat, there may be many sets of solutions leading to very similar values of the minimum OFV. In this setting we may get a minimimization failure, or the $COV may fail due to a non-singular R-matrix, or if $COV does run some of the parameters will have very large SEs. In this sense the $COV does provide information on the precision or lack thereof (imprecision) in the estimation. One can't use the $COV to assess bias if the true model is unknown, however, if the model is unstable we should be concerned that we might have biased estimates. I have a somewhat anecdotal but nevertheless evidence based comment on this. I recently completed a PK model analysis using WT, AGE, SCR and SEX as covariates (697 subjects, 2567 concs). The model did not run $COV, in fact it didn't even minimize successfully. Other evidence convinced me it was not far away from an appropriate minimum and because it had a more biologically sound basis than its more successful neighbours I preferred this model. I bootstrapped the original data set using the preferred model and found 28% of 1055 bootstrap runs minimized successfully and 7.1% ran the $COV step. The mean of the parameters obtained from all bootstrap runs and the mean from those which ran the $COV step were all within 2%. I conclude that $COV does not indicate lower bias compared with runs that do not minimize. [Ken Kowalski] It concerns me that such a low proportion minimized successfully. What happens if you substantially change your starting values? You might find than one or more of the bootstrap parameter means are substantially different. We can't assess bias from bootstrapping a real data set where the true model for the data is unknown. Just because the $COV step ran that does not mean your model is not over-parameterized. Those runs where the $COV was successful may be numerically nonsingular but still nearly singular (output from $COV can help assess this). Thus, it doesn't surprise me that the mean estimates aren't different between those where the $COV was successful and all the runs. The value of the $COV goes well beyond a simple success/failure flag. When the $COV is successful we still need to inspect the output from the $COV to assess the stability. When the $COV fails we don't have this luxury but we get warning messages regarding invertability problems with the hessian that indicate we have a stability problem. Again, if we have a stability problem we may continue to proceed with the over-parameterized model but we should do so cautiously. To assess imprecision I computed the ratio of the mean standard error from the $COV successful runs to the bootstrap standard error obtained from all runs. For THETA:se estimates the $COV SE was on average 3% smaller but for OMEGA:se the $COV SE was 58% larger than the overall bootstrap SE. I conclude from this that the imprecision of THETA:se was negligibly different when the $COV step was successful. The difference in the OMEGA :se may reflect the intrinsic difficulty in obtaining estimates of OMEGA and OMEGA:se. Perhaps the asymptotic assumptions involved in $COV produce an upward bias. [Ken Kowalski] I believe Mats Karlsson has shown that asymptotic SEs for elements of Omega are not very good. 95% confidence intervals obtained from all the bootstrap runs were very similar to those obtained from minimization successful and $COV successful runs. The 95% CI predicted from the asymptotic SE was on average 21% larger (range 15-35%) than the bootstrap CI. In order to explore the issue a bit further I simulated a data set using the mean bootstrap parameter estimates from all runs. I then bootstrapped this simulated data set (1772 runs). The minimization success rate was double (56%) that of the original real data bootstrap runs and 12.5% ran $COV. Because the true parameter values for the simulation are known the absolute bias can be computed. Only 3 out of 29 parameters had an absolute bias larger than 10%. There were negligible differences between the absolute bias using estimates from all runs, minimization successful runs or $COV successful runs. [Ken Kowalski] The fact that you get 44% minimization failures and 87.5% $COV failures when you bootstrapped your simulated data set, based on the model developed from your original data set, provides evidence that your model is unstable. This is what I claim the $COV step failure from your original model fit was diagnosing. You indicate that 3 out of 29 parameters had an absolute bias greater than 10%. Perhaps the instability in your model is related to the estimates of these parameters. If they are not important parameters then perhaps I wouldn't be concerned. To illustrate my point consider the simple example where we have very little sampling in the absorption phase. Perhaps in each individual the observed Tmax corresponds to the first observation. In this setting we can have convergence and/or $COV failures and wildly biased estimates for ka (e.g., an estimate of ka >>100 hr^-1). Fortunately, we may find that even though ka is not well estimated we can still get relatively accurate estimates of CL/F which we may be more interested in. Thus, the instability in the model is related to the estimation of ka and the limitations of the design at early time points. I would be inclined to fix ka at some prior estimate (if I had one) to remove the instability and obtain successful minimization and $COV and acknowledge the limitations of the design/model to estimate ka. Alternatively, we could use simulation/bootstrapping to verify that poor (biased) estimation of ka is not likely to unduly bias CL/F. Further evaluation perhaps using simulation and bootstrapping is a cautious way to proceed when considering over-parameterized models. For me, I like to fit alternative models (often reduced hierarchical models) as a set of diagnostic runs and inspect the $COV output so that I can understand where the limitations are with the design/data/model. Both approaches can help us stay out of trouble. This means the $COV step is not a guide to reduced bias. [Ken Kowalski] The $COV should not be used as a guide to reduce bias. One can fit a mispecified reduced (smaller) model that is quite stable with a successful $COV and get biased estimates just as one can fit the true model and get biased estimates if the design/data doesn't support fitting all the parameters of the true model (ie., the true model may be over-parameterized resulting in a failed $COV). If the reduced model is not very plausible we may discard it or recognize its limitations particularly for extrapolation. However, it is naive to simply trust an over-parameterized model fit simply because the model is more plausible and proceed without caution. A true but over-parameterized model fit may have difficulty in estimation due to the limitations of the design to support the model. If the fit converges to a local minimum we still need to be concerned about bias even though we are fitting the true model. If you have a strong belief that the over-parameterized model is more plausible this is where explicity incorporating prior information on parameters that may be difficult to estimate from the existing design/data may be helpful. The imprecision pattern was similar with the simulated data but the magnitude of differences between the mean $COV SE and the mean bootstrap SE were larger than those seen with the original real dataset. For $COV SE the THETA:se estimates were about 50% smaller while OMEGA:se were 400% larger than the bootstrap SE. There were no real differences depending on whether all runs, minimization successful or $COV successful runs were used ($COV successful runs tended to be a bit larger). 95% confidence intervals obtained from all the bootstrap runs on the simulated dataset were very similar to those obtained from minimization successful and $COV successful runs. The 95% CI predicted from the asymptotic SE was on average 22% larger (range 14-46%) than the bootstrap CI. My conclusion from this empirical exploration of one data set and model suggests that a successful $COV is of no value for selection of models with improved bias or imprecision. It is a quicker way of obtaining some idea of the parameter 95% confidence interval but it is upwardly biased compared with the bootstrap estimate. I am not typically interested in parameter CIs for every model I run. I am happy to leave that until I have finished model building and prefer to rely on bootstrap CIs. I think we are in agreement on almost all issues that you raise except for the diagnostic value of the $COV in relation to the thing you call "stability". I dont know what stability means so perhaps you would like to offer a definition and some evidence for your assertion. [Ken Kowalski] Hopefully the above responses give you some sense of what I mean by stability. Over-parameterization of the model (too many parameters estimated relative to the information content of the data/design), ill-conditioning of the hessian (R-matrix) which can lead to numerically unstable inversion of the R-matrix ($COV failures) and instability of the paramater estimation (overly sensitive to the starting values) are all symptoms of a stability problem with the model. _______________________________________________________