Nlmefit covb and PSI

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Andres on 15 Sep 2012
When using nlmefit or nlmefitsa for estimating a mixed effects model, the algorithm gives me 2 covariance matrices as result: PSI and stats.covb. Anyone can tell me what is the difference between these two? The help says this: PSI, an r-by-r estimated covariance matrix for the random effects. covb — The estimated covariance matrix of the parameter estimates
I know that if I want to simulate a population using nlmefitsa's results I should generate random samples using beta and PSI. But, how can I use covb?
Thanks if anyone can help me.

Accepted Answer

Star Strider
Star Strider on 15 Sep 2012
Edited: Star Strider on 15 Sep 2012
I haven't done much with nmlefitsa, so I don't have much experience with it. As a general rule however, COVB (the covariance matrix of the parameter estimates), provides one rather important bit of information on the parameters themselves: it allows you to calculate the confidence intervals on the parameters.
With Beta a column vector of the parameter values, and CovB the covariance matrix, the 95% confidence intervals are given by:
CI95 = [Beta-1.96*sqrt(diag(CovB)) Beta+1.96*sqrt(diag(CovB))];
The 1.96 value is the ‘critical value’ (the z-statistic or z-score) corresponding to the 95% confidence interval:
cv = norminv(0.975, 0, 1);
The usual interpretation of the confidence intervals on the parameters is that if the confidence interval for a particular parameter includes zero, that parameter is not needed in the model.
Andres on 16 Sep 2012
Thanks! this clarifies a lot the meaning of covb. I was thinking it could be used to simulate some intrinsic variability, but apparently that is not the case.
Star Strider
Star Strider on 16 Sep 2012
My pleasure!
Your question stimulated my curiosity as well to see if there was other information in the parameter covariance matrix and other information about the estimated parameters. I discovered a good discussion in ‘A Sensitivity Matrix Based Methodology for Inverse Problem Formulation’. It did not change what I stated in my answer, but it is the most thorough discussion of parameter characteristics and their interpretation that I've seen recently.
I am not certain what you mean by ‘intrinsic variability’, but expressions of the effect of the parameters on the model are variously the Jacobian and the sensitivity matrix, and there may be others. The Jacobian is the partial derivatives of the function with respect to each parameter. This becomes a problem in models that are ‘nonlinear in the parameters’, in which some or all of the partial derivatives of the function with respect to each parameter are functions of at least one other parameter. An interesting discussion of that is ‘Traditional and Generalized Sensitivity Functions and Their Uses in Biomathematical Models’.

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