R help - Jul 2006 - non positive-definite G matrix in mixed models: bootstrap?

Dear list,
In a mixed model I selected I find a non positive definite random effects 
variance-covariance matrix G, where some parameters are estimated close to 
zero, and related confidence intervals are incredibly large.

Since simplification of the random portion is not an option, for both 
interest in the parameters and significant increase in the model fit, I 
would like to collect "unbiased" random effects estimates.

I used bootstrap to this purpose, creating a linear model for each cluster 
and bootstraping the variance of the coefficients. Is this procedure 
reasonable? Would it be reasonable in this case to keep the marginal portion 
of the mixed model?
Note that in presence of positive-definite G matrix this bootstrap approach 
and the mixed effect model give highly similar estimates and that in the non 
positive-definite model the bootstrap and mixed model marginal-model 
estimates are highly similar as well.

Thank you
    Bruno

There is a paper by Rogosa and Saner which shows some equivalences in
what you are doing under certain conditions. They show similarities
between bootstrapping with linear models and how the estimates might be
similar to those obtained from a mixed model.

Rogosa, D. R., and Saner, H. M. (1995). Longitudinal data analysis
examples with random coefficient models. 
Journal of Educational and Behavioral Statistics, 20, 149-170. 

Harold

> -----Original Message-----
> From: r-help-bounces at stat.math.ethz.ch 
> [mailto:r-help-bounces at stat.math.ethz.ch] On Behalf Of Bruno 
> L. Giordano
> Sent: Tuesday, July 11, 2006 9:31 AM
> To: r-help at stat.math.ethz.ch
> Subject: [R] non positive-definite G matrix in mixed models: 
> bootstrap?
> 
> Dear list,
> In a mixed model I selected I find a non positive definite 
> random effects variance-covariance matrix G, where some 
> parameters are estimated close to zero, and related 
> confidence intervals are incredibly large.
> 
> Since simplification of the random portion is not an option, 
> for both interest in the parameters and significant increase 
> in the model fit, I would like to collect "unbiased" random 
> effects estimates.
> 
> I used bootstrap to this purpose, creating a linear model for 
> each cluster and bootstraping the variance of the 
> coefficients. Is this procedure reasonable? Would it be 
> reasonable in this case to keep the marginal portion of the 
> mixed model?
> Note that in presence of positive-definite G matrix this 
> bootstrap approach and the mixed effect model give highly 
> similar estimates and that in the non positive-definite model 
> the bootstrap and mixed model marginal-model estimates are 
> highly similar as well.
> 
> Thank you
>     Bruno
> 
> ______________________________________________
> R-help at stat.math.ethz.ch mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide! 
> http://www.R-project.org/posting-guide.html
>

Have you considered 'simulate.lme'?  I believe that this is what 
Bates included in the 'nlme' package for obtaining confidence intervals,
joint confidence regions, etc., when there were questions about the 
results for whatever reason.  I have not tried it with a singular model, 
but I believe Bates has.  In particular, have you reviewed ch. 2 in 
Pinheiro and Bates (2000) Mixed-Effects Models in S and S-Plus (Springer)?

	  I am not a fan of bootstrapping.  In mixed-effects applications, 
bootsrapping needs to incorporate the constraints imposed by the 
mixed-effects model.  For example, if you have several samples per batch 
and several batches per lot, you need to bootstrap lots as well as 
batches within lot and samples within batch.

	  The advantage of bootstrapping is that if the normality assumptions 
behind the mixed model do not hold, the bootstrap results will still 
have some validity.  However, the range of extrapolability of bootstrap 
results is limited to other situations whose distribution is plausibly 
like the particular sample you bootstrapped, and I wouldn't know how to 
evaluate that.  By contrast, I know how to extrapolate simulation 
results.  To decide whether such results apply, I make normal 
probability plots of the data, random effects, and residuals.  If they 
all seem normal, I feel it is reasonable to use the simulation results.

	  Others may offer a different perspective (or a correction, as the 
case may be).  However, you asked for comments about bootstrapping mixed 
models.  At least you've got one.

	  Hope this helps.
	  Spencer Graves

Bruno L. Giordano wrote:
> Dear list,
> In a mixed model I selected I find a non positive definite random effects 
> variance-covariance matrix G, where some parameters are estimated close to 
> zero, and related confidence intervals are incredibly large.
> 
> Since simplification of the random portion is not an option, for both 
> interest in the parameters and significant increase in the model fit, I 
> would like to collect "unbiased" random effects estimates.
> 
> I used bootstrap to this purpose, creating a linear model for each cluster 
> and bootstraping the variance of the coefficients. Is this procedure 
> reasonable? Would it be reasonable in this case to keep the marginal
portion
> of the mixed model?
> Note that in presence of positive-definite G matrix this bootstrap approach
> and the mixed effect model give highly similar estimates and that in the
non
> positive-definite model the bootstrap and mixed model marginal-model 
> estimates are highly similar as well.
> 
> Thank you
>     Bruno
> 
> ______________________________________________
> R-help at stat.math.ethz.ch mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide!
http://www.R-project.org/posting-guide.html