Hello,

I am running a complex level 2 variation model (variance at level 2 as a function of predictors). I know how to get the level 2 variance from this model and graph it. Is there a way to get confidence (or credible, if using MCMC) intervals around the variance function? I can't find any manual that does this.

In IGLS methods, I get Wald type standard error (I think) by default for level 2 random parameters (two variance terms and a covariance term). But I know this standard error may not be appropriate since variance usually does not follow a normal distribution. Anyhow, how can I use this standard error or other more appropriate standard error to get the confidence intervals for level 2 variance formed by the three parameters?

In MCMC methods, I get the standard error and credible intervals. How can I use these to form credible intervals around the variance function?

If you have advice or if there is a manual for this, let me know. Thank you.

Sun

## variance function and its uncertainty

### Re: variance function and its uncertainty

Hi Sun,

With regard MCMC my suggestion is to look at my MCMC manual where I cover how one works out confidence intervals for derived quantities - here I cover in section 4.7 the difference between 2 schools, 4.8 their ranks with CIs and most relevant for you the ICC/VPC in section 4.9. Here you can see that by using the split command you can put the chains of all parameters into separate columns and then use these to construct derived quantities like the VPC. I guess for variance functions you would want to choose a load of X values to plug into the formula to construct the curve. You should be able to write macro code to loop over these X values and extract the 2.5% / 97.5% percentiles as well as the median and thus get the data you need for the plot.

Best wishes,

Bill.

With regard MCMC my suggestion is to look at my MCMC manual where I cover how one works out confidence intervals for derived quantities - here I cover in section 4.7 the difference between 2 schools, 4.8 their ranks with CIs and most relevant for you the ICC/VPC in section 4.9. Here you can see that by using the split command you can put the chains of all parameters into separate columns and then use these to construct derived quantities like the VPC. I guess for variance functions you would want to choose a load of X values to plug into the formula to construct the curve. You should be able to write macro code to loop over these X values and extract the 2.5% / 97.5% percentiles as well as the median and thus get the data you need for the plot.

Best wishes,

Bill.

### Re: variance function and its uncertainty

Any advice on IGLS? To get confidence intervals on the variance functions, perhaps using standard error, or other methods.

### Re: variance function and its uncertainty

Hi Sun,

Not really as as you say you don't have normality so any use of the SEs would be very approximate. you might consider bootstrapping I guess but easiest with MCMC.

Best wishes,

Bill.

Not really as as you say you don't have normality so any use of the SEs would be very approximate. you might consider bootstrapping I guess but easiest with MCMC.

Best wishes,

Bill.