Customised predictions including Level 1 variance

[OliverTS]

I am creating a longitudinal growth model in which a biometric variable at time 't' (level 1) is modeled as a polynomial function of the age of the individual. The data relates to twins, and so each individual (level 2) is furthermore nested within another set (level 3). I have created models with random intercept and random slope terms at both levels 2 and 3, and the level 1 variance also increases as a function age. Using the 'customised prediction' option and selecting a coverage interval for level 2 seems to me to give a 95% prediction interval for the growth curves of individuals. However, I am not sure whether the output takes account of the level 1 variance, would anyone be able to let me know?

Ideally, I would like to be able to create reference intervals for the expected range of individual observations (i.e. accounting for variance at all levels) at any given age.

Any guidance regarding this would be much appreciated!

[ChrisCharlton]

I just checked this with Professor Kelvyn Jones and his response is below:

The customised predictions is primarily a tool for making predictions with confidence intervals for the fixed part of the model; although as you noted it also provides coverage intervals (also with confidence intervals) for the higher-level level variance functions. It does not provide the coverage intervals for the level 1 variance function. However, you can get the level 1 variance function as follows: click on Model on Main Menu, choose Variance function and keep Level to be level 1. You can then put some selected values of the X's in the columns of the table to get the estimated variance at level 1 for these values (these could be the same values as you used in the Customised predictions). Alternatively you can choose to store the variance function and associated SE's in columns at the bottom of the window; these will then be for all the observed level 1 units to which you have fitted the model. Once you have derived the level 1 variance function in this way, you can get 95% coverage intervals as +/- 1.96* Square Root of the Variance function - these are given as departures (offsets is the term used in the Customised graphics facility) from the mean values. The variance function window is covered in Chapter 7 of the MLwiN User Manual.


Kelvyn Jones

[OliverTS]

Thank you for the reply, that's very helpful. As I understand it, the variance function window only gives the variance for a single level of the model at a time, whereas I am looking to create a summary of the variance across all levels of the model. As such, I think that I may need to use the model structure and parameter estimates from MLwiN to create overall prediction intervals using another piece of software.

Cheers

Oliver

[joneskel]

You asked specifically about level 1 variance so that is why I responded about that. However, the variance function can be calculated at any level separately (eg level 1 and level 2 and level 3) and they are additive; so you could do each, add then up to get a total variance function around the mean, and then use this to calculate 95% (say) coverage intervals

[OliverTS]

Thank you for the follow-up. I was initially thinking that this problem would require something more complicated than adding up the variance function at each level (I'm still getting started on this type of modelling). However, I had just about come around to the conclusion that this method (i.e. adding up the variance functions) would be fine in this situation so thank you for the confirmation!

[joneskel]

Yes, this works quite simply for Normal-theory models as the variances are additive.