Mixed-effects Location Scale Analysis

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does anyone know a way to model standard deviation as a dependent variable?

I have a number of samples and I explained the mean differences of my dependent variable. It's quite easy given the multilevel regression.

Now I would like to explain differences in variance (or SD of mean).  For example, there is a high variance of GPA in larger schools and a small variance of GPA in smaller schools. How to model it with or without multilevel approach given I have a school size variable as an independent and GPA as a dependent variable?

Any advice is welcome!

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关键词：Location Analysis effects Analysi Analys explained standard schools example without

you can give the variance function regression approach of Western and Bloome (2009) a try. Its not multilevel, but maybe it is a start.

http://onlinelibrary.wiley.com/d ... 1.2009.01222.x/full

There have been several papers on this, and also a nice description of this in Goldstein's Multilevel Statistical Models (4th edition).
i worked on a couple of papers and sofware in this area, including:

• Hedeker, D., Mermelstein, R.J., & Demirtas, H. (2008). An application of a mixed-effects location scale model for analysis of Ecological Momentary Assessment (EMA) data. Biometrics, 64, 627-634. <http://www.uic.edu/classes/bstt/bstt513/Biometrics08.pdf>(pdf file) <http://www.uic.edu/classes/bstt/bstt513/LSappendix.pdf>(SAS code)
• Hedeker, D. & Nordgren, R. (2013). MIXREGLS: A program for mixed-effects location scale analysis. Journal of Statistical Software, 52(12), 1-38. <http://www.jstatsoft.org/v52/i12>(link to manual and program)
  

Some of the references in these sources describe similar approaches, many using Bayesian methods, but the above are likelihood based.
Don Hedeker

As I understand it, with Bayesian estimation in Mplus TWOLEVEL, you could set up a "factor" at level-1 for GPA and have a random slope leading from the factor to the observed GPA value.  Fix the residual error variance to zero.

Your random slope would now represent the standard deviation of the GPA within each school and would appear on level 2 of your model for inclusion in a level 2 model.

I am just beginning to work with this modeling framework so we have now tapped the entire extent of my knowledge.  More is available here on this approach (in models general, not specifically to this situation):
http://www.statmodel.com/download/NCME_revision2.pdf

Do you want to explain the within-school variances or the between school variances? If your focus is on the within-school variances, then the fact that GPA has absolute upper and lower bounds suggests that beta regression is an option. The first two papers below introduce the basic single-level model, then Zimprich (2010) illustrates extending to include a random effect. The Hallgren et al. (2014) paper is an application of a mixed effects beta regression with a few fixed effects and a random intercept in the location submodel, plus a dispersion submodel for the level 1 variance that includes two main effects and an interaction.

• Ferrari, S., & Cribari-Neto, F. (2004). Beta regression for modeling rates and proportions. Journal of Applied Statistics, 31(7), 799-815. doi: 10.1080/0266476042000214501
  

• Smithson, M., & Verkuilen, J. (2006). A better lemon squeezer? Maximum likelihood regression with beta-distributed dependent variables. Psychological Methods, 11(1), 54-71. doi: 10.1037/1082-989X.11.1.54
  

• Zimprich, D. (2010). Modeling change in skewed variables using mixed beta regression models. Research in Human Development, 7(1), 9-26. doi: 10.1080/15427600903578136
  

• Hallgren, R. C., Pierce, S. J., Prokop, L. L., Rowan, J. J., & Lee, A. S. (2014). Electromyographic activity of rectus capitis posterior minor muscles associated with voluntary retraction of the head. The Spine Journal, 14(1), 104-112. doi: 10.1016/j.spinee.2013.06.011
  

I think Western and Bloome (2009) and Mplus twolevel approaches are actually what I am looking for.
And I need more time to get aquainted with the other suggested models.

If you have raw data there is always the low tech approach of modelling the absolute deviations from the median. (If you do this  in a one-way ANOVA you have a fairly robust version of the Levene test prodecure for equality of variances.)