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  • TUI H. NOLAN (a1) and MATT P. WAND (a1)


We define and solve classes of sparse matrix problems that arise in multilevel modelling and data analysis. The classes are indexed by the number of nested units, with two-level problems corresponding to the common situation, in which data on level-1 units are grouped within a two-level structure. We provide full solutions for two-level and three-level problems, and their derivations provide blueprints for the challenging, albeit rarer in applications, higher-level versions of the problem. While our linear system solutions are a concise recasting of existing results, our matrix inverse sub-block results are novel and facilitate streamlined computation of standard errors in frequentist inference as well as allowing streamlined mean field variational Bayesian inference for models containing higher-level random effects.


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[1]Baltagi, B. H., Econometric analysis of panel data (John Wiley & Sons, Chichester, 2013).
[2]Fitzmaurice, G., Davidian, M., Verbeke, G. and Molenberghs, G. (eds), Longitudinal data analysis (Chapman & Hall/CRC, Boca Raton, FL, 2008); doi:10.1201/9781420011579.
[3]Gentle, J. E., Matrix algebra (Springer, New York, 2007); doi:10.1007/978-0-387-70873-7.
[4]Goldstein, H., Multilevel statistical models, 4th edn (John Wiley & Sons, Chichester, 2010); doi:10.1002/9780470973394.
[5]Harville, D. A., Matrix algebra from a statistician’s perspective (Springer, New York, 2008).
[6]Henderson, C. R., “Best linear unbiased estimation and prediction under a selection model”, Biometrics 31 (1975) 423447; doi:10.2307/2529430.
[7]Hołubowski, W., Kurzyk, D. and Trawiński, T., “A fast method for computing the inverse of symmetric block arrowhead matrices”, Appl. Math. Inf. Sci. 9 (2015) 319324; doi:10.12785/amis/092L06.
[8]Longford, N. T., “A fast scoring algorithm for maximum likelihood estimation in unbalanced mixed models with nested random effects”, Biometrika 74 (1987) 817827; doi:10.1093/biomet/74.4.817.
[9]McCulloch, C. E., Searle, S. R. and Neuhaus, J. M., Generalized, linear, and mixed models, 2nd edn (John Wiley & Sons, Hoboken, NJ, 2008).
[10]Nolan, T. H., Menictas, M. and Wand, M. P., Streamlined computing for variational inference with higher level random effects. arXiv:1903.06616v3 (2020).
[11]Pinheiro, J. C. and Bates, D. M., Mixed-effects models in S and S-PLUS (Springer, New York, 2000); doi:10.1007/978-1-4419-0318-1.
[12]Rao, J. N. K. and Molina, I., Small area estimation, 2nd edn (John Wiley & Sons, Hoboken, NJ, 2015); doi:10.1002/9781118735855.
[13]Saberi Nejafi, S., Edalatpanah, S. A. and Gravvanis, G. A., “An efficient method for computing the inverse of arrowhead matrices”, Appl. Math. Lett. 33 (2014) 15; doi:10.1016/j.aml.2014.02.010.
[14]Stanimirović, P. S., Katsikis, V. N. and Kolundžija, D., “Inversion and pseudoinversion of block arrowhead matrices”, Appl. Math. Comput. 341 (2019) 379401; doi:10.1016/j.amc.2018.09.006.
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  • TUI H. NOLAN (a1) and MATT P. WAND (a1)


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