zbMATH — the first resource for mathematics

Spectral parameterization for linear mixed models applied to confounding of fixed effects by random effects. (English) Zbl 1421.62090
Summary: Linear mixed models (LMMs) are a commonly-used tool for fitting linear models with correlated errors across a wide variety of fields. However, adding random effects to a linear model can cause unexpected large changes in fixed effect estimates relative to the same model without random effects, and the process by which such changes occur is not well understood. We present the spectral parameterization of LMMs as a tool for understanding such confounding, and develop diagnostics for evaluating the effect on fixed effect estimates of collinearities between columns of the fixed effect and reparameterized random effect design matrices, as well as individual observations. We also present spectral parameterizations of ANOVA-like models, intrinsic conditional autoregressive models, and approximations to Gaussian processes.
62J05 Linear regression; mixed models
62J10 Analysis of variance and covariance (ANOVA)
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
60G15 Gaussian processes
GMRFLib; spectralGP
Full Text: DOI
[1] Anselin, L., Spatial Econometrics: Methods and Models, (1988), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht, The Netherlands
[2] Bailey, R., General balance: artificial theory or practical relevance?, (Proceedings of the International Conference on Linear Statistical Inference LINSTAT93, (1994), Springer), 171-184 · Zbl 0813.62067
[3] Banerjee, S.; Roy, A., Linear Algebra and Matrix Analysis for Statistics, (2014), CRC Press · Zbl 1309.15002
[4] Belsley, D. A.; Kuh, E.; Welsch, R. E., Regression Diagnostics: Identifying Influential Data and Sources of Collinearity, (2004), John Wiley and Sons · Zbl 0479.62056
[5] Besag, J.; Kooperberg, C., On conditional and intrinsic autoregressions, Biometrika, 82, 4, 733-746, (1995) · Zbl 0899.62123
[6] Fallat, S. M.; Johnson, C. R., Totally Nonnegative Matrices, (2011), Princeton University Press · Zbl 1390.15001
[7] Hanks, E. M.; Schliep, E. M.; Hooten, M. B.; Hoeting, J. A., Restricted spatial regression in practice: geostatistical models, confounding, and robustness under model misspecification, Environmetrics, 26, 4, 243-254, (2015)
[8] Hodges, J. S., Richly Parameterized Linear Models: Additive, Time Series, and Spatial Models Using Random Effects, (2013), CRC Press
[9] Hodges, J. S.; Reich, B. J., Adding spatially-correlated errors can mess up the fixed effect you love, Amer. Statist., 64, 4, 325-334, (2010) · Zbl 1217.62095
[10] Paciorek, C. J., Bayesian smoothing with Gaussian processes using Fourier basis functions in the spectralGP package, J. Stat. Softw., 19, 2, nihpa22751, (2007)
[11] Parlett, B. N.; Wu, W. D., Eigenvector matrices of symmetric tridiagonals, Numer. Math., 44, 1, 103-110, (1984) · Zbl 0542.15001
[12] Reich, B. J.; Hodges, J. S., Identification of the variance components in the general two-variance linear model, J. Statist. Plann. Inference, 138, 6, 1592-1604, (2008) · Zbl 1131.62022
[13] Reich, B. J.; Hodges, J. S.; Zadnik, V., Effects of residual smoothing on the posterior of the fixed effects in disease-mapping models, Biometrics, 62, 4, 1197-1206, (2006) · Zbl 1114.62124
[14] Royle, J. A.; Wikle, C. K., Efficient statistical mapping of avian count data, Environ. Ecol. Stat., 12, 2, 225-243, (2005)
[15] Rue, H.; Held, L., Gaussian Markov Random Fields: Theory and Applications, (2005), CRC press · Zbl 1093.60003
[16] Speed, T., General balance, Encycl. Stat. Sci, (1983)
[17] Van Mieghem, P., Graph Spectra for Complex Networks, (2010), Cambridge University Press
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.