×

Straightforward intermediate rank tensor product smoothing in mixed models. (English) Zbl 1322.62197

Summary: Tensor product smooths provide the natural way of representing smooth interaction terms in regression models because they are invariant to the units in which the covariates are measured, hence avoiding the need for arbitrary decisions about relative scaling of variables. They would also be the natural way to represent smooth interactions in mixed regression models, but for the fact that the tensor product constructions proposed to date are difficult or impossible to estimate using most standard mixed modelling software. This paper proposes a new approach to the construction of tensor product smooths, which allows the smooth to be written as the sum of some fixed effects and some sets of i.i.d. Gaussian random effects: no previously published construction achieves this. Because of the simplicity of this random effects structure, our construction is useable with almost any flexible mixed modelling software, allowing smooth interaction terms to be readily incorporated into any Generalized Linear Mixed Model. To achieve the computationally convenient separation of smoothing penalties, the construction differs from previous tensor product approaches in the penalties used to control smoothness, but the penalties have the advantage over several alternative approaches of being explicitly interpretable in terms of function shape. Like all tensor product smoothing methods, our approach builds up smooth functions of several variables from marginal smooths of lower dimension, but unlike much of the previous literature we treat the general case in which the marginal smooths can be any quadratically penalized basis expansion, and there can be any number of them. We also point out that the imposition of identifiability constraints on smoothers requires more care in the mixed model setting than it would in a simple additive model setting, and show how to deal with the issue. An interesting side effect of our construction is that an ANOVA-decomposition of the smooth can be read off from the estimates, although this is not our primary focus. We were motivated to undertake this work by applied problems in the analysis of abundance survey data, and two examples of this are presented.

MSC:

62J12 Generalized linear models (logistic models)
62J10 Analysis of variance and covariance (ANOVA)
62G05 Nonparametric estimation
65C60 Computational problems in statistics (MSC2010)

Software:

gamair; MEMSS; lme4; gss; R; SemiPar; S-PLUS
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Bates, D., Maechler, M.: lme4: Linear mixed-effects models using S4 classes (2010). http://CRAN.R-project.org/package=lme4
[2] Belitz, C., Lang, S.: Simultaneous selection of variables and smoothing parameters in structured additive regression models. Comput. Stat. Data Anal. 53(1), 61–81 (2008) · Zbl 1452.62029 · doi:10.1016/j.csda.2008.05.032
[3] Borchers, D.L., Buckland, S.T., Priede, I.G., Ahmadi, S.: Improving the precision of the daily egg production method using generalized additive models. Can. J. Fish. Aquat. Sci. 54, 2727–2742 (1997) · doi:10.1139/f97-134
[4] Breslow, N.E., Clayton, D.G.: Approximate inference in generalized linear mixed models. J. Am. Stat. Assoc. 88, 9–25 (1993) · Zbl 0775.62195
[5] Davison, A.C.: Statistical Models. Cambridge University Press, Cambridge (2003) · Zbl 1044.62001
[6] Eilers, P.H.C.: Discussion of Verbyla, A.P., B.R. Cullis, M.G. Kenward and S.J. Welham (1999). The analysis of designed experiments and longitudinal data by using smoothing splines. J. R. Stat. Soc. C 48(3), 307–308 (1999) · Zbl 0956.62062
[7] Eilers, P.H.C., Marx, B.D.: Flexible smoothing with B-splines and penalties. Stat. Sci. 11(2), 89–102 (1996) · Zbl 0955.62562 · doi:10.1214/ss/1038425655
[8] Eilers, P.H.C., Marx, B.D.: Multivariate calibration with temperature interaction using two-dimensional penalized signal regression. Chemom. Intell. Lab. Syst. 66, 159–174 (2003) · doi:10.1016/S0169-7439(03)00029-7
[9] Fahrmeir, L., Kneib, T., Lang, S.: Penalized structured additive regression for space time data: A Bayesian perspective. Stat. Sin. 14, 731–761 (2004) · Zbl 1073.62025
[10] Gu, C.: Smoothing Spline ANOVA Models. Springer, Berlin (2002) · Zbl 1051.62034
[11] Gu, C., Kim, Y.-J.: Penalized likelihood regression: general formulation and efficient approximation. Can. J. Stat. 30(4), 619–628 (2002) · Zbl 1018.62032 · doi:10.2307/3316100
[12] Harville, D.A.: Matrix Algebra from a Statisticians Perspective. Springer, Berlin (1997) · Zbl 0881.15001
[13] Hastie, T., Tibshirani, R.: Generalized additive models (with discussion). Stat. Sci. 1, 297–318 (1986) · Zbl 0955.62603 · doi:10.1214/ss/1177013604
[14] Kim, Y.J., Gu, C.: Smoothing spline Gaussian regression: More scalable computation via efficient approximation. J. R. Stat. Soc. B 66, 337–356 (2004) · Zbl 1062.62067 · doi:10.1046/j.1369-7412.2003.05316.x
[15] Kimeldorf, G., Wahba, G.: A correspondence between Bayesian estimation of stochastic processes and smoothing by splines. Ann. Math. Stat. 41, 495–502 (1970) · Zbl 0193.45201 · doi:10.1214/aoms/1177697089
[16] Lee, D.-J., Durbán, M.: P-spline ANOVA type interaction models for spatio-temporal smoothing. Stat. Model. 11(1), 49–69 (2011) · doi:10.1177/1471082X1001100104
[17] Lin, X., Zhang, D.: Inference in generalized additive mixed models using smoothing splines. J. R. Stat. Soc. B 61, 381–400 (1999) · Zbl 0915.62062 · doi:10.1111/1467-9868.00183
[18] Parker, R., Rice, J.: Discussion of Silverman (1985). J. R. Stat. Soc. B 47(1), 41–42 (1985)
[19] Pinheiro, J.C., Bates, D.M.: Mixed-Effects Models in S and S-PLUS. Springer, Berlin (2000) · Zbl 0953.62065
[20] R Development Core Team: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna (2010). ISBN 3-900051-07-0, www.R-project.org
[21] Ruppert, D., Wand, M.P., Carroll, R.J.: Semiparametric Regression. Cambridge University Press, Cambridge (2003) · Zbl 1038.62042
[22] Silverman, B.W.: Some aspects of the spline smoothing approach to non-parametric regression curve fitting (with discussion). J. R. Stat. Soc. B 47, 1–53 (1985) · Zbl 0606.62038
[23] Verbyla, A.P., Cullis, B.R., Kenward, M.G., Welham, S.J.: The analysis of designed experiments and longitudinal data by using smoothing splines. J. R. Stat. Soc. C 48(3), 269–311 (1999) · Zbl 0956.62062 · doi:10.1111/1467-9876.00154
[24] Wahba, G.: Spline bases, regularization and generalized cross validation for solving approximation problems with large quantities of noisy data. In: Cheney, E. (ed.) Approximation Theory III. Academic Press, London (1980) · Zbl 0485.41012
[25] Wahba, G.: Spline Models for Observational Data. SIAM, Philadelphia (1990) · Zbl 0813.62001
[26] Wood, S.N.: Stable and efficient multiple smoothing parameter estimation for generalized additive models. J. Am. Stat. Assoc. 99, 673–686 (2004) · Zbl 1117.62445 · doi:10.1198/016214504000000980
[27] Wood, S.N.: Low-rank scale-invariant tensor product smooths for generalized additive mixed models. Biometrics 62(4), 1025–1036 (2006a) · Zbl 1116.62076 · doi:10.1111/j.1541-0420.2006.00574.x
[28] Wood, S.N.: Generalized Additive Models: An Introduction with R. Taylor & Francis/CRC Press, London (2006b) · Zbl 1087.62082
[29] Wood, S.N.: Fast stable direct fitting and smoothness selection for generalized additive models. J. R. Stat. Soc. B 70(3), 495–518 (2008) · Zbl 05563356 · doi:10.1111/j.1467-9868.2007.00646.x
[30] Wood, S.N.: Fast stable restricted maximum likelihood and marginal likelihood estimation of semiparametric generalized linear models. J. R. Stat. Soc. B 73(1), 3–36 (2011) · doi:10.1111/j.1467-9868.2010.00749.x
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.