Efficient two-dimensional smoothing with P-spline ANOVA mixed models and nested bases
Low-rank smoothing techniques have gained much popularity in non-standard regression modeling. In particular, penalized splines and tensor product smooths are used as flexible tools to study non-parametric relationships among several covariates. The use of standard statistical software facilitates t...
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| Abstract | Low-rank smoothing techniques have gained much popularity in non-standard regression modeling. In particular, penalized splines and tensor product smooths are used as flexible tools to study non-parametric relationships among several covariates. The use of standard statistical software facilitates their use for several types of problems and applications. However, when interaction terms are considered in the modeling, and multiple smoothing parameters need to be estimated standard software does not work well when datasets are large or higher-order interactions are included or need to be tested. In this paper, a general approach for constructing and estimating bivariate smooth models for additive and interaction terms using penalized splines is proposed. The formulation is based on the mixed model representation of the smooth-ANOVA model by Lee and Durbán (in press), and several nested models in terms of random effects components are proposed. Each component has a clear interpretation in terms of function shape and model identifiability constraints. The term PS-ANOVA is coined for this type of models. The estimation method is relatively straightforward based on the algorithm by Schall (1991) for generalized linear mixed models. Further, a simplification of the smooth interaction term is used by constructing lower-rank basis (nested basis). Finally, some simulation studies and real data examples are presented to evaluate the new model and the estimation method. |
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| AbstractList | Low-rank smoothing techniques have gained much popularity in non-standard regression modeling. In particular, penalized splines and tensor product smooths are used as flexible tools to study non-parametric relationships among several covariates. The use of standard statistical software facilitates their use for several types of problems and applications. However, when interaction terms are considered in the modeling, and multiple smoothing parameters need to be estimated standard software does not work well when datasets are large or higher-order interactions are included or need to be tested. In this paper, a general approach for constructing and estimating bivariate smooth models for additive and interaction terms using penalized splines is proposed. The formulation is based on the mixed model representation of the smooth-ANOVA model by Lee and Durbán (in press), and several nested models in terms of random effects components are proposed. Each component has a clear interpretation in terms of function shape and model identifiability constraints. The term PS-ANOVA is coined for this type of models. The estimation method is relatively straightforward based on the algorithm by Schall (1991) for generalized linear mixed models. Further, a simplification of the smooth interaction term is used by constructing lower-rank basis (nested basis). Finally, some simulation studies and real data examples are presented to evaluate the new model and the estimation method. Low-rank smoothing techniques have gained much popularity in non-standard regression modeling. In particular, penalized splines and tensor product smooths are used as flexible tools to study non-parametric relationships among several covariates. The use of standard statistical software facilitates their use for several types of problems and applications. However, when interaction terms are considered in the modeling, and multiple smoothing parameters need to be estimated standard software does not work well when datasets are large or higher-order interactions are included or need to be tested. In this paper, a general approach for constructing and estimating bivariate smooth models for additive and interaction terms using penalized splines is proposed. The formulation is based on the mixed model representation of the smooth-ANOVA model by Lee and Durbán (in press), and several nested models in terms of random effects components are proposed. Each component has a clear interpretation in terms of function shape and model identifiability constraints. The term PS-ANOVA is coined for this type of models. The estimation method is relatively straightforward based on the algorithm by Schall (1991) for generalized linear mixed models. Further, a simplification of the smooth interaction term is used by constructing lower-rank basis (nested basis). Finally, some simulation studies and real data examples are presented to evaluate the new model and the estimation method. |
| Author | Eilers, Paul Durbán, María Lee, Dae-Jin |
| Author_xml | – sequence: 1 givenname: Dae-Jin surname: Lee fullname: Lee, Dae-Jin email: dae-jin.lee@csiro.au organization: CSIRO Mathematics Informatics and Statistics, Private Bag 33, Clayton, VIC 3169, Australia – sequence: 2 givenname: María surname: Durbán fullname: Durbán, María email: mdurban@est-econ.uc3m.es organization: Department of Statistics, Universidad Carlos III de Madrid, Escuela Politécnica Superior, Leganés 28911 Madrid, Spain – sequence: 3 givenname: Paul surname: Eilers fullname: Eilers, Paul email: p.eilers@eramusmc.nl organization: Department of Biostatistics, Erasmus Medical Centre, Rotterdam, The Netherlands |
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| Cites_doi | 10.18637/jss.v009.i01 10.1007/s11222-012-9314-z 10.1198/1061860043010 10.1111/j.1541-0420.2006.00574.x 10.1214/ss/1038425655 10.1016/j.csda.2007.10.022 10.1016/j.csda.2008.05.032 10.1093/biomet/60.2.255 10.1016/j.csda.2004.07.008 10.1111/j.1467-9868.2008.00695.x 10.1198/106186002844 10.1191/1471082X04st080oa 10.1177/1471082X1001100104 10.1111/j.1467-9868.2010.00749.x 10.1093/biomet/78.4.719 10.1214/aos/1176344136 10.1080/01621459.1993.10594284 10.1111/j.2517-6161.1993.tb01939.x 10.1111/j.1467-9868.2006.00543.x 10.1111/j.2517-6161.1993.tb01917.x |
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| Title | Efficient two-dimensional smoothing with P-spline ANOVA mixed models and nested bases |
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