On the estimation of variance parameters in non-standard generalised linear mixed models: application to penalised smoothing

Abstract

We present a novel method for the estimation of variance parameters in generalised linear mixed models. The method has its roots in Harville (1977)'s work, but it is able to deal with models that have a precision matrix for the random-effect vector that is linear in the inverse of the variance parameters (i.e., the precision parameters). We call the method SOP (Separation of Overlapping Precision matrices). SOP is based on applying the method of successive approximations to easy-to-compute estimate updates of the variance parameters. These estimate updates have an appealing form: they are the ratio of a (weighted) sum of squares to a quantity related to effective degrees of freedom. We provide the sufficient and necessary conditions for these estimates to be strictly positive. An important application field of SOP is penalised regression estimation of models where multiple quadratic penalties act on the same regression coefficients. We discuss in detail two of those models: penalised splines for locally adaptive smoothness and for hierarchical curve data. Several data examples in these settings are presented. * This is a pre-print of an article published in Statistics and Computing. The final authenticated version is available online at: https://doi.org/10.1007/s11222-018-9818-2. arXiv:1801.07278v2 [stat.ME] 12 Jun 2018 The SOP methodRegardless of the structure of G, estimation of β and α, and, when necessary, the dispersion parameter φ, does not pose a problem, and can be done as discussed in Section 2 above (see also Appendix C for a detailed description of the estimating algorithm). Recall that estimates for α k are obtained as α k = G k Z k P z. The hat matrix associated with the k-th random component α k is, once again, H k = Z k G k Z k P , and the effective degrees of freedom of this component is ED k = trace (H k ). The variance parameters σ 2 k l (l = 1, . . . , p k and k = 1, . . . , c), however, cannot be estimated by means of Harville's approach. This is a consequence of G not being linear in the variance parameters.

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