Flexible Mixed Models: Regression Splines and Thin-Plate Regression Splines in a Mixed Model Framework

Abstract

Regression splines and thin-plate regression splines were fitted inside generalized linear mixed models with good results. Their role in prediction and as exploratory tools are examined. Regression splines were specified in advance using biological information and compared with knot positions chosen using the data available. A forwards selection procedure was used to choose knots for thin-plate regression splines, and both cross-validation and fit statistics were used to discriminate between competing models. Parameter bias was assessed using a parametric bootstrap in the generalized mixed model setting, and bias for both high and low variance data was compared. Model-based, bootstrap, and robust inference methods were used to assess parameter inference, and the impact of peculiar individuals on the models were examined. Forestry growth and mortality data is used for the modelling throughout. Model specification using biological information returned good results, and models with a relatively small number of well chosen knots outperformed models with larger numbers of relatively poorly placed knots. The generalized mixed model fixed effects estimates were found to be unbiased, but the model-based variance estimates were consistently too small. While variance estimates for terms with random effects were more realistic, robust measures of inference were consistently more reliable. For the normal errors models, model-based inference was only valid when complex covariance structures were specified or robust inference was used Generalized mixed models were found to be relatively robust to influential individuals while cross-validation enabled problematic individuals to be identified.