Error of estimation and sample size in the linear mixed model

The linear mixed model is increasingly used in psychological applications. Whereas the model was once only applied rarely, because designs that would require the model were either avoided, or analyzed improperly, the model has found such favor in psychology that study designs are being contrived so as to allow use of the model. A potential complication of this trend is the intersection of the small sample sizes routine in psychological applications and the potential sensitivity of the model to small sample sizes. In truth, little is known about the small sample properties of the linear mixed model (Demidenko, 2004). Over the past three decades a handful of articles have attempted to understand the finite sample properties of linear mixed model estimators. Each of these studies has contributed to our understanding of the behavior of point estimation in the linear mixed model. A limitation of these preceding studies has been: 1. The simulation of balanced data, which rarely occurs in psychological applications 2. The use of simple models, which are also rarely used in psychological applications. In this study both of these limitations are addressed. Outcomes of interest include model convergence rates, point estimate bias and point estimate root mean squared error (RMSE). Findings indicate that high rates of non-convergence are observed under unbalanced data when both few independent sampling units (ISUs) are sampled and few observations are sampled per ISU. Fixed effect point estimates are nearly unbiased in all sample sizes. Consistent with theory (Demidenko, 2004), for some (co)variance parameters, when few ISUs are sampled, no matter how many observations are sampled per ISU, problematic bias remains. Complex models exhibit greater bias than simpler models. (Co)variance parameter estimates exhibit more bias when population values are small than when they are large. When population (co)variance parameters generating values are small, RMSE is smaller than when population values are large. Consistent with theory and previous studies, full maximum likelihood (FML) estimates are more biased than restricted maximum likelihood (REML) estimates. I conclude with implications for the use of these models in applied research.