1 Introduction

Linear Mixed Models (LMM) are advanced statistical methods for continuous types of data where the data generation process involves complex experimental designs or there is an inherent clustered structure in the data. We come across several examples of such datasets in Biological Sciences, Medical Sciences, Social Sciences, Ecological Sciences etc. The methodologies relevant to standard linear regression models cannot handle such complex data. Inadequate modelling consideration for analysing these data may also result in biased and misleading estimates.

In a balanced dataset, the outcomes from LMM are identical to the outcomes of the analysis of variance (ANOVA) with blocking structure. However, LMMs are more flexible and powerful tools to account for very complex data designs allowing inclusions of multiple sources of variations. For example, although the primary interest of this modelling approach is to infer the magnitude, direction and precision of effect size for predictors of interest like regression models, LMM allows to model the grouping structure of the data, accounts for the complex design structure and provides the magnitude of variability at the cluster level. Hence, an understanding of the principles and applications of LMM is essential to handle data obtained from wide-ranging experimental designs.

LMMs are known by several names depending on the modelling scope and application areas - linear mixed model, multilevel model, hierarchical linear model, random effects model, random coefficient models, random parameter models, nested data model etc. All modelling frameworks are driven by similar principles and theories, although applications of some of the above models may be restricted to a specific data structure. In this session, we will use the term Linear Mixed Models (LMM) and mainly focus on the application of these models in multilevel data.

Principles of LMM can also be extended beyond continuous types of data (i.e. errors following a normal distribution) to data for which the underlying error distributions follow a binomial, Poisson, negative binomial, gamma or so-called the exponential family of distribution. Such models is termed as Generalised Linear Mixed Models (GLMM). We will not cover GLMM here, but the concept gained from LMM could be extended to the GLMM scenarios.

We will provide an overview of the experimental design and data structure suitable to apply the linear mixed models, explain the concepts of fixed and random effects and theoretical framework of the mixed model, and finally, explore fitting such models in the SPSS environment.