Nested by design: model fitting and interpretation in a mixed model era
- Nested data structures are ubiquitous in the study of ecology and evolution, and such structures need to be modelled appropriately. Mixed-effects models offer a powerful framework to do so. Nested effects can usually be fitted using the syntax for crossed effects in mixed models, provided that the coding reflects implicit nesting. But the experimental design (either nested or crossed) affects the interpretation of the results.
- The key difference between nested and crossed effects in mixed models is the estimation and interpretation of the interaction variance. With nested data structures, the interaction variance is pooled with the main effect variance of the nested factor. Crossed designs are required to separate the two components. This difference between nested and crossed data is determined by the experimental design (thus by the nature of data sets) and not by the coding of the statistical model.
- Data can be nested by design in the sense that it would have been technically feasible and biologically relevant to collect the data in a crossed design. In such cases, the pooling of the variances needs to be clearly acknowledged. In other situations, it might be impractical or even irrelevant to apply a crossed design. We call such situations naturally nested, a case in which the pooling of the interaction variance will be less of an issue.
- The interpretation of results should reflect the fact that the interaction variance inflates the main effect variance when dealing with nested data structures. Whether or not this distinction is critical depends on the research question and the system under study.
- We present mixed models as a particularly useful tool for analysing nested designs, and we highlight the value of the estimated random variance as a quantity of biological interest. Important insights can be gained if random-effect variances are appropriately interpreted. We hope that our paper facilitates the transition from classical anovas to mixed models in dealing with categorical data.