Analysis of linear mixed models with an extension to three or more factors each having both fixed and random levels

Abstract

Studies or experiments involving three or more factors, each having both fixed and random levels, usually require the use of linear mixed models on treatments arranged in completely randomised design (CRD), randomised complete block design (RCBD), or any other design. These scenarios require analysts to be more accurate when measuring some of the effects of factors. When independent factors have a dichotomous composition in factor levels (either fixed or random effects) two issues that need careful attention immediately emerge: (i) the assumed linear mixed model under the partitioning approach will involve a design matrix that is either a full-rank or less-than-full-rank form, (ii) the approach leaves the partitioned data subsets vulnerable to outlier contamination, which might subsequently compromise the level of accuracy and precision of the selected partitioned models. Traditional statistical approaches have to be reoriented in order to extract all the variations in the data sufficiently. This study builds upon the partitioning approach by Njuho and Milliken (2005, 2009), and extends the concept to the case of contaminated linear mixed model Estimation (Koller and Stahel, 2011), and the issue of characterising treatment effect variation (Dixon, 2016; Ding et al., 2019) in various experimental designs that involve three or more factors, each having both fixed and random levels. The robust package, available in the Comprehensive R Archive (CRAN), was used to robustly fit linear mixed models when considerably little outlier contamination exists in the data set. To circumvent the tedious process of creating partitions of experimental data based on targeted factor levels (data scrapping), a SAS code was proposed for generating partitioned and combined analyses. The partitioning approach effectively offered alternative ways of getting more accurate estimation and analysis of fixed effects and variance components in the case of a non-full-rank design matrix scenario by considering the partitioned and combined analysis of experimental data based on the targeted factor level combinations and the desired inference scope. The study confirmed that the partitioning approach is compatible with the use of robust estimation methods, which resulted in improved precision in the model estimates. In addition, the partitioning approach was considered for multi-stratum experimental designs where randomization at different levels is necessary to achieve better model precision at different levels of the experiment such as the split-split-plot treatment structure, where all the three factors, each with both fixed and random levels, are laid in an RCBD. The essence of the approach was in manipulating the appropriate factor combinations in order to allow for narrow, intermediate and broad inferential space on the levels of each of the factors as well as their associated interactions. Furthermore, the approach proved to be useful beyond the fundamental consideration of homogeneous and uncorrelated error variance in the estimation process of linear mixed models. In essence, the study provides solutions for regaining the information that could be lost in various experimental designs if traditional analysis approaches are not improved.