We define and solve classes of sparse matrix problems that arise in multilevel modelling and data analysis. The classes are indexed by the number of nested units, with two-level problems corresponding to the common situation, in which data on level-1 units are grouped within a two-level structure. We provide full solutions for two-level and three-level problems, and their derivations provide blueprints for the challenging, albeit rarer in applications, higher-level versions of the problem. While our linear system solutions are a concise recasting of existing results, our matrix inverse sub-block results are novel and facilitate streamlined computation of standard errors in frequentist inference as well as allowing streamlined mean field variational Bayesian inference for models containing higher-level random effects.

The frost susceptibility of Australian commercial cereal crops, in particular wheat and barley, has become an economically devastating issue for growers. The relative risk to frost damage of the currently available varieties is obtained through testing varieties in a series of field experiments at locations susceptible to frost events (FEs). The experimental design, measurement protocols and resultant data from these frost expression experiments (FEEs) are complex due to the unpredictability of the timing and severity of FEs, and the maturity of the plants at the time of the events. Design and protocol complexities include the use of multiple sowing dates and the recording of plant maturity. Data difficulties include a high degree of unbalance, and in the instance of multiple frosts in a FEE, there is a longitudinal aspect. A linear mixed model analysis was adopted to accommodate these characteristics of individual FEEs and the multi-environment trial analysis of 17 FEEs. Finally, an approach is demonstrated for dissemination of results that are of use to both growers and breeders.

This study investigated bidirectional associations between intake of food groups and depressive symptoms in 1058 Italian participants (aged 20–102 years) of the Invecchiare in Chianti study. Dietary intake, assessed with a validated FFQ, and depressive symptoms, measured with the Center for Epidemiologic Studies Depression scale (CES-D), were assessed at baseline and after 3, 6 and 9 years. Associations of repeated measurements of intakes of thirteen food groups with 3-year changes in depressive symptoms, and vice versa, were analysed using linear mixed models and logistic generalised estimating equations. Fish intake was inversely (quartile (Q)4 v. Q1, B=–0·97, 95 % CI –1·74, –0·21) and sweet food intake positively (Q4 v. Q1, B=1·03, 95 % CI 0·25, 1·81) associated with subsequent CES-D score. In the other direction, higher CES-D scores were associated with decreases in intakes of vegetables (ratio: 0·995, 95 % CI 0·990, 0·999) and red and processed meat (B=–0·006, 95 % CI –0·010, –0·001), an increase in dairy product intake (ratio: 1·008, 95 % CI 1·004, 1·013), and increasing odds of eating savoury snacks (OR: 1·012, 95 % CI 1·000, 1·024). Fruit, nuts and legumes, potatoes, wholegrain bread, olive oil, sugar-sweetened beverages, and coffee and tea were not significantly associated in either direction. Our study confirmed bidirectional associations between food group intakes and depressive symptoms. Fish and sweet food intakes were associated with 3-year improvement and deterioration in depressive symptoms, respectively. Depressive symptoms were associated with 3-year changes in vegetable, meat, dairy product and savoury snack intakes. Trials are necessary to examine the causal associations between food groups and depression.

When herbicides are applied in mixture, and infestation by weeds is less than expected compared with when herbicides are applied alone, a synergistic effect is said to exist. The inverse response is described as being antagonistic. However, if the expected response is defined as a multiplicative, nonlinear function of the means for the herbicides when applied alone, then standard linear model methodology for tests of hypotheses does not apply directly. Consequently, nonlinear mixed-model methodology was explored using the nonlinear mixed-model procedure (PROC NLMIXED) of SAS System®. Generality of the methodology is illustrated using data from a randomized block design with repeated measures in time. Nonlinear mixed-model estimates and tests of synergistic and antagonistic effects were more sensitive in detecting significance, and PROC NLMIXED was a versatile tool for implementation.

One of the procedures used most recently with longitudinal data is linear mixed models. In the context of health research the increasing number of studies that now use these models bears witness to the growing interest in this type of analysis. This paper describes the application of linear mixed models to a longitudinal study of a sample of Spanish adolescents attending a mental health service, the aim being to investigate their knowledge about the consumption of alcohol and other drugs. More specifically, the main objective was to compare the efficacy of a motivational interviewing programme with a standard approach to drug awareness. The models used to analyse the overall indicator of drug awareness were as follows: (a) unconditional linear growth curve model; (b) growth model with subject-associated variables; and (c) individual curve model with predictive variables. The results showed that awareness increased over time and that the variable ‘schooling years’ explained part of the between-subjects variation. The effect of motivational interviewing was also significant.