Mixed models for multicategorical repeated response: modelling the time effect of physical treatments on strawberry sepal quality
Introduction
Postharvest and horticultural research often involves measuring quality change during storage and shelf life of horticultural commodities. In these types of experiments, quality characteristics of the same subject are often repeatedly and non-destructively assessed over time, typically resulting in a longitudinal data structure. The repeated measures over time are strongly correlated within subjects, but independent between the subjects. Ordinary regression techniques assume independence between the repeated measures and therefore are not appropriate to analyse the data (Verbeke and Molenberghs, 2000). Mixed-effects models allow compensation for the within-subject correlation structure, and even more, allow inclusion of between-subject variability in the model. The latter is of great benefit in describing the biological variability in a batch of fruit. A mixed-effects model generally contains random effects in addition to the fixed effects. For longitudinal data, the random component of the mixed model approach allows for subject-specific intercepts and slopes across time. Moreover, it allows for the presence of missing data and time-varying or invariant experimental variables (Pinheiro and Bates, 2000, Verbeke and Molenberghs, 2000).
Until now, quality measures followed over time have been assumed to be continuous (e.g. firmness), but in postharvest research the quality measure often takes on discrete values (e.g. fruit exhibiting strong, mild or no symptomology of a given disorder). Consequently, mixed-effects models for multicategorical or ordinal outcomes are recommended to analyse these data. Recently, an increasing amount of work has focused on generalised linear mixed models for non-continuous or multicategorical response data (GLMM) (Hedeker and Gibbons, 1994, Sheu, 2002). Although these statistical models have already been applied in biomedical and psychological studies, no literature was found on their application in postharvest research. In postharvest research the quality measurements are often characterised by a large inherent biological variability, and therefore GLMM is perfectly suited to extract more information from the repeated non-destructive quality measurements than have been done until now.
The objective of this paper is to illustrate, by means of a case study in postharvest research, the usefulness of generalised linear mixed models to analyse repeated quality measurements with a multicategorical response. GLMM was applied to study the effect of physical treatments, UV-C treatment and pulsed light treatment, on the visual quality of strawberry sepals. These treatments have proven beneficial in reducing microbial spoilage during storage of strawberries (Marquenie et al., 2002b, Marquenie et al., 2003). However, when the physical treatment is too intense it may affect the visual quality of the fruit. Fresh green sepals make the strawberries look much more attractive than brown dehydrated sepals.
Section snippets
Fruit material
The strawberry cultivar used for all experiments was Fragaria ananassa cv. Elsanta, since this is the most widely cultured variety in Belgium. The fruit was harvested at commercial harvesting time at the research centre Proeftuin Aardbeien en Houtig Kleinfruit, PCF Tongeren (Belgium). To enable almost year round experiments with strawberries, berries produced according to different culture methods (substrate in greenhouse, soil under plastic tunnels and field grown berries) were used.
Inoculation of the fruit
The
UV-C treatment
In Fig. 1, 3 sepal quality versus time profiles are shown for non-inoculated strawberries treated with a dose of 0.01 J/cm2. The large biological variability between the individual strawberries at day 1 increases with time and justifies the implementation of mixed models including random intercepts and slopes. To explore the data the average quality profile of 20 strawberries is shown for each of the different UV-C treatments (Fig. 2). The strawberry quality profiles for the highest UV-doses (1
Conclusions
Generalised linear mixed models are very well suited to describe the change with time of quality characteristics of individual fruit. The inclusion of random intercepts and slopes, allowed a description of the biological variability inherently present in batches of fruit. Based on the threshold concept formulated in the literature, these models were adapted for multicategorical response variables, which often occur in postharvest research. This statistical technique was applied to study the
Acknowledgements
This research is supported by the Ministry of Small Enterprises, Traders and Agriculture, Directorate of research and Development (Project S-5856). The research at the Flanders Centre of Postharvest Technology is supported by the Flemish Government. Jeroen Lammertyn is Postdoctoral Researcher of the Fund for Scientific Research Flanders (F.W.O.-Vlaanderen). Bart De Ketelaere is Postdoctoral Fellow with the Research Council of the Catholic University of Leuven.
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