The Analysis of Designed Experiments and Longitudinal Data by Using Smoothing Splines

Abstract

In designed experiments and in particular longitudinal studies, the aim may be to assess the effect of a quantitative variable such as time on treatment effects. Modelling treatment effects can be complex in the presence of other sources of variation. Three examples are presented to illustrate an approach to analysis in such cases. The ®rst example is a longitudinal experiment on the growth of cows under a factorial treatment structure where serial correlation and variance heterogeneity complicate the analysis. The second example involves the calibration of optical density and the concentration of a protein DNase in the presence of sampling variation and variance heterogeneity. The ®nal example is a multienvironment agricultural ®eld experiment in which a yield±seeding rate relationship is required for several varieties of lupins. Spatial variation within environments, heterogeneity between environments and variation between varieties all need to be incorporated in the analysis. In this paper, the cubic smoothing spline is used in conjunction with ®xed and random effects, random coef®cients and variance modelling to provide simultaneous modelling of trends and covariance structure. The key result that allows coherent and¯exible empirical model building in complex situations is the linear mixed model representation of the cubic smoothing spline. An extension is proposed in which trend is partitioned into smooth and nonsmooth components. Estimation and inference, the analysis of the three examples and a discussion of extensions and unresolved issues are also presented.1999 Royal Statistical Society 0035±9254/99/48269 Appl. Statist. (1999) 48, Part 3, pp. 269^311 treatments, e.g. time in the longitudinal setting, the interaction of treatments with the quantitative variable is generally of interest. The following experiments illustrate such situations and provide the motivation for the methods discussed in this paper. Data sets for the examples can be accessed at http://www.blackwellpublishers.co.uk/rss/ 1.1. Example 1: live-weights of cows An experiment was carried out to study the eects of altered dietary iron intake in dairy cattle infected with Mycobacterium paratuberculosis (see Lepper et al. (1989)). Previous studies on mice had established that the multiplication of the organism was inhibited by low rather than high dietary iron intake. The aim of the study on cows was to examine the long-term eects of iron dosing on the progress of the organism. Several variables were measured in the experiment, with the focus on the live-weight data in this paper. At the start of the experiment, 28 female calves, at most 2 weeks old, were subdivided into two groups of eight and 20 animals and accommodated in two yards. 4 weeks later, the group of 20 was inoculated with the organism (the infected group) and this is taken as time 0. The ®rst live-weights were taken 122 days later and were recorded to the nearest 5 kg. Iron dosing began on day 155 (5 days after the second measurement of live-weight). The two groups were subdivi...

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