## Random Coefficients Models 651 Introduction

A random coefficients model is an alternative approach to modelling repeated measures data. Here, a model is devised to describe arithmetically the relationship of a measurement with time. The statistical properties of random coefficients models have already been introduced in Sections 1.4.2 and 2.1.4. Here, we will consider in more depth the practical details of fitting these models and the situations in which they are most appropriate.

The most common applications are those in which a linear relationship is assumed between the outcome variable of interest and time. The main question of interest is then likely to be whether the rate of change in this outcome variable differs between the 'treatment' groups. Such an example was reported by Smyth et al. (199 7). They carried out a randomised controlled trial of glutathione versus placebo in patients with ovarian cancer who were being treated with cis-platinum. This drug has proven efficacy in the treatment of ovarian cancer, but has a number of adverse effects as well. Amongst these is a toxic effect on the kidneys. This effect can be monitored by the creatinine levels in the patients' blood. One of the hoped-for secondary effects of glutathione was to reduce the rate of decline of renal (kidney) function. This was assessed using a random coefficients model, but analysis showed no statistically significant difference between the rates of decline in the two treatment arms. Such an analysis may find widespread application in the analysis of 'safety' variables in clinical trials, because it is important to establish what effect new drugs may have on a range of biochemical and haematological variables. If these variables are measured serially, analysis is likely to be more efficient if based on all observations, using a method which will be sensitive to a pattern of rise or decline in the 'safety' variables. A further example in which the rate of decline of CD4 counts is compared in two groups of HIV-infected haemophiliacs will be presented in detail in Section 6.6.1.

In fitting linear random coefficients models, as described above, we will wish to fit fixed effects to represent the average rate of change of our outcome variables over time (i.e. a time effect) and we will assess the extent to which treatments differ in the average rate of change by fitting a treatment-time interaction. We will also require fixed effects to represent the average intercepts for each treatment (i.e. a treatment effect). In addition to the fixed effects representing average slopes and intercepts, the random coefficients model allows the slopes and intercepts to vary randomly between patients and cause a separate regression line to be fitted for each patient. This is achieved by fitting patient effects as random (to allow intercepts to vary) and patient-time as random to allow slopes to vary. These effects are used in the calculation of the standard errors of the time and treatment-time effects, which are our main focus of interest. Our basic model is therefore

Fixed effects: time, treatment, treatment-time,

Random effects: patient, patient-time.

The effects described above represent a minimum set of effects which will be considered in the model. Other patient characteristics, such as age and sex and their interactions with time, can readily be incorporated into the model, and we will see later that polynomial relationships and the effect of baseline levels can also be incorporated.

When the repeated measures data are obtained at fixed points in time, there will be a choice between the use of covariance pattern models and random coefficients models. This choice may be influenced by how well the dependency of the observations on time can be modelled, and whether interest is centred on the changing levels of the outcome variable over time, or on its absolute levels.

In many instances, the random coefficients model will be the 'natural' choice, as in the examples presented. If the times of observation are not standardised, or if there are substantial discrepancies between the scheduled times and actual time of observation, then random coefficients models are more likely to be the models of choice.