## Some Useful Definitions

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We conclude this introductory chapter with some definitions. The terms we are introducing here will recur frequently within subsequent chapters, and the understanding of these definitions and their relevance should increase as their applications are seen in greater detail. The terms we will introduce are containment, balance and error strata. In the analyses we will be presenting, we usually wish to concentrate on estimates of treatment effects. With the help of the definitions we are introducing we will be able to distinguish between situations where the treatment estimates are identical whether fixed effects models or mixed models are fitted. We will also be able to identify the situations where the treatment estimates will coincide with the simple average calculated from all observations involving that treatment. The first term we need to define is containment.

### 1.6.1 Containment

Containment occurs in two situations. First, consider the repeated measures data encountered in Section 1.4. In that hypertension trial, DBP was recorded at four visits after treatment had been started. In the analysis of that study, the residual variance will reflect variation within patients at individual visits. However, in this trial the patients receive the same treatment throughout, and so all the observations on a patient will reflect the effect of that one treatment on the patient. It can therefore perhaps be appreciated intuitively that it is the variation in response between patients which is appropriate for assessing the accuracy of the estimates of treatment effects rather than the residual or within-patient variation. We can see this more dramatically with a totally artificial set of data which might have arisen from this trial.

 Patient Treatment 1 2 3 4 1 A 80 80 80 80 2 B 85 85 85 85 3 B 85 85 85 85 4 A 91 91 91 91

In this situation there is no within-patient variation and the residual variance is zero. Thus, if the residual variance were used in the determination of the precision of treatment estimates, we would conclude that these data showed convincingly that treatment B produced lower DBPs than treatment A. Common sense tells us that this conclusion is ridiculous with these data, and that between-patient variation must form the basis for any comparison.

In the above situation we say that treatment effects are contained within patient effects.

The second situation where we can meet containment can also be illustrated with data from the hypertension trial, this time concentrating on the multi-centre aspect of the design. In Section 1.3 we actually met containment for the first time when dealing with Model E, and both centre effects and centre-treatment effects were fitted as random. we say in this context that the treatment effects are contained within centre-treatment effects. In fact, there is no requirement for the centre-treatment effects to be random for the definition of containment to hold. Thus, similarly in Model D, where the centre-treatment effects were regarded as fixed, we can still refer to the treatment effects as being contained within centre-treatment effects. It applies in general to any data with a hierarchical structure in which the fixed effects (treatment) appears in interaction terms with other effects.