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# Chapter 1The need for more than one random-effect term when fitting a regression line

## 1.1 A data set with several observations of variable Y at each value of variable X

One of the commonest, and simplest, uses of statistical analysis is the fitting of a straight line, known for historical reasons as a regression line, to describe the relationship between an explanatory variable, X and a response variable, Y. The departure of the values of Y from this line is called the residual variation, and is regarded as random. It is natural to ask whether the part of the variation in Y that is explained by the relationship with X is more than could reasonably be expected by chance: or more formally, whether it is significant relative to the residual variation. This is a simple regression analysis, and for many data sets it is all that is required. However, in some cases, several observations of Y are taken at each value of X. The data then form natural groups, and it may no longer be appropriate to analyse them as though every observation were independent: observations of Y at the same value of X may lie at a similar distance from the line. We may then be able to recognize two sources of random variation, namely

• variation among groups
• variation among observations within each group.

This is one of the simplest situations in which it is necessary to consider the possibility that there may be more than a single stratum of random variation—or, in the language of mixed modelling, that a model ...

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