## Estimation or prediction of random effects

In the previous model the patient terms were regarded as random effects. That is, they were defined as realisations of samples from a normal distribution with mean equal to zero, and with variance ap2. Thus, their expected values are zero.

We know, however, that patients may differ from one another, and the idea that all have the same expected value is counter-intuitive. We resolve this paradox by attempting to determine for each individual patient a prediction of the location within the normal distribution from which that patient's observations have arisen. This prediction will be affected by the prediction for all other patients, and will differ from the corresponding estimate in the fixed effects model. The predictions will be less widely spread than the fixed effects estimates and because of this they are described as shrunken. The extent of this shrinkage depends on the relative sizes of the patient and residual variance components. In the extreme case where the estimate of the patient variance component is zero, all patients will have equal predictions. Shrinkage will also be relatively greater when there are fewer observations per patient. Shrinkage occurs for both balanced and unbalanced data and the relevant formula is given in Section 2.2.3. Although, on technical grounds, it is more accurate to refer to predictions of random effects categories (e.g. of individual patients), in this book we will use the more colloquial form of expression, and refer to estimates of patient effects.

In our example, using the complete trial data, the random effects estimates can be obtained computationally using proc mixed. They are listed below along with the fixed effects patient means.

Patient number |
1 |
2 |
3 |
4 |
5 |
6 |

Fixed patients Random patients |
16.0 17.2 |
23.9 |
20.6 |

We observe that the mean estimates are indeed 'shrunken' towards the grand mean of 20.8. Shrinkage has occurred because patients are treated as a sample from the overall patient population.

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