Leverage and Confidence in Models

An important experimental question relates to how well quantitative information can be predicted after a series of experiments has been carried out. For example, if observations have been made between 40 and 80 °C, what can we say about the experiment at 90 °C It is traditional to cut off the model sharply outside the experimental region, so that the model is used to predict only within the experimental limits. However, this approaches misses much information. The ability to make a prediction often reduces smoothly from the centre of the experiments, being best at 60 °C and worse the further away from the centre in the example above. This does not imply that it is impossible to make any statement about the response at 90 °C, simply that there is less confidence in the prediction than at 80 °C, which, in turn, is predicted less well dian at 60 °C. It is important to be able to visualise how the ability to predict a response (e.g. a synthetic yield or a concentration) varies as the independent factors (e.g. pH, temperature) are changed. 

When only one factor is involved in the experiment, the predictive ability is often visualised by confidence bands. The size of these confidence bands depends on the magnitude of the experimental error. The shape , however, depends on the experimental design, and can be obtained from the design matrix (Section 2.2.3) and is influenced by the arrangement of experiments, replication procedure and mathematical model. The concept of leverage is used as a measure of such confidence. The mathematical definition is given by 

Calculate (D D) l. Note that this matrix is also used in the /-test, as discussed in Section 2.2.4.3. 

These diagonal values are the values of leverage for each experiment. 

This numerical value of leverage has certain properties. 

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