chapter  2
Simple Models: Definitions of Error and Parameter Estimates
Pages 15

Although we will sometimes want to consider a speciļ¬ed or hypothesized value of B0 in order to ask an explicit question about data, it is much more common to consider the equation:

Yi = Ī²0 + Īµi

where Ī²0 is a true parameter that is estimated from the data. Continuing with the medical example, suppose that the body temperatures were all from people who had taken a certain drug. We might suspect that, except for error, they all have the same body temperature, but it is not the usual body temperature of 37Ā°C. We use Ī²0 to represent whatever the body temperature might be for those who

have taken the drug. It is important to realize that Ī²0 is unknowable; we can only estimate it from the data. In terms of these true values, Īµi is the amount by which Yi diļ¬ers from Ī²0 if we were ever to know Ī²0 exactly. We use b0 to indicate the estimate of Ī²0 that we derive from the data. Then the predicted value for the ith observation is:

YĖi = b0

and

DATA = MODEL + ERROR

becomes

Yi = b0 + ei

where ei is the amount by which our prediction misses the actual observation. Thus, ei is the estimate of Īµi. The goal of tailoring the model to provide the best ļ¬t to the data is equivalent to making the errors:

ei = Yi ā b0

as small as possible. We have only one parameter, so this means that we want to ļ¬nd the estimate b0 for that one parameter Ī²0 that will minimize the errors. However, we are really interested not in each ei but in some aggregation of all the individual ei values. There are many diļ¬erent ways to perform this aggregation. In this chapter, we consider some of the diļ¬erent ways of aggregating the separate ei into a summary measure of the error. Then we show how each choice of a summary measure of the error leads to a diļ¬erent method of calculating b0 to estimate Ī²0 so as to provide the best ļ¬t of the data to the model. Finally, we consider expressions that describe the ātypicalā error.