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**Unformatted text preview: **179 Multiple Regression in Practical Applications In Chapters 11–12 we defined and described the computa- tions and statistical inference procedures associated with t linear regression analysis. In this chapter suggestions are given about how to proceed to formulate regression models and progress through an analysis. Certainly the multiple linear regression tool is used to help accomplish one or more objectives:. • Often a major objective is to explain as much of the variation in Y as possible . Part of the variance in y values is due to the fact that the means of the Y populations are not all the same. By mod- elling the relationship among the Y population means, we “explain” that part of the overall variance in observed y ’s which is caused by differences in population means. The remainder of the variance will be “unexplained” or due to random error. • Certainly, the better the model the more of the vari- ance that will be explained, and that translates into better precision in prediction. So, a second objective – motivating the effort to explain as much variation, is predicting y ’s at unobserved levels of x ’s and esti- mating the Y population mean there. 180 Naturally, we must select the set of independent variables, the levels of each, and the specific functions of these at which to take observations. Then the observed data are used to refine the model, possibly adding or deleting func- tions of the x ’s, to arrive at the most satisfactory model. • Often a third objective in multiple regression analysis is simply determining which independent variables are most strongly related to Y , i.e., which ones in combination do the best job of predicting. Your textbook gives a good description of the steps once usually takes when conducting an analysis. We will fol- low the outline of the text. Selecting the Variables(Step 1) Normally, designating one or more Y variables is nat- ural. These are key variables of interest and learning more about their behavior is the main purpose of the study. Identifying potentially useful independent variables is not always so easy. Usually, people who are experts in the substantive area of the study will be called on to provide a list of possible x variables. Then data ( y ) will be col- lected at selected levels of these x variables. Usually many more x variables are identified than are actually needed to form a good model. Some subsets of the x ’s are often so closely related to one another that each provides about the same information about y , hence 181 only one variable from the subset is needed in the model. Identifying such redundancies is the major challenge in variable selection. One selection procedure which may be used is to do all possible regressions . This means fit every possi- ble submodel of the full model involving all independent variables. Then a “good” submodel is selected based on some criterion related to the model fits. Some criteria often used are: 1. Small s...

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