Introduction to Linear Regression Analysis. Douglas C. Montgomery

Introduction to Linear Regression Analysis - Douglas C. Montgomery


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of whether the patient is a surgical or medical patient (0 = surgical, 1 = medical), and an index measuring the patient’s anxiety level. We start by building a simple linear regression model relating the response variable satisfaction to severity.

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      Low values for R2 occur occasionally in practice. The model is significant, there are no obvious problems with assumptions or other indications of model inadequacy, but the proportion of variability explained by the model is low. Now this is not an entirely disastrous situation. There are many situations where explaining 30 to 40% of the variability in y with a single predictor provides information of considerable value to the analyst. Sometimes, a low value of R2 results from having a lot of variability in the measurements of the response due to perhaps the type of measuring instrument being used, or the skill of the person making the measurements. Here the variability in the response probably arises because the response is an expression of opinion, which can be very subjective. Also, the measurements are taken on human patients, and there can be considerably variability both within people and between people. Sometimes, a low value of R2 is a result of a poorly specified model. In these cases the model can often be improved by the addition of one or more predictor or regressor variables. We see in Chapter 3 that the addition of another regressor results in considerable improvement of this model.

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      The purpose of this section is to introduce readers to SAS and to R. Appendix D gives more details about using SAS, including how to import data from both text and EXCEL files. Appendix E introduces the R statistical software package. R is becoming increasingly popular since it is free over the Internet.

      Table 2.7 gives the SAS source code to analyze the rocket propellant data that we have been analyzing throughout this chapter. Appendix D provides detail explaining how to enter the data into SAS. The statement PROC REG tells the software that we wish to perform an ordinary least-squares linear regression analysis. The “model” statement specifies the specific model and tells the software which analyses to perform. The variable name to the left of the equal sign is the response. The variables to the right of the equal sign but before the solidus are the regressors. The information after the solidus specifies additional analyses. By default, SAS prints the analysis-of-variance table and the tests on the individual coefficients. In this case, we have specified three options: “p” asks SAS to print the predicted values, “clm” (which stands for confidence limit, mean) asks SAS to print the confidence band, and “cli” (which stands for confidence limit, individual observations) asks SAS to print the prediction band.

      Table 2.8 gives the SAS output for this analysis. PROC REG always produces the analysis-of-variance table and the information on the parameter estimates. The “p clm cli” options on the model statement produced the remainder of the output file.

       TABLE 2.7 SAS Code for Rocket Propellant Data

data rocket;
input shear age;
cards;
2158.70 15.50
1678.15 23.75
2316.00 8.00
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