Linear regression - 9 Linear regression Linear regression...

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9 Linear regression Linear regression modeling is an extremely powerful data analysis tool, useful for a variety of inferential tasks such as prediction, parameter estimation and data description. In this section we give a very brief introduction to the lin- ear regression model and the corresponding Bayesian approach to estimation. Additionally, we discuss the relationship between Bayesian and ordinary least squares regression estimates. One difficult aspect of regression modeling is deciding which explanatory variables to include in a model. This variable selection problem has a natural Bayesian solution: Any collection of models having different sets of regressors can be compared via their Bayes factors. When the number of possible regres- sors is small, this allows us to assign a posterior probability to each regression model. When the number of regressors is large, the space of models can be explored with a Gibbs sampling algorithm. 9.1 The linear regression model Regression modeling is concerned with describing how the sampling distribu- tion of one random variable Y varies with another variable or set of variables x = ( x 1 , . . . , x p ). Specifically, a regression model postulates a form for p ( y | x ), the conditional distribution of Y given x . Estimation of p ( y | x ) is made using data y 1 , . . . , y n that are gathered under a variety of conditions x 1 , . . . , x n . Example: Oxygen uptake (from Kuehl (2000)) Twelve healthy men who did not exercise regularly were recruited to take part in a study of the effects of two different exercise regimen on oxygen uptake. Six of the twelve men were randomly assigned to a 12-week flat-terrain running program, and the remaining six were assigned to a 12-week step aerobics program. The maximum oxygen uptake of each subject was measured (in liters per minute) while running on an inclined treadmill, both before and P.D. Hoff, A First Course in Bayesian Statistical Methods , Springer Texts in Statistics, DOI 10.1007/978-0-387-92407-6 9, c Springer Science+Business Media, LLC 2009
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150 9 Linear regression after the 12-week program. Of interest is how a subject’s change in maximal oxygen uptake may depend on which program they were assigned to. However, other factors, such as age, are expected to affect the change in maximal uptake as well. 20 22 24 26 28 30 -10 -5 0 5 10 15 age change in maximal oxygen uptake aerobic running Fig. 9.1. Change in maximal oxygen uptake as a function of age and exercise program. How might we estimate the conditional distribution of oxygen uptake for a given exercise program and age? One possibility would be to estimate a population mean and variance for each age and program combination. For example, we could estimate a mean and variance from the 22-year-olds in the study who were assigned the running program, and a separate mean and variance for the 22-year-olds assigned to the aerobics program. The data from this study, shown in Figure 9.1, indicate that such an approach is problematic.
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