notes509fall11sec63 - STAT 509 Sections 6.3-6.4 More on...

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STAT 509 Sections 6.3-6.4: More on Regression Simple linear regression involves using only one independent variable to predict a response variable. • Often, however, we have data on several independent variables that may be related to the response. In that case, we can use a multiple linear regression model: Y i = 0 + 1 x i 1 + 2 x i 2 + …+ k x i k + i Y i = response value for i th individual x ij = value of the j -th independent variable for the i th individual 0 = Intercept of regression equation j = coefficient of the j -th independent variable i = i th random error component Example (Table 6.34): Data are measurements on 25 coal specimens. Y = coking heat (in BTU/pound) for i th specimen X 1 = fixed carbon (in percent) for i th specimen X 2 = ash (in percent) for i th specimen X 3 = sulfur (in percent) for i th specimen Y i = 0 + 1 x i 1 + 2 x i 2 + 3 x i 3 + i We assume the random errors i have mean 0 (and variance 2 ), so that E( Y ) = 0 + 1 x 1 + 2 x 2 + 3 x 3 .
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We again estimate 0 , 1 , 2 , 3 , etc., from the sample data using the principle of least squares. • For multiple l inear regression, we will always use software to get the estimates b 0 , b 1 , b 2 , b 3 , etc. Fitting the Multiple Regression Model Given a data set, we can use R to obtain the estimates b 0 , b 1 , b 2 , b 3 , …that produce the prediction equation with the smallest possible SS res = R code for example: > my.data <- read.table(file = "http://www.stat.sc.edu/~hitchcock/cokingheatdata.txt", col.names=c('x1','x2','x3','y'), header=FALSE) > attach(my.data) > lm(y ~ x1 + x2 + x3) Least squares prediction equation here: We interpret the estimated coefficient b j as estimating the predicted change in the mean response for a one- unit increase in X j , given that all other independent
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