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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 = β 0 + β 1 x i 1 + β 2 x i 2 + …+ β k x i k + ε i Y i = response value for i th individual x ij = value of the j -th independent variable for the i th individual β 0 = Intercept of regression equation β j = coefficient of the j -th independent variable ε 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 = β 0 + β 1 x i 1 + β 2 x i 2 + β 3 x i 3 + ε i • We assume the random errors ε i have mean 0 (and variance σ 2 ), so that E( Y ) = β 0 + β 1 x 1 + β 2 x 2 + β 3 x 3 .
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• We again estimate β 0 , β 1 , β 2 , β 3 , etc., from the sample data using the principle of least squares. • For multiple linear 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 variables are held constant .
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