# Math 322 Biostatistics
# Lecture, April 13, 2009
# R examples for chapter 11.9 (Multiple regression)
# Geir Arne Hjelle (hjelle@math.wustl.edu)

# Read in data about bloodpressure, birthweight and age (in days)
bp <- read.table("http://www.math.wustl.edu/~hjelle/m322/r/bloodpressure.txt",
                 header = TRUE)
attach(bp)

# One way to get a quick overview of the data is to plot them. If you try
# to plot a dataframe containing more than 2 variables, R will make a matrix
# of plots of all the possible pairs of variables.
plot(bp)

# To carry out the least squares method, we specify a linear model. To say
# that we have two explanatory variables we use +.
lm(BloodPressure ~ Birthweight + Age)

# Note that these partial regression coefficients are different from the
# ones we obtain by doing simple regression on one of the variables.
lm(BloodPressure ~ Birthweight)
lm(BloodPressure ~ Age)

# To compare the coefficients to find which is most important, we need
# to standardize them, so that they are on the same scale. This is done
# by multiplying and dividing by the standard deviations of the data.
coeff.Birthweight <- coefficients(lm(BloodPressure ~ Birthweight + Age)
                                  )["Birthweight"]
coeff.Birthweight * sd(Birthweight) / sd(BloodPressure)

coeff.Age <- coefficients(lm(BloodPressure ~ Birthweight + Age))["Age"]
coeff.Age * sd(Age) / sd(BloodPressure)

# These can be thought of as correlation coefficients where we have adjusted
# for the other variable(s). They will not equal the correlation coefficients
# from simple regression.
cor(Birthweight, BloodPressure)
cor(Age, BloodPressure)

# Tests for the regression is again done by applying summary to our linear
# model
summary(lm(BloodPressure ~ Birthweight + Age))