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#
#	Topic: Generalized Linear Models 
#
#
#	Script: 20101122.R
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#	Date: 22 November 2010
#
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# Read in the data from the file "crime.csv"
# The 'header=TRUE' parameter tells R that the 
# first row contains variable names

gd <- read.csv("http://oforsber.kvasaheim.com/courses/stat4073/data/crime.csv", header=TRUE)
attach(gd)

# This tells you the names of the variables in the data set

names(gd)

# Let us now try to determine how much the violent crime rate
# in 1990 is dependent on the wealth of the state in 1990 (gsp90),
# the numbers of citizens' initiatives used (inituse), the inherent
# level of republicanism in the state (repub) and the total 
# population in 1990.

# First, a linear model (which assumes Normality)
m1 <- glm(vcrime90 ~ gsp90 + inituse + repub + pop90)
summary(m1)
# How would you determine if the Normality assumption was violated?


# Second, a linear model (which assumes Normality), but transforming the
# dependent variable (as it was bounded below)
m2 <- glm(vcrime90 ~ gsp90 + inituse + repub + pop90, family=gaussian(link=log) )
summary(m2)


# Now, let's predict initiative use from these four varaibles
m3 <- glm( inituse ~ gsp90 + vcrime90 + repub + pop90, family=poisson(link=log) )
summary(m3)
# Note that inituse is a count variable. Thus, we use a different family
# and a different link function (why the log link?)


# Finally, let us predict 'South=ness' from the other four variables. In
# reality this would not make sense, but I wanted to demonstrate how one
# would model a binary dependent variable
m4 <- glm(census4=="South" ~ gsp90 + inituse + repub + pop90, family=binomial(link=logit) )
summary(m4)
# Here, you are actually just modeling the underlying probability that a person
# is Southern. The link takes that 0-1 bounded probability variable and unbounds
# it. Other links available include loglog, cloglog, and probit. Of these, only
# the logit, probit, and cauchit are symmetric. Also, of these, only the logit 
# and probit are frequently used.