## Chapter 11 R examples:  Simple Linear Regression

# Reading the data into a temporary file called "my.datafile":

my.datafile <- tempfile()
cat(file=my.datafile, "  
1 1
2 1
3 2
4 2
5 4
", sep=" ")


# Name the data set and give the variable (column) a name:
stimulus <- read.table(my.datafile, header=FALSE, col.names=c("drug", "time"))

# Alternatively, could type:


# stimulus <- read.table("http://people.stat.sc.edu/hitchcock/reactiontime.txt", header=TRUE)

attach(stimulus)  # attaching the data frame

# A Scatterplot of the data (in R, the X-variable goes first):

plot(drug, time)

# Finding the least squares estimates for the simple linear regression:
# Make sure to put the response variable first!

SLR.stimul <- lm(time ~ drug)
SLR.stimul

# A more detailed output:

summary(SLR.stimul)

# The estimated values of the y-intercept and slope are given under "Estimates".
# We see that beta_0-hat is -0.1 and beta_1-hat is 0.7 
# so the estimated SLR model can be written as 
# Y-hat = -0.1 + 0.7 X.  
# Now this model can be used for predicting Reaction Time 
# for any particular drug amount. 

# Suppose we want to do a test for model usefulness by testing H_0: beta_1 = 0. 
# The test statistic for this test is given under "t value", in the "DRUG_X" row. 
# We see that t=3.66 for this test.  The p-value (for the TWO-TAILED alternative) 
# is given in the next column over, labeled "Pr(>|t|)".  
# We see the P-value is 0.0354.  So if our significance level alpha is .05, 
# we would reject H_0 and conclude the model is useful. 

# Our estimate of sigma (which we call s) is given next to "Residual standard error". 
# Note that in this example, s=0.60553.  This matches what we found in class. 

# Plotting the least squares line on top of the scatterplot of points:

abline(SLR.stimul)

# Finding the correlation coefficient between drug amount and reaction time:

cor(drug, time)


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