Lecture 11 Chapter 6. Correlation and Linear Regression

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Lecture 11Chapter 6. Correlation and Linear
Regression
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• 6.1 Introduction
• This chapter is concerned with relationships
  between continuous variables.
• Example (see Handout 11)
• During the 1950s radioactive water leaked into
  the Columbia river in Washington DC. Data were
  collected on an exposure index (X), and the
  cancer mortality rate (Y) (deaths per 100,000
  per year) for the years 1959-1964, for each of
  nine counties downstream
• Exposure (x) 8.3 6.4 3.4 3.8
  2.6 11.6 1.2 2.5 1.6
• Mortality (y) 210 180 130 170
  130 210 120 150 140

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• Both the variables X and Y are measurements on a
  continuous scale.
• We are interested in how these two variables are
  related, or associated.
• As usual, the sensible thing to do first is to
  have a look at the data. The best thing to do
  here is to plot the mortality rate against the
  exposure index....

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• The plot suggests that there is a clear
  relationship (association) between the mortality
  rate and the exposure index. The relationship
  looks approximately linear (like a straight
  line).
• In this chapter we do two things
• Use a measure called correlation to describe the
  strength of the association between two
  variables.
• 2. Use a method called linear regression to
  model the relationship between two variables
  which are associated in a way which is
  approximately linear.

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• 6.2 Correlation
• There are a several different measures of
  association in usage, but we will only consider
  the most common, which is called Pearsons
  product moment correlation coefficient or more
  briefly the sample linear correlation coefficient
  or just the Pearson correlation. It is usually
  denoted by the letter r.

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Additional Notes (Slide 1 of 2)

• The value of r always lies between -1 and 1
• Values of r near to 1 indicate a strong positive
  linear relationship
• Values of r near to -1 indicate a strong negative
  linear relationship
• Values of r near to 0 indicate there is very
  little linear relationship.

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Additional Notes (Slide 2 of 2)

• Lets see what Minitab tells us about the Pearson
  correlation for our example above. We use
• StatgtBasic StatisticsgtCorrelation...
• Minitab tells us two things
• the Pearson correlation is r 0.917
• the P-value is 0.000

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• Note that this correlation is close to 1,
  indicating a strong positive linear relationship.
• What about the p-value?
• This is the result of the hypothesis test of the
  null hypothesis
• H0 The linear correlation in the population is
  zero.
• Our value of p 0.000 indicates that we reject
  the null hypothesis. There does appear to be a
  strong positive linear relationship between
  exposure and mortality.

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• The correlation coefficient r is a very useful
  summary measure, but it us often misused. Some
  points to remember are as follows
• 1. A high correlation does not necessarily imply
  a a cause-and-effect relationship.
• Although a value of r close to 1 does indicate a
  strong positive linear association, a linear
  relationship is not always the most appropriate.
  Always produce a plot of y against x.
• 3. A value close to zero indicates no linear
  relationship. That does not necessarily mean
  there is no relationship!

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For the data plotted below, r 0.020, and the
p-value is 0.854. This correctly identifies there
is no linear relationship, but there clearly is a
relationship!
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• 6.3 Simple Linear Regression
• The correlation coefficient tells us about the
  strength of a linear relationship, but it doesnt
  allow us to do things like make predictions about
  new data.
• For this we need a model for the data. If we
  think there is an approximately linear
  relationship, we use the equation of a straight
  line, which relates X and Y
• Y a ßX
• Here the values of a (alpha) and ß (beta) are the
  intercept and the slope of the straight line
  respectively. The slope, ß, is usually of much
  more interest, because it tells us how Y changes
  with X.

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• Since we dont expect the data to lie exactly on
  a straight line, we always add a random error
  component, e (epsilon), so the equation becomes
• Y a ßX e (Equation 1)
• Equation 1 is the equation of a simple linear
  regression. In order to use it to model our data,
  we need to choose the values of a and ß which
  work best.
• E.g. for the exposure-mortality data, we might
  obtain....

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• Notice that in the plot above, a has been chosen
  as 118.4, and ß as 9.03.
• This indicates that in our model, the mortality
  rate increases by 9.03 for every unit increase in
  the exposure index, and the mortality rate when
  the exposure index is zero is 118.4.
• But how were these values chosen?
• The usual criterion, and the one used above is to
  use the least squares estimates for a and ß...

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• We obtain these in Minitab using
• StatgtRegressiongtRegression...
• if we want the equation etc., and...
• StatgtRegressiongtFitted Line Plot...
• if we want the graph with the fitted line
  superimposed.