Examining Linear Relationships: Correlation and Regression

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Examining Linear Relationships Correlation and
Regression
Topics Scatterplots Correlation review Least
squares regression line Interpretation of
regression model
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Examining Linear Relationships Correlation and
Regression

• We can examine the relationship between two
  numerical variables by observing a scatterplot.
• The simplest relationship is linear

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Creating the Scatterplot

• In regression analysis the variable of main
  interest is called the response variable
• The variable that we believe affects or drives
  the response variable is called the explanatory
  variable

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Creating the Scatterplot

• We always put the explanatory variable on the
  horizontal axis and the response variable on the
  vertical axis.
• If a store manager believes that large
  promotional expenditures cause larger values of
  sales he should put Sales on the vertical axis
  and Expenditure on the horizontal axis.

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Examining Linear Relationships Correlation and
Regression

• The relationship is strong if the points in a
  scatterplot cluster tightly around some straight
  line. If this line rises form left to right then
  the relationship is positive. If it falls from
  left to right then the relationship is negative.

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Examining Linear Relationships Correlation and
Regression

• The correlation coefficient is a numerical
  measure of the strength of the linear
  relationship between two variables.
• r 1 for perfect positive
• r -1 for perfect negative

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Assessing Linear Relationships with Correlation
Coefficient r
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Regression Line

• Whenever the correlation is high enough to
  indicate at least a moderately strong linear
  relationship between the two variables we usually
  draw a line through the scatterplot points to
  model the relationship.
• The best fitting line through the points is
  called the regression line.

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Regression Line

• The regression line is a straight line that
  describes how the response variable Y changes as
  an explanatory variable X changes.
• The ultimate goal is often to use the regression
  line to predict the value of Y for a given value
  of X

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Least Squares Estimation

• The best fitting (regression) line through the
  points is obtained by satisfying the following
  condition
• The sum of the squares of the vertical distances
  from the sample points to the line must be as
  small as possible.
• Method known as Least squares estimation.

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Scenario for Regression Example

• To see how effective their advertising and other
  promotional activities are, the Pharmex drugstore
  collected data from 50 randomly selected
  metropolitan regions.
• In each region it compared Pharmexs promotional
  expenditures and sales to those of the leading
  competitor in the region over the past year.

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Regression Scenario Variable Names

• Promote Pharmexs promotional expenditures as a
  percentage of those of the leading competitor
• Sales Pharmexs sales as a percentage of those
  of the leading competitor

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Regression Scenario Scatterplot
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Regression Scenario Least Squares Equation
predicted Sales 25.126 0.762Promote
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Generic Equation of a Straight Line
Y a bX Slope, b the change in the mean
value of Y for each unit increase in X Intercept,
a (theoretically) the mean value of Y when X
equals zero. May not have a practical meaning.
Valid only if data includes observations with X
0.
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Interpretation of Slope and Intercept Pharmex
Regression
predicted Sales 25.126 0.762Promote
intercept slope Slope For each unit increase in
the promotional expenses index, sales index is
expected to increase by about 0.76 Intercept In
a region that does no promotions the sales index
is expected to be about 25.1 (theoretical)
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Causation

• Unless the data is obtained in a carefully
  controlled experiment we should never make
  definitive statements about causation in
  regression analysis.
• Reason - we can almost never rule out the
  possibility that some other (lurking) variable is
  causing the variation in both of the observed
  variables.