Relationships Between Two Measurement Variables

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Relationships Between Two Measurement Variables

• Correlation and Linear Regression

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Examples

• Relationship between degrees Celsius and degrees
  Fahrenheit?
• Relationship between left foot size and right
  foot size?
• Relationship between Eric McCoos rushing yards
  and PSUs total points?
• Relationship between SAT score and GPA?

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Types of Relationships

• Deterministic if we know the value of one
  variable, we can determine the exact value of the
  second variable. A perfect relationship.
• Statistical If we know the value of one
  variable, we can estimate the typical, or
  average, value of the second variable. An
  approximate or not-so-perfect relationship.

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A Deterministic Relationship
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A Statistical Relationship
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Three Methods

• Graphical description using scatter plots
• Numerical description using correlation
• Numerical description using linear regression

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Scatter Plots

• Summarizes the relationship between two
  measurement variables.
• Horizontal axis represents one variable and
  vertical axis represents second variable.
• Plot one point for each pair of measurements.

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Scatter plot
(63,28)
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Scatter plot
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Scatter plot
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Correlation coefficient (r)

• A number, between -1 and 1, that measures how
  closely the values of two measurement variables
  fall on a straight line.
• Not appropriate for summarizing nonlinear
  relationships.

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Interpretation of r

• r 1 a perfect line with both variables
  increasingly simultaneously
• r -1 a perfect line with one variable
  decreasing as the other variable increases
• r 0 no linear relationship
• r gt 0 variables increase together in less than
  perfect line
• r lt 0 as one variable increases, the other
  decreases in less than perfect line.

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r 1
14
r 0
15
r 0.17
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r 0.91
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Correlation not appropriate
But r 0.71, deceivingly smaller than r for
females.
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r 0.50
With this game removed, r 0.28. One point can
greatly impact r.
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r - 0.84
Norway Finland
U.S.
Italy
France
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Linear Regression

• Procedure used to determine the formula for the
  line that best describes the relationship between
  the two measurement variables.
• The estimated line is called the least squares
  line, and has the form y a bx

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Least Squares Estimation
Deaths 260.5 - 23 liters
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Least Squares Line

• The line, y a bx, that minimizes the sum of
  the squared distances between all of the points
  and the line.
• Let statistical software find the line for you.

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Interpretation of Line

• Deaths 260.5 - 23 Liters
• Slope -23 For every additional liter of wine
  consumed per person, country reduces, on average,
  number of heart disease deaths by 23.
• Intercept 260.5 If number of liters of wine
  consumed per person is 0, country will have 260.5
  heart disease deaths.

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Use of Line for Prediction

• Deaths 260.5 - 23 Liters
• Line predicts average value of y for a given
  value of x.
• If 8 liters consumed, predict that country will
  have average of 260.5-23 8, or 76.5 deaths per
  100,000 people.

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Do not extrapolate!

• Do not use the least squares line to predict
  values outside of the range of the original data.
  The relationship may not hold outside of this
  range.
• Deaths 260.5 - 23 Liters
• If 15 liters of wine consumed per person, heart
  disease deaths is 260.5 - (23 15), or -85
  deaths per 100,000 ?!?

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Least Squares Estimation
After 23.6 0.87 Before
r 0.69
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Least Squares Estimation
After 23 0.80 Before
r 0.84
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Comparison of Pulse Rates

• Mens relationship (r0.84) appears to be
  stronger than female relationship (r0.69).
• Increase in pulse rate appears to be greater in
  females (b 0.87) than males (b 0.80)
• Before pulse rate of 70
• Predicted After Pulse for Males 79
• Predicted After Pulse for Females 84.5

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Cautionary Notes

• We have only studied how to describe sample data.
  We did not learn how to test to see if the
  relationships hold in the population.
• Correlation does not mean causation!