Sec 3.8: The Coefficient of Determination

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Sec 3.8 The Coefficient of Determination
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Recall some facts about the correlation
coefficient

• It tells you whether or not two variables are
  linearly related to each other.
• It tells you whether that relationship is
  positive or negative.
• It indicates the strength of that relationship.

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A warning about the correlation coefficient

• Correlation does not imply causation.
• To be correlated means the two
• variables are related. Correlation tells
• you that as one variable changes the
• other seems to change in a predictable
• way. If you want to show one variable
• causes change in another you need to use
• a different kind of statistic.

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• The correlation coefficient also tells you how
  much variation in one variable is related to
  changes in the other variable.
• It is NOT a percentage. A correlation coefficient
  is a ratio. The coefficient of determination,
  denoted by r2 translates the correlation
  coefficient into a percentage.

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Definition 3.4 (p.124)

• The coefficient of determination is
• It represents the proportion of the sum of
• squares of deviations of the y values about
• their mean that can be attributed to a linear
• relationship between y and x.

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Practical Interpretation of the Coefficient of
Determination, r2

• About 100(r2) of the sample variation
• in y (measured by the total sum of
• squares of deviations of the sample y
• values about their mean) can be
• explained by (or attributed to) using x to
• predict y in the straight-line model.

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Example

• Suppose you compute a correlation
• coefficient and get r.9.
• What does that tell you about the relationship
  between x and y?
• The coefficient of determination is r2.81. This
  tells us that 81 of the variation in y can be
  explained by using x to predict y in the straight
  line model.

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Why calculate the coefficient of determination?

• Its easier for most people to understand
  percents.
• For example if the correlation coefficient on one
  set of data is r.80 and the correlation
  coefficient on another set of data is r.40 you
  cant say that the first set of data has a
  relationship that is twice as strong as the
  second set (because r is not a percentage).

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Comparing r and r2
r r2
1.00 1.00
0.90 0.81
0.80 0.64
0.70 0.49
0.60 0.36
0.50 0.25
0.40 0.16
0.30 0.09
0.20 0.04
0.10 0.01
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There is some relationship between the variables.
So if r.4, then r2.16.
There is a stronger relationship between the
variables. So if r.8, then r2.64.
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• Suppose a linear model is used to relate
• cost of mechanical work in construction
• (heating, ventilating, plumbing) to floor
• area. Suppose we have evidence to
• support the conclusion that floor area
• and mechanical cost are linearly related.
• And we find r2.35. This tells us
• that only 35 of the variation in
• mechanical cost is accounted for by
• differences in floor area. This could lead us
• to include other independent variables in the
• model to help account for the remaining 65
• of the variation in the mechanical cost not
• explained by the variation in floor area.

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Summary

1. The coefficient of correlation r is a measure of
   the strength of the linear relationship between
   two variables x and y.
2. The coefficient of determination r2 is a measure
   of percent of variation in one variable that is
   accounted for by the other variable.