Correlation

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Correlation

• Pearsons r

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CORRELATION
Pearson's r  
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Questions Asked by Regression Correlation
Analyses

• REGRESSION
• What is the estimated effect of X on Y?
• Is the slope steep or shallow?
• CORRELATION
• How good a fit are the data points to the
  regression line?
• How strong is the relationship between X and Y?

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Correlation Coefficient

• Produces a summary measure of how close points
  fall to the line.
• If points close to line there is little deviation
  ( ) little residual variance
• Conversely, the greater the deviation the lower
  the correlation.
• Measures how well do the IV and DV co-vary

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Positive Correlation
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Negative Correlation
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Interpreting the Correlation Coefficient
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Co-Variance How well do X and Y go together,
co-vary?What other variables co-vary?
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Heritability of Height
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What other variables Co-Vary?

• Education and income?
• Political interest and voter turnout?
• Robberies and poverty?

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How about the co-variation between the number
of churches in a city and the number of crimes?
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Why?
Correlation is not Causation!
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The Definitional Formula for Pearsons r
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Understanding the Correlation Coefficient
(r)Numerator degree to which X and Y vary
together.Denominator degree to which X and Y
vary separately (total amount of variance in X
and Y).
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r Ratio of Variances

• actual amount of variance
• r
• maximum possible amount of variance
• Key Question How well does the Independent
  Variable predict the Dependent Variable?

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If you did not know the relationship between the
X and Y variables, what would be the best
predictor for Y ?
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Two Forms of Residual Variance

• If we did not know beta e.g., we did not know
  the relationship between poverty and robberies,
  the mean of Y would be best predictor
• If we know the relationship know beta then
  best predictor is beta, the residual variance is

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Regression Line
Mean of Y
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Example

• Dependent Variable Robberies per 100,000
• Independent VariablePercentage of families
  below poverty line

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Interpreting r the correlation coefficient of
.359

• .359 is positive
• it is moderately strong
• The key Square the correlation coefficientr
  .359 ? r2 .129

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Interpreting r2

• r2 explains 12.9 of the variance of X on Y.
• Beta (knowing poverty level) improves prediction
  of robberies by 12.9 over the prediction using
  mean of Y.

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Next Example

• Dependent Variable Murders per 100,000
• Independent Variable Percentage of families
  below poverty line

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Last Example

• Dependent Variable Number of prior convictions
• Independent Variable Sentence length in years

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Interpretation 72 of variance explained,
while 28 remain unexplained.