Correlation

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Correlation
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Two variables Which test?
X
Contingency analysis
Logistic regression
Y
Correlation Regression
t-test
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Two variables Which test?
X
Contingency analysis
Logistic regression
Y
Correlation Regression
t-test
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Relationship Between Two Numerical Variables
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Relationship Between Two Numerical Variables
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Correlation

• What is the tendency of two numerical variables
  to co-vary (change together)?

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Correlation

• What is the tendency of two numerical variables
  to co-vary (change together)?
• Correlation coefficient r measures the strength
  and direction of the linear association between
  two numerical variables

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Correlation

• What is the tendency of two numerical variables
  to co-vary (change together)?
• Correlation coefficient r measures the strength
  and direction of the linear association between
  two numerical variables
• Population parameter r (rho)
• Sample estimate r

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Sum of squares X and Y
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Sum of products
Sum of squares X and Y
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Shortcuts
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r
r
r
r
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Correlation assumes...

• Random sample
• X is normally distributed with equal variance for
  all values of Y
• Y is normally distributed with equal variance for
  all values of X

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Correlation assumes...

• Random sample
• X is normally distributed with equal variance for
  all values of Y
• Y is normally distributed with equal variance for
  all values of X

Bivariate normal distribution
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Correlation coefficient facts

• -1 lt r lt 1 -1 lt r lt 1

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Correlation coefficient facts

• -1 lt r lt 1 -1 lt r lt 1
• Positive r variables increase together
• Negative r when one variable increases, the
  other decreases, and vice-versa

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Correlation coefficient facts

• -1 lt r lt 1 -1 lt r lt 1
• Positive r variables increase together
• Negative r when one variable increases, the
  other decreases, and vice-versa

uncorrelated
positive
negative
r0
r 1
r -1
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Correlation coefficient facts

• Coefficient of determination r2
• Describes the proportion of variation in one
  variable that can be predicted from the other

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Standard error of r
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Confidence Limits for r
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Example

• Are the effects of new mutations on mating
  success and productivity correlated?
• Data from Drosophila melanogaster
• n 31 individuals

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X is productivity, Y is the mating success

• Sum of products 2.796
• Sum of squares for X 16.245
• Sum of squares for Y 1.6289

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X is productivity, Y is the mating success
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Confidence Limits for r
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Confidence Limits for r
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Confidence Limits for r
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Confidence Limits for r
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Confidence Limits for r
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Confidence Limits for r
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Example Why Sleep?
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Example Why Sleep?

• 10 experimental subjects
• Measured increase in slow-wave activity during
  sleep
• Measured improvement in task after sleep -
  hand-eye coordination activity

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Example Why Sleep?
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Why sleep?

• Sum of products 1127.4
• Sum of squares X 2052.4
• Sum of squares Y 830.9
• Calculate a 95 C.I. for ?

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Hypothesis Testing for Correlations

• Can test hypotheses relating to correlations
  among variables
• Closely related to regression - the topic for
  next Tuesdays lecture

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Hypothesis Testing for Correlations

• H0 r 0
• HA r ? 0

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If r 0,...
r is normally distributed with mean 0
with df n -2
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Example

• Are the effects of new mutations on mating
  success and productivity correlated?
• Data from Drosophila melanogaster

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Hypotheses

• H0 Mating success and productivity are not
  related (r 0)
• HA Mating success and productivity are
  correlated (r ? 0)

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X is productivity, Y is the mating success

• Sum of products 2.796
• Sum of squares for X 16.245
• Sum of squares for Y 1.6289

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df n-231-229
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df n-231-229
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Why sleep?

• Sum of products 1127.4
• Sum of squares X 2052.4
• Sum of squares Y 830.9
• Test for a correlation different from zero in
  these data.

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Checking Assumptions for Correlation

• Bivariate normal distribution
• Relationship is linear (straight line)
• Cloud of points in scatter plot is circular or
  elliptical
• Frequency distributions of X and Y are normal

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Linear Relationship?
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Maximum correlation possible
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Maximum correlation possible
Correlation of zero
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Maximum correlation possible
Correlation of zero
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Cloud of points elliptical?
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X and Y normal?

• Use usual techniques for both X and Y separately
• Be wary of outliers

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Quick Reference Guide - Correlation Coefficient

• What is it for? Measuring the strength of a
  linear association between two numerical
  variables
• What does it assume? Bivariate normality and
  random sampling
• Parameter ?
• Estimate r
• Formulae

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Quick Reference Guide - t-test for zero linear
correlation

• What is it for? To test the null hypothesis that
  the population parameter, ?, is zero
• What does it assume? Bivariate normality and
  random sampling
• Test statistic t
• Null distribution t with n-2 degrees of freedom
• Formulae

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T-test for correlation
Null hypothesis ?0
Sample
Test statistic
Null distribution t with n-2 d.f.
compare
How unusual is this test statistic?
P gt 0.05
P lt 0.05
Reject Ho
Fail to reject Ho