Linear%20Correlation

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Linear Correlation
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Perfect Correlation

• 2 variables x and y are perfectly correlated if
  they are related by an affine transform
• y ax b
• The correlation is positive if agt0 and negative
  if alt0.
• By corollary, 2 variables are perfectly
  positively correlated if and only if each pair of
  corresponding values has the same z-score.
• If the 2 variables are perfectly negatively
  correlated, corresponding z-scores will be equal
  in magnitude but opposite in sign.

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Pearsons r
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Scatterplots
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Pearsons r only measures linear dependence

• Two variables can have low correlation and still
  be highly dependent.

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Higher-Order Models
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Pearsons r depends on the range of the variables
under study

• r2 measures the proportion of variance in one
  variable accounted for by the other.
• If the range of variable X is restricted, it will
  account for less of the variance in Y.

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Pearsons r is Sensitive to Outliers
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Standard Definition of Correlation (Population)
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Standard Definition of Correlation (Sample)
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Alternative (Equivalent) Formula
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Computational Formula
For a population
For a sample
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Example 6130A 2005-2006 Assignment Marks
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End of Lecture 7

• Wed, Oct 29 2008

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Correlation and the Power of Matched Tests
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Correlation and the Power of Matched t-tests

• Now that we understand correlation, we can better
  understand the power of matched t-tests when
  scores in the two conditions are correlated.

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Recall formulae for standard error for
independent and matched tests

• Independent t-test

• Matched t-test

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Knowing the expected std error, we can estimate
the expected t-value

• Independent t-test

• Matched t-test

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The power of matched t-tests

• Large positive correlations between scores in the
  two conditions will mean a greater expected
  t-score for the matched design.
• But keep in mind that the critical value for the
  matched design will be somewhat larger as well,
  due to a smaller df.
• Which test is more powerful is decided by the
  exact tradeoff between these two effects.

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Applying Correlation Analysis
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Adjusted Correlation Coefficient
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Testing Pearsons r for Significance
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Underlying Assumptions (For Inference)

• Independent random sampling
• Bivariate normal distribution

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Applications of Pearsons r

• Measuring reliability and validity
• Examples
• e.g., test-retest reliability
• Split-half reliability
• Inter-rater reliability
• Criterion validity of self-report (correlate
  self-report against behavioural measure)
• Correlation between tests that are supposed to
  measure the same thing.
• Correlation between algorithmic model and human
  responses in behavioural studies.
• Measuring relationships between variables
  (correlational studies)
• e.g., frequency of cannabis and alcohol use
• Measuring relationships between IVs and DVs
  (experimental studies, when IV on interval/ratio
  scale
• e.g., exam performance as a function of alcohol
  consumption on previous night.

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Power Analysis for Pearsons r
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Confidence Intervals for Pearsons r

• Pearsons r is bounded on -1..1.
• Consequently, sampling distribution for r is not
  normal.
• Sampling distribution for rgt0 is negatively
  skewed.
• Sampling distribution for rlt0 is positively
  skewed.
• Thus confidence intervals are generally not
  symmetric.

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Fisher Transform

• Fisher transform (Appendix r') Method for
  symmetrizing r to facilitate calculation of
  confidence interval using standard normal table.

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Confidence Intervals on r
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End of Lecture 8

• Nov 5 2008

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Testing Difference of Pearson Correlations from 2
Independent Samples

• Converting the skewed r distribution to an
  (approximately) normal distribution allows
  straightforward two-sample testing

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Example
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