Correlation

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Correlation

• Overview
• Correlation is a useful statistical procedure
  that can be used to evaluate the relation between
  two variables
• In the case of Pearson correlation, which we will
  be focusing on, the two variables being
  investigated are continuous variables that are
  measured on an interval or ratio scale
• Correlation is frequently used when an
  observational experimental design is employed

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Correlation

• Overview
• e.g., the relation between years of education and
  income
• e. g., the relation between amount of coffee
  consumed and hours of sleep
• e.g., the relation between income and age (in
  years) at death

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Correlation

• Overview
• In each example, there are two variables, and
  each observation consists of a value in each
  variable
• e.g., take education and income. For each person
  in a sample it is necessary to record her/his
  education (say in years) and annual income (say
  in )

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Correlation

• Overview
• this information can be recorded in a table or a
  graph
• graphs are useful because they depict the nature
  of the relation between two variables
• Direction of relation
• Positive () correlation variables move in the
  same direction
• Negative (-) correlation variables move in the
  opposite direction

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Correlation

• Overview
• Form of relation
• Linear versus non-linear. Can you fit a straight
  line through the graph so that the magnitude of
  the error does not vary systematically? Pearson
  correlation measures the degree to which there is
  a linear relation between the two variables
• Degree of relation
• Pearson correlation 1 means data are fit
  perfectly by straight line going up -1 means
  data are fit perfectly by straight line going
  down 0 means there is no relation between two
  variables

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• When to use correlation
• Prediction.
• e.g. use test scores to predict job/university
  performance.
• Validity
• Validate a score on a test by calculating the
  correlation between the score on that test to a
  second test that has been validated

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Correlation

• When to use correlation
• Reliability
• Determine stability of a test by computing the
  correlation between the two tests
• Theory verification
• Determine whether predicted relation between two
  variables is found

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Correlation

• Pearson correlation
• Pearson correlation r, measures the magnitude of
  the linear relation between two variables
• r (covariability of X and Y) / (variability of
  X and Y separately)
• r SP / vSSX SSY
• Where sum of products SP S (X MX) (Y MY)
• And SSX S(X MX) 2 and SSY S (Y MY)2

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Correlation

• Pearson correlation
• sum of products SP S (X MX) (Y MY)
• Exploring SP
• What happens to SP if X, Y relation is positive?
  How about negative? How about no relation?

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Correlation

• Pearson correlation
• r SP / vSSX SSY
• 14/ v644
• 14/ 82
• .875

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• Interpreting Pearson correlation
• Correlation investigates the relation between two
  variables. It does not imply causality
• Eg., foot size and height are correlated
• Correlation is affected by the range of scores
  represented in the data
• E.g., SATs and success in post secondary
  education are strongly related, but GREs and
  success in graduate school are weakly correlated

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Correlation

• Interpreting Pearson correlation
• Extreme values sometimes called outriders or
  outliers can have a dramatic effect on
  correlations
• Coefficient of determination r2 measures the
  proportion of variability in one variable that
  can be determined from the relationship with
  another variable
• E.g., if r .70 then r2 .49 about 49 of
  the variability in the Y score can be predicted
  from the X score