Correlation and Regression

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Correlation and Regression

• A BRIEF overview

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Correlation Coefficients

• Continuous IV DV
• or dichotomous variables (code as 0-1)
• mean interpreted as proportion
• Pearson product moment correlation coefficient
  range -1.0 to 1.0

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Interpreting Correlations

• 1.0, or - indicates perfect relationship
• 0 correlations no association between the
  variables
• in between - varying degrees of relatedness
• r2 as proportion of variance shared by two
  variables
• which is X and Y doesnt matter

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Positive Correlation

• regression line is the line of best fit
• With a 1.0 correlation, all points fall exactly
  on the line
• 1.0 correlation does not mean values identical
• the difference between them is identical

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Negative Correlation

• If r-1.0 all points fall directly on the
  regression line
• slopes downward from left to right
• sign of the correlation tells us the direction of
  relationship
• number tells us the size or magnitude

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Zero correlation

• no relationship between the variables
• a positive or negative correlation gives us
  predictive power

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Direction and degree
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Direction and degree (cont.)
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Direction and degree (cont.)
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Correlation Coefficient

• r Pearson Product-Moment Correlation
  Coefficient
• zx z score for variable x
• zy z score for variable y
• N number of paired X-Y values
• Definitional formula (below)

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Raw score formula
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Interpreting correlation coefficients

• comprehensive description of relationship
• direction and strength
• need adequate number of pairs
• more than 30 or so
• same for sample or population
• population parameter is Rho (?)
• scatterplots and r
• more tightly clustered around linehigher
  correlation

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Examples of correlations

• -1.0 negative limit
• -.80 relationship between juvenile street crime
  and socioeconomic level
• .43 manual dexterity and assembly line
  performance
• .60 height and weight
• 1.0 positive limit

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Describing rs

• Effect size index-Cohens guidelines
• Small r .10, Medium r .30, Large r
  .50
• Very high .80 or more
• Strong .60 - .80
• Moderate .40 - .60
• Low .20 - .40
• Very low .20 or less
• small correlations can be very important

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Correlation as causation??
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Nonlinearity and range restriction

• if relationship doesn't follow a linear pattern
  Pearson r useless
• r is based on a straight line function
• if variability of one or both variables is
  restricted the maximum value of r decreases

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Linear vs. curvilinear relationships
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Linear vs. curvilinear (cont.)
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Range restriction
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Range restriction (cont.)
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Understanding r
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Simple linear regression

• enables us to make a best prediction of the
  value of a variable given our knowledge of the
  relationship with another variable
• generate a line that minimizes the squared
  distances of the points in the plot
• no other line will produce smaller residuals or
  errors of estimation
• least squares property

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Regression line

• The line will have the form Y'ABX
• Where Y' predicted value of Y
• A Y intercept of the line
• B slope of the line
• X score of X we are using to predict Y

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Ordering of variables

• which variable is designated as X and which is Y
  makes a difference
• different coefficients result if we flip them
• generally if you can designate one as the
  dependent on some logical grounds that one is Y

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Moving to prediction

• statistically significant relationship between
  college entrance exam scores and GPA
• how can we use entrance scores to predict GPA?

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Best-fitting line (cont.)
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Best-fitting line (cont.)
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Calculating the slope (b)

• Nnumber of pairs of scores, rest of the terms
  are the sums of the X, Y, X2, Y2, and XY columns
  were already familiar with

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Calculating Y-intercept (a)

• b slope of the regression line
• the mean of the Y values
• the mean of the X values

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Lets make up a small example

• SAT GPA correlation
• How high is it generally?
• Start with a scatter plot
• Enter points that reflect the relationship we
  think exists
• Translate into values
• Calculate r regression coefficients