Lecture 4: Correlation and Regression

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Lecture 4Correlation and Regression

• Laura McAvinue
• School of Psychology
• Trinity College Dublin

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Correlation

• Relationship between two variables
• Do two variables co-vary / co-relate?
• Is mathematical ability related to IQ?
• Are depression and anxiety related?
• Does variable Y vary as a function of variable X?
• Does error awareness vary as a function of
  ability to sustain attention?
• Does accuracy of memory decline with age?

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Correlation

• Direction
• Do both variables move in the same direction?
• Do they move in opposite directions?
• Degree
• What is the degree or strength of the
  relationship?
• Analysis
• Scatterplot
• Correlation Coefficient
• Statistical significance

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Scatterplot

• Describe the relationship between the two
  variables using a scatterplot
• Visual representation of the relationship between
  the variables
• Plot each observation in the study, displaying
  its value on variable X and variable Y
• Place the predictor variable on the X axis
• The independent variable, which is making the
  prediction
• Place the criterion variable on the Y axis
• The dependent variable, which is being predicted

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• Participant Anxiety Depression
• 1 1
• 3 3
• 6 6

?
?
?
?
?
?
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No Relationship
Random Scatter
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Positive Relationship
Direction in scatter
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Negative Relationship
Direction in Scatter
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Sometimes, the direction of the relationship
might not be as obvious
What is the relationship between verbal coherence
and the number of pints of beer consumed?
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Regression Line

• Useful to add a regression line
• Model of the relationship
• Straight line that best represents the
  relationship between the two variables
• The line of best fit
• Helps us to understand the direction of the
  relationship

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Adding the regression line helps us see the
direction of the relationship
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Direction of Relationship

• Positive
• Two variables tend to move in the same direction
• As X increases, Y also increases
• As X decreases, Y also decreases
• Negative
• Two variables tend to move in opposite directions
• As X increases, Y decreases
• As X decreases, Y increases

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A Positive Relationship
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A Negative Relationship
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Degree of Relationship

• Degree or strength of relationship
• Calculate a correlation coefficient
• Pearson Product-Moment Correlation Coefficient
  (r)
• Statistic that varies between -1 and 1
• r 0, no relationship between the variables
• Change in X is not associated with systematic
  change in Y
• r 1, perfect positive correlation
• Increase in X associated with systematic increase
  in Y
• r -1, perfect negative correlation
• Increase in X associated with systematic decrease
  in Y

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Interpretation of r

• Perfect
• Negative
• relationship

Perfect Positive relationship
-1 0 1
Absolutely No relationship
Closer Pearson r is to one of the extremes, the
stronger the relationship between the variables
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Calculation of Pearson r

• Based on the covariance
• A statistic representing the degree to which two
  variables vary together
• Based on how an observation deviates from the
  mean on each variable

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Calculation of Pearson r

• Covariance is not suitable as measure of degree
  of relationship
• Absolute value is a function of standard
  deviations
• Scale the covariance by the standard deviations
• Pearson r

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Assessing Magnitude of r

• Cohens (1988) standards
• Small Medium Large
• .1 - .29 .3 - .49 .5 - 1
• Statistical Significance
• Test the null hypothesis that the true
  correlation in the population (rho) is zero
• Ho ? 0
• Calculate the probability of obtaining a
  correlation of this size if the true correlation
  is zero
• If p lt .05, reject Ho and conclude that it is
  unlikely that the results are due to chance, the
  correlation obtained represents a true
  correlation in the population

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Summary

• Interested in the relationship between two
  variables
• Direction and degree of relationship
• Scatterplot regression line
• Direction
• Correlation Coefficient
• Magnitude
• Statistical significance

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As temperature increases, ice-cream consumption
increases
r .73 (large) n 12 p .007
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As temperature increases, hot whiskey consumption
decreases
r -.908 (large) n 12 p lt.001
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Issues to consider

• Assumption of linearity
• Pearson correlation assumes there is a linear
  relationship between the two variables
• Assumes the relationship can be represented by a
  straight line
• It is possible that the relationship might be
  better represented by a curved line
• Examine scatterplot
• Curve-fitting procedures

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Linear?
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Non-linear?
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Non-linear
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Issues to consider

• Correlation can be affected by
• Range restrictions
• Heterogeneous subsamples
• Extreme observations
• Correlation does not mean causation

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Regression

• The regression line
• A straight line that represents the relationship
  between two variables
• Useful to add to the scatterplot to help us see
  the direction of the relationship
• But its much more than this
• Prediction
• Regression line enables us to predict Variable Y
  on the basis of Variable X

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Regression

• If you have an equation of the line that
  represents the relationship between Variables X
  Y, you can use it to predict a value of Y given a
  certain value of X.

Y 45
X 63
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Regression Equation
Regression Coefficients
Predicted value of Y
Predicting value of X
The basic equation of a line
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Regression Equation

• b
• The slope of the regression line
• The amount of change in Y associated with a
  one-unit change in X
• a
• The intercept
• The point where the regression line crosses the Y
  axis
• The predicted value of Y when X 0

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Regression Equation
Y
b
a
X
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Same intercept, different slopes
Same slope, different intercepts
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Summary

• The relationship between two variables, X Y
• Correlation
• Degree and direction of relationship
• Regression
• Predict Y, given X
• More on regression next lecture