Correlation and Regression

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Chapter 3

• Correlation and Regression

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Overview

• Bivariate distributions two scores for each
  individual
• Scatter plot a visual method of looking at the
  association between two variables
• Correlation is not causation

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Correlation coefficient

• Describes the direction and magnitude of a
  relationship
• Negative correlation As X increases, Y
  decreases
• Positive correlation As X increases, Y
  increases
• Magnitude can vary from 0 to 1
• Usually describes linear relationships

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Regression line

• Use to predict scores on a second variable based
  on the first variable
• The regression line is the best-fitting straight
  line through a set of data points
• Uses the principle of least squares which
  minimizes the squared deviations around the
  regression line

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Regression coefficient or slope
The slope describes how much change is expected
in Y each time X increases by one unit The top
part of the equation calculates how much X and Y
covary The bottom part of the equation calculates
how much variability there is in X
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Intercept, a
The intercept, a, is the value of Y when X is
zero, i.e., it is where the regression line
crosses the Y axis
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The Regression Equation
Once b and a are known, for any value of X, Y can
be predicted When Z-scores are used, the
correlation coefficient is the same as b and a0
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Statistical significance of a correlation
coefficient
Tests the null hypothesis that there is no
relationship between the 2 variables Degrees of
freedom (df) N -1 N is the number of
subjects Can also use statistical tables for
critical values of r
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Other correlation coefficients

• Pearson r requires that both variables are
  continuous measures
• If one variable is continuous and the other is
  dichotomous, use biserial or point biserial r
• If both variables are dichotomous, use phi or
  tetrachoric r
• If variables are ranks, use Spearmans rho

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Coefficient of Determination

• r2 is the coefficient of determination
• Is the proportion of the total variation in
  scores on Y that we know as a function of
  information about X
• Example Correlation between ACT scores and
  freshman GPA is .30.
• .302 .09, i.e., 9 of the variability in
  freshman GPA can be accounted for by ACT scores

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Issues in correlation

• Shrinkage the regression equation created on
  one group doesnt work as well when applied to a
  second group
• Association does NOT imply causation
• Third variable problem
• Restriction of range