Psychology 820

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Psychology 820

• Correlation
• Regression Prediction

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Concept of Correlation

• A coefficient of correlation (r or ? rho) is a
  statistical summary of the degree and direction
  of relationship or association between two
  variables (X and Y)
• Degree of Relationship
• Correlations range from 0 to 1.00
• Direction of Relationship
• Positive () relationship High score on X goes
  with a High score on Y
• Negative (-) relationship High score on X goes
  with Low score on Y

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The Bivariate Normal Distribution

• A family of three dimensional surfaces

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Scatterplots

• The chief purpose of the scatterplot is for the
  study of the nature of the relationship between
  two variables.
• Components of r
• Pearson Product Moment Correlation

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Additional Measures of Relationships

• Spearman Rank Correlation
• Both X and Y are ranks
• Phi Coefficient
• Both X and Y are dichotomies
• Point-Biserial Coefficient
• One dichotomous variable and one continuous
  measure
• Biserial Correlation
• One artificial dichotomy and one continuous
  measure
• Tetrachoric Coefficient
• Both X and Y are artificial dichotomies

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Linear and Curvilinear Relationships

• Only the degree of linear relationship is
  described by r or ?
• If there is a substantial nonlinear relationship
  between two variables, a different correlation
  coefficient (such as eta ?) should be used

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Linear Transformations and Correlation

• Any transformation of X or Y that is linear does
  not affect the correlation coefficient
• This includes transformations to z-scores,
  T-scores, addition of a constant to all values,
  subtracting multiplying or dividing by non-zero
  constants

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Effects of Variability on Correlation

• The variability (heterogeneity) of the sample has
  an important influence on r
• Range restriction

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Causation and Correlation

• Correlation must be carefully distinguished from
  causation.
• Third Variable Factor
• Effect of Outliers

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Regression and Prediction

• Prediction and correlation are opposite sides of
  the same coin
• Regression is usually the statistical method of
  choice when the predicted variable is an ordinal,
  interval, or ratio scale.
• Simple linear regression (1 IV 1 DV) extends to
  multiple regression (more than 1 IV)

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The Regression Effect

• The sons of tall fathers tend to be taller than
  average, but shorter than their fathers.
• The sons of short fathers tend to be shorter than
  average, but taller than their fathers.
• Regression to the Mean

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Regression Equation

• Y b X c (the equation of a straight line)
• Line of best fit
• Line of least-squares
• Prediction equation

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Proportion of Variance Interpretation of
Correlation

• The coefficient of determination (r2) is the
  proportion of variance in Y that can be accounted
  for by knowing X and, conversely, the proportion
  of variance in X that can be accounted for by
  knowing Y.
• The coefficient of nondetermination (k2) is the
  proportion of variance not accounted for

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Homoscedasticity
In a bivariate normal distribution the variance
of scores on Y will be the same for all values of
X (equal variance of Y scores for each value of
X) is known as homoscedasticity.

• This assumption means that the variance around
  the regression line is the same for all values of
  the predictor variable (X). The plot on the right
  shows a violation of this assumption. For the
  lower values on the X-axis, the points are all
  very near the regression line. For the higher
  values on the X-axis, there is much more
  variability around the regression line.

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Part Correlation

• It is the correlation of X1 (IQ) with X2
  (achievement posttest) after the portion of the
  posttest that can be predicted from the pretest
  has been removed.

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Partial Correlation

• Simple extension of part correlation
• The correlation of X1 and X2 with X3 held
  constant, removed, or partialed out is a partial
  correlation.

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Multiple Regression

• Multiple regression is the statistical method
  most commonly employed for predicting Y from two
  or more independent variables.

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Multiple Correlation

• The correlation between Y and Ypredicted when the
  prediction is based on two or more independent
  variables is termed multiple correlation