BHS 307 Statistics for the Behavioral Sciences

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BHS 307 Statistics for the Behavioral Sciences

• Chapter Regression

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Regression Line

• A way of making a somewhat precise prediction
  based upon the relationships between two
  variables.
• Predictor variable criterion variable
• The regression line is placed so that it
  minimizes the predictive error.
• When based upon the squared predictive error the
  line is called a least squares regression line.

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Demo

• This demo from the textbooks student website
  shows how different lines result in different
  MSEs (mean square error)
• http//www.ruf.rice.edu/lane/stat_sim/reg_by_eye/
  index.html

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Least Squares Equation

• Y bX a
• To obtain Y
• Solve for b and a using the data from the
  correlation analysis
• Substitute b and a into the regression equation
  and solve for Y.
• To find points along the line, substitute X
  values into the regression equation and calculate
  Y.

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Textbook Approach

• Prediction using Z scores
• Zy b(Zx) where b r
• b is called the standardized regression
  coefficient because it is being used for
  prediction.
• Prediction using raw scores
• Change the persons raw score to a z-score using
  the z-score formula.
• Multiple by b, then change the resulting z-score
  back to a raw score.

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Formula for Regression Line

• Solving for b
• Solving for a
• Then insert both into formula
• Y bX a
• Plug in values of X and solve for Y.

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Standard Error of the Estimate

• The average amount of predictive error.
• Average amount actual Y values deviate from
  predicted Y values.
• No predictive error when r 1
• Extreme predictive error when r 0
• Again, formulas vary.

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Error Bars show the Standard Error of the
Estimate (Regression Line)
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Squared Correlation Coefficient

• r2 the square of the correlation coefficient
• Also called coefficient of determination
• Measures the proportion of variance of one
  variable predictable from its relationship with
  the other variable.
• It is the variance of the errors from
  repetitively predicting the mean, minus error
  variance using least squares, expressed as a
  proportion.

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Interpretation of r2

• r2 not r is the true measure of strength of
  association and the proportion of a perfect
  relationship.
• Large values of r2 are unusual in behavioral
  research.
• Large values of r2 do not indicate causation.
• Explained variance refers to predictability not
  causality.

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Regression Toward the Mean

• The mean is a statistical default use the mean
  to predict when r is 0 or unknown.
• Smaller values of r move the prediction toward
  the mean.
• The smaller r is, the greater the predictive
  error, hedged by moving toward the mean.
• Chance results in a regression to the mean with
  repeated measures.

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Regression Fallacy

• The statistical regression of extreme values
  toward the mean occurs due to chance.
• Israeli pilots praised for landings do worse on
  next landing.
• It is a mistake (fallacy) to interpret this
  regression as a real effect.
• Praise did not cause the change in landings.

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Testing for Regression Fallacy

• Divide the group showing regression into two
  groups (1) manipulation, (2) control without
  manipulation.
• Underachievers could show improvement due to
  regression upward to mean.
• Always include a control group for regression to
  the mean.