Regression - Statistics for the Behavioral Sciences - Lecture Slides, Slides of Behavioural Science

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Statistics for the Behavioral

Sciences

Regression

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.

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’.

Error Bars show the Standard Error of

the Estimate (Regression Line)

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.

Calculating Predictive Error

2

( ) 2

2

−

− ′

−

= ∑^

n

Y Y n

SS s (^) y x y x

Definition Formula:

Computation Formula:

n

SS r

s y x y

Comparing the Regression Line to the

Mean

Mean of Y

Z Score Approach

 Prediction using Z scores:

 Z (^) y = β(Z (^) x ) where β = r  β is called the standardized regression coefficient because it is being used for prediction.

 Prediction using raw scores:

 Change the person’s raw score to a z- score using the z-score formula.  Multiple by β, then change the resulting z-score back to a raw score.

Interpretation of r^2

 r^2 – not r – is the true measure of strength of association and the proportion of a perfect relationship.

 Large values of r 2 are unusual in behavioral research.

 Large values of r 2 do not indicate causation.  “Explained variance” refers to predictability not causality.

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.

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.