Multiple Regression

1
Chapter 17

• Multiple Regression

2
Why multiple regression?

• One variable can be used to predict another.
• However, 2 or more variables may be able to
  predict even better.
• Uses
• Prediction
• Can also be used to determine which independent
  variable has the greatest effect
• Forming and confirming psychological theories

3
Terminology

• Criterion - Y, what it is that we are predicting
• Predictors - X1, X2, X3 etc.

4
The relationship between correlation and
(un)explained variation

• The scatter about the regression line can be
  represented by the variance at right.

5
The relationship between correlation and
(un)explained variation

• This ratio is the portion of the variance which
  remains unexplained
• It turns out that this portion is related to r by
  the following equation.

6
The relationship between correlation and
(un)explained variation

• Which means that r2 is the portion of the
  variance explained by the independent variable.

7
Example

• A group of schizophrenics are the subject of a
  study.
• We want to predict the time from the patients
  first psychotic episode to first relapse.
• We hope to do this based on two psychometrics
• Brief Psychiatric Rating Scale (positive
  subscale)
• Premorbid Adjustment Scale
• Generally, we expect that the time to relapse
  will decrease as either the Psychiatric Rating or
  the Premorbid Adjustment increase.

8
Example

• Suppose that the correlation r between
  Psychiatric Rating and time to relapse is -.4.
• This means that we have explained .42 .16 or
  16 of the variance of in time to relapse.
• 84 of the variance remains unexplained.

9
Example

• So we look to Premorbid Adjustment for further
  explanation.
• Say that the correlation r between Premorbid
  Adjustment and time to relapse is -.3.
• This means that we have explained another .32
  .09 or 9 of the variance of in time to relapse.
• So, now we are explaining 16 9 25 of time
  to relapse.
• Or, maybe not.

10
Example

• What if Psychiatric Rating and Premorbid
  Adjustment are highly correlated.
• This could mean that the two measures are largely
  the same thing, so we are just explaining the
  same thing twice.
• We can only add 16 and 9 if our 2 predictors
  are uncorrelated.
• r 0

11
Standardized regression equation

• Lets call the Psychiatric Rating X1.
• And, call the Premorbid Adjustment X2.
• Recall the simple form of the regression equation
  when there is only one predictor.

12
Standardized regression equation

• With 2 predictors it looks like this

13
Partial correlation

• We have considered the case with no correlation
  between predictors and the case with complete
  correlation between predictors.
• What about the case where there is partial
  correlation between predictors?

14
Notation

• We now have 3 correlation coefficients
• The correlation between X1 and Y
• The correlation between X2 and Y
• The correlation between X1 and X2.
• r1y and r2y are called validities.

15
Overlapping and non-overlapping explanations

• Suppose r12 .2.
• Then r122 .04.
• This is the overlapping area.
• Notice that we also lose .04 from each circle.

16
Overlapping and non-overlapping explanations

• Because region C is being explained twice, we
  need to reduce the weights ( r ) in our
  regression equation.
• These reduced weights are called beta weights.

17
Raw score prediction formula

• Follows the same form as in single regression.
• Where

18
Exercises

• Page 525
• 2 a, 7 a, 8 a

19
Overlapping and non-overlapping explanations

• Special cases
• r12 0, the regression equation reduces to the
  uncorrelated case.

20
Overlapping and non-overlapping explanations

• Special cases
• r12 1, r1y and r2y start to compete in the
  numerator.
• Eventually, one numerator hits 0.
• The denominator also moves toward 0 but not as
  fast.

21
How much variance have we explained with any
given multiple regression?

• The total amount of variance accounted for is
  denoted R2
• In the case where r12 0
• In general we must adjust for correlation between
  predictors.
• R2 is called the coefficient of multiple
  determination.
• R is called the multiple correlation coefficient.
  ( multiple R for short )

22
Geometric interpretation of the 2 predictor
regression equation

• Where before we had a line, now we have a plane.
• Notice that the plane can be very steep in one
  direction and not in the other.
• This depends on b1 and b2

23
Exercises

• Page 525
• 1, 2 (b c), 4 (a, b, d), 5 (a b),