1. Design and Data Analysis in Psychology II
LESSON 4.2. MULTIPLE LINEAR REGRESSION. SEMIPARTIAL AND PARTIAL CORRELATION
Design and Data Analysis in Psychology II
Susana Sanduvete Chaves
Salvador Chacón Moscoso

2. SEMIPARTIAL CORRELATION
Example:

3. SEMIPARTIAL CORRELATION
Example:
Because X1 and X2 correlate

4. SEMIPARTIAL CORRELATIONY
Semipartial correlation a c b X1 X2
When X2 is included, R2 increases 0.15

5. SEMIPARTIAL CORRELATION
The order in which the independent variables are included in the model, influences the results. Example:
X1 is included firstly:
X2 is included firstly:
It is explained by X2
It is explained by X1

6. SEMIPARTIAL CORRELATION
The variable will explain less from the model:
As more correlated is with other variables.
As later it is introduced.
There are no rules to specify the entrance order. Usual criterion: The first variable is which presents the highest rXY (in the example, X1 would be the first one because rY1 > rY2)

7. MULTIPLE SEMIPARTIAL CORRELATION (MORE THAN TWO INDEPENDENT VARIABLES)Y Y X1 X4 X1 X3 X3 X2 X2

8. Exercise 1 about semipartial correlation (February 1999, ex. 3)
The variable intelligence (X1) explains the 55% of the variability of scholar performance. When hours studied (X2) is included, the explained variability is the 90%. Using this information and what you have in the following Venn diagram:

9. Exercise 1 about semipartial correlation
Calculate r12, ry1, ry2, Ry(1.2), Ry(2.1)
Complete de Venn diagram Y
0.3
0.3 X2 X1

10. Exercise 2 about semipartial correlation
Taking into account the following data:
Calculate

11. STATISTIC SIGNIFICANCE OF THE SEMIPARTIAL CORRELATION COEFFICIENT
Example: significance of
k1 = 2
theoreticalF= F(α,k-k1,N-k-1)

12. STATISTIC SIGNIFICANCE OF THE SEMIPARTIAL CORRELATION COEFFICIENT
Example: ¿ is significant in the model?

13. STATISTIC SIGNIFICANCE OF THE SEMIPARTIAL CORRELATION COEFFICIENT
F(0.05,2,6) = 5.14 – H0

14. PARTIAL CORRELATION
Definition of the partial correlation squared: Proportion of shared variability by Xi and Y, having ruled out Xk variability completely.

15. PARTIAL CORRELATION
Amount of variability shared by X1 and Y, having ruled out X2:
Amount of variability shared by X2 and Y having ruled out X1:

16. PARTIAL CORRELATION: EXAMPLEY a c b
0.1 X2 X1
Partial correlations

17. PARTIAL CORRELATION: EXAMPLE

18. DIFFERENCES BETWEEN PARTIAL AND SEMIPARTIAL CORRELATIONS (SQUARED)
NOMENCLATURE
(Without brackets)
DEFINITION
The ‘non-studied’ variable is previously included in the model
The ‘non-studied’ variable is erased from the model
(Y variability is reduced as explained variability by the erased variable has been ruled out)
FORMULA
(numerator in partial correlation)