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

SEMIPARTIAL CORRELATION Example:

SEMIPARTIAL CORRELATION Example: Because X1 and X2 correlate

SEMIPARTIAL CORRELATION Y Semipartial correlation a c b X1 X2 When X2 is included, R2 increases 0.15

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

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)

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

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:

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

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

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

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

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

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

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:

PARTIAL CORRELATION: EXAMPLE Y a c b 0.1 X2 X1 Partial correlations

PARTIAL CORRELATION: EXAMPLE

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)