X Y

Variance Covariance Correlation

Scatter plot

Relations and Associations Y X

The purpose of regression is to explain the variability in Y from the information on X given that X and Y are linearly related. The distribution of Y is also called the unconditional distribution of Y is a sample estimate of the unconditional population mean. is a sample estimate of the conditional population mean.

X Y SS Y SS res SS reg

X Y SS Y SS res Objective of research Misses Imperfection of Theory Hits Theory or model

The distribution of Y at a given level of X is called the conditional distribution of Y. It should have smaller variance than the unconditional distribution. s 2 y is an estimate of the unconditional population variance. s 2 y.x is an estimate of the conditional population variance which is also called “residual variance.”

Fit a line to best represent the scatter points. ß0ß0 ß1ß1

ß 0 or intercept is the value of Y when X=0. ß 1 or regression coefficient is value change in Y associated with one unit change in X.

The line represents the predicted value of Y at a given level of X, The scatter points represent the actual value of Y at a given value of X Ordinary least Squares (OLS) method fit the line which minimizes

Standard error or regression: Average error of prediction Average deviation from the regression line Standard deviation: Average deviation from the mean

X

SS Total SS y, SS total SS Residual SS r, SS res =+ SS regression, SS reg,

Null Hypothesis: ß = 0 Assume Null is true, what is the probability that ? Sampling t distribution of under the Null: p<.05 ß = 0

Total Variability of Y. SS Y R2R2 X Variability of Y that is predicted by X. SS reg 1-R 2

Proportion of variance of Y that is predicted by X. Proportion of variance of Y that is not predicted by X.

Adjusted R 2 Small sample size Large number of predictors

X1X1 X2X2 Y Multiple Regression in Motion

Y X1X1 X2X2 R 2 y.12

Y X1X1 X2X2

Y X1X1 X2X2 Zero-Order

Y X1X1 X2X2

Y X1X1 X2X2 Semi-Partial  2

Y X1X1 X2X2 Semi-Partial  1

Y R 2 : due to X 1 X2X2 X3X3 X1X1 R 2 change: Unique of X 2, X 3 Controlling for X 1

Analysis Strategies Confirmatory –Enter predictors in sequence and examine R 2 change Exploratory –Forward –Background –Stepwise

Hierarchical Regression 1) Enter variables from existing theory (R 2 ) 2 ) Enter variables of your theory (R 2 increment) 1) Enter Demographic variables (R 2 ) 2 ) Enter variables of your theory (R 2 increment) 1) Enter variables of earlier time (R 2 ) 2 ) Enter variables of later time (R 2 increment) OR

Variable Names: Sex1Child’s gender, 1 = male, 0 = female Bul Child aggression in schools EmChild emotion regulation AChild activity level IChild reactivity or intensity Phy1(2) Father (mother) harsh parenting or physical punishment Dp1(2)Father (mother) depression Mary1(2)Father (mother) marital satisfaction

SPSS Commands: REGRESSION /STATISTICS COEFF CHANGE /DEPENDENT bul /METHOD=ENTER sex1 /METHOD=ENTER em a i /METHOD=ENTER dp1 mary1 dp2 mary2 /METHOD=ENTER phy1 phy2.

SPSS Output: Variables Entered/Removed ModelVariables Variables Method Entered Removed SEX1. Enter 2 I, A, EM. Enter 3 DP1, MARY2, DP2, MARY1. Enter 4 PHY1, PHY2. Enter a All requested variables entered. b Dependent Variable: BUL