X

Y

1

)X)(XX(Xs ii2

nVariance

1-n

)Y-)(YX-(XCov ii

xy Covariance

dx

xy

ss

Covr Correlation

Scatter plot

Y

X

Scatter plot

X

Y

X

Y

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.

Y

Y

X

Y

Y

Y

Y

SSY

SSres

SSreg

...

.. ..

..

..

.

.

X

Y

Y

Y

Y

...

.. ..

..

..

.

.SSreg

SSY

SSres

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.

• s2y is an estimate of the unconditional

population variance.

• s2y.x is an estimate of the conditional

population variance which is also called “residual variance.”

Fit a line to best represent the scatter points.

ß0

ß1

X

Y

• ß0 or intercept is the value of Y when X=0.

• ß1 or regression coefficient is value change

in Y associated with one unit change in X.

iXßßY 10i

XßYß 10

21ßx

xy

21ßx

xy

s

Cov

x

yxy s

sr1ß

• 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

2)Y(Y

Y

YY

YY

YY

X

Y

Standard error or regression: Average error of predictionAverage deviation from the regression line

1

)ˆ( 2

.

n

YYs ii

xy

1

)( 2

n

YYs i

Standard deviation:Average deviation from the mean

YYi

YYi

ii YY iY

iY

)Y(Y)YY(YY iiii

Y

X

22 )( YYy i SS TotalSSy, SStotal

22 )ˆ(. ii YYxy SS ResidualSSr, SSres

YSS regSS resSS= +

22 )ˆ(ˆ YYy i SS regression

, SSreg,YSS

• Null Hypothesis: ß = 0

• Assume Null is true, what is the probability that ?

• Sampling t distribution of under the Null:

Xß ß

ß

p<.05

ß = 0

2

y.xb

x

ss

Total Variability of Y. SSY

R2

X

Variability of Y that is predicted by X. SSreg

1-R2

Proportion of variance of Y that is predicted by X.

Yres2 SSSSR1

Yreg2 SSSSR

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

Adjusted R2

• Small sample size

• Large number of predictors

X1 X2

Y

Multiple Regression in Motion

Y

X1 X2

R2y.12

y.122R-1

Y

X1 X2

1y2r

Y

X1 X2

Zero-Order

2y2r

Y

X1 X2

Zero-Order

y(2.1)2r

Y

X1 X2

Semi-Partial2

y(1.2)2rY

X1 X2

Semi-Partial1

Y110 XßßY

R2: due to X1

X2X3

X1

3322110 XßXßXßßY

R2 change: Unique of X2, X3

Controlling for X1

Analysis Strategies

• Confirmatory– Enter predictors in sequence and examine R2

change

• Exploratory– Forward– Background– Stepwise

Hierarchical RegressionHierarchical Regression

1) Enter variables from existing theory (R2)

2 ) Enter variables of your theory (R2 increment)

1) Enter Demographic variables (R2)

2 ) Enter variables of your theory (R2 increment)

1) Enter variables of earlier time (R2)

2 ) Enter variables of later time (R2 increment)

OR

OR

Variable Names:Sex1 Child’s gender, 1 = male, 0 = femaleBul Child aggression in schoolsEm Child emotion regulationA Child activity levelI Child reactivity or intensityPhy1(2) Father (mother) harsh parenting or physical

punishmentDp1(2) Father (mother) depressionMary1(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

Model Variables Variables Method Entered Removed

----------------------------------------------------------------------------------1 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