x y. variance covariance correlation scatter plot
Post on 19-Dec-2015
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TRANSCRIPT
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