Regression Models with 2 quantitative variables (& Interaction)

2 quantitative variables main effects model with no interaction interaction model

2 centered quant vars

y’ = b0 + b1x1 + b2x2

“X1” is a centered quantitative variable

X1 X1 – X1mean

“X2” is a centered quantitative variable

X1 X1 – X1mean

2 centered quant vars

y’ = b0 + b1x1 + b2x2

b0 mean of those in Cz with X=0 (mean)

b1 slope of Y-X1 regression

- slope same for all values of X2 no interaction

b2 slope of Y-X2 regression

- slope same for all values of X1 no interaction

0 1

0

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4

0

50

60

y’ = b0 + b1X1 + b2X2

X2=0

+1std X2

-1std X2

-2 -1 0 1 2 X

b0 = ht of X2mean line

b1 = slope of X2mean line

b2 = htdifs among X2-lines

X2-lines all have same slp(no interaction)

3a Xs are centered quant variables (both are standardized std = 1)

2 centered quant var & their product term/interaction

y’ = b0 + b1x1 + b2x2 + b3xz

“X1” is a centered quantitative variable

X1 X1 – X1mean

“X2” is a centered quantitative variable

X1 X1 – X1mean

“XZ” represents the interaction of “X1” and”X2”

2 centered quant vars & their interaction

y’ = b0 + b1x1 + b2x2 + b3xz

b0 mean of those in Cz with X=0 (mean)

b1 slope of Y-X1 regression

- Simple Y-X1 slope for X2 mean=0

b2 slope of Y-X2 regression

- Simple Y-X2 slope for X1 mean=0

b3 interaction of X1 & X2

- how slope of Y-X1 reg lines change with X2 value

- how slope of Y-X2 reg lines change with X1 value

0 1

0

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4

0

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y’ = b0 + b1Xcen + b2Zcen + b3XZ

b0

b1

-b2Z=0+1std Z

-1std Z

b2

Zcen = Z – Xmean

3b 2 quantitative predictors w/ interaction (both are standardized std = 1)

-2 -1 0 1 2 Xcen

a = ht of Zmean line

b1 = slope of Zmean line

b2 = htdifs among Z-lines

Xcen = X – Xmean ZX = Xcen * Zcen

b3

b3 b3 = slpdifs among Z-lines