Introduction to the General Linear Model (GLM)
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Introduction to the General Linear Model (GLM) 1 quantitative variable & 1 2-group variable 1a main effects model with no interaction 1b interaction model 1 quantitative variable & 1 3-group variable 2a main effects model with no interaction 2b interaction model
description
Introduction to the General Linear Model (GLM). 1 quantitative variable & 1 2-group variable 1a main effects model with no interaction 1b interaction model 1 quantitative variable & 1 3-group variable 2a main effects model with no interaction 2b interaction model. - PowerPoint PPT Presentation
Transcript of Introduction to the General Linear Model (GLM)
Introduction to the General Linear Model (GLM)
1 quantitative variable & 1 2-group variable 1a main effects model with no interaction 1b interaction model
1 quantitative variable & 1 3-group variable 2a main effects model with no interaction 2b interaction model
There are two important variations of each of these models
1. Main effects model
• Centered or coded terms for each variable
• No interaction – assumes regression slope homogeneity
• b-weights for binary & quant variables each represent main effect of that variable
2. Interaction model
• Centered or coded terms for each variable
• Term for interaction - does not assume reg slp homogen !!
• b-weights for binary & quant variables each represent the simple effect of that variable when the other variable = 0
• b-weight for the interaction term represented how the simple effect of one variable changes with changes in the value of the other variable (e.g., the extent and direction of the interaction)
#1a centered quant variable & dummy coded 2-grp variable
y’ = b0 + b1x + b2z
“X” is a centered quantitative variable
X X – Xmean
“Z” is a dummy-coded 2-group variable (Cz = 0 & Tx = 1)
Z Tz = 1 Cz = 0
#1a centered quant variable & dummy coded 2-grp variable
y’ = b0 + b1x + b2z
b0 mean of those in Cz with X=0 (mean)
b1 slope of Y-X regression line for Cz (=0)- slope same for both groups no interaction
b2 group difference for X=mean (=0) - group different same for all values of X no interaction
0 1
0
20
30
4
0
50
60
y’ = b0 + b1X + b2Z
Cz
Tz
-2 -1 0 1 2 Xcen
b0 = ht of Cz line
b1 = slp of Cz line
b2 = htdif Cz & Tz
Z-lines have same slp(no interaction)
20 5 10
#1a quantitative (Xcen) & 2-group (Tz=1 Cz=0)
#1b centered quant var, dummy coded 2-group var & their product term/interaction
y’ = b0 + b1x + b2z + b3xz
“X” is a centered quantitative variable
X X – Xmean
“Z” is a dummy-coded 2-group variable
Z Tz = 1 Cz = 0
“XZ” represents the interaction of “X” and “Z”
XZ X * Z
#1b centered quant var, dummy coded 2-group var & their product term/interaction
y’ = b0 + b1x + b2z + b3xz
b0 mean of those in Cz with X= 0 (mean)
b1 slope of Y-X regression line for Cz (=0)*
b2 group difference for X=0 (mean)*
b3 how slope of y-x reg line for Tz (=1) differs from slope of y-x reg line for Cz (=0)
* Because the interaction is included, slopes may be different for different grps * Because the interaction is included, group differences may be different for
different X values
0 1
0
20
30
4
0
50
60
Cz
Tz
-2 -1 0 1 2 Xcen
b0 = ht of Cz line
b1 = slp of Cz line
b2 = htdif Cz & Tz
b3 = slpdif Cz & Tz
y’ = b0 + b1X + b2Z + b3XZ
30 -5 15 15
#1b quantitative (X) & 2-group (Tz Cz) predictors w/ interaction
0 1
0
20
30
4
0
50
60
-2 -1 0 1 2 Xcen
b0 = ht of Cz line
b1 = slp of Cz line
b2 = htdif Cz & Tz
b3 = slpdif Cz & Tz
y’ = b0 + b1X + b2Z + b3XZ
#1b quantitative (X) & 2-group (Tz Cz) predictors w/ interaction
0 1
0
20
30
4
0
50
60
-2 -1 0 1 2 Xcen
b0 = ht of Cz line
b1 = slp of Cz line
b2 = htdif Cz & Tz
b3 = slpdif Cz & Tz
y’ = b0 + b1X + b2Z + b3XZ
#1b quantitative (X) & 2-group (Tz Cz) predictors w/ interaction
0 1
0
20
30
4
0
50
60
-2 -1 0 1 2 Xcen
b0 = ht of Cz line
b1 = slp of Cz line
b2 = htdif Cz & Tz
b3 = slpdif Cz & Tz
y’ = b0 + b1X + b2Z + b3XZ
#1b quantitative (X) & 2-group (Tz Cz) predictors w/ interaction
0 1
0
20
30
4
0
50
60
-2 -1 0 1 2 Xcen
b0 = ht of Cz line
b1 = slp of Cz line
b2 = htdif Cz & Tz
b3 = slpdif Cz & Tz
y’ = b0 + b1X + b2Z + b3XZ
#1b quantitative (X) & 2-group (Tz Cz) predictors w/ interaction
0 1
0
20
30
4
0
50
60
-2 -1 0 1 2 Xcen
b0 = ht of Cz line
b1 = slp of Cz line
b2 = htdif Cz & Tz
b3 = slpdif Cz & Tz
y’ = b0 + b1X + b2Z + b3XZ
#1b quantitative (X) & 2-group (Tz Cz) predictors w/ interaction
0 1
0
20
30
4
0
50
60
-2 -1 0 1 2 Xcen
b0 = ht of Cz line
b1 = slp of Cz line
b2 = htdif Cz & Tz
b3 = slpdif Cz & Tz
y’ = b0 + b1X + b2Z + b3XZ
#1b quantitative (X) & 2-group (Tz Cz) predictors w/ interaction
0 1
0
20
30
4
0
50
60
-2 -1 0 1 2 Xcen
b0 = ht of Cz line
b1 = slp of Cz line
b2 = htdif Cz & Tz
b3 = slpdif Cz & Tz
y’ = b0 + b1X + b2Z + b3XZ
#1b quantitative (X) & 2-group (Tz Cz) predictors w/ interaction
0 1
0
20
30
4
0
50
60
-2 -1 0 1 2 Xcen
b0 = ht of Cz line
b1 = slp of Cz line
b2 = htdif Cz & Tz
b3 = slpdif Cz & Tz
y’ = b0 + b1X + b2Z + b3XZ
#1b quantitative (X) & 2-group (Tz Cz) predictors w/ interaction
0 1
0
20
30
4
0
50
60
-2 -1 0 1 2 Xcen
b0 = ht of Cz line
b1 = slp of Cz line
b2 = htdif Cz & Tz
b3 = slpdif Cz & Tz
y’ = b0 + b1X + b2Z + b3XZ
#1b quantitative (X) & 2-group (Tz Cz) predictors w/ interaction
0 1
0
20
30
4
0
50
60
-2 -1 0 1 2 Xcen
b0 = ht of Cz line
b1 = slp of Cz line
b2 = htdif Cz & Tz
b3 = slpdif Cz & Tz
y’ = b0 + b1X + b2Z + b3XZ
#1b quantitative (X) & 2-group (Tz Cz) predictors w/ interaction
#2a centered quant var & dummy coded 3-grp var
y’ = b0 + b1x + b2z1 + b3z2
“X” is centered quantitative variable
X X – Xmean
“Z1” & “Z2” are dummy-codes for the 3-group variable
Z1 Tz1 = 1 Tz2 = 0 Cz = 0
Z2 Tz1 = 0 Tz2 = 1 Cz = 0
#2a centered quant var & dummy coded 3-grp var
y’ = b0 + b1x + b2z1 + b3z2
b0 mean of those in Cz with X=0 (mean)
b1 slope of Y-X regression line for Cz (=0)- slope same for all groups no interaction
b2 Tz1 - Cz difference for X=mean (=0) - group different same for all values of X no interaction
b3 Tz2 - Cz difference for X=mean (=0) - group different same for all values of X no interaction
0 1
0
20
30
40
5
0
60
Cz
Tz2
Tz1
-2 -1 0 1 2 X
b0 = ht of Cz line
b2 = htdif Cz & Tz1
b3 = htdif Cz & Tz2
b1 = slp of Cz line
y’ = b0 + b1X + b2Z1 + b3Z2
Z-lines have same slp(no interaction)
35 -155 5
#2a quantitative (Xcen) & 3-group Z1 Tz1 = 1 Tz2 = 0 Cz = 0
Z2 Tz1=0 Tz2 = 1 Cz = 0
#2b centered quant var, dummy coded 3-group var & their product terms/interaction
y’ = b0 + b1x + b2z1 + b3z2 + b4xz1 + b5xz2
“X” is centered quantitative variableX X – Xmean
“Z1” & “Z2” are dummy-codes for the 3-group variable
Z1 Tz1 = 1 Tz2 = 0 Cz = 0
Z2 Tz1 = 0 Tz2 = 1 Cz = 0
“XZ1” & “XZ2” represent the interaction of “X” and “Z” XZ1 X * Z1
XZ2 X * Z2
#2b centered quant var, dummy coded 3-group var & their product terms/interaction
y’ = b0 + b1x + b2z1 + b3z2 + b4xz1 + b5xz2
b0 mean of those in Cz with X= 0 (mean)
b1 slope of Y-X regression line for Cz
b2 Tz1 - Cz difference for X=0 (mean)*
b3 Tz2 - Cz difference for X=0 (mean)*
b4 how slope of y-x reg line for Tz1 differs from slope of y-x reg line for Cz *b4 how slope of y-x reg line for Tz2 differs from slope of y-x reg line for Cz *
*Because the interaction is included, group differences may be different for different X values
* Because the interaction is included, slopes may be different for different grps
0 1
0 2
0 3
0
40
50
60
y’ = b0 + b1Xcen + b2Z1 + b3Z2 + b4XZ1 + b5XZ2
CxTx1
Tx2
-2 -1 0 1 2 Xcen
b0 = ht of Cz line
b2 = htdif Cz & Tz1
b3 = htdif Cz & Tz2
b1 = slp of Cz line
b4 = slpdif Cz & Tz1
b5 = slpdif Cz & Tz2
30 5 15 5 2 -8
#2b quantitative (Xcen) & 3-group Z1 Tz1 = 1 Tz2 = 0 Cz = 0
Z2 Tz1=0 Tz2 = 1 Cz = 0
and interactions XZ1 = X*Z1 XZ2 = X*Z2
0 1
0 2
0 3
0
40
50
60
y’ = b0 + b1Xcen + b2Z1 + b3Z2 + b4XZ1 + b5XZ2
-2 -1 0 1 2 Xcen
b0 = ht of Cz line
b2 = htdif Cz & Tz1
b3 = htdif Cz & Tz2
b1 = slp of Cz line
b4 = slpdif Cz & Tz1
b5 = slpdif Cz & Tz2
#2b quantitative (Xcen) & 3-group Z1 Tz1 = 1 Tz2 = 0 Cz = 0
Z2 Tz1=0 Tz2 = 1 Cz = 0
and interactions XZ1 = X*Z1 XZ2 = X*Z2
0 1
0 2
0 3
0
40
50
60
y’ = b0 + b1Xcen + b2Z1 + b3Z2 + b4XZ1 + b5XZ2
-2 -1 0 1 2 Xcen
b0 = ht of Cz line
b2 = htdif Cz & Tz1
b3 = htdif Cz & Tz2
b1 = slp of Cz line
b4 = slpdif Cz & Tz1
b5 = slpdif Cz & Tz2
#2b quantitative (Xcen) & 3-group Z1 Tz1 = 1 Tz2 = 0 Cz = 0
Z2 Tz1=0 Tz2 = 1 Cz = 0
and interactions XZ1 = X*Z1 XZ2 = X*Z2
0 1
0 2
0 3
0
40
50
60
y’ = b0 + b1Xcen + b2Z1 + b3Z2 + b4XZ1 + b5XZ2
-2 -1 0 1 2 Xcen
b0 = ht of Cz line
b2 = htdif Cz & Tz1
b3 = htdif Cz & Tz2
b1 = slp of Cz line
b4 = slpdif Cz & Tz1
b5 = slpdif Cz & Tz2
#2b quantitative (Xcen) & 3-group Z1 Tz1 = 1 Tz2 = 0 Cz = 0
Z2 Tz1=0 Tz2 = 1 Cz = 0
and interactions XZ1 = X*Z1 XZ2 = X*Z2
0 1
0 2
0 3
0
40
50
60
y’ = b0 + b1Xcen + b2Z1 + b3Z2 + b4XZ1 + b5XZ2
-2 -1 0 1 2 Xcen
b0 = ht of Cz line
b2 = htdif Cz & Tz1
b3 = htdif Cz & Tz2
b1 = slp of Cz line
b4 = slpdif Cz & Tz1
b5 = slpdif Cz & Tz2
#2b quantitative (Xcen) & 3-group Z1 Tz1 = 1 Tz2 = 0 Cz = 0
Z2 Tz1=0 Tz2 = 1 Cz = 0
and interactions XZ1 = X*Z1 XZ2 = X*Z2
0 1
0 2
0 3
0
40
50
60
y’ = b0 + b1Xcen + b2Z1 + b3Z2 + b4XZ1 + b5XZ2
-2 -1 0 1 2 Xcen
b0 = ht of Cz line
b2 = htdif Cz & Tz1
b3 = htdif Cz & Tz2
b1 = slp of Cz line
b4 = slpdif Cz & Tz1
b5 = slpdif Cz & Tz2
#2b quantitative (Xcen) & 3-group Z1 Tz1 = 1 Tz2 = 0 Cz = 0
Z2 Tz1=0 Tz2 = 1 Cz = 0
and interactions XZ1 = X*Z1 XZ2 = X*Z2
0 1
0 2
0 3
0
40
50
60
y’ = b0 + b1Xcen + b2Z1 + b3Z2 + b4XZ1 + b5XZ2
-2 -1 0 1 2 Xcen
b0 = ht of Cz line
b2 = htdif Cz & Tz1
b3 = htdif Cz & Tz2
b1 = slp of Cz line
b4 = slpdif Cz & Tz1
b5 = slpdif Cz & Tz2
#2b quantitative (Xcen) & 3-group Z1 Tz1 = 1 Tz2 = 0 Cz = 0
Z2 Tz1=0 Tz2 = 1 Cz = 0
and interactions XZ1 = X*Z1 XZ2 = X*Z2
0 1
0 2
0 3
0
40
50
60
y’ = b0 + b1Xcen + b2Z1 + b3Z2 + b4XZ1 + b5XZ2
-2 -1 0 1 2 Xcen
b0 = ht of Cz line
b2 = htdif Cz & Tz1
b3 = htdif Cz & Tz2
b1 = slp of Cz line
b4 = slpdif Cz & Tz1
b5 = slpdif Cz & Tz2
#2b quantitative (Xcen) & 3-group Z1 Tz1 = 1 Tz2 = 0 Cz = 0
Z2 Tz1=0 Tz2 = 1 Cz = 0
and interactions XZ1 = X*Z1 XZ2 = X*Z2
0 1
0 2
0 3
0
40
50
60
y’ = b0 + b1Xcen + b2Z1 + b3Z2 + b4XZ1 + b5XZ2
-2 -1 0 1 2 Xcen
b0 = ht of Cz line
b2 = htdif Cz & Tz1
b3 = htdif Cz & Tz2
b1 = slp of Cz line
b4 = slpdif Cz & Tz1
b5 = slpdif Cz & Tz2
#2b quantitative (Xcen) & 3-group Z1 Tz1 = 1 Tz2 = 0 Cz = 0
Z2 Tz1=0 Tz2 = 1 Cz = 0
and interactions XZ1 = X*Z1 XZ2 = X*Z2
0 1
0 2
0 3
0
40
50
60
y’ = b0 + b1Xcen + b2Z1 + b3Z2 + b4XZ1 + b5XZ2
-2 -1 0 1 2 Xcen
b0 = ht of Cz line
b2 = htdif Cz & Tz1
b3 = htdif Cz & Tz2
b1 = slp of Cz line
b4 = slpdif Cz & Tz1
b5 = slpdif Cz & Tz2
#2b quantitative (Xcen) & 3-group Z1 Tz1 = 1 Tz2 = 0 Cz = 0
Z2 Tz1=0 Tz2 = 1 Cz = 0
and interactions XZ1 = X*Z1 XZ2 = X*Z2
0 1
0 2
0 3
0
40
50
60
y’ = b0 + b1Xcen + b2Z1 + b3Z2 + b4XZ1 + b5XZ2
-2 -1 0 1 2 Xcen
b0 = ht of Cz line
b2 = htdif Cz & Tz1
b3 = htdif Cz & Tz2
b1 = slp of Cz line
b4 = slpdif Cz & Tz1
b5 = slpdif Cz & Tz2
#2b quantitative (Xcen) & 3-group Z1 Tz1 = 1 Tz2 = 0 Cz = 0
Z2 Tz1=0 Tz2 = 1 Cz = 0
and interactions XZ1 = X*Z1 XZ2 = X*Z2
