Introduction to the General Linear Model (GLM)

1
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

2

• There are two important variations of each of
  these models
• 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)

3

• 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

4

• 1a ? centered quant variable dummy coded
  2-grp variable
• y b0 b1x b2z

b0 ? mean of those in Cz with X0 (mean) b1 ?
slope of Y-X regression line for Cz (0) - slope
same for both groups ? no interaction b2 ? group
difference for Xmean (0) - group
different same for all values of X ? no
interaction
5
1a quantitative (Xcen) 2-group (Tz1 Cz0)
y b0 b1X b2Z
20
5
10
b0 ht of Cz line
b1 slp of Cz line
0 10 20 30 40 50 60
b2 htdif Cz Tz
Tz
Z-lines have same slp (no interaction)
Cz
-2 -1 0 1
2 ? Xcen
6

• 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

7

• 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 X0 (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
8
1b quantitative (X) 2-group (Tz Cz) predictors
w/ interaction
y b0 b1X b2Z b3XZ
30
15
15
-5
b0 ht of Cz line
Tz
b1 slp of Cz line
b2 htdif Cz Tz
0 10 20 30 40 50 60
b3 slpdif Cz Tz
Cz
-2 -1 0 1
2 ? Xcen
9
1b quantitative (X) 2-group (Tz Cz) predictors
w/ interaction
y b0 b1X b2Z b3XZ
b0 ht of Cz line
b1 slp of Cz line
b2 htdif Cz Tz
0 10 20 30 40 50 60
b3 slpdif Cz Tz
-2 -1 0 1
2 ? Xcen
10
1b quantitative (X) 2-group (Tz Cz) predictors
w/ interaction
y b0 b1X b2Z b3XZ
b0 ht of Cz line
b1 slp of Cz line
b2 htdif Cz Tz
0 10 20 30 40 50 60
b3 slpdif Cz Tz
-2 -1 0 1
2 ? Xcen
11
1b quantitative (X) 2-group (Tz Cz) predictors
w/ interaction
y b0 b1X b2Z b3XZ
b0 ht of Cz line
b1 slp of Cz line
b2 htdif Cz Tz
0 10 20 30 40 50 60
b3 slpdif Cz Tz
-2 -1 0 1
2 ? Xcen
12
1b quantitative (X) 2-group (Tz Cz) predictors
w/ interaction
y b0 b1X b2Z b3XZ
b0 ht of Cz line
b1 slp of Cz line
b2 htdif Cz Tz
0 10 20 30 40 50 60
b3 slpdif Cz Tz
-2 -1 0 1
2 ? Xcen
13
1b quantitative (X) 2-group (Tz Cz) predictors
w/ interaction
y b0 b1X b2Z b3XZ
b0 ht of Cz line
b1 slp of Cz line
b2 htdif Cz Tz
0 10 20 30 40 50 60
b3 slpdif Cz Tz
-2 -1 0 1
2 ? Xcen
14
1b quantitative (X) 2-group (Tz Cz) predictors
w/ interaction
y b0 b1X b2Z b3XZ
b0 ht of Cz line
b1 slp of Cz line
b2 htdif Cz Tz
0 10 20 30 40 50 60
b3 slpdif Cz Tz
-2 -1 0 1
2 ? Xcen
15
1b quantitative (X) 2-group (Tz Cz) predictors
w/ interaction
y b0 b1X b2Z b3XZ
b0 ht of Cz line
b1 slp of Cz line
b2 htdif Cz Tz
0 10 20 30 40 50 60
b3 slpdif Cz Tz
-2 -1 0 1
2 ? Xcen
16
1b quantitative (X) 2-group (Tz Cz) predictors
w/ interaction
y b0 b1X b2Z b3XZ
b0 ht of Cz line
b1 slp of Cz line
b2 htdif Cz Tz
0 10 20 30 40 50 60
b3 slpdif Cz Tz
-2 -1 0 1
2 ? Xcen
17
1b quantitative (X) 2-group (Tz Cz) predictors
w/ interaction
y b0 b1X b2Z b3XZ
b0 ht of Cz line
b1 slp of Cz line
b2 htdif Cz Tz
0 10 20 30 40 50 60
b3 slpdif Cz Tz
-2 -1 0 1
2 ? Xcen
18
1b quantitative (X) 2-group (Tz Cz) predictors
w/ interaction
y b0 b1X b2Z b3XZ
b0 ht of Cz line
b1 slp of Cz line
b2 htdif Cz Tz
0 10 20 30 40 50 60
b3 slpdif Cz Tz
-2 -1 0 1
2 ? Xcen
19

• 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

20

• 2a ? centered quant var dummy coded 3-grp
  var
• y b0 b1x b2z1 b3z2

b0 ? mean of those in Cz with X0 (mean) b1 ?
slope of Y-X regression line for Cz (0) - slope
same for all groups ? no interaction b2 ? Tz1 -
Cz difference for Xmean (0) - group
different same for all values of X ? no
interaction b3 ? Tz2 - Cz difference for Xmean
(0) - group different same for all
values of X ? no interaction
21
2a quantitative (Xcen) 3-group Z1 ? Tz1
1 Tz2 0 Cz 0

Z2 ? Tz10 Tz2 1 Cz 0
y b0 b1X b2Z1
b3Z2
35
-15
5
5
b0 ht of Cz line
Tz2
b1 slp of Cz line
Cz
b2 htdif Cz Tz1
0 10 20 30 40 50 60
Tz1
b3 htdif Cz Tz2
Z-lines have same slp (no interaction)
-2 -1 0 1
2 ? X
22

• 2b ? centered quant var, dummy coded
  3-group var
• their product
  terms/interaction
• y b0 b1x b2z1 b3z2 b4xz1 b5xz2
• 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
• XZ1 XZ2 represent the interaction of X
  and Z
• XZ1 ? X Z1
• XZ2 ? X Z2

23

• 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 X0 (mean) b3 ? Tz2 - Cz
difference for X0 (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
24
2b quantitative (Xcen) 3-group Z1 ? Tz1
1 Tz2 0 Cz 0

Z2 ? Tz10 Tz2 1 Cz 0 and
interactions XZ1 XZ1 XZ2 XZ2
y b0 b1Xcen b2Z1 b3Z2
b4XZ1 b5XZ2
-8
2
5
30
15
5
b0 ht of Cz line
b1 slp of Cz line
Tx2
b2 htdif Cz Tz1
0 10 20 30 40 50 60
b4 slpdif Cz Tz1
Tx1
b3 htdif Cz Tz2
Cx
b5 slpdif Cz Tz2
-2 -1 0 1
2 ? Xcen
25
2b quantitative (Xcen) 3-group Z1 ? Tz1
1 Tz2 0 Cz 0

Z2 ? Tz10 Tz2 1 Cz 0 and
interactions XZ1 XZ1 XZ2 XZ2
y b0 b1Xcen b2Z1 b3Z2
b4XZ1 b5XZ2
b0 ht of Cz line
b1 slp of Cz line
b2 htdif Cz Tz1
0 10 20 30 40 50 60
b4 slpdif Cz Tz1
b3 htdif Cz Tz2
b5 slpdif Cz Tz2
-2 -1 0 1
2 ? Xcen
26
2b quantitative (Xcen) 3-group Z1 ? Tz1
1 Tz2 0 Cz 0

Z2 ? Tz10 Tz2 1 Cz 0 and
interactions XZ1 XZ1 XZ2 XZ2
y b0 b1Xcen b2Z1 b3Z2
b4XZ1 b5XZ2
b0 ht of Cz line
b1 slp of Cz line
b2 htdif Cz Tz1
0 10 20 30 40 50 60
b4 slpdif Cz Tz1
b3 htdif Cz Tz2
b5 slpdif Cz Tz2
-2 -1 0 1
2 ? Xcen
27
2b quantitative (Xcen) 3-group Z1 ? Tz1
1 Tz2 0 Cz 0

Z2 ? Tz10 Tz2 1 Cz 0 and
interactions XZ1 XZ1 XZ2 XZ2
y b0 b1Xcen b2Z1 b3Z2
b4XZ1 b5XZ2
b0 ht of Cz line
b1 slp of Cz line
b2 htdif Cz Tz1
0 10 20 30 40 50 60
b4 slpdif Cz Tz1
b3 htdif Cz Tz2
b5 slpdif Cz Tz2
-2 -1 0 1
2 ? Xcen
28
2b quantitative (Xcen) 3-group Z1 ? Tz1
1 Tz2 0 Cz 0

Z2 ? Tz10 Tz2 1 Cz 0 and
interactions XZ1 XZ1 XZ2 XZ2
y b0 b1Xcen b2Z1 b3Z2
b4XZ1 b5XZ2
b0 ht of Cz line
b1 slp of Cz line
b2 htdif Cz Tz1
0 10 20 30 40 50 60
b4 slpdif Cz Tz1
b3 htdif Cz Tz2
b5 slpdif Cz Tz2
-2 -1 0 1
2 ? Xcen
29
2b quantitative (Xcen) 3-group Z1 ? Tz1
1 Tz2 0 Cz 0

Z2 ? Tz10 Tz2 1 Cz 0 and
interactions XZ1 XZ1 XZ2 XZ2
y b0 b1Xcen b2Z1 b3Z2
b4XZ1 b5XZ2
b0 ht of Cz line
b1 slp of Cz line
b2 htdif Cz Tz1
0 10 20 30 40 50 60
b4 slpdif Cz Tz1
b3 htdif Cz Tz2
b5 slpdif Cz Tz2
-2 -1 0 1
2 ? Xcen
30
2b quantitative (Xcen) 3-group Z1 ? Tz1
1 Tz2 0 Cz 0

Z2 ? Tz10 Tz2 1 Cz 0 and
interactions XZ1 XZ1 XZ2 XZ2
y b0 b1Xcen b2Z1 b3Z2
b4XZ1 b5XZ2
b0 ht of Cz line
b1 slp of Cz line
b2 htdif Cz Tz1
0 10 20 30 40 50 60
b4 slpdif Cz Tz1
b3 htdif Cz Tz2
b5 slpdif Cz Tz2
-2 -1 0 1
2 ? Xcen
31
2b quantitative (Xcen) 3-group Z1 ? Tz1
1 Tz2 0 Cz 0

Z2 ? Tz10 Tz2 1 Cz 0 and
interactions XZ1 XZ1 XZ2 XZ2
y b0 b1Xcen b2Z1 b3Z2
b4XZ1 b5XZ2
b0 ht of Cz line
b1 slp of Cz line
b2 htdif Cz Tz1
0 10 20 30 40 50 60
b4 slpdif Cz Tz1
b3 htdif Cz Tz2
b5 slpdif Cz Tz2
-2 -1 0 1
2 ? Xcen
32
2b quantitative (Xcen) 3-group Z1 ? Tz1
1 Tz2 0 Cz 0

Z2 ? Tz10 Tz2 1 Cz 0 and
interactions XZ1 XZ1 XZ2 XZ2
y b0 b1Xcen b2Z1 b3Z2
b4XZ1 b5XZ2
b0 ht of Cz line
b1 slp of Cz line
b2 htdif Cz Tz1
0 10 20 30 40 50 60
b4 slpdif Cz Tz1
b3 htdif Cz Tz2
b5 slpdif Cz Tz2
-2 -1 0 1
2 ? Xcen
33
2b quantitative (Xcen) 3-group Z1 ? Tz1
1 Tz2 0 Cz 0

Z2 ? Tz10 Tz2 1 Cz 0 and
interactions XZ1 XZ1 XZ2 XZ2
y b0 b1Xcen b2Z1 b3Z2
b4XZ1 b5XZ2
b0 ht of Cz line
b1 slp of Cz line
b2 htdif Cz Tz1
0 10 20 30 40 50 60
b4 slpdif Cz Tz1
b3 htdif Cz Tz2
b5 slpdif Cz Tz2
-2 -1 0 1
2 ? Xcen
34
2b quantitative (Xcen) 3-group Z1 ? Tz1
1 Tz2 0 Cz 0

Z2 ? Tz10 Tz2 1 Cz 0 and
interactions XZ1 XZ1 XZ2 XZ2
y b0 b1Xcen b2Z1 b3Z2
b4XZ1 b5XZ2
b0 ht of Cz line
b1 slp of Cz line
b2 htdif Cz Tz1
0 10 20 30 40 50 60
b4 slpdif Cz Tz1
b3 htdif Cz Tz2
b5 slpdif Cz Tz2
-2 -1 0 1
2 ? Xcen