• General Linear Model: Y = Xβ + e
• Efficiency: ability to estimate β, given X
• Efficiency 1 Var(X) XTX Var(β)
It ain’t gonna get technical
now is it?
A B C D 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0
XXT
A 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0B 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0C 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0D 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0
.=
A B C DA 5 0 0 0B 0 5 0 0C 0 0 5 4D 0 0 4 5
XT
X
Non-overlappingconditions
Overlappingconditions
•Efficiency 1 Var(X) Var(β)
1 1 1/Var(X) 1/XTX
A B C DA 5 0 0 0B 0 5 0 0C 0 0 5 4D 0 0 4 5
XT
X inv(XT
X) A B C DA 0.2 0 0 0B 0 0.2 0 0C 0 0 0.6 -0.4D 0 0 -0.4 0.6
•Efficiency is specific to condition or contrast
Efficiency 1
cT inv(XTX ) c When c is Simple Effect, e.g. [1 0 0 0]
A, B: Efficiency = 1/0.2 = 5
C, D: Efficiency = 1/0.6 = 1.7
When c is Contrast, e.g. [1 -1 0 0]
A-B: Efficiency = 1/0.4 = 2.5
C-D: Efficiency = 1/2 = 0.5
inv(XT
X) A B C DA 0.2 0 0 0B 0 0.2 0 0C 0 0 0.6 -0.4D 0 0 -0.4 0.6
A B C D E F 1 0 0 0 0 0 1 0 1 0 0 1 1 0 0 0 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 1 1 0 0 1 0 0 1 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0
Different Designs – Boxcar Events
inv(XT
X) A B C D E FA 0.2488 0.0377 -0.0297 -0.0396 -0.0012 -0.0873B 0.0377 0.2862 -0.0941 -0.0421 -0.0873 -0.0263C -0.0297 -0.0941 0.2871 0.0495 -0.0297 -0.0941D -0.0396 -0.0421 0.0495 0.2327 -0.0396 -0.0421E -0.0012 -0.0873 -0.0297 -0.0396 0.2488 0.0377F -0.0873 -0.0263 -0.0941 -0.0421 0.0377 0.2862
Efficiency Simple Effects: A, B = C,D = E,F = 4Efficiency Contrasts: A - B = C – D = E – F = 2
1 2 3 4 5 6
1
2
3
4
5
6
X
Blocked
Fixed Interleaved
Random
1.5
Different Designs – Haemodynamic Responses
Blocked
Fixed Interleaved
Random-Uniform
1 2 3 4 5 6 7 8
10
20
30
40
50
60
70
80
Random-Sinusoidal
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
inv(XT
X)X
5
2.8
3.5
Relative Efficiency
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
Different Designs – Haemodynamic Responses
2.5
Blocked
X
5
2.8
3.5
inv(XT
X)10
20
30
40
50
60
70
80
Relative Efficiency0 5 10 15 20 25 30 35 40
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25 30 35 40-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25 30 35 40-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0.5 1 1.5 2 2.5 3 3.5 4 4.5
100
200
300
400
500
600
700
800
900
Different SOA’s – Variable No. of Trials
X inv(XT
X)
4.2Random:Events =
25
2.1
Relative Efficiency
Random:Events =
50
0.5 1 1.5 2 2.5 3 3.5 4 4.5
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0.5 1 1.5 2 2.5 3 3.5 4 4.5
100
200
300
400
500
600
700
800
900
Different SOA’s – Variable Min SOA
X inv(XT
X)
10.0Random:Min SOA
= 5 secs
7.5Relative Efficiency
Random:Min SOA
= 0.5 secs
0.5 1 1.5 2 2.5 3 3.5 4 4.5
0.5
1
1.5
2
2.5
3
3.5
4
4.5
But as the SOA gets smaller, the HRF- linear
convolution model breaks down, and the
ability to estimate simple effects
vs. baseline diminshes
6400 6450 6500 6550 6600 6650 6700
-5
0
5
10
15
20
x 10-3
x
≠
6400 6450 6500 6550 6600 6650 6700
0
0.01
0.02
0.03
0.04
0.05
6400 6450 6500 6550 6600 6650 6700
0
0.01
0.02
0.03
0.04
0.05
1
1
2