Efficiency in Experimental Design

J. Winston

Starring …

P. Bentley

• 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

Different Designs – Calculated Efficiencies

I wish my

BlocksWere

BIGGER

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

Joel, can you