A

BY

EntityUnder

Experimentation

Number of factors (n) eachat 2 levels

Number of treatments infull factorial (2n)

2 4

3 8

4 16

5 32

6 64

7 128

8 256

9 512

10 1024

n 2n

Table 8.11

Setting Notation

Factor A set to its low level A-

Factor A set to its high level A+

Factor B set to its low level B-

Factor b set to its high level B+

Table 8.1

Treatment Combination Setting of A Setting of B

Notation(for each treatment)

1 A- B- (1)

2 A- B+ b

3 A+ B- a

4 A+ B+ ab

Table 8.2

Note: There are (# of Treatment Combinations)! number of possible ways to test these factors and levels (“Run Combinations”).

Pick one randomly to do your testing. Don’t always test in a pre-described order.

Notation in right columnof the table above Meaning

(1) Both A and B are set to their low levels

a Only A is set to its high level

b Only B is set to its high level

ab Both A and B are set to their high levels

Table 8.3

A B Y

― ― y(1) = (1)

― + yb = b

+ ― ya = a

+ + yab = ab

Table 8.4

B

A

aba

b(1)-

+

- +

Figure 8.2

2

)())1(( babaA

Eq. 1

“A main effect”

The “A main effect” is the average change in Ywhen factor A changes from low to high (- to +)

A is the average change from the bottom to the top of the square.

2

)())1(( aabbB

Eq. 2

“B main effect”

The “B main effect” is the average change in Ywhen factor B changes from low to high (- to +)

B is the average change from the left to the right side of the square.

2

)())1(( baabAB

Eq. 3

AB is the difference in the average of the responses across both diagonals of the square.

AB is a measure of the interaction between factors A and B.

TreatmentStimulus Type

(S)Electrode Type

(E)Reaction Time

(T) msec

1 Square Silver 160 (1)

2 Square Platinum 150 e

3 Sinusoidal Silver 190 s

4 SinusoidalPlatinum

170 es

Table 8.5

-

+

- +

E

S

190160

170

e

150

es

s(1)

Figure 8.3

152

2010

2

)190170()160150(

2

))()1((

seseE

252

2030

2

)150170()160190(

2

))())1((

eessS

52

340330

2

)190150()160170(

2

)())1((

seesES

A

- +

Y

(1)ab

b+

a -

B

Figure 8.4

Square

190

180

170

160

150

140

200

Silver

Platinum

E

SinusoidalS

T(ms)

Figure 8.5

AA

AAAX A

2

2

)(

AAXAAA A

Eq. 4

Eq. 5

XA XB XAXB Y

−1 −1 +1 Y(1) = (1)

−1 +1 −1 Yb = b

+1 −1 −1 Ya = a

+1 +1 +1 Yab = ab

Table 8.6

I XA XB XAXB Y

+1 −1 −1 +1 Y(1) = (1)

+1 −1 +1 −1 Yb = b

+1 +1 −1 −1 Ya = a

+1 +1 +1 +1 Yab = ab

Table 8.7

I XA XB XAXB

+(1) −(1) −(1) +(1)

+b −b +b −b

+a +a −a −a

+ab +ab +ab +ab

(1)+b+a+ab -(1)-b+a+ab -(1)+b-a+ab (1)-b-a+ab

Table 8.8

kI = ((1)+b+a+ab)/4 Eq. 6a

kA = (-(1)-b+a+ab)/4 Eq. 6b

kB = (-(1)+b-a+ab)/4 Eq. 6c

kAB = ((1)-b-a+ab)/4 Eq. 6d

Y = kI+kAXA+ kBXB+ kABXAXB Eq. 7

Resistance (ohms)Factor A

Capacitance (μF)Factor B Fall Time (μsec)

10500 0.05 360 = a

10000 0.06 420 = b

10000 0.05 350 = (1)

10500 0.06 440 = ab

Example: RC Circuit

Resistance (ohms)Factor A

Capacitance (μF)Factor B Fall Time (μsec)

10000 0.05 350 = (1)

10000 0.06 420 = b

10500 0.05 360 = a

10500 0.06 440 = ab

Example: RC Circuit

Example: RC Circuit

Resistance (ohms)Factor A

Capacitance (μF)Factor B Fall Time (μsec)

−1 −1 350 = (1)

−1 +1 420 = b

+1 −1 360 = a

+1 +1 440 = ab

Tables 8.7 and 8.8

I XA XB XAXB t

+1 −1 −1 +1 350

+1 −1 +1 −1 420

+1 +1 −1 −1 360

+1 +1 +1 +1 440

350+420+360+440 = 1570 -350-420+360+440 = 30 -350+420-360+440 = 150 350-420-360+440 = 10

Example: RC Circuit

I XA XB XAXB Y

+(1) −(1) −(1) +(1) Y(1) = (1)

+b −b +b −b Yb = b

+a +a −a −a Ya = a

+ab +ab +ab +ab Yab = ab

(1)+b+a+ab -(1)-b+a+ab -(1)+b-a+ab (1)-b-a+ab

km = ((1)+b+a+ab)/4 Eq. 6a

kA = (-(1)-b+a+ab)/4 Eq. 6b

kB = (-(1)+b-a+ab)/4 Eq. 6c

kAB = ((1)-b-a+ab)/4 Eq. 6d

t = 392.5 + 7.5XA+ 37.5XB+ 2.5XAXB msec.

km = ((1)+b+a+ab)/4 = (350+420+360+440)/4 = 392.5 msec.

kA = (-(1)-b+a+ab)/4 = (-350-420+360+440)/4 = 7.5 msec.

kB = (-(1)+b-a+ab)/4 = (-350+420-360+440)/4 = 37.5 msec.

kAB = ((1)-b-a+ab)/4 = (1570+30+150+10)/4 = 2.5 msec.

Y = km+kAXA+ kBXB+ kABXAXB Eq. 7

-1

-0.7

5

-0.5

-0.2

5

0

0.25

0.5

0.75 1

1

0.7

0.4

0.1

-0.2

-0.5

-0.8

300

330

360

390

420

450

t

Xa

Xb

420-450

390-420

360-390

330-360

300-330

Figure 8.6

XA XB Replicate 1 Replicate 2 Replicate 3 Totals!!!

−1 −1 (1) (3) (1) (5) (1) (12) (1)

−1 +1 b (6) b (9) b (11) b

+1 −1 a (1) a (4) a (8) a

+1 +1 ab (2) ab (7) ab (10) ab

km = ((1)+b+a+ab)/12 Eq. 8a

kA = (-(1)-b+a+ab)/12 Eq. 8b

kB = (-(1)+b-a+ab)/12 Eq. 8c

kAB = ((1)-b-a+ab)/12 Eq. 8d

Table 8.9

I XA XB XC XAXB XAXC XBXC XAXBXC TOTAL

+1 −1 −1 −1 +1 +1 +1 −1 (1)

+1 −1 −1 +1 +1 −1 −1 +1 c

+1 −1 +1 −1 −1 +1 −1 +1 b

+1 −1 +1 +1 −1 −1 +1 −1 bc

+1 +1 −1 −1 −1 +1 +1 +1 a

+1 +1 −1 +1 −1 −1 −1 −1 ac

+1 +1 +1 −1 +1 +1 −1 −1 ab

+1 +1 +1 +1 +1 −1 +1 +1 abc

Table 8.10

km = ((1)+c+b+bc+a+ac+ab+abc))/8m Eq. 9a

kA = (-(1)-c-b-bc+a+ac+ab+abc))/8m Eq. 9b

kB =(-(1)-c+b+bc-a-ac+ab+abc))/8m Eq. 9c

kC =(-(1)+c-b+bc-a+ac-ab+abc))/8m Eq. 9d

kAB =((1)+c-b-bc-a-ac+ab+abc))/8m Eq. 9e

kAC =((1)-c+b-bc-a+ac-ab+abc))/8m Eq. 9f

kBC =((1)-c-b+bc+a-ac-ab+abc))/8m Eq. 9g

kABC = (-(1)+c+b-bc+a-ac-ab+abc))/8m Eq. 9h

Y = km + kAXA + kBXB + kCXC + kABXAXB + kACXAXC +

kBCXBXC+kABCXAXBXC Eq. 10