Two-level Factorial Designs

Bacteria Example:– Response: Bill length– Factors:

B: Bacteria (Myco, Control) T: Room Temp (Warm, Cold) I: Inoculation (Eggs, Chicks)

Yandell, B. (2002) Practical Data Analysis for Designed Experiments, Chapman & Hall, London

Two-level Factorial Designs

Bacteria

Temp. Egg Chick

Control

Cold 39.77 40.23

Myco Cold 39.19 38.95

Control

Warm 40.37 41.71

Myco Warm 40.21 40.78

Cube Plot

+

Temp

Inoculation

W

C

CE

C

39.77

40.78

40.21

41.71

38.95

40.23

39.19

40.37

Bacteria

M

Estimated Effects

For a k-factor design with n replicates, the cell means are estimated as

We can write any effect as a contrast; interaction contrasts are obtained by element-wise multiplication of main effect contrast coefficients.

kk iiii Y 11

Estimated Effects

The resulting contrasts are mutually orthogonal.

The contrasts (up to a scaling constant) can be summarized as a table of ±1’s.

Orthogonal Contrast Coefficients

Run

B

T

I

BT

BI

TI

BTI

(1) -1 -1 -1 +1 +1 +1 -1

b +1 -1 -1 -1 -1 +1 +1

t -1 +1 -1 -1 +1 -1 +1

bt +1 +1 -1 +1 -1 -1 -1

i -1 -1 +1 +1 -1 -1 +1

bi +1 -1 +1 -1 +1 -1 -1

ti -1 +1 +1 -1 -1 +1 -1

bti +1 +1 +1 +1 +1 +1 +1

Estimated Effects

If we code contrast coefficients as ±1, the estimated effects are:

These effects are twice the size of our usual ANOVA effects.

.2222.111112

1 YcYc

k

Estimated Effects

The sum of squares for the estimated effect can be computed using the sum of squares formula we learned for contrasts

2

1

2

2

2

21

21

Effect1Effect

SS(Effect)

kk

in

cn

22 2effect) (EstimatedSS(effect) kn

Estimated Effects

Bacteria ExampleB effect=(39.19+38.95+40.21+40.78-

39.77-40.23-40.37-41.71)/4 =-.7375SSB=(-.7375)2x2=1.088 The entire ANOVA table for this

example can be constructed in this way

ANOVA df Effect SS

B 1 -.7375 1.0878

T 1 1.2325 3.0381

I 1 .5325 .5671

BT 1 .1925 .0741

BI 1 -.3675 .2701

TI 1 .4225 .3570

BTI 1 -.0175 .0006

Error 0 0

Total 7 5.395

Testing Effects

With replication (n>1)

Without replication (k large)– Claim higher-order interactions are

negligible and pool them– For k=6, if 3-way (and higher)

interactions are negligible, 42 d.f. would be available for error

FSSA

MSE~ F 1,2k n 1

Testing Effects

Without replication--Normal Probability Plots– If none of the effects is significant, the

effects are orthogonal normal random variables with mean 0 and variance

2

n2 k 2

Testing Effects

Because the effects are normal, they are also independent

IID normal effects can be “tested” using a normal probability plot (Minitab Example)

Yandell uses a half-normal plot You can pool values on the line as

error and construct an ANOVA table

Testing Effects

Lenth (1989) developed a more formal test of effects.

Denote the effects by ei, i=1,…,m. We say that the ei’s are iid N(0,t2),

where t is their common standard error.

Testing Effects

Lenth develops two estimates of the common standard error, t, of the ci’s:

ise

io

emedPSE

emeds

oi 5.25.1

5.1

Testing Effects

Though both are consistent estimates, PSE is more robust

The following terms are used to test effects

PSEmtSME

PSEmtME

m

3/,2

)1(1

3/,2

)1(1

/1

Testing Effects

The df term was developed from a study of the empirical distribution of PSE2

ME is a 1- a confidence bound for the absolute value of a single effect

SME is an exact (since the effects are independent) simultaneous 1-a confidence bound for all m effects