22-1

Factorial Treatments (§ 14.3, 14.4, 15.4)

Completely randomized designs - One treatment factor at t levels.Randomized block design - One treatment factor at t levels, one

block factor at b levels.Latin square design - One treatment factor at t levels, two block

factors, each at t levels.

Many other blocking structures are available.Check the literature on Experimental Design.

Now we move on to the situation where the t treatment levels are defined as combinations of two or more “factors”.

Factor – a controlled variable (e.g. temperature, fertilizer type, percent sand in concrete mix). Factors can have several levels (subdivisions).

22-2

Example 1 of Factors

What factors (characteristics, conditions) will make a wiring harness for a car last longer?

FACTOR LevelsNumber of strands 7 or 9.

Length of unsoldered, 0, 3, 6, or 12.uninsulated wire (in 0.01 inches)

Diameter of wire (gauge) 24, 22, or 20

A treatment is a specific combination of levels of the three factors.

T1 = ( 7 strand, 0.06 in, 22 gauge)

Response is the number of stress cycles the harness survives.

22-3

Example 2 of Factors

What is the effect of temperature and pressure on the bonding strength of a new adhesive?

Factor x1: temperature (any value between 30oF to 100oF)Factor x2: pressure (any value between 1 and 4 kg/cm2

Factors (temperature, pressure) have continuous levels, Treatments are combinations of factors at specific levels.

Response is bonding strength - can be determined for any combination of the two factors.

Response surface above the (x1 by x2) Cartesian surface.

22-4

Other Examples of Factors

• The effect of added Nitrogen, Phosphorus and Potassium on crop yield.

• The effect of replications and duration on added physical strength in weight lifting.

• The effect of age and diet on weight loss achieved.• The effect of years of schooling and gender on Math scores.• The effect of a contaminant dose and body weight on liver

enzyme levels.

Since many of the responses we are interested in are affected by multiple factors, it is natural to think of treatments as being constructed as combinations of factor levels.

22-5

One at a Time Approach

Consider a Nitrogen and Phosphorus study on crop yield. Suppose two levels of each factor were chosen for study: N@(40,60), P@(10,20) lbs/acre.

One-factor-at-a-time approach: Fix one factor then vary the other

Treatment N P Yield ParameterT1 60 10 145 1

T2 40 10 125 2

T3 40 20 160 3

H0: 1-2 = test of N-effect (20 unit difference observed in response).

H0: 2-3 = test of P-effect (35 unit difference observed in response).

If I examined the yield at N=60 and P=20 what would I expect to find?

E(Y | N=60,P=20) = - =160 + 20 = 180?E(Y | N=60,P=20) = +(1 - 2)+(3 - 2) = 125+20+35 = 180?

22-6

Interaction and Parallel Lines

We apply the N=60, P=20 treatment and get the following:

Treatment N P Yield ParameterT1 60 10 145

T2 40 10 125 2

T3 40 20 160 3

T4 60 20 130 4

10 20 P

Yield

120

150

160

170

140

130

180 Expected T4

Observed T4

N=40

N=6020

T2

T1T3

22-7

Parallel and Non-Parallel Profiles

10 20 P

120

150

160

170

140

130

180

N=40

N=6020

Parallel Lines => the effect of the two factors is additive (independent).

Non-Parallel Lines => the effect of the two factors interacts (dependent).

The effect of one factor on the response does not remain the same for different levels of the second factor. That is, the factors do not act independently of each other.

Without looking at all combinations of both factors, we would not be able to determine if the factors interact.

22-8

Factorial Experiment

Factorial Experiment - an experiment in which the response y is observed at all factor level combinations.

An experiment is not a design. (e.g. one can perform a factorial experiment in a completely randomized design, or in a randomized complete block design, or in a Latin square design.)

Design relates to how the experimental units are arranged, grouped, selected and how treatments are allocated to units.

Experiment relates to how the treatments are formed. In a factorial experiment, treatments are formed as combinations of factor levels.

(E.g. a fractional factorial experiment uses only a fraction (1/2, 1/3, 1/4, etc.) of all possible factor level combinations.)

22-9

General Data Layout Two Factor (a x b) Factorial

Column Factor (B)Row Factor(A) 1 2 3 … b Totals

1 T11 T12 T13 … T1b A1

2 T21 T22 T23 … T2b A2

3 T31 T32 T33 … T3b A3

… … … … … … ...a Ta1 Ta2 Ta3 … Tab Aa

Totals B1 B2 B3 … Bb G

yijk= observed response for the kth replicate (k=1,

…,n) for the treatment defined by the combination of the ith level of the row factor and the jth level of the column factor.

yyG

yTB

yTA

yyT

a

i

b

j

n

kijk

j

a

iijj

i

b

jiji

ij

n

kijkij

1 1 1

1

1

1

22-10

Model

ij = mean of the ijth table cell,

expected value of the response for the combination for the ith row factor level and the jth column factor level.

Overall Test of no treatment differences:

Ho: all ij are equalHa: at least two ij differ

Test as in a completely randomized design with a x b treatments.

ijkijijky

22-11

Sums of Squares

SSCellsTSSyySSE

abn

GT

nyynSSCells

abn

GyyyTSS

a

i

b

j

n

kijijk

a

i

b

jij

a

i

b

jij

a

i

b

j

n

kijk

a

i

b

j

n

kijk

2

1 1 1

2

1 1

22

1 1

1 1 1

22

2

1 1 1

1

)1(

1

1

nabdf

abdf

abndf

within

cells

total

withincells dfdfcells

within

FMSE

dfSSCells

F

df

SSEMSE

,

2

an follows ,

ˆ

22-12

After the Overall F test

As with any experiment, if the hypothesis of equal cell means is rejected, the next step is to determine where the differences are.

In a factorial experiment, there are a number of predefined contrasts (linear comparisons) that are always of interest.

• Main Effect of Treatment Factor A - Are there differences in the means of the factor A levels (averaged over the levels of factor B).

• Main Effect of Treatment Factor B - Are there differences in the means of the factor B levels (averaged over the levels of factor A).

• Interaction Effects of Factor A with Factor B - Are the differences between the levels of factor A the same for all levels of factor B? (or equivalently, are the differences among the levels of factor B the same for all levels of factor A? (Yes no interaction present; no interaction is present.)

22-13

Main Effects

aH 10 :

Column Factor (B)

Row Factor(A) 1 2 3 … b

1 11 12 13 … 1b 1

2 21 22 13 … 1b 2

3 31 32 13 … 1b 3

… … … … … … ...

a a1 a2 a3 … ab a

Totals 1 2 3 … b

Testing is via a set of linear comparisons.

b

jjb bbb 1

11111

111 Factor A main effects:

22-14

Testing for Main Effects: Factor A

1

,2

1

a

SSAMSAyynbSSA

a

ii

)1(,1an follows ,F nabaFMSE

MSA

There are a levels of Treatment Factor A. This implies that there are a-1 mutually independent linear contrasts that make up the test for main effects for Treatment Factor A. The Sums of Squares for the main effect for treatment differences among levels of Factor A is computed as the sum of the individual contrast sums of squares for any set of a-1 mutually independent linear comparisons of the a level means. Regardless of the chosen set, this overall main effect sums of squares will always equal the value of SSA below.

Reject H0 if F > F(a-1),ab(n-1),

a 10 :H

22-15

Profile Analysis for Factor A

53

Mean for level 1 of Factor A

Mean for level 5 of Factor A

Profile for level 2 of Factor B. Profile of mean of Factor A (main effect of A).

120

150

160

170

140

130

180 11

12

13

1

51

52

5

1 2 3 4 5Factor A Levels

22-16

Insignificant Main Effect for Factor A

120

150

160

170

140

130

180 11

12

13

1

51

52

5

1 2 3 4 5

Factor A Levels

Is there strong evidence of a Main Effect for Factor A?

SSA small (w.r.t. SSE) No.

22-17

Significant Main Effect for Factor A

120

150

160

170

140

130

180

1 2 3 4 5Factor A Levels

Is there strong evidence of a Main Effect for Factor A?

SSA large (w.r.t. SSE) Yes.

22-18

Main Effect Linear Comparisons - Factor A

0:

0:

0:

0:

:

54544

53533

52522

51511

10

L

L

L

L

H a

Column Factor (B)

Row Factor(A) 1 2 b=3

1 11 12 13 1

2 21 22 13 2

3 31 32 13 3

4 41 42 43 4

a=5 51 52 53 5

Totals 1 2 3

Testing via a set of linear comparisons.

Not mutually orthogonal, but together they represent a-1=4 dimensions of comparison.

jiijijkyE

1312111 3

1

3

1

3

1

22-19

Main Effect Linear Comparisons - Factor B

0:

0:

:

32322

31311

10

L

L

H b

Column Factor (B)

Row Factor(A) 1 2 b=3

1 11 12 13 1

2 21 22 13 2

3 31 32 13 3

4 41 42 43 4

a=5 51 52 53 5

Totals 1 2 3

Testing via a set of linear comparisons.

Not mutually orthogonal, but together they represent b-1=2 dimensions of comparison.

jiijijkyE

5121111 5

1

5

1

5

1

22-20

Testing for Main Effects: Factor B

1

,2

1

b

SSBMSByynaSSB

b

jj

)1(,1an follows ,F nabbFMSE

MSB

There are b levels of Treatment Factor B. This implies that there are b-1 mutually independent linear contrasts that make up the test for main effects for Treatment Factor B. The Sums of Squares for the main effect for treatment differences among levels of Factor B is computed as the sum of the individual contrast sums of squares for any set of b-1 mutually independent linear comparisons of the b level means. Regardless of the chosen set, this overall main effect sums of squares will always equal the value of SSB below.

Reject H0 if F > F(b-1),ab(n-1),

b 10 :H

22-21

Interaction

Two Factors, A and B, are said to interact if the difference in mean response for two levels of one factor is not constant across levels of the second factor.

120

160

140

180

1 2 3 4 5

Factor A Levels

120

160

140

180

1 2 3 4 5

Factor A Levels

Differences between levels of Factor B do not depend on the level of Factor A.

Differences between levels of Factor B do depend on the level of Factor A.

22-22

Interaction Linear Comparisons

Interaction is lack of consistency in differences between two levels of Factor B across levels of Factor A. 120

160

140

180

1 2 3 4 5

Factor A Levels

11

12

13

21

22

23

33

32

31

43

42

41

53

52

51

These four linear comparisons tested simultaneously is equivalent to testing that the profile line for level 1 of B is parallel to the profile line for level 2 of B.

Four more similar contrasts would be needed to test the profile line for level 1 of B to that of level 3 of B.

0

0

0

0

52514241

52513231

52512221

52511211

22-23

Model for Interaction

0)()()()(: 22211211222112110 H

ijkijjiijky

Tests for interaction are based on the ij terms exclusively.

If all ij terms are equal to zero, then there is no interaction.

)()(

)()(

121121

122111111211

22-24

Overall Test for Interaction

MSE

MSAB

MSEba

SSABF )1)(1(

H0: No interaction, HA: Interaction exists.

TS:

RR: F > F(a-1)(b-1),ab(n-1),

a

i

b

jij

a

i

b

jjiij

abn

GSSBSSAT

n

yyyyn

SSBSSASSCellsSSAB

1 1

22

2

1 1

1

22-25

Partitioning of Total Sums of Squares

TSS = SSR + SSE = SSA + SSB + SSAB + SSE

ANOVA Table

Source df SS MS F Between Cells ab-1 SSCells MSCells MSCells/MSE

Factor A a-1 SSA MSA MSA/MSE Factor B b-1 SSB MSB MSB/MSE

Interaction (a-1)(b-1) SSAB MSAB MSAB/MSE

Error(Within Cells) ab(n-1) SSE MSE Total (corrected) abn-1 TSS

22-26

Multiple Comparisons in Factorial Experiments

• Methods are the same as in the one-way classification situation i.e. composition of yardstick. Just need to remember to use: (i) MSE and df error from the SSE entry in AOV table; (ii) n is the number of replicates that go into forming the sample means being compared; (iii) t in Tukey’s HSD method is # of level means being compared.

• Significant interactions can affect how multiple comparisons are performed.

If Main Effects are significant AND Interactions are NOT significant:Use multiple comparisons on factor main effects (factor means).

If Interactions ARE significant:1) Multiple comparisons on main effect level means should NOT be done as they are meaningless.2) Should instead perform multiple comparisons among all factorial means of interest.

22-27

Two Factor Factorial Example: pesticides and fruit trees (Example 14.6 in Ott & Longnecker, p.896)

An experiment was conducted to determine the effect of 4 different pesticides (factor A) on the yield of fruit from 3 different varieties of a citrus tree (factor B). 8 trees from each variety were randomly selected; the 4 pesticides were applied to 2 trees of each variety. Yields (bushels/tree) obtained were:

49, 39 50, 55 43, 38 53, 48

55, 41 67, 58 53, 42 85, 73

66, 68 85, 92 69, 62 85, 99

Pesticide (A)

1 2 3 4Variety (B)

1

2

3

This is a completely randomized 3 4 factorial experiment with factor A at a=4 levels, and factor B at b=3 levels. There are t=34=12 treatments, each replicated n=2 times.

22-28

Example in MinitabA B yield1 1 491 1 391 2 551 2 411 3 661 3 682 1 502 1 552 2 672 2 582 3 852 3 923 1 433 1 383 2 533 2 423 3 693 3 624 1 534 1 484 2 854 2 734 3 854 3 99

Stat > ANOVA

> Two-way…

Two-way ANOVA: yield versus A, B

Analysis of Variance for yield Source DF SS MS F PA 3 2227.5 742.5 17.56 0.000B 2 3996.1 1998.0 47.24 0.000Interaction 6 456.9 76.2 1.80 0.182Error 12 507.5 42.3Total 23 7188.0

Interaction not significant; refit additive model:

Stat > ANOVA > Two-way > additive model

Two-way ANOVA: yield versus A, B

Source DF SS MS F PA 3 2227.46 742.49 13.86 0.000B 2 3996.08 1998.04 37.29 0.000Error 18 964.42 53.58Total 23 7187.96

S = 7.320 R-Sq = 86.58% R-Sq(adj) = 82.86%

22-29

Analyze Main Effects with Tukey’s HSD (MTB)

Stat > ANOVA > General Linear Model

Use to get factor or profile plots

MTB will use t=4 & n=6 to compare A main effects, and t=3 & n=8 to compare B main effects.

22-30

Tukey Analysis of Main Effects (MTB)

A = 1 subtracted from: Difference SE of AdjustedA of Means Difference T-Value P-Value2 14.833 4.226 3.5100 0.01223 -1.833 4.226 -0.4338 0.97194 20.833 4.226 4.9297 0.0006

A = 2 subtracted from: Difference SE of AdjustedA of Means Difference T-Value P-Value3 -16.67 4.226 -3.944 0.00484 6.00 4.226 1.420 0.5038

A = 3 subtracted from: Difference SE of AdjustedA of Means Difference T-Value P-Value4 22.67 4.226 5.364 0.0002

Summary:

A3 A1 A2 A4

All Pairwise Comparisons among Levels of BB = 1 subtracted from:

Difference SE of AdjustedB of Means Difference T-Value P-Value2 12.38 3.660 3.381 0.00893 31.38 3.660 8.573 0.0000

B = 2 subtracted from:

Difference SE of AdjustedB of Means Difference T-Value P-Value3 19.00 3.660 5.191 0.0002

Summary:

B1 B2 B3

22-31

Compare All Level Means with Tukey’s HSD (MTB)

MTB will use t=4*3=12 & n=2 to compare all level combinations of A with B.

If the interaction had been significant, we would then compare all level means…

22-32

Tukey Comparison of All Level Means (MTB)

A = 1, B = 1 subtracted from: Difference SE of AdjustedA B of Means Difference T-Value P-Value1 2 4.000 6.503 0.6151 0.99991 3 23.000 6.503 3.5367 0.09832 1 8.500 6.503 1.3070 0.96232 2 18.500 6.503 2.8448 0.26952 3 44.500 6.503 6.8428 0.00073 1 -3.500 6.503 -0.5382 1.00003 2 3.500 6.503 0.5382 1.00003 3 21.500 6.503 3.3061 0.13954 1 6.500 6.503 0.9995 0.99454 2 35.000 6.503 5.3820 0.00554 3 48.000 6.503 7.3810 0.0003

...etc....

A = 4, B = 2 subtracted from: Difference SE of AdjustedA B of Means Difference T-Value P-Value4 3 13.00 6.503 1.999 0.6882

There are a total of

t(t-1)/2=12(11)/2=66

pairwise comparisons here!

Note that if we wanted just the comparisons among levels of B (within each level of A), we should use t=3 & n=2. (Not possible in MTB.)

22-33

Example in R

ANOVA Table: Model with interaction> fruit <- read.table("fruit.txt",header=T)> fruit.lm <- lm(yield~factor(A)+factor(B)+factor(A)*factor(B),data=fruit)> anova(fruit.lm) Df Sum Sq Mean Sq F value Pr(>F) factor(A) 3 2227.5 742.5 17.5563 0.0001098 ***factor(B) 2 3996.1 1998.0 47.2443 2.048e-06 ***factor(A):factor(B) 6 456.9 76.2 1.8007 0.1816844 Residuals 12 507.5 42.3

Main Effects: Interaction is not significant so fit additive model> summary(lm(yield~factor(A)+factor(B),data=fruit))

Call: lm(formula = yield ~ factor(A) + factor(B), data = fruit)

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 38.417 3.660 10.497 4.21e-09 ***factor(A)2 14.833 4.226 3.510 0.002501 ** factor(A)3 -1.833 4.226 -0.434 0.669577 factor(A)4 20.833 4.226 4.930 0.000108 ***factor(B)2 12.375 3.660 3.381 0.003327 ** factor(B)3 31.375 3.660 8.573 9.03e-08 ***

375.12417.38ˆ

833.1417.38ˆ

833.14417.38ˆ

417.38ˆ

12

31

21

11

22-34

Profile Plot for the Example (R)

Level 3, Factor B

Level 2, Factor B

Level 1, Factor B

Averaging the 2 reps, in each A,B combination gives a typical point on the graph: ijy

interaction.plot(fruit$A,fruit$B,fruit$yield)

22-35

Pesticides and Fruit Trees Example in RCBD Layout

Suppose now that the two replicates per treatment in the experiment were obtained at different locations (Farm 1, Farm 2).

49, 39 50, 55 43, 38 53, 48

55, 41 67, 58 53, 42 85, 73

66, 68 85, 92 69, 62 85, 99

Pesticide (A)

1 2 3 4Variety (B)

1

2

3

This is now a 3 4 factorial experiment in a randomized complete block design layout with factor A at 4 levels, factor B at 3 levels, and the location (block) factor at 2 levels. (There are still t=34=12 treatments.)

The analysis would therefore proceed as in a 3-way ANOVA.