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PPT – Factorial Treatments ( PowerPoint presentation | free to download - id: 502589-ZGVmM

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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).

Example 1 of Factors

What factors (characteristics, conditions) will

make a wiring harness for a car last longer?

FACTOR Levels Number 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.

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.

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.

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_at_(40,60), P_at_(10,20) lbs/acre.

One-factor-at-a-time approach Fix one factor

then vary the other

Treatment N P Yield Parameter T1 60 10 145 m1 T

2 40 10 125 m2 T3 40 20 160 m3

H0 m1-m2 test of N-effect (20 unit difference

observed in response). H0 m2-m3 test of

P-effect (35 unit difference observed in

response).

If I examined the yield at N60 and P20 what

would I expect to find?

E(Y N60,P20) m3(m1 - m2) 160 20

180? E(Y N60,P20) m2(m1 - m2)(m3 - m2)

1252035 180?

Interaction and Parallel Lines

We apply the N60, P20 treatment and get the

following

Treatment N P Yield Parameter T1 60 10 145 m1 T

2 40 10 125 m2 T3 40 20 160 m3 T4 60 20 130 m4

Yield

Expected T4

180

170

T3

160

N40

T1

150

140

Observed T4

N60

20

130

120

T2

P

20

10

Parallel and Non-Parallel Profiles

Parallel Lines gt the effect of the two factors

is additive (independent). Non-Parallel Lines gt

the effect of the two factors interacts

(dependent).

180

170

160

N40

150

140

N60

20

130

120

P

20

10

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.

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.)

General Data Layout Two Factor (a x b) Factorial

Column Factor (B) Row Factor(A) 1 2 3 b Tota

ls 1 T11 T12 T13 T1b A1 2 T21 T22 T23 T2b A2

3 T31 T32 T33 T3b A3 ... a Ta1 T

a2 Ta3 Tab Aa Totals B1 B2 B3 Bb G

yijk observed response for the kth replicate

(k1,,n) for the treatment defined by the

combination of the ith level of the row factor

and the jth level of the column factor.

Model

mij 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 mij are equal Ha at least two mij differ

Test as in a completely randomized design with a

x b treatments.

Sums of Squares

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.)

Main Effects

Column Factor (B) Row Factor(A) 1 2 3 b 1 m

11 m12 m13 m1b m1? 2 m21 m22 m13 m1b m2? 3 m

31 m32 m13 m1b m3? ... a ma1 ma2 m

a3 mab ma? Totals m?1 m ?2 m ?3 m ?b m ??

Factor A main effects

Testing is via a set of linear comparisons.

Testing for Main Effects Factor A

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 gt F(a-1),ab(n-1),a

Profile Analysis for Factor A

Mean for level 5 of Factor A

Mean for level 1 of Factor A

m53

180

m11

m51

170

m5 ?

160

m ? ?

m1?

150

m13

140

m52

m12

130

120

5

4

1

3

2

Factor A Levels

Profile of mean of Factor A (main effect of A).

Profile for level 2 of Factor B.

Insignificant Main Effect for Factor A

m51

180

m11

170

m5 ?

m13

160

m ? ?

m1?

150

140

m52

m12

130

120

5

4

1

3

2

Factor A Levels

Is there strong evidence of a Main Effect for

Factor A?

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

Significant Main Effect for Factor A

180

170

160

m ? ?

150

140

130

120

5

4

1

3

2

Factor A Levels

Is there strong evidence of a Main Effect for

Factor A?

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

Main Effect Linear Comparisons - Factor A

Column Factor (B) Row Factor(A) 1 2 b3 1 m1

1 m12 m13 m1? 2 m21 m22 m13 m2? 3 m31 m32 m13 m3

? 4 m41 m42 m43 m4? a5 m51 m52 m53 m5? Totals

m?1 m ?2 m ?3 m ? ?

Testing via a set of linear comparisons.

Not mutually orthogonal, but together they

represent a-14 dimensions of comparison.

Main Effect Linear Comparisons - Factor B

Column Factor (B) Row Factor(A) 1 2 b3 1 m1

1 m12 m13 m1 ? 2 m21 m22 m13 m2

? 3 m31 m32 m13 m3 ? 4 m41 m42 m43 m4

? a5 m51 m52 m53 m5 ? Totals m?1 m ? 2 m ? 3 m

? ?

Testing via a set of linear comparisons.

Not mutually orthogonal, but together they

represent b-12 dimensions of comparison.

Testing for Main Effects Factor B

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 gt F(b-1),ab(n-1),a

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.

180

180

160

160

140

140

120

120

5

4

5

1

4

3

1

3

2

2

Factor A Levels

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.

Interaction Linear Comparisons

m51

m41

m21

180

m52

m31

Interaction is lack of consistency in differences

between two levels of Factor B across levels of

Factor A.

m22

m42

m53

160

m11

m32

m43

140

m23

m12

m33

120

m13

5

4

1

3

2

Factor A Levels

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.

Model for Interaction

Tests for interaction are based on the abij terms

exclusively.

If all abij terms are equal to zero, then there

is no interaction.

Overall Test for Interaction

H0 No interaction, HA Interaction exists.

TS

F gt F(a-1)(b-1),ab(n-1),a

RR

Partitioning of Total Sums of Squares

TSS SSR SSE SSA SSB SSAB SSE

ANOVA Table

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 Tukeys 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.

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

Pesticide (A) 1 2 3

4

Variety (B) 1 2 3

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

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

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

This is a completely randomized 3 ? 4 factorial

experiment with factor A at a4 levels, and

factor B at b3 levels. There are t3?412

treatments, each replicated n2 times.

Example in Minitab

Stat gt ANOVA gt Two-way

A B yield 1 1 49 1 1 39 1 2 55 1 2 41 1 3 66 1 3 6

8 2 1 50 2 1 55 2 2 67 2 2 58 2 3 85 2 3 92 3 1 43

3 1 38 3 2 53 3 2 42 3 3 69 3 3 62 4 1 53 4 1 48

4 2 85 4 2 73 4 3 85 4 3 99

Two-way ANOVA yield versus A, B Analysis of

Variance for yield Source DF

SS MS F P A

3 2227.5 742.5 17.56

0.000 B 2 3996.1 1998.0

47.24 0.000 Interaction 6

456.9 76.2 1.80 0.182 Error

12 507.5 42.3 Total

23 7188.0

Interaction not significant refit additive model

Stat gt ANOVA gt Two-way gt additive model

Two-way ANOVA yield versus A, B Source DF

SS MS F P A 3

2227.46 742.49 13.86 0.000 B 2

3996.08 1998.04 37.29 0.000 Error 18

964.42 53.58 Total 23 7187.96 S

7.320 R-Sq 86.58 R-Sq(adj) 82.86

Analyze Main Effects with Tukeys HSD (MTB)

Stat gt ANOVA gt General Linear Model

Use to get factor or profile plots

MTB will use t4 n6 to compare A main effects,

and t3 n8 to compare B main effects.

Tukey Analysis of Main Effects (MTB)

A 1 subtracted from Difference SE of

Adjusted A of Means

Difference T-Value P-Value 2 14.833

4.226 3.5100 0.0122 3 -1.833

4.226 -0.4338 0.9719 4 20.833

4.226 4.9297 0.0006 A 2 subtracted

from Difference SE of

Adjusted A of Means Difference T-Value

P-Value 3 -16.67 4.226 -3.944

0.0048 4 6.00 4.226 1.420

0.5038 A 3 subtracted from Difference

SE of Adjusted A of

Means Difference T-Value P-Value 4

22.67 4.226 5.364 0.0002

All Pairwise Comparisons among Levels of B B 1

subtracted from Difference SE of

Adjusted B of Means Difference

T-Value P-Value 2 12.38 3.660

3.381 0.0089 3 31.38 3.660

8.573 0.0000 B 2 subtracted

from Difference SE of

Adjusted B of Means Difference T-Value

P-Value 3 19.00 3.660 5.191

0.0002

Summary B1 B2 B3

Summary A3 A1 A2 A4

Compare All Level Means with Tukeys HSD (MTB)

If the interaction had been significant, we would

then compare all level means

MTB will use t4312 n2 to compare all level

combinations of A with B.

Tukey Comparison of All Level Means (MTB)

A 1, B 1 subtracted from Difference

SE of Adjusted A B of Means

Difference T-Value P-Value 1 2 4.000

6.503 0.6151 0.9999 1 3 23.000

6.503 3.5367 0.0983 2 1 8.500

6.503 1.3070 0.9623 2 2 18.500

6.503 2.8448 0.2695 2 3 44.500

6.503 6.8428 0.0007 3 1 -3.500

6.503 -0.5382 1.0000 3 2 3.500

6.503 0.5382 1.0000 3 3 21.500

6.503 3.3061 0.1395 4 1 6.500

6.503 0.9995 0.9945 4 2 35.000

6.503 5.3820 0.0055 4 3 48.000

6.503 7.3810 0.0003 ... etc. ... A 4, B

2 subtracted from Difference SE

of Adjusted A B of Means

Difference T-Value P-Value 4 3 13.00

6.503 1.999 0.6882

There are a total of t(t-1)/212(11)/266

pairwise comparisons here! Note that if we

wanted just the comparisons among levels of B

(within each level of A), we should use t3

n2. (Not possible in MTB.)

Example in R

ANOVA Table Model with interaction gt fruit lt-

read.table("fruit.txt",headerT) gt fruit.lm lt-

lm(yieldfactor(A)factor(B)factor(A)factor(B),d

atafruit) gt anova(fruit.lm)

Df Sum Sq Mean Sq F value Pr(gtF) 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 gt summary(lm(yieldfactor(A)factor

(B),datafruit)) Call lm(formula yield

factor(A) factor(B), data fruit) Coefficients

Estimate Std. Error t value

Pr(gtt) (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

Profile Plot for the Example (R)

Averaging the 2 reps, in each A,B combination

gives a typical point on the graph

interaction.plot(fruitA,fruitB,fruityield)

Level 3, Factor B

Level 2, Factor B

Level 1, Factor B

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).

Pesticide (A) 1 2 3

4

Variety (B) 1 2 3

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

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

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

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 t3?412 treatments.) The analysis

would therefore proceed as in a 3-way ANOVA.