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1.0 Factorial Experiment Design by Block .......................................... 3

1.1 Factorial Experiment in Incomplete Block ............................................ 3

1.2 Factorial Experiment with Two Blocks .................................................. 3

1.3 Factorial Experiment with Four Blocks ................................................. 5

Example 1 ..................................................................................................................... 6

2.0 Fractional Factorial Experiment .................................................... 8

2.1 Half Duplicate Type of One Half Fractional Factorial Design ............ 9

2.2 Quarter Duplicate Type of One Half Fractional Factorial Design .... 11

2.3 Designing Fractional Factorial Experiment ......................................... 13

Example 2 ................................................................................................................... 14 Example 3 ................................................................................................................... 15

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Figure 1: Experimental design of 23 factorial design............................................. 4

Figure 2: Table showing “-“ and “+” of defined contrast ABC ............................. 4

Figure 3: Design table of 24 factorial experiments with defining contrast ACD ... 5

Figure 4: Generalize interaction of 24 factorial experiment .................................. 5

Figure 5: Group treatment of 24 factorial experiment ............................................ 6

Figure 6: Design table and blocks of 24 factorial experiments using AB and CD

as defining contrasts ............................................................................... 6

Figure 7: Results of 23-1 two block experimental design ....................................... 6

Figure 8: Experimental block and confounded effect of 23-1 experiment .............. 7

Figure 9: Eight corner cube used to determine the block of 23-1 experiment ......... 7

Figure 10: Experimental results of two blocks for 23-1 experiment ......................... 8

Figure 11: 24 factorial design showing ABCD interaction factor ............................ 9

Figure 12: Design table of half duplicate blocks of 24 factorial experiments using

ABCD as defining contrast ................................................................... 10

Figure 13: Design table of half duplicate block 2 of 24 factorial experiments

shown in Fig. 12 ................................................................................... 11

Figure 14: Design table and blocks of 25 factorial experiments using ABD and

ACE as defining contrasts ..................................................................... 12

Figure 15: Contrasts for selected fractional factorial designs ............................... 14

Figure 16: Basic 23 design for example 2 .............................................................. 14

Figure 17: 27-4 fractional factorial design for example 2 ....................................... 15

Figure 18: Test factors and limits for hardness of a powdered metal component . 15

Figure 19: 27-4 fractional factorial experiment arrangement for example 3 ........... 15

Figure 20: Experimental results of the 27-4 factional factorial experiment for

example 3 .............................................................................................. 16

Figure 21: ANOVA results of the factors on hardness of a powdered metal

component ............................................................................................ 17

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1.0 Factorial Experiment Design by Block

Very often one needs to eliminate the influence of extraneous factors when running

an experiment. One can do it by block method. What one’s concern is one factor in

the presence of one or more unwanted factors? For example, one predicted a shift

will occur while an experiment is being carried out. This can happen when one has

to change to a new batch of raw material mid-way throughout the experiment due

to insufficient material or limited blender capacity. Thus, the objective is to

eliminate this factor influencing the data analysis.

1.1 Factorial Experiment in Incomplete Block

Let’s use a 23 factorial design to illustrate how blocking is being designed. In order

to make all eight experiments in 23 2-level full factorial design, eight experiments

are to be conducted under same conditions, which is as homogeneous as possible. It

requires batches of raw material to be used are sufficient for completing all eight

experiments. If it requires changing to new batch of material, all the eight

experiments will not have “identical” material. In this case, the 23 design can

arranged in two blocks of four experiments each to neutralize the effect of possible

blend difference. One block uses the old batch of material and the other block uses

new batch of material. It is equivalent to perform two 23-1 factorial experiment.

The disadvantage with such an experimental set-up is that certain effects are

completely confounded or mixed with the blocks. As the result of blocking in which

the number of effects confounded, it depends on the number of blocks.

1.2 Factorial Experiment with Two Blocks

One effect is confounded in an experiment with two blocks. Usually the highest

order interaction is selected to be confounded. Thus, the three-factor interaction

effect is confounded in a 23 factorial design with two blocks. In this scenario, only

the main effects and two-factor interactions can be studied. The method of

distributing the experimental combinations between the blocks for a 23 factorial

design is shown as follows.

1. Define the effect to be confounded called the defining contrast. In this case, the

logical defining contrast is the three-factor interaction ABC because ABC is the

highest interaction.

2. Write all the 23 combinations in a table with “-” representing low level and “+”

representing high level. The experimental combinations for this factorial design

are given in Fig. 3.28.

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3. All the combinations that have the sign “-” in column ABC in Fig. 1grouped

into one block, whereas the other combinations that have the sign “+” form the

second block shown in Fig. 2.

4. Perform the experiments using blended material 1 for the experiments in block

1, while blended material 2 is to be used for experiments of block 2.

Experiment

i

Factor

A B C ABC

1 - - - -

2 - - + +

3 - + - +

4 - + + -

5 + - - +

6 + - + -

7 + + - -

8 + + + +

Figure 1: Experimental design of 23 factorial design

Block 1 for ABC equal to “+” Block 2 for ABC equal to “-”

A B C A B C

- - + - - -

- + - - + +

+ - - + - +

+ + + + + -

Figure 2: Table showing “-“ and “+” of defined contrast ABC

Let’s look at how to divide the experimental combinations in a 24 factorial

experiment into two blocks using ACD as the defining contrast.

The experimental design is shown in Fig. 3. “*” indicates the selected

experimental combination for “-” and “+” blocks.

Experiment

i

Main Factor and Defining Contrast Block 1

“-”

Block 2

“+” A B C D ACD

1 - - - - - *

2 - - - + + *

3 - - + - + *

4 - - + + - *

5 - + - - - *

6 - + - + + *

7 - + + - + *

8 - + + + - *

9 + - - - + *

10 + - - + - *

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11 + - + - - *

12 + - + + + *

13 + + - - + *

14 + + - + - *

15 + + + - - *

16 + + + + + *

Figure 3: Design table of 24 factorial experiments with defining contrast ACD

1.3 Factorial Experiment with Four Blocks

If the treatment combinations of a 2k factorial experiment are to be divided into four

incomplete blocks then the experimenter can choose any two defining contrasts i.e.

those effects that will be confounded with the blocks. A third effect, called the

generalized interaction of the two defining contrasts, is automatically confounded

with the blocks. Thus, a total of three effects will be confounded with blocks in an

experiment with four incomplete blocks.

Let’s look at the procedure to divide a 24 factorial experiment into four

incomplete blocks.

1. The experimenter needs to choose two defining contrasts and two effects that

are to be confounded. Supposing the experimenter chooses AB and CD as the

defining contrasts.

2. The third effect, which is the generalized interaction that will be confounded

by multiplying both the defining contrasts and choosing the letters with odd

exponent only. In this case, ABxCD = ABCD is the generalized interaction,

because each of the letter A, B, C, and D has an exponent of one. More examples

defining contrast and generalized interactions of 24 factorials are given in Fig.

4.

3. Group the treatment combinations into four blocks based on the signs in the

defining contrasts selected is shown in Fig. 5. In this case, the table design and

blocks are shown in Fig. 6.

4. The experimental observations corresponding to the treatment combinations in

each block should be collected under identical conditions.

Defining Contrast Generalized Interaction

AB ABC C

ABD ABC CD

BCD AB ACD

Figure 4: Generalize interaction of 24 factorial experiment

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AB CD Block

- - 1

- + 2

+ - 3

+ + 4

Figure 5: Group treatment of 24 factorial experiment

Main Factor Contrast Block

A B C D AB CD 1 2 3 4

- - - - + + *

- - - + + - *

- - + - + - *

- - + + + + *

- + - - - + *

- + - + - - *

- + + - - - *

- + + + - + *

+ - - - - + *

+ - - + - - *

+ - + - - - *

+ - + + - + *

+ + - - + + *

+ + - + + - *

+ + + - + - *

+ + + + + + *

Figure 6: Design table and blocks of 24 factorial experiments using AB and CD as defining

contrasts

Example 1

The results of 23-1 two blocks experimental design are shown in Fig. 7. Experiment

1, 4, 6, and 7 use old batch of material while, experiment 2, 3, 5, and 8 use new

batch of material. Determine the significance of the factors using effect method.

# Factor

Results A B C

1 - 1 - 1 - 1 34

2 - 1 - 1 + 1 62

3 - 1 + 1 - 1 43

4 - 1 + 1 + 1 58

5 + 1 - 1 - 1 56

6 + 1 - 1 + 1 51

7 + 1 + 1 - 1 58

8 + 1 + 1 + 1 54 Figure 7: Results of 23-1 two block experimental design

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Solution

Using the block method mentioned the text which is using effect to be confounded

and naturally it is the highest interaction ABC. Thus, the experimental block are

shown in Fig. 8.

# Factor Confounded Effect

Block Results A B C ABC

1 - 1 - 1 - 1 - 1 1 34

2 - 1 - 1 + 1 + 1 2 62

3 - 1 + 1 - 1 + 1 2 43

4 - 1 + 1 + 1 - 1 1 58

5 + 1 - 1 - 1 + 1 2 56

6 + 1 - 1 + 1 - 1 1 51

7 + 1 + 1 - 1 - 1 1 58

8 + 1 + 1 + 1 + 1 2 54

Figure 8: Experimental block and confounded effect of 23-1 experiment

Alternatively, the blocks can be determined from the eight corner cube shown in

Fig. 9. Experiment 1, 4, 6, and 7, which are the white corner are designated for block

1, while experiment 2, 3, 5, and 8, which are the black corner are designated for

block 2.

Figure 9: Eight corner cube used to determine the block of 23-1 experiment

The response of two blocks are shown in Fig. 10.

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# Factor

Results A B C

1 - 1 - 1 - 1 34

4 - 1 + 1 + 1 58

6 + 1 - 1 + 1 51

7 + 1 + 1 - 1 58

(a) Experimental results of block 1

# Factor

Results A B C

2 - 1 - 1 + 1 62

3 - 1 + 1 - 1 43

5 + 1 - 1 - 1 56

8 + 1 + 1 + 1 54

(b) Experimental results of block 2

Figure 10: Experimental results of two blocks for 23-1 experiment

Using block 1 results, the effect due to factor A is (51 + 58)/2 - (34 + 58)/2 = 8.5

The effect due factor B is (58 +58)/2 - (34 + 51)/2 = 15.5.

The effect due to factor C is (58 + 51)/2 - (58 + 34)/2 = 8.5.

Using block 2 results, the effect due to factor A is (56 + 54)/2 - (62 + 43)/2 = 2.5

The effect due factor B is (43 +54)/2 - (62 + 56)/2 = - 10.6.

The effect due to factor C is (62 + 54)/2 - (53 + 54)/2 = 4.5.

Both blocks show that factor B has significant effective, while the significant effect

of factor A and C are about the same.

2.0 Fractional Factorial Experiment

According to Glossary & Tables for Statistical Quality Control published by The

American Society of Quality Control ASQC 1983, it defines fractional factorial

design as “A factorial experiment in which only an adequately chosen fraction of

the treatment combinations required for the complete factorial experiment is

selected to be run”.

The 2k factorial experiment can become quite large and involve large resource

if k value is large. In many experimental situations, certain higher order interactions

are assumed to negligible or even though they are not negligible. It would be a waste

of experimental effort to use the complete factorial experiment. Thus, when k is

large, the experimenter can make use of a fractional factorial experiment whereby

only one half, one fourth, or even one eighth of the total factorial experimental

design is actually carried out. It is desired that the chosen fractional factorial designs

experiments have the desirable properties of being both balanced and orthogonal.

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Alternative for k value larger than five, Plackett-Burman design is also a better

choice. In 1946, R. L. Plackett and J. P. Burman published their famous paper

entitled “The Design of Optimal Multifactorial Experiments” which described the

construction of very economical design with the run number a multiple of four

instead of a power of two. Plackett-Burman design is a very efficient screening

design when only main effects are the interested factors.

In fractional factorial experiment, there is confounded effect where the main

factors are used to estimate another main factor from the estimate of the interaction

effect between two or more main factors. This would mean confounding lead to the

loss of ability to estimate some effects and/or interactions.

2.1 Half Duplicate Type of One Half Fractional Factorial Design

The construction of a half duplicate design is same as the allocation of a 2k 2-level

factorial experiment into two blocks. Firstly, a defining contrast is selected to be

confounded then the two blocks are constructed with either one of them can be

selected as the design to be carried out the experiment.

Let’s consider a 24 factorial experimental design as shown in Fig. 11 showing

four factor ABCD interaction.

Experiment

i

Factor

A B C D ABCD

1 - - - - +

2 - - - + -

3 - - + - -

4 - - + + +

5 - + - - -

6 - + - + +

7 - + + - +

8 - + + + -

9 + - - - -

10 + - - + +

11 + - + - +

12 + - + + -

13 + + - - +

14 + + - + -

15 + + + - -

16 + + + + +

Note that column ABCD is obtaining by multiplying the sign of column A, B, C, and D

Figure 11: 24 factorial design showing ABCD interaction factor

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If one wishes to use a half duplicate design with the chosen defining contrast ABCD

then based on the 24 factorial design showing ABCD interaction factor shown in Fig.

11, the two block experimental designs can be formulated and is shown in Fig. 12.

Block 1 for ABCD equal to “-” Block 2 for ABCD equal to “+”

A B C D A B C D

- - - + - - - -

- - + - - - + +

- + - - - + - +

- + + + - + + -

+ - - - + - - +

+ - + + + - + -

+ + - + + + - -

+ + + - + + + +

Figure 12: Design table of half duplicate blocks of 24 factorial experiments using ABCD as

defining contrast

From Fig. 12, either block can be selected for experiment. If one selects block 2 then

experimental data needs to be collected following experimental combinations

shown Fig. 13 that contains eight combinations, with all possible main factors and

interactions in a 24 full factorial experimental design. Even though there are two or

more duplications like AD and BC, it allows us to calculate an explicit sum of square

for error with no increase in the number of sum of square due to main factors or

interactions. The number of sum of square with the above data is 8 - 1 = 7. The total

number of possible effects i.e. main factors and their interactions in a 24 experiment

is 15, out of which interaction ABCD is not present in block 2, because all the

combinations in this block have the same sign “+”. This leaves out 14 effects that

are present in the experiment, which means that each of the seven sum of squares is

shared by two effects. It can be seen in Fig. 3.40 that there are seven pairs of effects

i.e. main factors and interactions such that the effects in each pair have the same “−”

and “+” signs and the same sum of squares. All the pairs are A&BCD, B&ACD,

C&ABD, D&ABC, AB&CD AC&BD, and AD&BC. The effects in a pair are called

aliases. The aliases in each group can be obtained by deleting the letters with an

even exponent from the product of the effects i.e. main factor or interaction and the

defining contrast. For example, the alias of A is AxABCD = A2BCD = BCD. The

aliases in this one half fractional factorial design are (A + BCD), (B + ACD), (C +

ABD), (D + ABC), (AB + CD), (AC + BD), and (AD + BC).

In summary, in a one half fractional factorial design, the sum of squares of the

defining contrast cannot be calculated. In addition, there are exactly two effects,

which are the main factors and/or interactions are in each alias group. If the test

statistic obtained from the sum of squares of an alias group is significant, one cannot

determine which one of the members of that group is the significant factor without

supplementary statistical evidence. However, fractional factorial designs have their

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greatest use when k is quite large and there is some previous knowledge concerning

the interactions. It becomes evident that one should always be aware of what the

alias structure is for a fractional experiment before finally adopting the experimental

plan. Proper choice of defining contrasts and awareness of the alias structure are

important considerations before an experimental design is selected.

Main Factor Interaction Factor

A B C D AB AC AD BC BD CD ABC ABD ACD BCD ABCD

- - - - + + + + + + - - - - +

- - + + + - - - - + + + - - +

- + - + - + - - + - + - + - +

- + + - - - + + - - - + + - +

+ - - + - - + + - - + - - + +

+ - + - - + - - + - - + - + +

+ + - - + - - - - + - - + + +

+ + + + + + + + + + + + + + +

Figure 13: Design table of half duplicate block 2 of 24 factorial experiments shown in Fig. 12

2.2 Quarter Duplicate Type of One Half Fractional Factorial Design

The construction of a quarter duplicate design is identical to the allocation of a 2k

factorial experiment into four blocks. Two defining contrasts are specified to

partition the 2k combinations into four blocks. Any one of the four blocks can be

selected to perform the experiment and analysis. In this design, the defining

contrasts and the generalized interaction are not present because each of these will

have the same sign “−” or “+” in any block selected.

Let’s consider a one quarter fractional design of a 25 factorial design,

constructed using ABD and ACE as the defining contrasts. The generalized

interaction is BCDE. The experimental design combinations and assignment of

blocks are shown in Fig. 14.

Experiment

i

Main Factor Defining

Contrasts Block Assignment

A B C D E ABD ACE 1 2 3 4

1 - - - - - - - *

2 - - - - + - + *

3 - - - + - + - *

4 - - - + + + + *

5 - - + - - - + *

6 - - + - + - - *

7 - - + + - + + *

8 - - + + + + - *

9 - + - - - + - *

10 - + - - + + + *

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11 - + - + - - - *

12 - + - + + - + *

13 - + + - - + + *

14 - + + - + + - *

15 - + + + - - + *

16 - + + + + - - *

17 + - - - - + + *

28 + - - - + + - *

19 + - - + - - + *

20 + - - + + - - *

21 + - + - - + - *

22 + - + - + + + *

23 + - + + - - - *

24 + - + + + - + *

25 + + - - - - + *

26 + + - - + - - *

27 + + - + - + + *

28 + + - + + + - *

29 + + + - - - - *

30 + + + - + - + *

31 + + + + - + - *

32 + + + + + + + *

Figure 14: Design table and blocks of 25 factorial experiments using ABD and ACE as defining

contrasts

In this design, ABD, ACE, and BCDE are not present because each of these factors

will have the same “−” or “+” sign in any of the four blocks. This leaves out 25 - 1 -

3 = 28 effects, which consist of five main factors and 23 interactions factors in this

design.

Since the total number of experimental combinations of the design is 1/4 (25),

which is 8, only seven (8 - 1) sums of square can be calculated. This means that each

sum of squares is shared by 28/7 = 4 effects, which are main factors and interaction

factors. Thus, there are four aliases in each group. The aliases in each group can be

obtained by deleting the letters with even exponents from the products of any one

effect i.e. main factor or interaction with each defining contrast and the generalized

interaction. For example, the aliases of factor A are AxABD = A2BD = BD, AxACE

= A2CE = CE, and AxBCDE = ABCDE.

This means that factor A, and interaction BD, CE and ABCDE share the same

sum of square, mean square, and test statistics.

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2.3 Designing Fractional Factorial Experiment

The type of alias relationship presents in a fractional factorial design of experiment

is defined by its resolution, which are resolution III, IV, and V.

Resolution III design: In this type of design, no main factor is aliased with any

other main factor. The main factors are aliased with two-factor interactions and the

two-factor interactions are aliased with other two-factor interactions. Examples are

23-1 and 25-2 designs.

Resolution IV design: It is a design where no main factor is aliased with either

another main factor or a two-factor interaction. Two-factor interactions are aliased

with other two-factor interactions. Examples are 24-1 and 26-2 designs.

Resolution V design: In this design, no main factor is aliased with either

another main factor or a two-factor interaction. No two-factor is aliased with other

two-factor interactions and two-factor interactions are aliased with three-factor

interactions. Examples are 25-1 and 26-1 designs.

Figure 15 contains recommended defining contrasts for selected fractional

factorial designs and their resolutions. A basic design is a 2a full factorial design

where a = k - q. For example, the basic design of a 27-3 fractional factorial design is

a 24 full factorial design. The number of rows, which are treatment combinations in

a 2k-q fractional factorial design is equal to the number of rows which are treatment

combinations in the associated basic design.

Number

of Factor

k

Fractional

Design 2k-q Resolution

Experiment/Treatment

Combination

Defining

Contract

3 23-1 (1/2) III 4 ABC

4 24-1 (1/2) IV 8 ABCD

5 25-2 (1/4) III 8 ABD, ACE

25-1 (1/2) V 16 ABCDE

6 26-3 (1/8) III 8 ABD, ACE, BCF

26-2 (1/4) IV 16 ABCE, BCDF

7

27-4 (1/16) III 8 ABD, ACE, BCF,

ABCG

27-3 (1/8) IV 16 ABCE, BCDF,

ACDG

27-2 (1/4) IV 32 ABCDF, ABDEG

8

28-4 (1/16) IV 16 BCDE, ACDF,

ABCG, ABDH

28-3 (1/8) IV 32 ABCF, ABDG,

BCDEH

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9

29-5 (1/32) III 16

ABCE, BCDF,

ACDG, ABDH,

ABCDJ

29-4 (1/16) IV 32 BCDEF, ACDEG,

ABDEH, ABCEJ

29-3 (1/8) IV 64 ABCDG, ACEFH,

CDEFJ Figure 15: Contrasts for selected fractional factorial designs

Example 2

An experiment is to be conducted to test the effect of seven factors on some response

variables. The experimenter is satisfied with Resolution III. It is a 27-4 fractional

design.

Solution

From Fig. 15, the recommended defining contrasts for this example are ABD, ACE,

BCF, and ABCG.

Start with the basic design, which is a 2a full factorial design where a = k - q. Since

k = 7 and q = 4, thus, the basic design is 23 full factorial design, which contains

factor A, B, and C and its orthogonal array is shown in Fig. 16.

Experiment

i

Main Factor Interaction Factor

A B C AB AC BC ABC

1 - - - + + + -

2 - - + + - - +

3 - + - - + - +

4 - + + - - + -

5 + - - - - + +

6 + - + - + - -

7 + + - + - - -

8 + + + + + + +

Figure 16: Basic 23 design for example 2

Using the alias relationship, identify the columns for the remaining q factors, which

are D, E, F, and G. One sees the following aliases.

One alias of D is DxABD = AB, which means D and AB share the same column.

One alias of E is ExACE = AC, which means E and ACE share the same column.

One alias of F is FxBCF = BC, which means F and BCF share the same column.

One alias of G is GxABCG = ABC, which means G and ABC share the same column.

Since they are aliases, they can be replaced. Thus, after replacing interaction factor

AB, AC, BC, and ABC of Fig. 16 with main factor D, E, F, and G respectively, Fig.

17 shown the modified experimental combinations for the final 27-4 fractional

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factorial design. Note that in real-life applications, signs “+” and “−” in the design

table are replaced by the actual levels of the factors.

Experiment

i

Factor

A B C D E F G

1 - - - + + + -

2 - - + + - -

3 - + - - + - +

4 - + + - - + -

5 + - - - - + +

6 + - + - + - -

7 + + - + - - -

8 + + + + + + +

Figure 17: 27-4 fractional factorial design for example 2

Example 3

Figure 18 contains test factor and limit for conducting experiments to test the effects

of seven factors on the hardness of a powdered metal component by analysis of

variance of all the seven factors.

Factor Test Limits

Designator Components Lower Limit Upper Limit

A Material composition 5% (-) 10% (+)

B Binder type 1 (-) 2 (+)

C Position in the basket Bottom (-) Top (+)

D Temperature of heat treatment 800oF (-) 900oF (+)

E Quenching bath medium Water (-) Oil (+)

F Annealing temperature 300oF (-) 400oF (+)

G Speed of conveyor belt in

annealing oven 2ft/min (-) 4ft/min (+)

Figure 18: Test factors and limits for hardness of a powdered metal component

27-4 fractional factorial experiment arrangement for example 3 is shown in Fig. 19.

A

Material

Composition

B

Binder

Type

C

Position

in the

Basket

D

Temperature

of Heat

Treatment

E

Quenching

Bath

Medium

F

Annealing

Temperature

G

Speed of

Conveyor Belt

in Annealing

Oven 5 1 Bottom 900 Oil 400 2

5 1 Top 900 Water 300 4

5 2 Bottom 800 Oil 300 4

5 2 Top 800 Water 400 2

10 1 Bottom 800 Water 400 4

10 1 Top 800 Oil 300 2

10 2 Bottom 900 Water 300 2

10 2 Top 900 Oil 400 4

Figure 19: 27-4 fractional factorial experiment arrangement for example 3

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The experimental results – the hardness test including the mean of duplicate and

mean of the average of the duplicate for each experiment are shown in Fig. 20.

Experiment

i

Factor Duplicate

yik

2

1i

iky

2

12

1

i

iky

A B C D E 1 2

1 - - - + + 71 72 143 71.5

2 - - + + - 106 100 206 103.0

3 - + - - + 59 62 121 60.5

4 - + + - - 91 94 185 92.5

5 + - - - - 122 119 241 120.5

6 + - + - + 91 94 185 92.5

7 + + - + - 131 119 250 125.0

8 + + + + + 85 69 154 77.0

Grand Total 1,485.0

Figure 20: Experimental results of the 27-4 factional factorial experiment for example 3

Solution

The analysis of variance ANOVA begins with calculating the sum of square for all

the factor A, B, C, D, and E.

Total sum of square is 2x2

SS3

22

1

2

12

1

2

1

2

T

3

3

j k

ik

i k

ik

y

y

.4.546,76.826,137373,14516

14856985119131

9491119122949162591001067271SS

22222

222222222222

T

The sum of square due to factor A

is SSA =

16

1485

8

154250185241

8

185121206143 222

= 139,740.6 - 137,826.6

= 1,914.0.

The sum of square due to factor B

is SSB =

16

1485

8

154250185121

8

185241206143 222

= 138,090.6 - 137,826.6 = 264.1.

The sum of square due to interaction of factor C

is SSC =

16

1485

8

154185185206

8

250241121143 222

= 137,865.6 - 137,826.6 = 39.1.

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The sum of square due to factor D

is SSD =

16

1485

8

154250206143

8

185241185121 222

= 137,854.1 - 137,826.6 = 27.6.

The sum of square due to interaction of factor E

is SSE =

16

1485

8

154185121143

8

250241185206 222

= 142,691.6 - 137,8276.6 = 4,865.1.

The analysis of variance ANOVA table for the hardness test is shown in Fig. 21.

Factor

Sum of

Square

of Factor

Degree of

Freedom of

Factor

Mean

Square of

Factor

Calculated

F-value for

Factor

F-value

from F-

Table for

= 0.05

p-value

A 1,914.0 1 1,914.0 43.8 F0.05(1, 10)

= 4.96 < 0.001

B 264.1 1 264.1 6.0 F0.05(1, 10)

= 4.96 < 0.050

C 39.1 1 39.1 0.9 F0.05(1, 10)

= 4.96 > 0.100

D 27.6 1 27.6 0.6 F0.05(1, 10)

= 4.96 > 0.100

E 4,865.1 1 4,865.1 111.3 F0.05(1, 10)

= 4.96 < 0.001

Error 436.5 10 43.7 - - -

Total 7,546.4 15 - - - -

Figure 21: ANOVA results of the factors on hardness of a powdered metal component

Results show that factor A, B, and E are significant at = 0.05. They have effects

on the hardness of a powdered metal component.

Soo King Lim

- 18 -

A

Analysis of variance ......................................... 16, 17

B

Burman, J. P. ............................................................ 9

C

Confounding ............................................................. 9

D

Defining contrast ...................................................... 3

F

Factorial Experiment design by blocking ................. 3

Fractional factorial experiment ................................. 8

G

Generalized interaction ............................................. 5

P

Plackett, R. L. ........................................................... 9

Plackett-Burman design ............................................ 9

p-value .................................................................... 17

R

Resolution III design ............................................... 13

Resolution IV design............................................... 13

Resolution V design ................................................ 13

S

Sum of square of error ............................................ 10

T

The American Society of Quality Control ................ 8