General Factor Factorial Design

General factor factorial designs
MEMBER’S NAME :
NIK NORAISYAH BT NIK ABD RAHMAN
NORHAIZAL BT MAHUSSAIN
NOR HAFIZA BT ISMAIL
NORAZIAH BT ISMAIL
GROUP:
D2CS2215B

BASIC DEFINITIONS AND PRINCIPLES OF THE
DESIGN
 Factorial designs are most efficient for the
experiments involve the study of the effects
of two or more factors.
 By a factorial design, we mean that in each
complete trial or replication of the
experiment all possible combination of the
levels of the factors are investigated.
 When factors are arranged in a factorial
design, they are often said to be crossed.

ADVANTAGES AND DISADVANTAGES OF
FACTORIALS DESIGN
Advantages of factorial designs :
i) There are more efficient than one-factor-at-
a time experiments.
ii) Factorial design is necessary when
interactions may be present to avoid
misleading conclusions.
iii) Factorial designs allow the effects of a
factor to be estimated at several levels of
the other factors.

Disadvantages of factorials design:
i) Size of experiment will increase if the
numbers of factors increase
ii) It is difficult to make sure the experimental
units are homogeneous if the numbers of
treatments are large.
iii) Difficult to interpret the large size of factorial
experiment especially when the interaction
between factors are exist.

CHARACTERISTICS
 The treatment must be amenable to being
administered in combination without changing dosage
in the presence of each other treatment.
 It must be acceptable not administer the individual
treatment,(i.e. placebo is ethical) or administer them
at lower doses if that will be required for the
combination.
 It must be genuinely interested in learning about
treatment combination require for the factorial design.
Otherwise some of the treatment combinations are
unnecessary, yet without them the advantages of the
factorial design are diminished.
 The therapeutic question must be chosen
appropriately, e.g., treatment that use different
mechanisms of action are more suitable candidates
for a factorial clinical trial.

WHEN TO USE
 Use when involve two or more factors that
have multiple levels. If there are many
multiple level factors, the size of a general
factor factorial design will be prohibitively
large.

LINEAR MODELS
 Fixed Effect Model Of A Two-Factor CRD
Mean model:
yijk = µijk + εijk i= 1,2,...,a
j= 1,2,...,b
k = 1,2,...,n
An alternative way to write the model for the data
is to define µijk = µ + τi + βj+(τβ)iji=1,2...,a so that
mean model become an effect model.

Effect model:
yijk = µ + τi + βj+(τβ)ij+ εijk i = 1,2,...,a
j = 1,2,...,b
k = 1,2,...,n
where:
yijk is the ijkth observtion
µ is the overall mean effect
τiis the ith level of the row factor A.
βj is the jth level of column factor B.
(τβ)ijis the interaction effect between factor A and factor B
εijkis a random error component

Blocking Factorial Design (RCBD)
Effect model:
yijk = µ + τi + βj+γk+ (τβ)ij + δk+ εijk i = 1,2,...,a
j = 1,2,...,b
k = 1,2,...,n
where:
yijk is the ijkth observation
µ is the overall mean effect
τi is the ith level of the row factor A.
βj is the jth level of column factor B.
(τβ)ij is the interaction effect between factor A and factor B.
δkis the effect of the kth block.
εijk is a random error component.

Designing a CRD Two-Factor
Factorial Experiment.
Steps:
1)Identify the treatment combination ab = 6 treatment
combination
i-a1b1 iv-a2b2
ii-a1b2 v- a3b1
iii-a2b1 vi- a3b2
2)Label the experimental units with number 1 to 24
3)Find 24 digit random number from random number table.
4)Rank the random number from the smallest to the largest
(ascending number)
5)Allocate first treatment combination to the first 4 experimental
unit, second treatment to the next 4 experimental units and
so on.

Random number Ranking
(experimental unit)
Treatment
combination
150 4 a1b1
465 11 a1b1
483 12 a1b1
930 21 a1b1
399 9 a1b2
069 1 a1b2
729 18 a1b2
919 20 a1b2
143 3 a2b1
368 8 a2b1
695 17 a2b1
409 10 a2b1
939 22 a2b2
611 16 a2b2

Random number Ranking
(experimental unit)
Treatment
combination
973 23 a2b2
127 2 a2b2
213 5 a3b1
540 14 a3b1
539 13 a3b1
976 24 a3b1
912 19 a3b2
584 15 a3b2
323 7 a3b2
270 6 a3b2

1
a1b2
2
a1b2
3
a2b1
4
a1b1
5
a3b1
6
a3b2
7
a3b2
8
a2b1
9
a1b2
10
a2b1
11
a1b1
12
a1b1
13
a3b1
14
a3b1
15
a3b2
16
a2b2
17
a2b1
18
a1b2
19
a3b1
20
a1b2
21
a1b1
22
a2b2
23
a2b2
24
a3b1
The CRD Two Factor-Factorial Design

EXAMPLE QUESTION
A manufacturing researcher wanted to determine if age
or gender significantly affect the time required to learn
an assembly line task. He randomly selected 24 adults
aged 20 to 64 years old, of whom 8 were 20 to 34 years
old ( 4 males, 4 females), 8 were 3 to 49 years old (4
males, 4 females ), 8 ere 50 to 64 years old ( 4 males, 4
females). He then measured the time (minutes )
required to complete a certain task. The data obtained
are shown below :

GENDER
AGE (years) Yi..
20 - 34 35 - 49 50 - 64
Male
5.2
5.1
5.7
6.1
{22.1}
4.8
5.8
5.0
4.8
{20.4}
5.2
4.3
5.5
4.7
{19.7}
62.2
Female
5.3
5.5
4.9
5.6
{21.3}
5.0
5.4
5.6
5.1
{21.1}
4.9
5.5
5.5
5.0
{20.9}
63.3
Y.j.
43.4 41.5 40.6 Y..= 125.5

ANOVA TABLE
Source Of
Variation
Sum of
Square
Degrees
Of
freedom
Mean
Square
F
Gender
Age
Gender*Ag
e
Error
Total
0.0504
0.5108
0.2709
2.9975
3.8296
1
2
2
18
23
0.0504
0.2554
0.1355
0.1665
0.3027
1.5339
0.8138

Hypothesis:
H0 : There is no interaction between age and gender.
H0 : There is interaction between age and gender.
Significant value: α=0.05
Test statistics: F0= = = 0.8138
Critical Value : F0.05,2,18= 3.55
Decision rule : Since F0(0.8138) < Fc (3.55) ,therefore
fail to reject Ho
Coclusion : There is no interaction between age and
gender.

DESIGNING A RCBD TWO-FACTOR
FACTORIAL EXPERIMENT
EXAMPLE:
 The procedure is shown for 3 x 2 factorial experiment
run in a randomized complete block design with n=4(4
days)
Step 1:
Identify the treatment combinations arbitrarily ab=6
treatment combination
1-a1b1 2-a1b2 3-a2b1
4-a2b2 5-a3b1 6-a3b2

Step 2 :
Randomized the sequence of the 4 blocks conducting in
the experiment.( Read the first 3-digits of the random
number block 4. Rank the random number from the
smallest to the largest as follows.)
Random Number Ranking Block/Day
909 4 1
903 3 2
212 1 3
631 2 4

Step 3:
Randomized the sequence of running/testing the 6
treatment combination for block 3(Day 3).
( Read the next 6 three digit random number from random
number table)
Random
Number
Ranking
(Experimental
Units)
Treatment
Combination
369 1 1
712 2 2
777 3 3
969 6 4
866 4 5
958 5 6

Step 4:
Randomized the sequence of running/testing the 6 treatment
combination for block 4(Day 4).
number table)
Random Number Ranking
(Experimental
Units)
Treatment
Combination
608 3 1
262 2 2
023 1 3
916 5 4
990 6 5
698 4 6

Step 5:
number table)
(Experimental
Units)
Treatment
Combination
392 3 1
877 6 2
024 1 3
876 5 4
799 4 5
032 2 6

Step 6:
number table)
(Experimental
Units)
Treatment
Combination
924 6 1
186 2 2
699 4 3
790 5 4
182 1 5
479 3 6

The following table shows the plans of the experiment with the
treatments have been allocated to experimental units according
to RCBD.
Day 1 Day 2 Day 3 Day 4
1
5
1
3
1
1
1
3
2
2
2
6
2
2
2
2
3
6
3
1
3
3
3
1
4
3
4
5
4
5
4
6
5
4
5
4
5
6
5
4
6
1
6
2
6
4
6
5

A randomized block design experiment was conducted to
investigated the effects of two factors on the number of
grass shoots. The following table summarizes the data
observed per 2.5 x 2.5cm grass area after spraying with
maleic hydrazide herbicide. Factors involve are maleic
hydrazide application rates (R) with three levels : 0,5 and
10 kg per hectare and days delay in cultivation after spray
(D) with two levels:3 and 10 days.
EXAMPLE RCBD TWO FACTOR FACTORIAL DESIGN

BLOCK
D R 1 2 3 4 TOTAL
3 0 15.7 14.6 16.5 14.7 61.5
5 9.8 14.6 11.9 12.4 48.7
10 7.9 10.3 9.7 9.6 37.5
10 0 18.0 17.4 15.1 14.4 64.9
5 13.6 10.6 11.8 13.3 49.3
10 8.8 8.2 11.3 11.2 39.5
TOTAL 73.8 75.7 76.3 75.6 301.4

General Factor Factorial Design

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