Post on 27-Jan-2015
description
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 increaseii) 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 CRDMean 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 effectτi 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- a3b22)Label the experimental units with number 1 to 243)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 a1b1465 11 a1b1483 12 a1b1930 21 a1b1399 9 a1b2069 1 a1b2729 18 a1b2919 20 a1b2143 3 a2b1368 8 a2b1695 17 a2b1409 10 a2b1939 22 a2b2611 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
1a1b2
2a1b2
3a2b1
4a1b1
5a3b1
6a3b2
7a3b2
8a2b1
9a1b2
10a2b1
11a1b1
12a1b1
13a3b1
14a3b1
15a3b2
16a2b2
17a2b1
18a1b2
19a3b1
20a1b2
21a1b1
22a2b2
23a2b2
24a3b1
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 AgeGender*AgeErrorTotal
0.05040.51080.27092.99753.8296
1 2 2
18 23
0.05040.25540.13550.1665
0.30271.53390.8138
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 1712 2 2777 3 3969 6 4866 4 5958 5 6
Step 4:Randomized the sequence of running/testing the 6 treatment combination for block 4(Day 4).( Read the next 6 three digit random number from random number table)
Random Number
Ranking (Experimental
Units)
Treatment Combination
608 3 1262 2 2023 1 3916 5 4990 6 5698 4 6
Step 5:Randomized the sequence of running/testing the 6 treatment combination for block 2(Day 2).( Read the next 6 three digit random number from random number table)
Random Number
Ranking (Experimental
Units)
Treatment Combination
392 3 1877 6 2024 1 3876 5 4799 4 5032 2 6
Step 6:Randomized the sequence of running/testing the 6 treatment combination for block 1(Day 1).( Read the next 6 three digit random number from random number table)
Random Number
Ranking (Experimental
Units)
Treatment Combination
924 6 1186 2 2699 4 3790 5 4182 1 5479 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
13
11
13
22
26
22
22
36
31
33
31
43
45
45
46
54
54
56
54
61
62
64
65
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
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