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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