# FACTORIAL DESIGNS (Treatment Design)

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FACTORIAL DESIGNS (Treatment Design)
Erlina Ambarwati

Parts of Experimental Design
Set of experimental units. Set of treatments. Rules by which treatments are assigned to experimental units. Measurements made on experimental units following application of treatment. Erlina Ambarwati 4/3/2017

Experimental Units (e.g.)
Patients with heart disease in a drug study. Volunteers in a marketing study. Corn seeds in an agricultural study. Erlina Ambarwati 4/3/2017

Types of Treatment Structures
One-Way Treatment Structure Factorial Arrangement Treatment Structure Fractional Factorial Arrangement Treatment Structures Erlina Ambarwati 4/3/2017

Assignment Rules Completely Randomized Design
Randomized Complete Block Design Latin Squares Design Erlina Ambarwati 4/3/2017

Mortality in a health outcomes study. Survey score in marketing study.
Measurements (e.g.) Mortality in a health outcomes study. Survey score in marketing study. Plant size at time x for agricultural study. Erlina Ambarwati 4/3/2017

Definition of Factorial Design
An experiment in which the effects of multiple factors are investigated simultaneously. The treatments consist of all combinations that can be formed from the different factors. e.g. an experiment with 5 2-level factors would result in 32 treatments. Erlina Ambarwati 4/3/2017

Definition of Factorial Design
A set of factorial teratments consists of all combinations of all levels of two or more factors. Each treatment combination must contain one level of every factor. Erlina Ambarwati 4/3/2017

Definition of Factorial Design
The treatments are assigned randomly to the pool of experimental units with an equal number of units in each treatment. The number of experimental units assigned to each treatment is referred to as the number of replications. Erlina Ambarwati 4/3/2017

Problem of Factorial Experiments
The uniformity of experimental material in large number of treatment Factors A, B, C and D having levels a, b, c and d, there are t = abcd different treatments. With many factors and/or many levels, the number of treatments can get prohibitively large. Erlina Ambarwati 4/3/2017

2 Factor Model Specification
Yi = B0 + B1X1i + B2X2i + B3X1iX2i + ei Yi – Outcome for ith unit B0 – Intercept coefficient B1 – Effect 1 coefficient B2 – Effect 2 coefficient B3 – Interaction coefficient X1i – Level of factor 1 for ith unit X2i – Level of factor 2 for ith unit ei – Error term for ith unit Erlina Ambarwati 4/3/2017

Analysis of Factorial Design
Main Effects – effects of each factor independent of the remaining factors. Interaction Effects – 2- to n-way interaction effects between all combinations of factors. Design provides a lot more information than a single factor experiment with potentially not much more work. Erlina Ambarwati 4/3/2017

Example Experimental units – 100 patients with depression.
Set of factors – drug therapy (y/n) and psychotherapy (y/n) Rules - Randomly assign 25 patients to each of the possible combinations in (2). Measurement – Beck Depression Scale Erlina Ambarwati 4/3/2017

Two-Way Factorial Design
Column Treatment Row Treatment . . Cells . . . . . . . Erlina Ambarwati 4/3/2017 43

Purchase of Fashion Clothing By Income and Education
Low Income High Income Purchase Purchase High Low High Low 122 (61%) 78 (39%) High 241 (80%) 59 (20%) 300 High 200 (100%) Education Education 171 (57%) 129 (43%) 151 (76%) 49 (24%) 200 300 (100%) Low Low Erlina Ambarwati 4/3/2017

Factorial Design Amount of Humor Amount of Store No Medium High
Information Humor Humor Humor Low A B C Medium D E F High G H I Erlina Ambarwati 4/3/2017

The 2 x 2 Factorial Experiments
Block IV Aa Ba Ab Bb Block III Bb Aa Ba Ab Block II Ba Bb Ab Aa Block I Ab Aa Ba Bb Erlina Ambarwati 4/3/2017

Kombinasi Perlakuan N1: 25 kg P1: 25 kg N2: 50 kg P2: 40 kg
N1P1 N2P2 N3P3 P2 N1P2 N3P2 P3 N1P3 N2P3 4/3/2017 Erlina Ambarwati

Contoh: pembuatan plat elektroda dengan disepuh menggunakan dua arus listrik berbeda dan dua temperatur larutan. Masing-masing kombinasi ada 6 buah. Temperatur (B) Amper (A) Rendah Tinggi Total 19.8 X =3.3 23.4 X = 3.9 43.2 28.2 X = 4.7 18.6 X = 3.1 46.8 48.0 42.0 90 4/3/2017 Erlina Ambarwati

Pengaruh sederhana faktor A pada level rendah dari faktor B
Temperatur (B) Amper (A) Rendah Tinggi Total 90 Pengaruh sederhana faktor A pada level rendah dari faktor B Pengaruh sederhana faktor A pada level tinggi dari faktor B 4/3/2017 Erlina Ambarwati

Pengaruh utama faktor A
Temperatur (B) Amper (A) Rendah Tinggi Total 90 Pengaruh utama faktor A

Pengaruh sederhana faktor B pada A rendah A tinggi
Pengaruh utama faktor B: Temperatur (B) Amper (A) Rendah Tinggi Total 90 4/3/2017 Erlina Ambarwati

Kemungkinan dalam kombinasi perlakuan
Tidak ada interaksi Ada interaksi Ada interaksi Arus lemah Arus lemah Arus lemah ketebalan Arus tinggi Arus tinggi Arus tinggi Temperatur Erlina Ambarwati 4/3/2017

Perbandingan Kontras Ortogonal
Temperatur (B) Amper (A) Rendah Tinggi Total 90 Untuk mengetahui efek utama dan interaksinya 4/3/2017 Erlina Ambarwati

Erlina Ambarwati 4/3/2017

Formulas for Computing a Two-Way ANOVA
Erlina Ambarwati 4/3/2017 45

The Linear Model for a Two-Factor Factorial
Erlina Ambarwati 4/3/2017

Erlina Ambarwati 4/3/2017

Erlina Ambarwati 4/3/2017

db total = abr-1 dbperlk = ab-1 db A = a-1 Erlina Ambarwati 4/3/2017

dbB= b-1 db AxB = (a-1)(b-1) db error = (r-1)(ab-1) 4/3/2017

ANOVA OF FACTORIAL Souce of variation Degrees of freedoma
Sums of squares (SSQ) Mean square (MS) F Blocks (B) b-1 SSQB SSQB/(b-1) MSB/MSE First factor (F1) f-1 SSQF1 SSQF1/(f-1) MSF1/MSE Second factor (F2) s-1 SSQF2 SSQF2/(s-1) MSF2/MSE First X Second (FxS) (f-1)*(s-1) SSQFxS SSQFxS/((f-1)*(s-1)) MSFxS/MSE Error (E) (f*s-1)*(b-1) SSQE SSQE/((f*s-1)*(b-1)) Total (Tot) f*s*b-1 SSQTot awhere f=number of treatments in the first factor. s=number of treatments in the second factor and b=number of blocks or replications. 4/3/2017 Erlina Ambarwati

Another example Factor B Total Ai.. B1 B2 B3 B4 Factor A A1 A2 A3 11
12 32 9 13 11 38 14 8 10 28 10 10 34 12 10 35 12 30 11 31 11 30 9 26 7 11 24 6 125 128 111 Total B.j. 97 91 96 80 364 4/3/2017 Erlina Ambarwati

continued 4/3/2017

Bagaimana jika dipecah
ANOVA SR df SS MS Fhit Ftab A B A*B Sesatan 2 3 6 24 13.73 20.23 24.27 59.33 6.86 6.74 4.04 2.47 Total 35 117.56 2.77 2.72 1.63 3.42 3.03 2.51 Bagaimana jika dipecah Erlina Ambarwati 4/3/2017

Factorial 3 x 2 +1 control replication Total 1 2 Control A1B1 A1B2
4 5 6 7 8 40 4/3/2017 Erlina Ambarwati

B1 B2 B3 A1 A2 5 7 6 4 8 15 21 12 Erlina Ambarwati 4/3/2017

Erlina Ambarwati 4/3/2017

ANOVA SV df SS MS Fstat Ftab Ctrl vs treat Treatment Error 1 6-1 2-1
3-1 (3-1)(2-1) 5 1.71 3 2 1.09 0.28 7.84* 4.58ns 13.7* 4.58 6.61 5.05 5.79 Total 3x2x2-1 7.8 4/3/2017 Erlina Ambarwati

Factorial 3 x 4 x 2 Sum of treatment based on 5 replication B C A 1 2
10 9 12 19 14 16 6 8 11 15 7 13 18 4/3/2017 Erlina Ambarwati

Level A Level B Total A (Yi…) 1 2 3 4 29 23 28 22 25 20 21 96 90 102 Y.j.. 80 70 63 75 Level C Level B Total A (Y..k.) 1 2 3 4 31 49 25 45 23 40 24 51 103 185 Level A Level C 1 2 3 31 41 65 59 61

Erlina Ambarwati 4/3/2017

ANOVA SV df SS A B C AB AC BC ABC Error 2 3 1 6 23 1.8 5.27 56.03 4.93
2.47 2.03 0.67 73.20 4/3/2017 Erlina Ambarwati

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