FACTORIAL DESIGNS Treatment Design Erlina Ambarwati Parts of

FACTORIAL DESIGNS (Treatment Design) Erlina Ambarwati

Parts of Experimental Design 1. Set of experimental units. 2. Set of treatments. 3. Rules by which treatments are assigned to experimental units. 4. Measurements made on experimental units following application of treatment. 2 Erlina Ambarwati 12/2/2020

Experimental Units (e. g. ) Patients with heart disease in a drug study. Volunteers in a marketing study. Corn seeds in an agricultural study. 3 Erlina Ambarwati 12/2/2020

Types of Treatment Structures One-Way Treatment Structure Factorial Arrangement Treatment Structure Fractional Factorial Arrangement Treatment Structures 4 Erlina Ambarwati 12/2/2020

Assignment Rules Completely Randomized Design Randomized Complete Block Design Latin Squares Design 5 Erlina Ambarwati 12/2/2020

Measurements (e. g. ) Mortality in a health outcomes study. Survey score in marketing study. Plant size at time x for agricultural study. 6 Erlina Ambarwati 12/2/2020

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. 7 Erlina Ambarwati 12/2/2020

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. 8 Erlina Ambarwati 12/2/2020

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. 9 Erlina Ambarwati 12/2/2020

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. 10 Erlina Ambarwati 12/2/2020

2 Factor Model Specification Yi = B 0 + B 1 X 1 i + B 2 X 2 i + B 3 X 1 i. X 2 i + ei Yi – Outcome for ith unit B 0 – Intercept coefficient B 1 – Effect 1 coefficient B 2 – Effect 2 coefficient B 3 – Interaction coefficient X 1 i – Level of factor 1 for ith unit X 2 i – Level of factor 2 for ith unit ei – Error term for ith unit 11 Erlina Ambarwati 12/2/2020

Analysis of Factorial Design Main Effects – effects of each factor 12 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 12/2/2020

Example 1. Experimental units – 100 patients with depression. 2. Set of factors – drug therapy (y/n) and psychotherapy (y/n) 3. Rules - Randomly assign 25 patients to each of the possible combinations in (2). 4. Measurement – Beck Depression Scale 13 Erlina Ambarwati 12/2/2020

Two-Way Factorial Design Column Treatment. . Row Treatment Cells . . . . 14 Erlina Ambarwati 12/2/2020

Purchase of Fashion Clothing By Income and Education Low Income High Income Purchase High Low 15 High Low 122 (61%) 78 (39%) 200 (100%) 171 (57%) 129 (43%) 300 (100%) Erlina Ambarwati Education High Purchase Low High 241 (80%) 59 (20%) 300 Low 151 (76%) 49 (24%) 200 12/2/2020

Factorial Design Amount of Store Information Low Medium High 16 Erlina Ambarwati No Humor A D G Amount of Humor Medium Humor B E H High Humor C F I 12/2/2020

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 17 Erlina Ambarwati 12/2/2020

Kombinasi Perlakuan N 1: 25 kg N 2: 50 kg N 3: 75 kg P 1: 25 kg P 2: 40 kg P 3: 60 kg N 1 N 2 N 3 P 1 N 1 P 1 N 2 P 2 N 3 P 3 P 2 N 1 P 2 N 2 P 2 N 3 P 2 P 3 N 1 P 3 N 2 P 3 N 3 P 3 12/2/2020 Erlina Ambarwati 18

Contoh: pembuatan plat elektroda dengan disepuh menggunakan dua arus listrik berbeda dan dua temperatur larutan. Masing-masing kombinasi ada 6 buah. Temperatur (B) Rendah Amper (A) Tinggi Total 12/2/2020 Erlina Ambarwati Tinggi 19. 8 23. 4 X =3. 3 X = 3. 9 28. 2 18. 6 X = 4. 7 X = 3. 1 48. 0 42. 0 Total 43. 2 46. 8 90 19

Temperatur (B) Amper (A) Rendah Tinggi Total Rendah 19. 8 3. 3 23. 4 3. 9 43. 2 3. 6 Tinggi 28. 2 4. 7 18. 6 3. 1 46. 8 3. 9 Total 48. 0 42. 0 3. 5 90 Pengaruh sederhana faktor A pada level rendah dari faktor B Pengaruh sederhana faktor A pada level tinggi dari faktor B 12/2/2020 Erlina Ambarwati 20

Temperatur (B) Amper (A) Pengaruh utama faktor A Rendah Tinggi Total Rendah 19. 8 3. 3 23. 4 3. 9 43. 2 3. 6 Tinggi 28. 2 4. 7 18. 6 3. 1 46. 8 3. 9 Total 48. 0 42. 0 3. 5 90 21

Pengaruh sederhana faktor B pada A rendah A tinggi Pengaruh utama faktor B: Temperatur (B) Rendah Tinggi Total Rendah 19. 8 3. 3 23. 4 3. 9 43. 2 3. 6 Tinggi 28. 2 4. 7 18. 6 3. 1 46. 8 3. 9 Total 48. 0 42. 0 3. 5 90 Amper (A) 12/2/2020 Erlina Ambarwati

Kemungkinan dalam kombinasi perlakuan Ada interaksi Arus lemah Tidak ada interaksi ketebalan Arus lemah Arus tinggi Temperatur 23 Erlina Ambarwati 12/2/2020

Perbandingan Kontras Ortogonal Temperatur (B) Untuk mengetahui efek utama dan interaksinya Amper (A) 12/2/2020 Erlina Ambarwati Rendah Tinggi Total Rendah 19. 8 3. 3 23. 4 3. 9 43. 2 3. 6 Tinggi 28. 2 4. 7 18. 6 3. 1 46. 8 3. 9 Total 48. 0 42. 0 3. 5 90 24

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Formulas for Computing a Two-Way ANOVA 26 Erlina Ambarwati 12/2/2020

The Linear Model for a Two. Factorial 27 Erlina Ambarwati 12/2/2020

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db total = abr-1 dbperlk = ab-1 db A = a-1 30 Erlina Ambarwati 12/2/2020

db. B= b-1 db Ax. B = (a-1)(b-1) 31 db error = (r-1)(ab-1) 12/2/2020

ANOVA OF FACTORIAL Degrees of freedoma Sums of square s (SSQ) Blocks (B) b-1 SSQB/(b-1) MSB/MSE First factor (F 1) f-1 SSQF 1/(f-1) MSF 1/MSE Second factor (F 2) s-1 SSQF 2/(s-1) MSF 2/MSE (f-1)*(s-1) SSQFx. S/((f-1)*(s-1)) MSFx. S/MSE (f*s-1)*(b-1) SSQE/((f*s-1)*(b-1)) f*s*b-1 SSQTot Souce of variation First X Second (Fx. S) Error (E) Total (Tot) Mean square (MS) F awhere f=number of treatments in the first factor. s=number of treatments in the second factor and b=number of blocks or replications. 12/2/2020 Erlina Ambarwati 32

Another example Factor B Factor A A 1 A 2 A 3 Total B. j. 12/2/2020 Total Ai. . B 1 B 2 B 3 B 4 11 12 32 9 8 10 28 10 12 10 35 13 9 11 30 10 125 13 11 38 14 14 10 34 10 8 12 30 10 9 9 26 8 128 9 9 27 9 10 8 29 11 11 11 31 9 7 11 24 6 111 97 91 96 80 364 Erlina Ambarwati 33

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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 2. 72 1. 63 3. 42 3. 03 2. 51 Total 35 117. 56 Bagaimana jika dipecah 35 Erlina Ambarwati 12/2/2020

Factorial 3 x 2 +1 control replication Total 1 2 Control A 1 B 1 A 1 B 2 A 1 B 3 2 3 3 2 2 2 3 2 4 5 6 4 A 2 B 1 A 2 B 2 A 2 B 3 4 3 3 4 7 6 8 40 12/2/2020 Erlina Ambarwati 36

A 1 A 2 37 B 1 B 2 B 3 5 7 6 6 4 8 12 12 12 Erlina Ambarwati 15 21 12/2/2020

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ANOVA SV df SS MS Fstat Ftab 1. 71 5 3 0 2 1. 09 1. 71 1 3 0 1 0. 28 7. 84* 4. 58 ns 13. 7* 6. 61 5. 05 6. 61 Error 1 6 -1 2 -1 3 -1 (3 -1)(2 -1) 5 4. 58 5. 79 Total 3 x 2 x 2 -1 7. 8 Ctrl vs treat Treatment -A -B -A*B 12/2/2020 Erlina Ambarwati 39

Factorial 3 x 4 x 2 Sum of treatment based on 5 replication B C A 1 2 3 4 1 1 2 3 10 9 12 6 8 11 8 7 7 10 2 2 2 1 2 3 19 14 16 16 15 14 12 15 13 18 15 18 12/2/2020 Erlina Ambarwati 40

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

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ANOVA SV df SS A B C AB AC BC ABC Error 2 3 1 6 2 3 6 23 1. 8 5. 27 56. 03 4. 93 2. 47 2. 03 0. 67 73. 20 12/2/2020 Erlina Ambarwati 43