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Erlina Ambarwati FACTORIAL DESIGNS (Treatment Design)

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1 Erlina Ambarwati FACTORIAL DESIGNS (Treatment Design)

2 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. 7/2/ Erlina Ambarwati

3 Experimental Units (e.g.)  Patients with heart disease in a drug study.  Volunteers in a marketing study.  Corn seeds in an agricultural study. 7/2/ Erlina Ambarwati

4 Types of Treatment Structures  One-Way Treatment Structure  Factorial Arrangement Treatment Structure  Fractional Factorial Arrangement Treatment Structures 7/2/ Erlina Ambarwati

5 Assignment Rules  Completely Randomized Design  Randomized Complete Block Design  Latin Squares Design 7/2/ Erlina Ambarwati

6 Measurements (e.g.)  Mortality in a health outcomes study.  Survey score in marketing study.  Plant size at time x for agricultural study. 7/2/ Erlina Ambarwati

7 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. 7/2/ Erlina Ambarwati

8 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. 7/2/ Erlina Ambarwati

9 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. 7/2/ Erlina Ambarwati

10 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. 7/2/ Erlina Ambarwati

11 2 Factor Model Specification Yi = B 0 + B 1 X 1i + B 2 X 2i + B 3 X 1i X 2i + e i Y i – Outcome for i th unit B 0 – Intercept coefficient B 1 – Effect 1 coefficient B 2 – Effect 2 coefficient B 3 – Interaction coefficient X 1i – Level of factor 1 for i th unit X 2i – Level of factor 2 for i th unit e i – Error term for i th unit 7/2/ Erlina Ambarwati

12 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. 7/2/ Erlina Ambarwati

13 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 7/2/ Erlina Ambarwati

14 Cells Column Treatment Row Treatment..... Two-Way Factorial Design 7/2/ Erlina Ambarwati

15 Purchase of Fashion Clothing By Income and Education Low Income Purchase HighLow High Low Education 200 (100%) 300 (100%) (61%) 171 (57%) 78 (39%) 129 (43%) High Income Purchase High Low 241 (80%) 151 (76%) 59 (20%) 49 (24%) Education 7/2/ Erlina Ambarwati

16 Factorial Design Amount of Humor Amount of Store No Medium High Information Humor Humor Humor Low A B C Medium D E F HighG H I 7/2/ Erlina Ambarwati

17 Block IV Aa Ba Ab Bb Block III Bb Aa Ba Ab Block II Ba Bb Ab Aa Block I Ab Aa Ba Bb The 2 x 2 Factorial Experiments 7/2/ Erlina Ambarwati

18 Kombinasi Perlakuan N 1 : 25 kgP 1 : 25 kg N 2 : 50 kgP 2 : 40 kg N 3 : 75 kgP 3 : 60 kg N1N1 N2N2 N3N3 P1P1 N1P1N1P1 N2P2N2P2 N3P3N3P3 P2P2 N1P2N1P2 N2P2N2P2 N3P2N3P2 P3P3 N1P3N1P3 N2P3N2P3 N3P3N3P3 7/2/201418Erlina Ambarwati

19 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) RendahTinggiTotal Rendah 19.8 X = X = Tinggi 28.2 X = X = Total /2/201419Erlina Ambarwati

20 Pengaruh sederhana faktor A pada level rendah dari faktor B Pengaruh sederhana faktor A pada level tinggi dari faktor B Temperatur (B) Amper (A) RendahTinggiTotal Rendah Tinggi Total /2/201420Erlina Ambarwati

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

22 Pengaruh sederhana faktor B pada A rendah A tinggi Pengaruh utama faktor B: Temperatur (B) Amper (A) RendahTinggiTotal Rendah Tinggi Total /2/2014Erlina Ambarwati

23 Kemungkinan dalam kombinasi perlakuan Arus lemah Arus tinggi ketebalan Ada interaksi Arus tinggi Arus lemah Ada interaksi Tidak ada interaksi Temperatur 7/2/ Erlina Ambarwati

24 Perbandingan Kontras Ortogonal Untuk mengetahui efek utama dan interaksinya Temperatur (B) Amper (A) RendahTinggiTotal Rendah Tinggi Total /2/201424Erlina Ambarwati

25 7/2/ Erlina Ambarwati

26 Formulas for Computing a Two-Way ANOVA 7/2/ Erlina Ambarwati

27 The Linear Model for a Two-Factor Factorial 7/2/ Erlina Ambarwati

28 7/2/ Erlina Ambarwati

29 7/2/ Erlina Ambarwati

30 db total = abr-1 dbperlk = ab-1 db A = a-1 7/2/ Erlina Ambarwati

31 dbB= b-1 db AxB = (a-1)(b-1) db error = (r-1)(ab-1) 7/2/

32 ANOVA OF FACTORIAL Souce of variation Degrees of freedom a Sums of square s (SSQ) Mean square (MS) F Blocks (B)b-1SSQ B SSQ B /(b-1)MS B /MS E First factor (F1)f-1SSQ F1 SSQ F1 /(f-1)MS F1 /MS E Second factor (F2)s-1SSQ F2 SSQ F2 /(s-1)MS F2 /MS E First X Second (FxS)(f-1)*(s-1)SSQ FxS SSQ FxS /((f-1)*(s-1))MS FxS /MS E Error (E)(f*s-1)*(b-1)SSQ E SSQ E /((f*s-1)*(b-1)) Total (Tot)f*s*b-1SSQ Tot a where f=number of treatments in the first factor. s=number of treatments in the second factor and b=number of blocks or replications. 7/2/201432Erlina Ambarwati

33 Factor B Total A i.. B1B2B3B4 Factor AA1 A2 A Total B.j Another example 7/2/201433Erlina Ambarwati

34 continued 7/2/

35 ANOVA SRdfSSMSFhitFtab A B A*B Sesatan Total Bagaimana jika dipecah 7/2/ Erlina Ambarwati

36 Factorial 3 x 2 +1 control replication Total 12 Control A 1 B 1 A 1 B 2 A 1 B 3 A 2 B 1 A 2 B 2 A 2 B /2/201436Erlina Ambarwati

37 B1B1 B2B2 B3B3 A1A2A1A /2/ Erlina Ambarwati

38 7/2/ Erlina Ambarwati

39 ANOVA SVdfSSMSFstatFtab Ctrl vs treat Treatment -A -B -A*B Error (3-1)(2-1) * 4.58 ns 13.7* Total 3x2x /2/201439Erlina Ambarwati

40 Factorial 3 x 4 x 2 Sum of treatment based on 5 replication B CA /2/201440Erlina Ambarwati

41 Level A Level B Total A (Y i… ) Y.j Level C Level B Total A (Y..k. ) Level A Level C

42 7/2/ Erlina Ambarwati

43 ANOVA SVdfSS A B C AB AC BC ABC Error /2/201443Erlina Ambarwati


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