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Erlina Ambarwati FACTORIAL DESIGNS (Treatment Design)

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Presentasi berjudul: "Erlina Ambarwati FACTORIAL DESIGNS (Treatment Design)"— Transcript presentasi:

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. 7/2/2014 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. 7/2/2014 3 Erlina Ambarwati

4 Types of Treatment Structures  One-Way Treatment Structure  Factorial Arrangement Treatment Structure  Fractional Factorial Arrangement Treatment Structures 7/2/2014 4 Erlina Ambarwati

5 Assignment Rules  Completely Randomized Design  Randomized Complete Block Design  Latin Squares Design 7/2/2014 5 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. 7/2/2014 6 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. 7/2/2014 7 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. 7/2/2014 8 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. 7/2/2014 9 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. 7/2/2014 10 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 7/2/2014 11 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. 7/2/2014 12 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 7/2/2014 13 Erlina Ambarwati

14 Cells........................ Column Treatment Row Treatment..... Two-Way Factorial Design 7/2/2014 14 Erlina Ambarwati

15 Purchase of Fashion Clothing By Income and Education Low Income Purchase HighLow High Low Education 200 (100%) 300 (100%) 300 200 122 (61%) 171 (57%) 78 (39%) 129 (43%) High Income Purchase High Low 241 (80%) 151 (76%) 59 (20%) 49 (24%) Education 7/2/2014 15 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 7/2/2014 16 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 7/2/2014 17 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 =3.3 23.4 X = 3.9 43.2 Tinggi 28.2 X = 4.7 18.6 X = 3.1 46.8 Total48.042.090 7/2/201419Erlina 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 Rendah19.8 3.3 23.4 3.9 43.2 3.6 Tinggi28.2 4.7 18.6 3.1 46.8 3.9 Total48.0 4.0 42.0 3.5 90 7/2/201420Erlina Ambarwati

21 Pengaruh utama faktor A Temperatur (B) Amper (A) RendahTinggiTotal Rendah19.8 3.3 23.4 3.9 43.2 3.6 Tinggi28.2 4.7 18.6 3.1 46.8 3.9 Total48.0 4.0 42.0 3.5 90 21

22 Pengaruh sederhana faktor B pada A rendah A tinggi Pengaruh utama faktor B: Temperatur (B) Amper (A) RendahTinggiTotal Rendah19.8 3.3 23.4 3.9 43.2 3.6 Tinggi28.2 4.7 18.6 3.1 46.8 3.9 Total48.0 4.0 42.0 3.5 90 7/2/2014Erlina Ambarwati

23 Kemungkinan dalam kombinasi perlakuan Arus lemah Arus tinggi ketebalan Ada interaksi Arus tinggi Arus lemah Ada interaksi Tidak ada interaksi Temperatur 7/2/2014 23 Erlina Ambarwati

24 Perbandingan Kontras Ortogonal Untuk mengetahui efek utama dan interaksinya Temperatur (B) Amper (A) RendahTinggiTotal Rendah19.8 3.3 23.4 3.9 43.2 3.6 Tinggi28.2 4.7 18.6 3.1 46.8 3.9 Total48.0 4.0 42.0 3.5 90 7/2/201424Erlina Ambarwati

25 7/2/2014 25 Erlina Ambarwati

26 Formulas for Computing a Two-Way ANOVA 7/2/2014 26 Erlina Ambarwati

27 The Linear Model for a Two-Factor Factorial 7/2/2014 27 Erlina Ambarwati

28 7/2/2014 28 Erlina Ambarwati

29 7/2/2014 29 Erlina Ambarwati

30 db total = abr-1 dbperlk = ab-1 db A = a-1 7/2/2014 30 Erlina Ambarwati

31 dbB= b-1 db AxB = (a-1)(b-1) db error = (r-1)(ab-1) 7/2/2014 31

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 A3 11 12 32 9 13 11 38 14 9 9 27 9 8 10 28 10 14 10 34 10 8 29 11 12 10 35 13 8 12 30 10 11 11 31 9 11 30 10 9 9 26 8 7 11 24 6 125 128 111 Total B.j. 97919680364 Another example 7/2/201433Erlina Ambarwati

34 continued 7/2/2014 34

35 ANOVA SRdfSSMSFhitFtab 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 35117.56 3.42 3.03 2.51 3.42 3.03 2.51 2.77 2.72 1.63 2.77 2.72 1.63 Bagaimana jika dipecah 7/2/2014 35 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 3 23324342332434 22323342232334 45647684564768 40 7/2/201436Erlina Ambarwati

37 B1B1 B2B2 B3B3 A1A2A1A2 5757 6666 4848 15 21 12 7/2/2014 37 Erlina Ambarwati

38 7/2/2014 38 Erlina Ambarwati

39 ANOVA SVdfSSMSFstatFtab Ctrl vs treat Treatment -A -B -A*B Error 1 6-1 2-1 3-1 (3-1)(2-1) 5 1.71 5 3 0 2 1.09 1.71 1 3 0 1 0.28 7.84* 4.58 ns 13.7* 4.58 6.61 5.05 6.61 5.79 Total 3x2x2-17.8 7/2/201439Erlina Ambarwati

40 Factorial 3 x 4 x 2 Sum of treatment based on 5 replication B CA1234 111222111222 123123123123 10 9 12 19 14 16 6 8 11 16 15 14 8 7 8 12 15 13 7 10 18 15 18 7/2/201440Erlina Ambarwati

41 Level A Level B Total A (Y i… ) 1234 123123 29 23 28 22 23 25 20 22 21 25 22 28 96 90 102 Y.j.. 80706375 Level C Level B Total A (Y..k. ) 1234 1212 31 49 25 45 23 40 24 51 103 185 Level A Level C 12 123123 31 41 65 59 61 41

42 7/2/2014 42 Erlina Ambarwati

43 ANOVA SVdfSS 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 7/2/201443Erlina Ambarwati


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