FACTORIAL DESIGNS (Treatment Design)

1. FACTORIAL DESIGNS(Treatment Design) Erlina Ambarwati

2. 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

3. 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. Types of Treatment Structures • One-Way Treatment Structure • Factorial Arrangement Treatment Structure • Fractional Factorial Arrangement Treatment Structures Erlina Ambarwati

5. Assignment Rules • Completely Randomized Design • Randomized Complete Block Design • Latin Squares Design 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. 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. 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. 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. 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. Erlina Ambarwati

11. 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

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. Erlina Ambarwati

13. 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

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

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

16. Amount of Humor Amount of Store No Medium High Information Humor HumorHumor Low A B C Medium D E F High G H I Factorial Design Erlina Ambarwati

17. 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

18. Kombinasi Perlakuan N1: 25 kg P1: 25 kg N2: 50 kg P2: 40 kg N3: 75 kg P3: 60 kg Erlina Ambarwati

19. Contoh: pembuatan plat elektroda dengan disepuh menggunakan dua arus listrik berbeda dan dua temperatur larutan. Masing-masing kombinasi ada 6 buah. Erlina Ambarwati

20. Pengaruh sederhana faktor A pada level rendah dari faktor B Pengaruh sederhana faktor A pada level tinggi dari faktor B Erlina Ambarwati

21. Pengaruh utama faktor A

22. Pengaruh sederhana faktor B pada A rendah A tinggi Pengaruh utama faktor B: Erlina Ambarwati

23. 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

24. Perbandingan Kontras Ortogonal Untuk mengetahui efek utama dan interaksinya Erlina Ambarwati

25. Erlina Ambarwati

26. Formulas for Computing a Two-Way ANOVA Erlina Ambarwati

27. The Linear Model for a Two-Factor Factorial Erlina Ambarwati

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29. Erlina Ambarwati

30. db total = abr-1 dbperlk = ab-1 db A = a-1 Erlina Ambarwati

31. dbB= b-1 db AxB = (a-1)(b-1) db error = (r-1)(ab-1)

32. ANOVA OF FACTORIAL Erlina Ambarwati

33. Another example Erlina Ambarwati

34. continued

35. ANOVA 2.77 2.72 1.63 3.42 3.03 2.51 Bagaimana jika dipecah Erlina Ambarwati

36. Factorial 3 x 2 +1 control Erlina Ambarwati

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39. ANOVA Erlina Ambarwati

40. Factorial 3 x 4 x 2 Sum of treatment based on 5 replication Erlina Ambarwati

42. Erlina Ambarwati

43. ANOVA Erlina Ambarwati