Two factor factorial_design_pdf
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This document describes a two-factor factorial design experiment involving two factors, each with two levels. It defines what a factorial design is, how it is more efficient than single-factor experiments, and allows investigating interactions between factors. It provides an example of a 2x2 factorial design measuring the effects of two materials and two temperatures on battery life. Analysis of variance is performed on the example to investigate main effects and interactions between the factors.
Two factor factorial_design_pdf
- 1. TWO-‐FACTOR FACTORIAL DESIGN PREPARED BY: SITI AISYAH BT NAWAWI
- 2. Basic Definition and Principles ▪ Factorial designs ➢ most efficient in experiments that involve the study of the effects of two or more factors. ➢2k means there are k factors in the experiment and each factor has two levels ➢Factor levels: ❖ Quantitative ❖ All combinations of factor levels will be investigated ❖ Number of treatment combinations = 2k ! E.g.: a levels of factor A, b levels of factor B; each replicate contains all ab treatment combinations. !!!!!!!!! !!! 32 =2 factors with each factor has 3 levels
- 3. THE ADVANTAGE OF FACTORIALS ! • The factorial designs can be easily illustrated. • More efficient than one-‐factor-‐at-‐a-‐time experiments. • A factorial design is necessary when interactions may be present to avoid misleading conclusions. • Factorial designs allow the effects of a factor to be estimated at several levels of the other factors, yielding conclusions that are valid over a range of experimental conditions. 35 26 18 9 0 no alch. one alch no barb. one barb. Interaction exist
- 4. THE TWO-‐FACTOR FACTORIAL DESIGN ! • The effects model
- 5. General arrangement for a Two-‐Factor Factorial Design ! • General arrangement for a Two-‐Factor Factorial Design table it comes from yijk where i=1,2,3,…,a (level of factor A) j=1,2,3,…,b (level of factor B) k=1,2,3,…,n (replication) See table on next page…
- 6. General arrangement for a Two-‐Factor Factorial Design ! ! ! yijk
- 7. ANOVA Table Source of Variation Sum of Squares Degrees of Freedom Mean Square F A treatments SS a – 1 B treatments SS b – 1 Interaction SS (a – 1) (b – 1) Error SS ab (n – 1) Total SS abn – 1 MS SSA −1 = a A A E F MS 0 = MS MS SSB −1 = b B B E F MS 0 = MS MS SSAB = a b ( −1)( −1) AB AB MS E F MS 0 = MS SSE = ab n ( −1) E
- 8. Cont.. a ΣΣΣ − = = = − ⋅⋅⋅ Total ijk abn i b j n k y SS y 1 1 1 2 2 y 1 ⋅⋅⋅ abn y = Σ − A i bn SS a i 2 1 2. . = y 1 ⋅⋅⋅ abn y = Σ − B j an SS b j 2 1 2 . . = AB total A B SS = SS − SS − SS E Total A B AB SS = SS − SS − SS − SS
- 9. Example
- 10. Cont..
- 11. cont.. • 2) ANOVA Now, calculate ANOVA table using formula given in previous slide
- 12. Cont.. Source of Variation Sum of Squares Degrees of Freedom Mean Square F Material types 10683.72 2 5,341.86 7.91 Temperature 39118.72 2 19,559.36 28.97 Interaction (Material*Temperature) 9613.78 4 2,403.44 3.56 Error 18230.75 27 675.21 Total 77646.97 35
- 13. cont..
- 14. Cont.. Do you get the same answer? If YES, lets continue..
- 15. Cont..
- 18. ! • Conclusion : This analysis indicates that at the temperature level 700F, the mean battery life is the same for material types 2 and 3