Two-Level Factorial and Fractional Factorial Designs in Blocks of Size Two

1. Two-Level Factorial and Fractional Factorial Designs in Blocks of Size Two • NORMAN R. DRAPER • Journal of Quality Technology; Jan 1997; 29, 1;pg. 71 • 報告者：謝瑋珊

2. Outlines • Introduction • Factorial Estimates with Paired Comparisons • Factorial Fractional with Paired Comparisons • An Example

3. Introduction • Experimental situations are necessary to work with blocks of a given size… • Size of two. • Assume that… • Interested factorial effects are estimable… • There are no interactions of blocks with factors. • Mirror-image(or foldover) pairs… • Levels of the factor are changed completely… • Are commonly used, but…

4. Factorial Estimates with Paired Comparisons-Two Factors • Six two-per-block factorial combinations: • (1,2), (1,3), (1,4), (2,3), (2,4), and (3,4) • Each pairing causes a different block effect. • Block-free comparison: • C12=Y2-Y1 C13=Y3-Y1 • C14=Y4-Y1 C23=Y3-Y2 • C24=Y4-Y2 C34=Y4-Y3

5. Main effects: L1 and L2Two factor interaction: L12 • 2L1 = -Y1+Y2-Y3+Y4 = C12+C34 = C14-C23 • 2L2 = -Y1-Y2+Y3+Y4 = C13+C24 = C14+C23 • 2L12 = Y1-Y2-Y3+Y4 = -C12+C34 = -C13+C24 • (C12,C13,C24,C34)or(C13,C23,C14,C24)or (C12,C34,C14,C23)

6. Combining mirror-image pairs in blocks of size two permits only main effects to be estimated free of blocks. • (C14,C23) • The set (C12,C13,C24,C34) requires changes of only one factor within pairing.

7. Factorial Estimates with Paired Comparisons-Three Factors • Possible paired comparisons: • Only 12 are needed to estimate all main effects and interactions. • An example: mirror-image pairs (C18,C27,C36,C45) • To add (C12,C13,C24,C34) and (C56,C57,C68,C78)

8. In general, putting together faces like those of Figure1(a), 1(b), and 1(c) without creating repeated pairs(using any pairing Cij only once) will also work. • One choice, for example, C12, C13, C15, C24, C26, C34, C37, C48, C56, C57, C68, and C78, which are the edges of the cube.

9. Factorial Estimates with Paired Comparisons-Four or More Factors • For four factors, for example, obtained by splitting the points of the 16 into two sets where any chosen factor is at its high or low level. • 12+12+8=32 pairings are needed. • In general, a full factorial two-level design in k factor has n= , points with possible pairings.

10. Let be the number of pairings for a design. Then • The actual number of individual runs needed is twice this,that is • More by a factor of k than for the design.

11. Factorial Fractional with Paired Comparisons-The Design • Consider a design, defined by I=123. • It is still possible to perform a fractional factorial in blocks of size 2.

12. The conventional contrasts: • 2L1 = -Y1+Y2-Y3+Y4 = C12+C34 = C14-C23 • 2L2 = -Y1-Y2+Y3+Y4 = C13+C24 = C14+C23 • 2L12 = Y1-Y2-Y3+Y4 = -C12+C34 = -C13+C24 Which estimate 1+23, 2+13, 3+12 effects by (C12,C13,C24,C34) or (C13,C23,C14,C24) or (C12,C34,C14,C23)

13. Factorial Fractional with Paired Comparisons-The Design • The design defined by I=1234, and there are possible pairings. • For example, the estimate of (1+234) is (-Y1+Y2-Y3+Y4-Y5+Y6-Y7+Y8)/4; and so on.

14. General Fractional Factorials • Use the same pattern of requirement developing as for factorials. • For a fractional factorial design with runs we need pairings. • runs are needed.

15. Illustrative Example • Two manufacturers, U and G, each offer two types of stockings, an economy(E) and a better(B) model. • Possible combinations: (1)UE, (2)GE, (3)UB, (4)GB.

16. The factorial effects are main effect U to G: (C12+C34)/2=15 main effect E to B: (C13+C24)/2=63 two-factor interaction: (-C12+C34)/2=11 • G-type stocking : 15 days longer • Better stocking : 63 days longer

17. Conclusion This method is useful to know in situation where runs are cheap but the response varies over time.

18. The end~ Thanks for your attention.