TwoLevel Factorial and Fractional Factorial Designs in Blocks

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

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

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…

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: – C 12=Y 2 -Y 1 C 13=Y 3 -Y 1 – C 14=Y 4 -Y 1 C 23=Y 3 -Y 2 – C 24=Y 4 -Y 2 C 34=Y 4 -Y 3 Run No. x 1 x 2 Y 1 2 3 4 -1 1 -1 -1 1 1 Y 2 Y 3 Y 4

Main effects: L 1 and L 2 Two factor interaction: L 12 • 2 L 1 = -Y 1+Y 2 -Y 3+Y 4 = C 12+C 34 = C 14 -C 23 • 2 L 2 = -Y 1 -Y 2+Y 3+Y 4 = C 13+C 24 = C 14+C 23 • 2 L 12 = Y 1 -Y 2 -Y 3+Y 4 = -C 12+C 34 = -C 13+C 24 • (C 12, C 13, C 24, C 34)or(C 13, C 23, C 14, C 24)or (C 12, C 34, C 14, C 23)

• Combining mirror-image pairs in blocks of size two permits only main effects to be estimated free of blocks. – (C 14, C 23) • The set (C 12, C 13, C 24, C 34) requires changes of only one factor within pairing.

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 (C 18, C 27, C 36, C 45) • To add (C 12, C 13, C 24, C 34) and (C 56, C 57, C 68, C 78)

• In general, putting together faces like those of Figure 1(a), 1(b), and 1(c) without creating repeated pairs(using any pairing Cij only once) will also work. • One choice, for example, C 12, C 13, C 15, C 24, C 26, C 34, C 37, C 48, C 56, C 57, C 68, and C 78, which are the edges of the cube.

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.

• 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.

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.

The conventional contrasts: – 2 L 1 = -Y 1+Y 2 -Y 3+Y 4 = C 12+C 34 = C 14 -C 23 – 2 L 2 = -Y 1 -Y 2+Y 3+Y 4 = C 13+C 24 = C 14+C 23 – 2 L 12 = Y 1 -Y 2 -Y 3+Y 4 = -C 12+C 34 = -C 13+C 24 Which estimate 1+23, 2+13, 3+12 effects by (C 12, C 13, C 24, C 34) or (C 13, C 23, C 14, C 24) or (C 12, C 34, C 14, C 23)

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 (-Y 1+Y 2 -Y 3+Y 4 -Y 5+Y 6 -Y 7+Y 8)/4; and so on.

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.

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.

• The factorial effects are main effect U to G: (C 12+C 34)/2=15 main effect E to B: (C 13+C 24)/2=63 two-factor interaction: (-C 12+C 34)/2=11 • G-type stocking : 15 days longer • Better stocking : 63 days longer

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

The end~ Thanks for your attention.