# 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.

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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: –C12=Y2-Y1 C13=Y3-Y1 –C14=Y4-Y1 C23=Y3-Y2 –C24=Y4-Y2 C34=Y4-Y3 Run No.x1x2Y 12341234 1 1 1 Y1 Y2 Y3 Y4

Main effects: L1 and L2 Two 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)

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.

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) Factorial Estimates with Paired Comparisons-Three Factors

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.

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. Factorial Estimates with Paired Comparisons-Four or More Factors

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.

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

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)

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. Factorial Fractional with Paired Comparisons-The Design

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

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. Illustrative Example

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

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

The end~ Thanks for your attention.

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