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There are 8 degrees of freedom between the nine blocks erectile dysfunction solutions pump cheap apcalis sx 20 mg with mastercard, so 4 more degrees of freedom must be confounded along with the two defining splits smoking erectile dysfunction statistics order apcalis sx paypal. These additional degrees of freedom are from the generalized interactions of the defining splits erectile dysfunction youtube order apcalis sx 20 mg on-line. If P1 and P2 are the defining splits, then the generalized interactions are P1 P2 and 2 P1 P2. Recall that we always write these two-degree-of-freedom splits in a three series with exponents of 0, 1, or 2, with the first nonzero exponent always being a 1. Second, if the leading nonzero exponent is not a 1, then square the term and reduce exponents modulo three again. The net effect of this second step is to leave zero exponents as zero and swap ones and twos. Confounded effects are P1, P2, 2 P1 P2 and P1 P2 Rearrange to get a leading exponent of 1 Confounding a 33 in nine blocks, continued the defining splits in Example 15. When we confound into 27 blocks using defining splits P1, P2, and P3, there are 26 degrees of freedom between blocks, comprising thirteen twodegree-of-freedom splits. Sup- 408 Factorials in Incomplete Blocks-Confounding pose that there are q defining contrasts, P1, P2. Applying this to q = 3, we get the following confounded terms: P1, P2, P3, P1 P2, 2 2 2 2 2 2 2 P1 P2, P1 P3, P1 P3, P2 P3, P1 P3, P1 P2 P3, P1 P2 P3, P1 P2 P3, and P1 P2 P3. First remove variation between blocks, then remove any treatment variation that can be estimated; any remaining variation is used as error. When there is only one replication, the highest-order interaction is typically used as an estimate of error. When we use partial confounding, we can estimate all treatment effects, but we will only have partial information on those effects that are partially confounded. Again consider two replications of a 32, but confound A1 B 1 in the first replication and A1 B 2 in the second. We can estimate A1 B 1 in the second replication and A1 B 2 in the first, so we have 4 degrees of freedom for interaction. However, the effective sample size for each of these interaction effects is nine, rather than eighteen. Interactions containing completely confounded splits have fewer than nominal degrees of freedom 15. Derivation and methods for some of these other designs takes some (abstract) algebra. For example, we have stated that multiplying two elements of the principal block together gives another element in the principal block, and that multiplying the principal block by any element not in the principal block yields an alternate block. These are a consequence of the facts that the factor-level combinations form an (algebraic) group, the principal block is a subgroup, and the alternate blocks are cosets. Confounding sk designs when s is prime is the straightforward generalization of the 0/1 and 0/1/2 methods we used for 2k and 3k designs. For example, when s = 5 and k = 4, represent the factor levels by 0, 1, 2, 3, and 4. Block into five blocks of size 125 using the defining split ArA B rB C rC DrD by computing L = rA xA + rB xB + rC xC + rD xD mod 5 and splitting into groups based on L. If you have two defining splits P1 and 4 3 2 P2, the confounded effects are P1, P2, P1 P2, P1 P2, P1 P2, and P1 P2. Now use standard methods for confounding a pmk, but take care that none of the generalized interactions that get confounded are actually main effects. All three of these degrees of freedom are in the 9-degree-of-freedom interaction for the four-series design. It is straightforward to choose sq 1 k k blocks of size s11-q sk2 or sq blocks of size sk1 s22-q. Divide the factor-level combinations in a 33 factorial into three groups of nine according to the A1 B 1 C 2 interaction term. Suppose that we have a partially confounded 33 factorial design run in four replicates, with A1 B 1 C 1, A1 B 1 C 2, A1 B 2 C 1, and A1 B 2 C 2 confounded in the four replicates.

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