1. An experimenter wishes to construct an incomplete block design for a experiment with blocks of size 4. Suppose the 4 factors are denoted by A, B, C, and D. Construct such a design by confounding...

hw


1. An experimenter wishes to construct an incomplete block design for a experiment with blocks of size 4. Suppose the 4 factors are denoted by A, B, C, and D. Construct such a design by confounding effects AD and BC with blocks. You may use the following table to facilitate your computations. Then, on the next page, indicate the treatment combinations that will go into each of the 4 blocks. Treatment Combination N-tuple representation of the treatment Combination Equation for confounding AD Equation for confounding BC Mod(2) reduction of 3rd column Mod(2) reduction of 4th column (1) a b ab c ac bc abc d ad bd abd cd acd bcd abcd Block 1 Block 2 Block 3 Block 4 2. A experiment on surface finishing of an overhead cam block auxiliary drive was reported by Sirvanci and Durmaz in 1993. The following is a version of this experiment. The three factors used in the experiment are: type of insert (Factor A), speed in rpm (Factor B), and feed rate in millimeters per minute (Factor C). The response variable is the roughness of the finished surface. The objective is to find the combination of factors that will minimize the roughness. Assume that the experiment was run using a CR design. The resulting data are shown below. Note that -1 indicates low level of the factor and 1 indicates the high level. A B C Treat Comb. Replicate 1 Replicate 2 Replicate 3 -1 -1 -1 (1) 54.6 73..0 55.4 1 -1 -1 a 86.2 66.2 86.0 -1 1 -1 b 41.4 51.2 58.6 1 1 -1 ab 62.8 64.8 74..6 -1 -1 1 c 59.6 52.8 61.0 1 -1 1 ac 82.0 72.8 73.4 -1 1 1 bc 43.4 49.0 49.6 1 1 1 abc 65.6 65.0 60.8 Note: -1 indicates low level of factor and +1 indicates high level of the factor. The above shows the experiment conducted in a CR design. Now suppose we have to run the experiment in blocks of size 4. Then each replication has to be run in two blocks, with the complete experiment needing 6 blocks. (a) Construct an incomplete block design with the interaction effect AB confounded with blocks in Replication 1, the interaction effect AC confounded with blocks in Replication 2, and the interaction effect BC confounded with blocks in Replication 3. Please show all derivation you did to determine the assignment of treatment combinations to each of the two blocks in each replication. Then indicate the treatment assignments in the table on the next page. Replication 1 Replication 2 Replication 3 Block 1 Block 3 Block 5 Block 2 Block 4 Block 6 3. Carry out the analysis of variance of the above data using the incomplete block assignment you derived in Part (a). Then copy and paste the ANOVA table (including the sum of squares due to main effects and interaction) below. 1 3 2 4 2 11223344 axaxaxax +++