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Chapter 9 Incomplete Block Designs: An Introduction 1 Incomplete Blocks of Treatments to Reduce Block Size Section 9.1-9.3 pp 310-315 These are used when you don’t have enough experimental units to run a complete replication in a block. They are also useful in that they reduce the estimate of the experimental error variance. Note that block is no longer synonymous with replication. Chamber Chamber Run 1 1 2 3 Run 2 1 2 3 Temp 25 30 40 Temp 35 30 25 o o o o o o Chamber Chamber Run 3 1 2 3 Run 4 1 2 3 Temp 40 25 35 Temp 40 30 35 o o o o o o Incomplete block designs may or may not be balanced. A balanced design has each treatment paired an equal number of times with every other treatment in the same blocks somewhere in the experiment. A partially balanced design is when different treatment pairs occur in the same blocks an unequal number of times or some treatment pairs never occur together in the same block.

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Chapter 9 Incomplete Block Designs: An Introduction 2 Chapter 9 - Balanced Incomplete Block Designs (BIB) These compare all treatments with equal precision. The incomplete block design has r replications of t treatments in b blocks of k experimental units and k<t . The total number of experimental units N=rt=bk . If r=3, t=4, b=4, k=3 we’d have N=12 as we see here. N=3(4)=4(3) The number of block in which each pair of treatments occurs together is given the symbol 8 . In the above example 8 =2 and we can calculate it directly as 8 =r(k-1)/(t-1) . 8 =3(3-1)/(4-1)=2 A balanced incomplete block design can be made by assigning the appropriate combinations of treatments to each of b= blocks or fewer and at times many fewer.
Chapter 9 Incomplete Block Designs: An Introduction 3 You actually get these designs from either reference books along the lines of Appendix 9A.1 or software. SAS will construct these with Proc QC and Proc Plan. There are also some very good programs on the market that will make these sorts of designs. For example for this design of t=4 and k=3 you would see something like (1,2,3), (1,2,4), (1,3,4), (2,3,4). This can be shown below as the left table. Following tables show the randomization scheme. Original Step 1. Randomize runs. Block Run Original Block 1 1 2 3 1 1 2 4 2 2 1 2 4 2 2 3 4 4 3 1 3 4 3 1 2 3 1 4 2 3 4 4 1 3 4 3

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Chapter 9 Incomplete Block Designs: An Introduction 4 Step 2. Randomize the treatment code to the chambers. Chamber Run A B C 1 2 4 1 2 3 4 2 3 1 2 3 4 1 4 3 Step 3. Randomize the treatment values to the codes. Chamber Run A B C 1 25 30 40 2 35 30 25 3 40 25 35 4 40 30 35
Chapter 9 Incomplete Block Designs: An Introduction 5 Analysis of BIB Designs Section 9.4 pp 315-320 Treatments and blocks are not orthogonal because all treatments don’t appear in each block. So SS done in the same manner as complete blocks will not be right and observed treatment means are biased. Blocks are fixed in this example.

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Chapter 9 Incomplete Block Designs: An Introduction 6 Vinylation of Methyl Glucoside. Example 9.2 Data a; input BLOCK PRESSNUM PERCENT @@; IF PRESSNUM = 1 THEN PRESSURE = 250; IF PRESSNUM = 2 THEN PRESSURE = 325; IF PRESSNUM = 3 THEN PRESSURE = 400;
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