Chemical yield may be influenced by the temperature, pressure, and/or time in th

Question

Chemical yield may be influenced by the temperature, pressure, and/or
time in the reactor vessel. Each of these factors may be set at a high or a low
level. Thus we have a 23 experiment. Unfortunately, the process feedstock
is highly variable, so batch to batch differences in feedstock are expected;
we must start with new feedstock every day. Furthermore, each batch of
feedstock is only big enough for seven runs (experimental units). We have
enough money for eight batches of feedstock. We decide to use a balanced incomplete block design, with
each of the eight factor-level combinations missing from one of the blocks.

Give a skeleton ANOVA (source and degrees of freedom only), and describe
an appropriate randomization scheme.

Explanation / Answer

<Please rate the answer and comment if you need more explanation>

Let us call the 23 treatments as T1, T2, T3, ... , T8.

To have a BIBD (balanced incomplete block design) our distribution of teatment should look like following:

So specific pattern of treatment is indicated here and that will be attained by Randomisation:

Randomization scheme:

Step 1: Select a single digit random number. If number >6 then ignore the draw and re-draw until you get a number <=6. If selected number is 'i' then remove the (i+1)th treatment from Block 1. i=0,1,2,3,4,5,6

Again select a random number between 0 to 5 following the same process as above except now ignoering any draw above 5. Remove the (j+1)th treatment from the 2nd block, where jth treatment is one of the 8 - (i+1)th treatment used above.

Follow the same procedure until we know exactly which treatment to remove from which block.

Step 2: For each block, now we know which all treatment to use. Choose a random number between 1 to 7. If chosen number if 'k' then assign kth treatment to first reactor vessel. Now again number the remaining treatments as 1 to 6 and choose a random number between 1 to 6 and assign the lth treatment to second chemical reactor and so on.

ANOVA Skeleton:

Block Treatments used Block 1 T1,T2,T3,T4,T5,T6,T7 Block 2 T1,T2,T3,T4,T5,T6,T8 Block 3 T1,T2,T3,T4,T5,T7,T8 Block 4 T1,T2,T3,T4,T6,T7,T8 Block 5 T1,T2,T3,T5,T6,T7,T8 Block 6 T1,T2,T4,T5,T6,T7,T8 Block 7 T1,T3,T4,T5,T6,T7,T8 Block 8 T2,T3,T4,T5,T6,T7,T8

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