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Last updated on Tue May 25 14:31:16 IST 2010.

Factorial design (part 3)

ANOVA

Here we shall consider a (2n ,2k ) experiment in r replicates. Constructing the ANOVA table is not entirely trivial. We split the process in 3 steps:
  1. Constructing Yate's SD table
  2. Constructing Treatment SS table
  3. Constructing the ANOVA table
Step 1: Yate's SD table
The first step is to compute the SD table. Here SD stands for ``Sum and Difference''. The SD table gives the ''effect totals'' for all the effects (totally confounded, partially confounded, not confounded). Recall that each effect has value either 0 or 1 for each run. The effect total is just the total of all observations for runs with value 1 minus the total for runs with value 0. Yate's SD table is an efficient way to compute these by hand. We shall discuss this table in the next section.

Let us denote the effect total for an effect X by CX. For example, we shall have CA, CAB etc.

To understand the SD table we have to learn a way to produce a list of number from a given list of numbers. The technique (which we shall call ``SD operation'')works only with lists of even length. It is best explained throgh an example:

To construct Yate's SD table using this technique we have to list all the 2n runs in Yate's standard order. Then we have to write down the totals for all observations for each run. Thus, if there are r replicates then for each run you have r observations (one per replicate). Add these r numbers and write the total against the run. This gives you a list of 2n totals. Let's take an example. Suppose that you have 3 factors and 2 replicates. The first replicate gives you the following observations:
Now we shall apply the SD operation on the last column n times (i.e., 3 times in our case.
The final columns give the CX's. For example, CA = -39. Notice that there is also a C1 from the first row, though there is not effect called 1. This C1 is nevertheless useful for crosschecking. It will always be the total of all the observations.
Step 2: Treatment SS
Here The treatment SS is obtained by adding SS due to each effect that is not totally confounded (let the number of such effects be e):
Now we shall learn how to find the SS dues to X (which we shall call SSX) from CX. There are two cases:
  1. If X is not confounded:
    ···(notconf)
  2. If X is partially confounded: then the formula must take into account which replicates confound X and which don't. Cx is the sum total for all the replicates. We have to remove the contribution of the replicates where X is confounded. This is done as
    CX* = CX - DX,
    where
    Here
    1. ∑'j is over replicates j confounding X,
    2. ∑i is over the 2n-k blocks in the j-th replicate.
    3. nX= number of factors in X.
    4. cji = number of factors common between X and any run in the i-th block of the j-th replicate.
    5. Bji = the total of observations in the i-th block of the j-th replicate.
    Now we shall use a formula much like (notconf):
    ···(parconf)
    where r* = number of replicates not confounding X.
Step 3: The ANOVA table
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© Arnab Chakraborty (2010)