Definition of blocking factors
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Table of randomized block designs

Nuisance variables are those which may affect the measured result, but are not considered of primary interest.  For example, in applying a treatment, nuisance variables might be the specific operator who prepared the treatment, the time of day the experiment was run, and the room temperature.  All experiments have nuisance factors. The experimenter will typically need to spend some time deciding which nuisance factors are important enough to keep track or control, if possible, during the experiment.

When we can control nuisance factors, an important technique known as blocking can be used to reduce or eliminate the contribution to experimental error contributed by nuisance factors. The basic concept is to create homogeneous blocks where the nuisance factors are held constant and the factor of interest is allowed to vary. Within blocks, it is possible to assess the effect of different levels of the factor of interest without having to worry about variations to to changes of the block factors. 

A nuisance factor is a blocking factor if every level of the primary factor occurs the same number of times with each level of the nuisance factor. The analysis of the experiment will focus on the effect of varying levels of the primary factor within each block of the experiment.

The general rule is:

  "Block what you can, randomize what you cannot"

Blocking is used to remove the effects of a few of the most important nuisance variables.  Randomization is then used to reduce the contaminating effects of the remaining nuisance variables.

One useful way to look at a randomized block experiment is is to consider it as a collection of completely randomized experiments, each run within one of the blocks of the total experiment.

where

    L₁ = number of levels (settings) of factor 1
    L₂ = number of levels (settings) of factor 2
    L₃ = number of levels (settings) of factor 3
    L₄ = number of levels (settings) of factor 4

.
.
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    L_k = number of levels (settings) of factor k

Suppose engineers at a semiconductor manufacturing facility want to test whether different wafer implant material dosages have a significant effect on resistivity measurements after a diffusion process taking place in a furnace. They have four different dosages they want to try and enough experimental wafers from the same lot to run three wafers at each of the dosages. 

The nuisance factor they are concerned with is "furnace run", since it is known that each furnace run differs from the last and impacts many process parameters.

One ideal way to run this experiment would be to run all the 4x3=12 wafers in the same furnace run. That would eliminate the nuisance furnace factor completely. However, regular production wafers have furnace priority, and only a few experimental wafers are allowed into any furnace run at the same time. 

A non-blocked way to run this experiment would be to run each of the twelve experimental wafers, in random order, one per furnace run. That would increase the experimental error of each resistivity measurement by the run to run furnace variability and make it more difficult to study the effects of the different dosages. The blocked way to run this experiment, assuming you can convince manufacturing to let you put four experimental wafers in a furnace run, would be to put four wafers with different dosages in each of three furnace runs. The only randomization would be choosing which of the three wafers with dosage 1 would go into furnace run 1, and similarly for the wafers with dosages 2, 3 and 4. 

Let X1 be dosage "level" and X2 be the blocking factor furnace run. Then the experiment can be described as follows:

   k  = 2 factors (1 primary factor X1 & 1 blocking factor X2)
   L₁ = 4 levels of factor X1
   L₂ = 3 levels of factor X2
    n  = 1 replication per cell
   N =L₁ * L₂   =  4 * 3 = 12 runs

Before randomization, the design trials look like:

An alternate way of summarizing the design trials would be to use a 4x3 matrix whose 4 rows are the levels of the treatment X1 and whose columns are the 3 levels of the blocking variable X2. The cells in the matrix have would have indices that match the X1, X2 combinations above.
 

(1,1)	(1,2)	(1,3)
(2,1)	(2,2)	(2,3)
(3,1)	(3,2)	(3,3)
(4,1)	(4,2)	(4,3)

By extension, note that the trials for any K factor randomized block design are simply the cell indices of a K dimensional matrix.

Estimating factor effects for a randomized block design

The model for a randomized block design with one nuisance variable is 

Y_i,j  = m + T_i + B_j + random error

   Y_i,j is any observation for which X1 = i and X2 = j
   X1 is the primary factor
   X2 is the blocking factor
   m is the general location parameter
   T_i is the effect for being in treatment i (of factor X1)
   B_j is the effect for being in block j (of factor X2)

Estimates for a Randomized Block Design

   Estimate for m:   ybarall = the average of all the data
   Estimate for T_i:  ybar(i) - ybarall
                 where ybar(i) = average of all Y for which X1 = i
   Estimate for B_j:  ybar(j) - ybarall
                 where ybar(j) = average of all Y for which X2 = j