5.3.3.1. Completely randomized designs

5. Process Improvement
5.3. Choosing an experimental design
5.3.3. How do you select an experimental design?

5.3.3.1. Completely randomized designs

These designs are for studying the effects of one factor without the need to take other nuisance factors into account

Here we consider completely randomized designs that have only 1 factor variable. The experiment compares the the values of a response variable based on the different levels of that factor variable.

For completely randomized designs, the levels of the factor variable are randomly assigned. By randomization, we mean that the run sequence of the experimental units is randomly assigned. For example, if there are 3 levels of the factor variable with each level to be run 2 times, then there are 6 factorial possible run sequences (or 6! ways to order the experimental trials). Because of the replication, the number of unique orderings is 90 ( since 90 = 6!/2!*2!*2!). An example of an unrandomized design would be to always run 2 replications for the first level, then 2 for the second level, and finally 2 for the third level. To randomize the runs, one way would be to put 6 slips of paper in a box with 2 having level 1, 2 having level 2, and 2 having level 3. Before each run, one of slips would be drawn blindly from the box and the level selected would be used for the next run of the experiment.

In practice, the randomization is typically performed by a computer program (in Dataplot, see the Generate Random Run Sequence menu under the main DEX menu). However, the randomization can also be generated from random number tables or by some physical mechanism (e.g., drawing the slips of paper).

All completely randomized designs with one factor are defined by 3 numbers:

   k = number of factors (= 1 for these designs)
   L = number of levels
   n = number of replications

and the total sample size (number of runs) is N = k x L x n.

Balance dictates that the number of replications be the same at each level of the factor (this will maximize the sensitivity of subsequent statistical t (or F) tests).

A typical example of a completely randomized design is the following:

      k = 1 factor (X1)
      L = 4 levels of that single factor (called "1", "2", "3", and "4")
      n = 3 replications per level
      N = 4 levels * 3 replications per level = 12 runs

The randomized sequence of trials might look like:

       X1
        3
        1
        4
        2
        2
        1
        3
        4
        1
        2
        4
        3

Note that in this example, there are 12!/3!*3!*3!*3! = 369,600 ways to run the experiment, all equally likely to be picked by a randomization procedure.

Model for a completely randomized design

Estimating and testing model factor levels

The model for the response is

Y_i,j = m+ T_i + random error
where

Y_i,j is any observation for which X1 = i
m (or mu) is the general location parameter
T_i is the effect for having treatment level i

Estimates and Statistical Tests

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

Statistical tests for levels of X1 are shown in the section on 1way ANOVA in Chapter 7.