Randomized Design: Single-Factor

In the design of experiments, completely randomized designs are for studying the effects of one primary factor without the need to take into account other nuisance variables. The experiment under a completely randomized design compares the values of a response variable based on the different levels of that primary factor. For completely randomized designs, the levels of the primary factor are randomly assigned to the experimental units.

Randomization

In complete random design, the run sequence of the experimental units is determined randomly. For example, if there are 3 levels of the primary factor with each level to be run 2 times, then there are $6!$ (where "!" denotes factorial) possible run sequences (or ways to order the experimental trials). Because of the replication, the number of unique orderings is 90 (since $90=\frac{6!}{2!2!2!}$). 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 the slips would be drawn blindly from the box and the level selected would be used for the next run of the experiment.

Three Key Numbers

All completely randomized designs with one primary factor are defined by three numbers: $k$ (the number of factors, which is always 1 for these designs), $L$ (the number of levels), and $n$ (the number of replications). The total sample size (number of runs) is $N=k\cdot L \cdot n$. 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). An example of a completely randomized design using the three numbers is:

• $k$ : 1 factor ($X_1$)
• $L$: 4 levels of that single factor (called 1, 2, 3, and 4)
• $n$: 3 replications per level
• $N$: 4 levels multiplied by 3 replications per level gives 12 runs