Completely randomized design

Three key numbers

All completely randomized designs with one primary 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 × L × 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).

Example

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

Sample randomized sequence of trials

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

The model for the response is ${\displaystyle Y_{i,j}=\mu +T_i+\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}}$

with

• Y_i,j being any observation for which X₁ = i (i and j denote the level of the factor and the replication within the level of the factor, respectively)
• μ (or mu) is the general location parameter
• T_i is the effect of having treatment level i

Estimates and statistical tests

Estimating and testing model factor levels

• Estimate for μ : ${\displaystyle \overline{Y}}$ = the average of all the data
• Estimate for T_i : ${\displaystyle {\overline{Y}}_i-\overline{Y}}$

with ${\displaystyle {\overline{Y}}_i}$ = average of all Y for which X₁ = i.

Statistical tests for levels of X₁ are those used for a one-way ANOVA and are detailed in the article on analysis of variance.