7.2 - Completely Randomized Design (CRD)

After identifying the experimental unit and the number of replications that will be used, the next step is to assign the treatments (i.e. factor levels or factor level combinations) to the experimental units.

In a completely randomized design (CRD), treatments are assigned to experimental units at random. This is typically done by listing the treatments and assigning a random number to each.

In the greenhouse experiment discussed in lesson 1, there was a single factor (fertilizer) with 4 levels (i.e. 4 treatments), six replications, and a total of 24 experimental units (potted plants). Suppose the image below is the greenhouse bench (viewed from above) that was used for the experiment.

Wall Open walkway

We need to be able to randomly assign each of the treatment levels to 6 potted plants. To do this, we first assign numbers to the physical position of the pots on the bench.

Wall Open walkway 2 3 4 5 6 8 9 10 11 12 13 1 4 15 16 18 1 9 20 21 22 23 24

Next, we randomly assign the 24 total treatments (4 treatment levels, replicated 6 times) to the potted plants. Examples of how this can be done using statistical software are found below.  

Using Technology Section

To make the assignments in SAS we can utilize the SAS surveyselect procedure seen below.

proc surveyselect data=greenhouse out=trtassignment outrandom
method=srs
samprate=1;
run;

The output would be as below. In practice, it is recommended to specify a seed to ensure the results are reproducible. 

 
Obs Fertilizer
1 F3
2 F2
3 Con
4 F2
5 F3
6 Con
7 F2
8 F2
9 F3
10 F1
11 F1
12 F3
13 F2
14 F1
15 F3
16 F3
17 F1
18 Con
19 Con
20 F2
21 Con
22 F1
23 Con
24 F1
In Minitab, this assignment can be done by manually creating two columns: one with each treatment level repeated 6 times (order not important) and the other with a position number 1 to N, where N is the total number of experimental units to be used (i.e. N = 24 in this example). The third column will store the treatment assignment.
Minitab table of data for Fert and position

Next, select Calc > Sample from Columns, fill in the dialog box as seen below, and click OK.

Minitab window for Sample from Columns
Note! Be sure to have the "Sample with replacement" box unchecked so that all treatment levels will be assigned to the same number of pots giving rise to a proper CRD for the specified number of replicates.

This will result in a completely random assignment.

Minitab table of date with Fert position and Fert_trt

This assignment can then be used to apply the treatment levels appropriately to pots on the greenhouse bench.

Wall Open walkway 2 3 4 5 6 8 9 10 11 12 13 1 4 15 16 18 1 9 20 21 22 23 24 F2 F2 F2 F2 F2 F2 F1 F1 F1 F1 F1 F3 F3 F3 F3 F3 F3 C C C C F1 C C Fertilizer 1 (Blue) Fertilizer 2 (Red) Fertilizer 3 (Black) No Fertilizer/Control (White) F2 F1 F3 C

To randomly assign treatment levels to each of our plants in R, we first create a vector containing all the possible treatments, then randomly sample from the vector to create the assignment.

trt_levels = c("F1","F2","F3","Control")
treatments = rep(trt_levels,6)

set.seed(1)
trt_assign = sample(treatments)
data.frame(trt_assign)
   trt_assign
1     Control
2          F3
3          F1
4          F2
5          F3
6          F2
7          F2
8          F2
9          F1
10    Control
11         F2
12         F2
13         F3
14         F3
15         F1
16         F3
17    Control
18         F1
19    Control
20    Control
21         F1
22         F1
23         F3
24    Control