Orthogonal designs with unequal replicates

Experiments in which two factors are varied are usually designed to have an equal number of replicates for each combination of factor levels (treatments). This design simplifies analysis of the data since the sums of squares in the analysis of variance table are not affected by the order of adding the two factors to the model.

There are other designs for two factors that share this feature. If the numbers of replicates for factor B are in the same proportion for each level of factor A, then the two factors are called orthogonal and again the analysis of variance table has the same sums of squares and p-values whichever order the factors are added to the model.

Orthogonal designs with unequal replicates are often used if either or both of the factors has a control level for which we want more replicates.

Replicates for orthogonal designs

The diagram below shows the number of replicates for all treatments in orthogonal designs in which one factor (A) has 3 levels and the other factor (B) has 4 levels.

The table initially describes a balanced design with a single replicate for each treatment. (The cells in the body of the table show the number of replicates for the treatments — all 1 here.)

Type the value 2 into the text-edit box above the table for factor level A1. The table now shows replicates for an orthogonal design in which treatments involving A1 have double the replicates of those involving A2 or A3.

Now type 3 into the text-edit box to the left of the table corresponding to factor level B1. The table now shows replicates for an orthogonal design in which each treatment involving B1 has three times the replicates of the correponding treatments involving B2, B3 or B4.

Any integers can be typed into the text-edit boxes at the top and left of the table. If the replicates equal the product of these row and column factors, the design will be orthogonal.

Randomisation

As with all other experimental designs, after the number of replicates has been chosen for each treatment, it is important to randomise allocation of the treatments to the experimental units.

Click Randomise treatments to randomly allocate them to the experimental units on the right for a balanced experiment with equal replicates.

Select Orthogonal with unequal replicates to use a design with unequal replicates and again observe how the treatments are randomly allocated to the experimental units.