Factorial design

An efficient design for experiments with two or more factors uses each possible combination of factor levels (called treatments) in the same number of experimental units. The repeat measurements for each treatment are called replicates and the design is called a factorial design.

The design on the previous page was an example of a factorial design for two factors but similar designs are also used for three or more factors.

Soybean yield and trace elements

An experiment was conducted to assess how different applications of manganese (Mn) and copper (Cu) affect the yield of soybeans.

In the experiment, a large field was subdivided into 32 plots and two were randomly allocated to each combination of Mn rate and Cu rate — i.e. there were 2 replicates. Soybeans were planted in rows 1 metre apart and the yield of soybeans (in kg per hectare) was recorded from each plot.

  Rate of Mn
Rate of Cu 20 50 80 110
1 1558
1578
2003
2033
2490
2470
2830
2810
2 1590
1610
2020
2051
2620
2632
2841
2841
5 1558
1550
2003
2010
2490
2690
2830
2910
7 1328
1427
2010
2031
2887
2832
2960
2941

Note that the rates of application of the two trace elements are numerical not categorical. However when only a few values are used, such factors can be treated as categorical for analysis purposes.

Legume germination

A greenhouse experiment was conducted to determine the rate of emergence of seeds of three species of legumes, treated and not treated with a fungicide, and planted in three soil types. The responses in the table below are the number of plants emerging out of 300 seeds.

  No fungicide   Fungicide
Legume species Silt loam Sand Clay   Silt loam Sand Clay
Alfalfa 266 286 66   276 271 215
Red clover 252 298 167   275 292 203
Sweet clover 152 197 52   178 219 121