Factorial experiments for two factors

A factorial experiment is the most efficient way to estimate the separate effects of two factors on a response. Usually factorial experiments are designed with the same number of experimental units used for each combination of factor levels. Using the terminology of experimental design, this is expressed as the same number of replicates for each treatment.

The design of factorial experiments should also include random allocation of the treatments (factor combinations) to the pool of experimental units.

When designing experiments, never forget to randomise the allocation of treatments.

A few examples of data from factorial experiments are given below.

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 for each of the 16 treatments. 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 layout of the data in the table does not correspond to the layout of the plots in the field — the treatments were randomly allocated to the plots.

Television screen brightness

An experiment was conducted to study the effect of glass type and phosphor type on the brightness of a television screen. The variable measured is the current in microampleres necessary to produce a certain brightness; the larger this current, the poorer the screen characteristics.

  Phosphor type
Glass type     A         B         C    
Type 1 280
290
285
300
310
295
270
285
290
Type 2 230
235
240
260
240
235
220
225
230

The 18 screen samples were tested in random order.