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.

Catalyst surface area

Researchers conducted an experiment to investigate how the mole contents of cobalt and the calcination temperature affected the surface area of an ison-cobalt hydroxide catalyst. Four cobalt levels and five temperatures were used in the experiment and one sample of catalyst was created and tested at each of the 20 combinations of a cobalt level and temperature. The table below shows the results of the experiment.

Temperature (°F)
  Cobalt (mole)   200 300 400 500 600
0.6   90.6     82.7     58.7     43.2     25.0  
1.0 127.1 112.3 19.6 17.8 9.1
2.6 53.1 52.0 43.4 42.4 31.6
3.0 40.9 37.9 27.5 27.3 19.0

Note that layout of the data in the table does not correspond to the order of the experimental runs — the treatments were used in random order.

Torque of locknuts

A manufacturer was finding unwanted differences in the torque values of a locknut that it made. Torque is the work (i.e. force × distance) required to tighten the nut. An experiment was therefore conducted to determine what factors affected the torque values. The type of plating process was isolated as the most probably factor to impact torque. Researchers also wanted to assess the difference in torque between threading the locknut onto a bolt or a mandrel (like a bolt but harder). Twenty locknuts were manufactured with different types of plating: cadmium and wax (C&W), no plating (HT) and phosphate and oil (P&O); ten were tested on bolts and ten on mandrels. A manual torque wrench was used to determine the torque of each.

Plating type
     C&W HT P&O
Bolt   20   16  
16   19
17   14
18   15
15   24
  26   30  
40   26
28   38
38   45
38   38
  25   45  
40   49
30   33
17   30
16   20
Mandrel   24   23  
18   14
17   18
17   12
15   11
  32   28  
22   27
30   28
35   30
32   30
  10   14  
13   11
17   14
16   15
15   16

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.