Simple experiments with one factor
Industrial experiments often try to assess the effects of several factors on a response and their design can be complex. We will build up to this complexity gradually, starting with relatively simple experiments. The most fundamental aspect of all experiments is that we apply treatments to experimental units, so we initially concentrate on the simplest type of experiment in which:
In laboratory and engineering experiments, it is often possible to use fairly homogeneous experimental units, so the second requirement is often met.
The concepts and methods that are introduced in this chapter are the basis for design and analysis of more complex experiments in later chapters.
Experimental design
If nothing is known about differences between the experimental units, the best way to conduct the experiment is to allocate the factor levels (treatments) at random to the experimental units. We showed earlier that non-random allocation of treatments can result in misleading estimates of their effects. In many experiments, the same number of experimental units is used for each level of the factor but this is not essential.
This type of experimental is called a completely randomised experiment.
The repeat measurements at each factor level are called replicates.
Randomisation
A key aspect of a completely randomised experiment is the random allocation of treatments to experimental units. The diagram below illustrates this randomisation for an experiment with 6 replicates for each of the 5 factor levels.
Click Randomise experiment to randomly allocate the treatments to experimental units.
Notice that the six replicates of Level 1 are equally likely to be used in any six of the experimental units. If the treatments are allocated to the experimental units in any other way, the design is not a completely randomised one. For example, if treatments are allocated to ensure that all three units in a row get the same treatment, the design is not completely randomised.
Examples
A few examples of completely randomised experiments are shown below.
Problem | Experiment | Randomisation |
---|---|---|
An engineer wants to assess how the surface finish of a metal part is affected by the feed rate of the machine that manufactures it. | Four different feed rates are used and 10 different parts are manufactured and tested at each feed rate. The surface finish of each part is recorded. | The feed rate was randomly varied from part to part so that the 10 replicates for each feed rate were in random order. The measurements of surface finish were also made in this order. |
A dairy company wants to compare the firmness of yoghurt manufactured with three different bacterial cultures. | Sixty 200ml containers of milk are heated and each culture is added to twenty of them. After 24 hours, the firmness of the yoghurt is measured with a vane test. | The 20 containers getting each culture were randomly selected from the 60. |
The yield of a chemical process is thought to depend on the temperature of the reaction and a chemical engineer wants to determine the temperature that maximises yield. | Five temperatures were chosen that cover a range of temperatures that are likely to include the optimum. The reaction is repeated 8 times at each temperature and the yield is recorded. | The 40 runs of the experiment are conducted in random order. |