Assessing differences between treatments
The treatment means in a completely randomised experiment summarise the effect of the factors. However the experimental results are variable — if the experiment was repeated with different experimental units, different means would be obtained.
In order to assess whether the differences between the treatment means are more than could be expected by chance, we must use a statistical model for the data that takes into account both the effect of the factors (explained variation) and random variation between experimental units getting the same treatment (unexplained variation).
From this model, it is possible to obtain confidence intervals for differences between treatments and also to test whether the factor does affect the response.
Reasons for variability in the response
In experimental data, there are two potential reasons for variability in the response.
Modelling the response
These two types of variation are modelled in different ways. The response, yi, for the i'th of the experimental units is written as the sum of two components,
yi = |
(explained) µi |
+ |
(unexplained) εi |
µi = f (xi, zi, ..., )
εi ∼ normal (0, σ)
Cereal bowl-life
The bowl-life of a breakfast cereal is defined to be the amount of time that the ceral will retain its crunchiness when milk is added and this depends on the temperature of the milk. An experiment was conducted in which milk was added to 5 ounces of the six portions of a particular cereal at each of 40, 45, 50 and 55°C. Bowl-life (seconds) was recorded for each portion.
Temperature of milk (°C) | |||
---|---|---|---|
40 | 45 | 50 | 55 |
13.8 14.8 11.1 11.3 9.7 8.7 |
12.5 12.7 10.9 10.5 6.5 7.1 |
10.5 10.2 8.7 8.2 6.5 5.2 |
8.7 8.7 6.8 6.6 3.6 4.3 |
The diagram below shows the data from the experiment with the crosses jittered a little (randomly moved) to separate them in the scatterplot.
Click in the centre of the diagram and drag towards the top left to display histograms of the response measurement at each value of x.
Select Model from the pop-up menu to display a model that could potentially underly the data.
Cement packing machines
A cement manufacturing company has a packing plant containing a large number of packing machines. To test the consistency of packing performance across machines, three machines (denoted by M/C 1, M/C 2 and M/C 3) were selected from the plant and 20 bags (each suposed to be of weight 50 lb) were filled in each of the three machines. The diagram below shows the actual weight (lb) of each of the 60 bags.
Since the explanatory variable (machine) is categorical, there is no requirement for a 'smooth' relationship between the mean bag weight and machine. A model can allow complete flexibility in the mean packing weight for the different machines.
Again select Model from the pop-up menu to display one such model.