Missing treatments

Up to this point, we have only considered experiments in which all treatments are present, even if replicated in an unbalanced way. If there are missing treatments, some model terms cannot be estimated but, except in extreme situations, it is only high-order interactions that are inestimable.

For example, if a completely randomised experiment for two factors A (with 4 levels) and B (with 5 levels) results in the replicates shown in the table below, there is no information about the response when A is at level 4 and B is at levels 4 or 5, so we cannot estimate the interaction between the factors at these levels.

  Number of replicates
   A level 1   A level 2   A level 3   A level 4 
B level 1   2 2 2 2
B level 2   2 2 1 2
B level 3   2 1 2 2
B level 4   2 2 1 0
B level 5   2 2 1 0

Analysis can proceed but since only 10 interaction parameters can be estimated (instead of the 12 interaction parameters that could be estimated if data were available from all treatments), the interaction sum of squares in the analysis of variance table will only have 10 degrees of freedom.

A more extreme example occurs with the replicates shown in the table below. Now there are so many missing treatment combinations that no interaction parameters can be estimated and we can only estimate the main effects of the two factors and show their explained sums of squares in an analysis of variance table. (There are no degrees of freedom for the interaction.)

  Replicates
   A level 1   A level 2   A level 3   A level 4 
B level 1   3 0 3 0
B level 2   3 0 0 2
B level 3   0 2 0 0
B level 4   0 0 3 0
B level 5   0 3 0 3

Warping of copper plates

An experiment was conducted to investigate warping of copper plates. The two factors studied in the experiment were the temperature and the copper content of the plates. The response variable measured the amount of warping and is modelled by:



yijk  =  µ


 + 
(explained
by Copper
)
βi


 + 
(explained
by Temp
)
γj


 + 

(interaction)
δij


 + 

(unexplained)
εijk

where parameters involving the baseline levels of the factors are defined to be zero.

The diagram below shows the resulting parameter estimates for temperature, copper and their interaction.

The experiment is initially balanced with equal replicates for all treatments. Drag the slider on the right to delete some of the observations. Observe that:

When 15 observations are missing, there is no information at all in the data about whether there is any interaction.

Missing treatments and degrees of freedom

A consequence of inestimable parameters caused by missing treatments is that the corresponding sum of squares in the analysis of variance table has fewer degrees of freedom.

Warping of copper plates

Again use the slider to delete some of the response measurements. Observe that:

The residual sum of squares reflects the unexplained variation in the data and the pairs of replicate observations within any treatment each supply one 'piece of information' about this. Observe that: