Mean residual sum of squares

All normal models can be expressed as:

where the error term has a normal distribution,

εi   ∼   normal (0, σ)

In any model, the mean residual sum of squares is an estimate of the error variance, σ2. This is an important summary of the model since it quantifies the amount of variation that is unexplained by the controlled factors — the 'randomness' of the data.

The mean residual sum of squares therefore provides a baseline measure of variability against which the explained sums of squares can be compared (with F ratios) to assess their significance.

Replicates and residual degrees of freedom

Since the mean residual sum of squares is the denominator of all F ratios, the more accurate it is as an estimate of σ2, the more powerful the test. Since the accuracy of this estimate depends on the residual degrees of freedom, we ideally want a large number of residual degrees of freedom.

Replicates of the experiment — two or more response values at each combination of factor levels — are the best way to get a reasonable number of residual degrees of freedom.

Data with a single replicate

Unfortunately experiments are not always conducted with replication. There are often so many treatment combinations that it would be too expensive to repeat each two or more times.

The need to have residual degrees of freedom to perform hypothesis tests means that either:


Bond strength of thermoplastic composite

Researchers conducted an experiment that dealt with the laser-assisted manufacture of a thermoplastic composite. Two factors were varied in the experiment: the laser power was set at 40W, 50W or 60W and the tape speed was set at 6.42, 13 and 27 m/s. (The tape speeds were roughly evenly spaced on a log scale, but we will treat both factors as categorical here so the spacing does not affect the results.)

A single replicate was used for each of the 9 treatments and the table below shows the interply bond strength of the composite as measured by a short-beam-shear test.

  Laser Power
Tape Speed 40 W 50 W 60 W
6.42  25.66   29.15   35.73 
13 28.00 35.09 39.56
27 20.65 29.79 35.66

The diagram below shows the data.

Drag the red arrow to add terms for the two factors and an interaction. Since there are no replicates, the model with interaction fits the data perfectly, so all residuals are zero. The residual sum of squares and degrees of freedom are therefore zero and we cannot test the significance of the interaction.

Drag the red arrow up to remove the interaction from the model. We have no way to assess whether there is an interaction between laser power and tape speed, but if we are willing to assume that they do not interact, the interaction sum of squares and degrees of freedom provide an estimate of unexplained variation and allow us to test the main effects of the factors.

If we assume that there is no interaction, we can conclude that there is strong evidence that the laser power affects bond strength but only moderate evidence of an effect of tape speed.