Error variance

All models used in this e-book have the general form


yi  =  
(explained)
µi

 + 
(unexplained)
εi

where

εi   ∼   normal (0, σ)

One important requirement of this model is that the error standard deviation is a value σ. This means that:

The response values have the same standard deviation for all treatments.

Transformations of the response

In some experiments, the response is more variable for some treatments than for others. This often happens when the response is a quantity that must, by definition, be greater than zero, such as weights and concentrations.

Non-constant response variation can often be cured by analysing the logarithms of the values.

Although it is not a certain way to spot when a logarithmic transformation will help, if the distribution of the raw response measurements is skew with a long tail of large values, a logarithmic transformation often helps.

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.

The diagram above shows the data and displays a model for the data (fitted by least squares). Observe that:

The torques are more variable when the P&O plating was used on bolts than for the other five treatments.

Again click Least squares then click the y-z rotation button. The three coloured sets of lines now represent the three observers and they are also parallel as a consequence of our additive model. Again drag the red arrows to see that this is always true for models without interaction.

Analysis of log(torque)

The diagram below is similar but uses the natural logarithms of the torques as the response.

Although the variation of log(torque) is still a little on the high side for the P&O plating on bolts but the assumption of equal treatment standard errors is much more reasonable for log(torque) than for torque itself.

Analyse a transformed response if it conforms to the assumption of equal treatment variation better then the raw data.