17.0 The Test for ‘Maximum Information”

An important question to ask and answer is whether or not the current problem formulation has extracted all of the information about the important assignable causes of variation affecting the performance measurement. In this case we say that the problem formulation and associated analyses have extracted the “maximum information” from the available data.^[49]

Theoretically, if the current problem formulation has been rigorous and complete, then there are two conditions that must hold to satisfy the definition of “maximum information”.

1. The “unexplained” variation will be controlled about 0.

2. The distribution for the “unexplained” variation will be a single, unimodal distribution.^[50]

It is clear that the first condition requires that the “unexplained” variation be produced primarily by the unexplainable causes of variation $\left\{ \varepsilon_{ktn} \right\}$ producing random variation about 0. This condition cannot be met if the unexplained model components $\left\{ \lambda_{ktn} \right\}$ and $\left\{ \eta_{ktn} \right\}$ are numerous and practically significant creating unstable and unpredictable variation in the performance measurement. In practice, however, it is expected that there may be some variation due to the unexplained causes $\left\{ \lambda_{ktn} \right\}$ and $\left\{ \eta_{ktn} \right\}$ which cannot be removed from the current data.^[51] Therefore, the two conditions above are theoretical or “gold standard” requirements that should be closely, but not necessarily perfectly, met if the maximum amount of information has been extracted from the available data.

The R2 residuals are used to test the two requirements for maximum information. If the maximum amount of information has been extracted from the original data Y, then the R2 residuals should vary in a controlled manner about 0, and the distribution for these residuals should be unimodal. The Individual Chart for the R2 residuals is used to test the requirement that the unexplained variation is controlled about 0. The histogram for the R2 residuals is used to test for the requirement of a single unimodal distribution. These two analyses are presented in Figure 17-1 and Figure 17-2.