# Statistics for Environmental Engineers

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min ^( Уа — У а )2    or    min ^( ув у в )2    or    min ^( ус у с )2

There are three concerns when using the determinant criterion (Box et al., 1973). All responses must be measured independently. If we know, for example from a mass balance, that the reaction was started with 1 mole of A and that the total amount of A, B, and C must equal 1 mole throughout the course of the experiment, it is tempting to reduce the experimental burden by measuring only A and B and then computing C by difference; that is, ус = 1 — уА — ув. If we then proceed as though three concentrations had been measured independently, we invite trouble. The computed Ус is not new information. It is redundant because it is linearly dependent on the sum of уА + ув. The mathematical consequence is that the V matrix will be singular and severe computational problems will arise. The practical consequence is that the experimenter has failed to collect data that could demonstrate model adequacy by verifying the mass balance constraint.

The second concern is linear dependence among the expected values of the responses. This happens if one of the model equations is a linear combination of one or more of the other model equations. This situation is more subtle than the similar one described above with data, but its consequence is the same: namely, a singular V matrix.

A third concern is always present and it involves model lack-of-fit. The analysis assumes the model (the set of equations) is correct. Model inadequacy will inflate the residual errors, which will cause the off-diagonal elements in |V| to be large even when true correlation among the measurement errors is small.

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