Statistics for Environmental Engineers

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Case Study: Bacterial Growth

Linearization may be helpful, as shown in Figure 45.1. The plotted data show the geometric growth of bacteria; x is time and y is bacterial population. The measurements are more variable at higher populations. Taking logarithms gives constant variance at all population levels, as shown in the right-hand panel of Figure 45.1. Fitting a straight line to the log-transformed data is appropriate and correct.

Case Study: A First-Order Kinetic Model

The model for BOD exertion as a function of time, assuming the usual first-order model, is yt = в1[1 —exp(-02t;)] + e. The rate coefficient (62) and the ultimate BOD (01) are to be estimated by the method of least squares. The residual errors are assumed to be independent, normally distributed, and with constant variance. Constant variance means that the magnitude of the residual errors is the same over the range of observed values of y. It is this property that can be altered, either beneficially or harmfully, by linearizing a model.

One linearization of this model is the Thomas slope method (TSM). The TSM was a wonderful shortcut calculation before nonlinear least squares estimates could be done easily by computer. It should never be used today; it usually makes things worse rather than better.

Why is the TSM so bad? The method involves plotting Y = y1/3 on the ordinate against X = yJb on the abscissa. The ordinate Y = y1/3 is badly distorted, first by the reciprocal and then by the cube root. The variance of the transformed variable Y is Var(Y) =    Var( y;). Suppose that the measured values are

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