# Statistics for Environmental Engineers

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44.5    Temperature Effect. Assuming the model for adjusting rate coefficients for temperature (k = Q1Q2T-20) is correct, design an experiment to estimate 01 and 02.

44.6    Monod Model. You are designing an experiment to estimate the parameters 01 and Q2 in the

biological model p = s+q- when someone suggests that the model should be p =12.

Would this change your experimental plans? If so, explain how.    S + Q2 + q

44.7    Oxygen Transfer. Manufacturers of aeration equipment test their products in large tanks of clean water. The test starts with a low initial dissolved oxygen concentration C0 and runs until the concentration reaches 8 to 12 mg / L. Recommend a sampling plan to obtain precise parameter estimates of the overall oxygen transfer coefficient (K) and the equilibrium dissolved oxygen concentration (CJ. The value of K is expected to be in the range of 1 to 2 h-1. The model is:

C = C„ — (C„ — C0) exp(-Kt)

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## Why Linearization Can Bias Parameter Estimates

KEY WORDS bias, biological kinetics, linear model, linearization, Lineweaver-Burke, Michaelis-Menten, nonlinear least squares, nonlinear model, parameter estimation, precision, regression, transformations, Thomas slope method.

An experimenter, having invested considerable care, time, and money, wants to extract all the information the data contain. If the purpose is to estimate parameters in a nonlinear model, we should insist that the parameter estimation method gives estimates that are unbiased and precise. Generally, the best method of estimating the parameters will be nonlinear least squares, in which variables are used in their original form and units. Some experimenters transform the model so it can be fitted by linear regression. This can, and often does, give biased or imprecise estimates. The dangers of linearization will be shown by example.

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