Statistics for Environmental Engineers

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FIGURE 7.1 An example of how a transformation can create constant variance. Constant variance at all levels is important so each data point will carry equal weight in locating the position of the fitted curve.


FIGURE 7.2 An example of how a transformation could create nonconstant variance.


value has roughly equal weight in determining the position of the line. The log transformation is used to achieve this equal weighting and not because it gives a straight line.


A word of warning is in order about using transformations to obtain linearity. A transformation can turn a good situation into a bad one by distorting the variances and making them unequal (see Chapter 45). Figure 7.2 shows a case where the constant variance of the original data is destroyed by an inappropriate logarithmic transformation.


In the examples above it was easy to check the variances at the different levels of the independent variables because the measurements had been replicated. If there is no replication, this check cannot be made. This is only one reason why replication is always helpful and why it is recommended in experimental and monitoring work.


Lacking replication, should one assume that the variances are originally equal or unequal? Sometimes the nature of the measurement process gives a hint as to what might be the case. If dilutions or concentrations are part of the measurement process, or if the final result is computed from the raw measurements, or if the concentration levels are widely different, it is not unusual for the variances to be unequal and to be larger at high levels of the independent variable. Biological counts frequently have nonconstant variance. These are not justifications to make transformations indiscriminately. Do not avoid making transformations, but use them wisely and with care.

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