# Statistics for Environmental Engineers

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The analysis can be generalized for к components, but it is more convenient to explain it specifically for a three-factor analysis (note that the case study involves three factors). In general, there are nb batches, ns specimens, and nt chemical tests, for a total of n = nbnsnt observations. The nested experimental design shown in Figure 25.2 consists of three batches, with two chemical tests on each of the four specimens from each batch, giving a total of n = (3)(4)(2) = 24 observations.

The overall error of any particular measurement yi will be ei = yi — n, where n is the true mean of the population of specimens. In practice, we estimate this mean by computing the average of all the measurements in the variance components experiment. A measurement on any one of the 24 test specimens produced by the design shown in Figure 25.2 will reflect variability contributed by each component, so:

ei = eb + es + et

where eb, es, and et are the error contributions from the batch, specimen, and test, respectively. Assuming these errors are random and independent, their variances will add to give the total population variance a2.

2 2 2 2

ay = ab + as + at

where the subscripts b, s, and t identify the variance components of the batches, specimens, and chemical tests, respectively. The aggregation of the error and variance components is diagrammed in Figure 25.3.

The variation among replicate chemical tests on each specimen provides an estimate of a,2. The variation among specimen averages reflects both test and specimen variance and provides an estimate of the quantity nta2s + a,2. The variation among batches embodies all three sources of variance and provides an estimate of the quantity nsntal + n,a2s + at2. The case study provides an opportunity to demonstrate the calculations to estimate these variances. A similar example with additional explanation is given by Box et al. (1978).

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