Statistics for Environmental Engineers

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This arithmetic is reminiscent of that used to estimate main effects and interactions. One difference is that in estimating the effects, division is by 4 instead of by 8. This is because there were four differences used to estimate each effect. The effects indicate how much the response is changed by moving from the low level to the high level (i.e., from -1 to +1). The regression model coefficients indicated how much the response changes by moving one unit (i.e., from -1 to 0 or from 0 to +1). The regression coefficients are exactly half as large as the effects estimated using the standard analysis of two-level factorial designs.

Precision of the Estimated Parameters

The variance of the coefficients is:

Var( b) = a2/16

The denominator is 16 is because there are n — 16 observations. In this replicated experiment, a2 is estimated by s2, which is calculated from the logarithms of the duplicate observations (Table 30.2). If there were no replication, the variance would be Var(b) = a 2/8 for a 23 experimental design, and awould be estimated from data external to the design.

The variances of the duplicate pairs are shown in the table below. These can be averaged to estimate the variance for each method.




(x103) of Duplicate Pairs

SMethod — /4(X10 )



0.7122 0.5747 0.6613

sA — 0.641



2.0826 1.9544 0.2591

sB — 2.369

The variances of A and B can be pooled (averaged) to estimate the variance of the entire experiment if they are assumed to come from populations having the same variance. The data suggest that the variance of Method A may be smaller than that of Method B, so this should be checked.

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