# Statistics for Environmental Engineers

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The assumptions of independence, normality, and constant variance are not equally important to the ANOVA. Scheffe (1959) states, “In practice, the statistical inferences based on the above model are not seriously invalidated by violation of the normality assumption, nor,… by violation of the assumption of equality of cell variances. However, there is no such comforting consideration concerning violation of the assumption of statistical independence, except for experiments in which randomization has been incorporated into the experimental procedure.”

If measurements had been replicated, it would be possible to make a direct estimate of the error sum of squares (a2). In the absence of replication, the usual practice is to use the higher-order interactions as estimates of a2. This is justified by assuming, for example, that the fourth-order interaction has no meaningful physical interpretation. It is also common that third-order interactions have no physical significance. If sums of squares of third-order interactions are of the same magnitude as the fourth-order interaction, they can be pooled to obtain an estimate of a that has more degrees of freedom.

Because no one is likely to manually do the computations for a four-factor analysis of variance, we assume that results are available from some commercial statistical software package. The analysis that follows emphasizes variance decomposition and interpretation rather than model specification.

The first requirement for using available statistical software is recognizing whether the problem to be solved is one-way ANOVA, two-way ANOVA, etc. This is determined by the number of factors that are considered. In the example problem there are four factors: S, P, DF, and CL. It is therefore a four-way ANOVA.

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