# Statistics for Environmental Engineers

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There are several common situations where linear regression must be used to analyze data from a factorial experiment. The factorial design may not have been executed precisely as planned. Perhaps one run has failed so there is a missing data point. Or perhaps not all runs were replicated, or the number of replicates is different for different runs. This makes the experiment unbalanced, and matrix multiplication and inversion cannot be done by inspection as in the case of a balanced two-level design.

 Xo X, Хг x^xf A ■+1 -1 -1 +1 +1 +T +1 +1 -1 -1 +1 +1 +1 -1 -О -1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +a 0 0 a2 0 +1 -a 0 0 a2 0 +1 0 +a 0 0 a2 +1 0 -a 0 0 a2 +1 0 0 0 0 0 +1 0 0 0 0 0.

— а -1    0    +1 +а

+1

-1

FIGURE 30.2 Experimental design and matrix of independent variables for a composite design with star points and center points. This design allows a quadratic model to be fitted by regression.

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