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Analyzing Factorial Experiments by Regression

KEY WORDS augmented design, center points, confidence interval, coded variables, cube plots, design
matrix, effects, factorial design, interaction, intercept, least squares, linear model, log transformation, main
effect, matrix, matrix of independent variables, inverse, nitrate, PMA, preservative, quadratic model, regres-
sion, regression coefficients, replication, standard deviation, standard error, transformation, transpose, star
points, variance, variance-covariance matrix, vector.

Many persons who are not acquainted with factorial experimental designs know linear regression. They
may wonder about using regression to analyze factorial or fractional factorial experiments. It is possible
and sometimes it is necessary.
If the experiment is a balanced two-level factorial, we have a free choice between calculating the
effects as shown in the preceding chapters and using regression. Calculating effects is intuitive and easy.
Regression is also easy when the data come from a balanced factorial design. The calculations, if done
using matrix algebra, are almost identical to the calculation of effects. The similarity and difference will
be explained.
Common experimental problems, such as missing data and failure to precisely set the levels of
independent variables, will cause a factorial design to be unbalanced or messy (Milliken and Johnson,
1992). In these situations, the simple algorithm for calculating the effects is not exactly correct and
regression analysis is advised.

Case Study: Two Methods for Measuring Nitrate

A large number of nitrate measurements were needed on a wastewater treatment project. Method A
was the standard method for measuring nitrate concentration in wastewater. The newer Method B was
more desirable (faster, cheaper, safer, etc.) than Method A, but it could replace Method A only if
shown to give equivalent results over the applicable range of concentrations and conditions. The
evaluation of phenylmercuric acetate (PMA) as a preservative was also a primary objective of the
experiment.
A large number of trials with each method was done at the conditions that were routinely being
monitored. A representative selection of these trials is shown in Table 30.1 and in the cube plots of
Figure 30.1. Panel (a) shows the original duplicate observations and panel (b) shows the average of
the log-transformed observations on which the analysis is actually done. The experiment is a fully
3
replicated 2 factorial design. The three factors were nitrate level, use of PMA preservative, and analytical
method.
The high and low nitrate levels were included in the experimental design so that the interaction of
concentration with method and PMA preservative could be evaluated. It could happen that PMA affects
one method but not the other, or that the PMA has an effect at high but not at low concentrations. The
low level of nitrate concentration (1–3 mg/L NO 3 -N) was obtained by taking influent samples from a
conventional activated sludge treatment process. The high level (20–30 mg/L NO 3 -N) was available in
samples from the effluent of a nitrifying activated sludge process.

© 2002 By CRC Press LLC