A two-level, full factorial design was implemented in these experiments to compare the effects of four different species on the corrosion parameters. A two-level, full factorial design consists of 2k experiments where k is the number of factors each with a high and low value. In this context a factor is an experimental variable, and a result is the quantitative measure of the parameter of interest. For example, in the study of corrosion , a factor may be the acid concentration in the solution, and a result would be the corrosion potential. The relevant statistical effects of each factor are found by comparing the results from all of the experiments with the high value of a factor to the results of all of the experiments with the low values. For example, for a four-factor analysis, the eight experiments with a high value of one factor are compared to the eight experiments with the low value. In this manner, it is also possible to quantify higher order (or combined) effects.
Table 1 shows a sample two-level three factor design. (Note: The design matrix and the given results are not all of the actual data collected in this experiment. The values have been simplified for the purpose of this example.)
Table 1
Experiment Factors Results
A B C Rep 1 Rep 2
(Water (Acid (Chloride (Corrosion (Corrosion
Content) conc.) conc.) Potential) Potential)
1 0 0 0 (1) -426 -314
2 1 0 0 a -226 -236
3 0 1 0 b -349 -514
4 1 1 0 ab -235 -287
5 0 0 1 c -388 -372
6 1 0 1 ac -469 -503
7 0 1 1 bc -218 -249
8 1 1 1 abc -289 -290
The mean effect of factor A alone for this three factor analysis can be found by
First order effect (A) = avg(a+)- avg(a-)
This main effect is the mean of all of the results where A was a factor minus the average of all of the results where A was not a factor. The second and third order effects can be found by the equation
Second order effect (AB)= [Mean effect of a(b+)- Mean effect of a(b-)]/2
= [Mean effect of b(a+)- Mean effect of b(a-)]/2
In the terms of our example , a second order effect would be the average water content effect with acid minus the average water content effect without acid divided by two. The other second and third order interactions are found in a similar manner. Figures 1a and 1b are the cube plots for this example. Figure 1a shows the factorial design, and Figure 1b shows the first, second and third order interactions. Table 2 gives the calculated effects.
Table 2
Experiment Mean Effect Error (Grand Mean) (-335.3) 13.3 a 37 26.5 b 63 26.5 ab 20.5 26.5 c -23.8 26.5 ac -117.8 26.5 bc 108.8 26.5 abc 4.8 26.5
When duplicate experiments are run, the variance can be determined for the entire set of data and the standard error may be found. Figure 2a shows a sample printout of the sum of the squares, the mean squares, the F ratios and the probabilities for this factorial analysis. From these values, the true effects can be separated from the effects which are within the experimental error. For this set of data, the acid content is a true effect. The second order acid effects are also strong effects. Furthermore, the same method employed to find the relevant effects in this three-factor analysis can be expanded and applied to a four-factor analysis.
Box, Hunter, and Hunter, Statistics for Experimenters. John Wiley & Sons, New York: 1978.
Hintze, Jerry L. Number Cruncher Statistical Software, Experimental Design, Version 5.4. Kaysville, Utah: 1989.
Statmost Statistical Analysis and Graphics, User's Handbook. DataMost Corporation, Salt Lake City, Utah: 1994.