The General factorial design is an extension of the two-level factorial design. Here the Factors can take more than two levels each. The design treats all Factors as categorical variables and includes all combinations of the factor levels. The Factors need not all have the same number of levels. A design with three factors: A at 2 levels, B at 4 levels and C at 5 levels, will have 2*4*5 = 40 runs or trials. In general, a design with K Factors where A has a levels, B has b levels ... K has k levels, will have a*b*c*...*k runs/trials.
The number of runs in a general factorial design quickly multiply and become prohibitively large, so two-level designs offer a more economical alternative, especially as Screening Designs when a large number of Factors are under consideration.
Two factors, Detergent Brand and Wash Cycle, are to be varied to see which combination yields the highest 'brightness score' for colored clothes. Four detergent brands are picked for the analysis and used for each of the three wash cycles: Delicate, Normal and Heavy Duty. Identical pieces of colored cloth are used for each run and a subjective 'brightness score' assigned on a scale of 1 to 10 for 'least' to 'most' bright. Each combination of the 4 detergent brands and 3 wash cycles is run, giving 12 runs. The design table is shown, along with the brightness scores for the 12 runs labeled o11 to o34. Analysis of the data should reveal both main effects and interactions that significantly influence the response.
Another Example of a General Factorial Design: - http://davidmlane.com/hyperstat/A134930.html