Technical note: Designing and analyzing quantitative factorial experiments

The response of a biological process to various factors is generally nonlinear, with many interactions among those factors. Although meta-analyses of data across multiple studies can help in identifying and quantifying interactions among factors, missing latent variables can result in serious misinterpretation. Eventually, all influential factors have to be studied simultaneously in one single experiment. Because of the curvature of the expected response and the presence of interactions among factors, the size of experiments grows very large, even when only 3 or 4 factors are fully arranged. There exists a class of experimental designs, named central composite designs (CCD), that considerably reduces the number of treatments required to estimate all the terms of a second-order polynomial equation without any loss of efficiency compared with the full factorial design. The objective of this technical note is to explain the construction of a CCD and its statistical analysis using the Statistical Analysis System. In short, a CCD consists of 2(k) treatment points (a first-order factorial design, where k represents the number of factors), augmented by at least one center point and 2 x k axial treatments. For 3 factors, the resulting design has 16 treatment points, compared with 27 for a full factorial design. For 4 factors, the CCD has 25 treatment points, compared with 81 for a full factorial design. The CCD can be made orthogonal (no correlation between parameter estimates) or rotatable (the variance of the estimated response is a function only of the distance from the design center and not the direction) by the location of the axial treatments. In spite of the reduced number of treatments compared with a full 3(k) factorial, the CCD is relatively efficient in estimation of the quadratic and interaction terms. Blocking of experimental units is often desirable and is sometimes required. Randomized block designs for CCD are found in some statistical design textbooks. The construction of incomplete, balanced (or near-balanced) designs for CCD experimental layouts is explained using an example. The Statistical Analysis System statements used to analyze a CCD, to identify the significant parameters in the polynomial equation, and to produce parameter estimates are presented and explained.