## 6 Other Approaches
### Analysis of Multivariate Designed Data  
The rational design of the experiments to be conducted is a fundamental step in any scientific discipline and the only way in which it can be guaranteed that the information sought be present in the collected data. The design of experiments involves four steps: (i) identifying some critical variables (_factors_) which could affect one or more properties of interest (_responses_), (ii) defining their range of variability that could be worth inspecting, (iii) planning the trials to be performed (so as to be at the same time as informative and as low in number as possible), and (iv) interpreting the final results [240, 241, 242, 243]. In this subsection, attention will be essentially focused on this last step, as it is the one where traditional statistical approaches commonly fail when instead of measuring a few (uncorrelated) responses for each experimental condition setting, a full spectroscopic signature is registered to characterize the concerned samples [244]. As summarized before, based on a rational design, experiments are conducted under different conditions, characterized by the fact that the factors whose effect is investigated are fixed (or controlled) at specific values (_levels_). In other words, each combination of factor levels is a so-called _design point_, at which the measurement of the responses is carried out. For instance, one could be interested in studying how the spectroscopic profile of a specific product varies as a function of temperature and pH, and then decide to run experimental trials only at three levels of temperature (25 \({}^{\circ}\)C, 40 \({}^{\circ}\)C and 55 \({}^{\circ}\)C) and at two levels of pH (5 and 9). In case all possible combinations of factor levels are explored, the design is called _full-factorial_. In the example sketched before, a full-factorial design would encompass six distinct experiments: 25 \({}^{\circ}\)C and pH = 5; 40 \({}^{\circ}\)C and pH = 5; 55 \({}^{\circ}\)C and pH = 5; 25 \({}^{\circ}\)C and pH = 9; 40 \({}^{\circ}\)C and pH = 9; 55 \({}^{\circ}\)C and pH = 9. At each condition setting, the desired responses should be measured; if the aim is a spectroscopic characterization, the response is highly multivariate and it is represented by the spectroscopic signature of the respective sample. Moreover, very often, the whole design is replicated at least twice to have an estimate of the variability not to be ascribed to the controlled factors. Given a particular design and having collected the data corresponding to every one of its points, the successive step is to analyze such data to first evaluate whether any of the controlled factors exhibits a statistically significant effect on the responses and (if this is the case) to interpret and describe in greater detail how this effect occurs (for instance, which spectral regions change with the increase of the temperature and how). The traditional statistical tool used in this type of contexts is the analysis of variance--univariate (ANOVA [245; 246; 247; 248]) or multivariate (MANOVA [58; 241; 244; 249; 250; 251; 252]), depending on the nature of the response(s) measured. However, when dealing with a high number of possibly very correlated descriptors (like spectral variables) and, therefore, with ill-conditioned data matrices, MANOVA suffers from the same drawbacks as MLR or LDA. To overcome these limitations, in the last 15 years, various approaches for the assessment of multivariate or megavariate data resulting from designed experiments have been proposed, all of them having in common the same initial partitioning of the experimental data array, which follows the classical ANOVA scheme based on the linear additivity of the effects.