Abstract
The aim of this chapter is to expose a multi-ways fuzzy analysis of variance (Mult-FANOVA) approach when the fuzziness is taken into consideration. This latter is based on a fuzzy regression model. As such, we expose two exploratory approaches: the first one by preserving the fuzzy nature of the calculated sums of squares using fuzzy approximations and afterward by defining a related heuristic decision rule to be able to decide, and the second one using a particular metric all along the calculations, namely the signed distance and the generalized one. For both approaches, the bootstrap technique is used. A procedure of calculating sequential sums of squares is also exposed when the ordering effect intervenes. For sake of illustration, we give two detailed empirical applications of our approach, where both are composed by multiple factors. By these applications, the objective is to present the Mult-FANOVA model where the generalized signed distance is used, and to compare it with the analysis using the same setups but in the classical environment. One of our main findings is that in both cases we get the same decision of rejecting or not the hypothesis of equality of the population mean vectors, but the decomposition of the sums is different. For the analysis on the residuals of the constructed model, the normality of their distributions is rejected, but stronger in the classical approach. This result can be seen as a disadvantage of the classical approach, and by consequence strengthen our support to the fuzzy one.
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Notes
- 1.
- 2.
Despite the complexity induced by the difference arithmetic in the decomposition of \(\tilde {X}_{i_1 i_2 j}\), remark that \(\overline {\tilde {X}}_{i_1 i_2 \bullet }\) is expressed by this form in accordance with the expression of the classical theory.
- 3.
The case with interactions can be similarly conceivable. Models which include interactions can be found in Berkachy (2018).
- 4.
For more information about this test, Tukey (1949) described the detailed procedure.
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Appendix
Appendix
This Appendix is devoted to the description of the Nestlé data set as follows:
1.1 D Description of the Nestlé data base
This data base is collected by the beverage section of the company Nestlé. An expert is asked to rate different aspects of a given confidential beverage. The collected data set is composed by 160 observations resulting from 32 trials of an experiment on the beverage. The full data base comprised of 13 variables among them 11 factors having 2 levels each given by “Up” and “Down,” and the variables FLOW and FILLING. A description of the variables constituting this data set is given in Table 8.12.
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Berkachy, R. (2021). Fuzzy Analysis of Variance. In: The Signed Distance Measure in Fuzzy Statistical Analysis. Fuzzy Management Methods. Springer, Cham. https://doi.org/10.1007/978-3-030-76916-1_8
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