## 2 Materials and Methods
### Data Analysis
#### 2.2.1 Lenth Statistics and Belief Functions
Since the experiments are replicated, the prediction of strong effects can be cast in simple probabilities, which are translated into masses according to the selected evidence combination rules. To form those masses, it is necessary to simply count the number of times that an effect appears to be labeled as strong, out of the D times that the OA-scheme has been replicated. To be declared as a strong effect, its Lenth statistic must be greater than the corresponding critical value. To test for strong evidence in rounding up the active factor group, only the effects that exceed the critical value of EER are retained from each individual OA dataset replication. This approach encompasses the idea of the independence of information sources (different replications) and not disengaging source reliability,while enabling the embedding of the potential variability in the level of conflict among the sources of evidence into the decision-making process. To revisit some rudimentary notions about belief functions, firstly, it is constructive to define the frame of discernment, \(\Omega\), which is the set of N exclusive and exhaustive hypotheses {H\({}_{i}\)} for 1 \(\leq\) i \(\leq\) N. Consequently, the power set, P(\(\Omega\)), consists of 2\({}^{\text{N}}\) hypotheses in the \(\Omega\): P(\(\Omega\)) = {\(\emptyset\), {H\({}_{1}\)}, {H\({}_{2}\)}, \(\ldots\), {H\({}_{N}\)}, {H\({}_{1}\cup\) H\({}_{2}\)}, \(\ldots\), \(\Omega\)}. The mass function, \(m\), is defined as the frequency of occurrences, in ratio form, for each of the \(\Omega\) elements in the closed range [0,1], i.e., \(m\): P(\(\Omega\))\(\rightarrow\)[0,1]. Given a subset A with A \(\subseteq\Omega\) and \(m\)(A) > 0, subset A is called a focal element. The focal element and its corresponding mass function constitute a belief structure. The mass function possesses two properties:  
\[\sum_{A\subseteq\Omega}m^{\Omega}(\text{A})=1\text{ and }m^{\Omega}(\emptyset)=0\]  
The definition for the basic belief function, \(bel(\text{X})\), \(\forall\) X \(\subseteq\)\(\Omega\) and A \(\subseteq\)\(\Omega\) (A \(\
eq\)\(\emptyset\)), is  
\[bel(X)=\sum_{A\subseteq X}m^{\Omega}(\text{A})\]  
Similarly, the definition for the plausibility function, \(pl(\text{X})\), is  
\[pl(X)=\sum_{A\cap X\
eq\varnothing}m^{\Omega}(\text{A})\]  
To effectuate the intermixing of several pieces of evidence, there are several combination rules that permit straightforward evaluation by incorporating information with respect to the degree of conflict among sources. Among the most popular combination rules, three alternatives will be employed: (1) the normalized rule (the Dempster-Shafer rule), (2) the conjunctive combination rule (the Smets rule) [65; 66; 67], and (3) [PERSON]'s rule [68; 69]. The Dempster-Shafer rule of combination for the two-source case, \(m^{\Omega}_{1\geq 2}(\text{A})\), is defined as  
\[m^{\Omega}_{1\geq 2}(\text{A})=\left\{ \begin{array}{ll}f_{12}(A)&\forall A \subseteq\Omega\text{ and }A\
eq\varnothing\\ &0\text{ if }A=\varnothing\end{array} \right.\]  
where \(f_{12}(A)\): 2\({}^{\Omega}\)\(\rightarrow\)[0,1] is the conjunctive rule function:  
\[f_{12}(A)=\sum_{X\cap Y=A}m^{\Omega}_{1}(\text{X})\ m^{\Omega}_{2}(\text{Y}) \ \forall A\subseteq\Omega\]  
The TBM conjunctive combination rule, \(m^{\Omega}_{1\cap 2}(X)\), therefore, is defined as  
\[m^{\Omega}_{1\leq 2}(X)=f_{12}(X)\]  
where the intersection symbol in subscript form denotes, only in this case, the conjunctive operator. Finally, [PERSON]'s combination rule, \(m^{\Omega}_{12}\), is  
\[m^{\Omega}_{12}(A)=\left\{ \begin{array}{ll}f_{12}(A)&\forall A\subseteq \Omega,\ A\
eq\Omega\text{ and }A\
eq\varnothing\\ f_{12}(\Omega)+f_{12}(\varnothing)&if\ A=\Omega\\ 0&if\ A=\varnothing\end{array} \right.\]  
To provide a broader perspective on the decision-making process, the application of the decision rules according to belief functions will be based on (1) the maximization of plausibility and (2) the maximization of credibility.