## 2 Materials and Methods
### Ordinal Logistic Regression Model  
As is shown, the experimental design presents the case of a categoric-type variable, which is a particular case that can be approached using generalized linear model tech \begin{table}
\begin{tabular}{c c} \hline \hline
**Observed Growth on Specimens** & **Rating** \\ \hline None & 0 \\ Traces of growth (less than 10\%) & 1 \\ Light growth (10 to 30\%) & 2 \\ Medium growth (30 to 60\%) & 3 \\ Heavy growth (60\% to complete coverage) & 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Rating according to the observed growth of the fungus on specimens. Adapted with oermission from [18] Copyrighted by ASTM International, 2013. Figure 6: Examples of the observed fungal growth are (**a**) traces of growth below 10% (1), (**b**) light growth from 10 to 30% (2), (**c**) medium growth from 30 to 60% (3), and (**d**) heavy growth from 60% to complete coverage (4). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Treatment** & \(\mathbf{X_{1}}\) & \(\mathbf{X_{2}}\) & \(\mathbf{X_{3}}\) & \(\mathbf{Y_{1}}\) & \(\mathbf{Y_{2}}\) & \(\mathbf{Y_{3}}\) & \(\mathbf{Y_{4}}\) & \(\mathbf{Y_{5}}\) \\ \hline
1 & 15 nm & 5 g & 0 s & 2 & 1 & 1 & 1 & 2 \\
2 & 45 nm & 5 g & 0 s & 2 & 2 & 2 & 3 & 1 \\
3 & 15 nm & 10 g & 0 s & 4 & 2 & 2 & 2 & 1 \\
4 & 45 nm & 10 g & 0 s & 1 & 1 & 1 & 2 & 2 \\
5 & 15 nm & 5 g & 10 s & 2 & 2 & 1 & 3 & 2 \\
6 & 45 nm & 5 g & 10 s & 4 & 2 & 1 & 1 & 3 \\
7 & 15 nm & 10 g & 10 s & 1 & 1 & 3 & 2 & 3 \\
8 & 45 nm & 10 g & 10 s & 3 & 2 & 3 & 2 & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Experimental results. niques [42]. The generalized linear models use a common algorithm for estimating the parameters using maximum likelihood, using weighted least squares with an adjusted dependent variable, and do not require preliminary guesses at the parameter values. When a categoric-type response is obtained, the best option among the generalized linear models are ordinal logistic regression models, which are appliable to both nominal and ordinal responses with more than two categories, in which a natural order can exist among the response categories [43, 44]  
The default link function for these models is logit, which accumulates the cumulative log odds. The cumulative log odds of the coefficient of a response belonging to a category with a value less than or equal to the category j, \(\mathrm{P}(y\leq\varsigma)\). The ordinal models are usually based on the assumption that the effects of the predictor variables are the same for every category on the logarithmic scale, i.e., the model has different intercepts but equal slopes. This model is called parallel regression on the proportional odds model. This is the default for ordinal responses, the proportional odds model is:  
\[ \begin{array}{l}\ln\left(\frac{\mathrm{P}(y\leq\varsigma_{1})}{\mathrm{P}(y >\varsigma_{1})}\right)=\ln\left(\frac{\pi_{1}}{\pi_{2}+\cdots+\pi_{k}}\right) =\alpha_{1}+\beta_{1}X_{1}+\beta_{2}X_{2}+\cdots+\beta_{\mathrm{P}}X_{ \mathrm{P}},\\ \\ \ln\left(\frac{\mathrm{P}(y\leq\varsigma_{2})}{\mathrm{P}(y>\varsigma_{2})} \right)=\ln\left(\frac{\pi_{1}+\pi_{2}}{\pi_{3}+\cdots+\pi_{k}}\right)=\alpha _{1}+\beta_{1}X_{1}+\beta_{2}X_{2}+\cdots+\beta_{\mathrm{P}}X_{\mathrm{P}},\\ \\ \vdots\\ \ln\left(\frac{\mathrm{P}(y\leq\varsigma_{0}-1)}{\mathrm{P}(y>\varsigma_{k-1} )}\right)=\ln\left(\frac{\pi_{1}+\pi_{2}+\cdots+\pi_{k-1}}{\pi_{k}}\right)= \alpha_{k-1}+\beta_{1}X_{1}+\beta_{2}X_{2}+\cdots+\beta_{\mathrm{P}}X_{ \mathrm{P}},\end{array}  \tag{1}\]  
where \(\pi_{j}=P(y=j)\) is the probability that an output is in the \(j\) category, \(k\) is the number of categories, and \(p\) is the number of predictor variables.