# Development of Predictive Equations for Thermal Conductivity of Compost Bedding
## 2 Materials and Methods
### Statistical Analysis  
To truly quantify the several parameters' impacts on the pack compost process, a quantitative assessment of each of its components must be performed, and the relevant physical processes must be included in this assessment. One aspect that has been typically neglected is the impact that material compaction has on the coupled water and heat fluxes across the pack layers. In this work, we developed some mathematical models that used empirical bedding material data to consider how compaction levels and moisture changed the substrate's thermal conductivity. A randomized experimental design with two replicates was analyzed in two different analyses. The first statistical analysis with each experimental unit consisted of three MC levels (30%, 45%, and 60%) and five DC\({}_{\rm o}\) levels (0.0; 0.1; 0.2; 0.3; and 0.4 MPa) with each dairy farm assessed as a repetition (42 farms). Therefore, each bedding sample contained 630 observations. Prior to analysis, the original dataset that consisted of 630 observations was split into a training set (70% of the data) and a test set (30% of the data). Variable selection and model fitting were performed on the training set while the test set was used for model validation. Statistical analysis was performed using the software R version 3.4.1 [33]. Analysis of variance (ANOVA) was performed considering the factorial scheme with two factors, to assess the possible dependence between the factors MC and DC\({}_{\rm o}\) in the description of the response variable \(k\). This ANOVA was performed using the \(lm\) function available on the basis of the statistical software R (R Core Team, 2019). The ANOVA was performed to assess the effects of MC and DC\({}_{\rm o}\) on the \(k\) of bedding materials and the possible interaction between the two factors (MC \(\times\) DC\({}_{\rm o}\)). The ANOVA was performed, using the package _stats_[33], to assess the effects of MC and DC\({}_{\rm o}\) on the \(k\) of bedding materials. The fixed effects of MC and DC\({}_{\rm o}\), as well as their interaction (MC \(\times\) DC\({}_{\rm o}\)), were included in the models. After this analysis, based on the significance of the statistical tests for MC, DC\({}_{\rm o}\), and their interactions, the appropriate regression models were chosen to describe \(k\). The selection of models was made, using the package _stats_[33], based on backward stepwise procedure starting from a complete model with all the predictive variables to the quadratic term. Non-significant predictors were removed from the model based on the relative reduction in sums of squares. In all analyses, a polynomial model was adjusted taking into account the interaction between the two explanatory variables (MC and DC\({}_{\rm o}\)), generating a response surface [34]. The selection of models was made based on backward stepwise procedure starting from a complete model with all the predictive variables to the quadratic term. Non-significant predictors were removed from the model based on the relative reduction in sums of squares. In view of this, a polynomial model was fitted, taking into account the dependent variables and the interaction between these variables, as described in Equation (1). \[k_{i}=b_{0}+b_{1}\cdot\mathrm{MC}+b_{2}\cdot DC_{o}+b_{3}\cdot SC^{2}+b_{4} \cdot\mathrm{MC}\cdot\mathrm{DC_{o}} \tag{1}\]  
where \(b_{0}\), \(b_{1}\), \(b_{2}\), \(b_{3}\), and \(b_{4}\) are the parameters of the polynomial regression model. The curve intercept is \(b_{0}\) and \(b_{1}\), \(b_{2}\), \(b_{3}\), and \(b_{4}\) represent the degree of influence of the respective variable on the thermal conductivity (\(k_{i}\)). The \(b_{1}\), \(b_{2}\), \(b_{3}\), and \(b_{4}\) are the linear coefficients, that is, keeping the others constant and adding a unit of measure in the explanatory variable associated with one of these coefficients, an increase or decrease in the response is expected. In order to obtain better precision in the modeling of \(k\), the analyses were evaluated for the three depths separately (H1, H2, and H3).