# Application of Logistic Regression for Production Machinery Efficiency Evaluation
## 3 Estimation of Markov Logistic Model Parameters  
The first stage of the study was to define possible explanatory variables in order to determine which of them could be used in the model. The following explanatory variables were selected: shift, device, occurrence of failure (yes or no) and no production order (yes or no). The shift predictor was analyzed first. First of all, the normality of distribution and homogeneity of the variance of the efficiency dependent variable during individual shifts was examined in order to determine the possible methods of statistical analysis. The distributions in all groups turned out to be inconsistent with the normal distribution, which is confirmed by the graphs in Figure 1 and the calculated chi-square test statistic values, presented in Table 1.  
Next, the homogeneity of variance in individual groups was checked; the Levene and Brown-Forsythe test was used for this purpose. The obtained results are presented in Table 2.  
Although the homogeneity of variance was confirmed in all groups, due to the lack of normality of distributions, the Mann-Whitney test was used to examine the significance of differences between individual averages, and the results thereof are presented in Table 3. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Chi-Square** & **Degrees of Freedom** & \(p\)**-Value** \\ \hline Shift = 1 st & 4355.45 & df = 8 & \(p\) = 0.00 \\ Shift = 2 nd & 5932.71 & df = 8 & \(p\) = 0.00 \\ Shift = 3 rd & 6474.96 & df = 8 & \(p\) = 0.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Chi-square test statistic values of the efficiency variable grouped by shifts. Figure 1: Graphs of normality of distribution of the efficiency variable grouped by shifts. The analyses showed that there were no significant differences between the efficiency of the second and third shift, so a decision was made to combine them. However, the values obtained for the first shift differ significantly from those obtained for the other shifts, therefore this group was left without interference. These conclusions are confirmed by Figure 2 showing the differences described. The same test was performed for the device variable. The machines analyzed were of a single type and came from a single production batch, which suggests that their productivity would be similar. In order to confirm the equality of averages, the analysis of distribution normality and variance equality in individual groups (this time defined by the device variable) was carried out again in order to select a proper statistical distribution. The results of the normality test did not confirm the conformity. All the calculated chi-square test statistic values did not allow the zero hypothesis of the compatibility of the examined distribution with the normal one to be accepted. A definite deviation is confirmed by Figure 3.  
The analysis of the equality of variance using the Levene and Brown-Forsythe tests showed that variances are not equal in some groups. Consequently, the Mann-Whitney test was used to check the difference between averages, the results of which are presented in Table 4.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & **H4** & **H5** & **H6** & **H12** & **H14** & **H21** & **H22** & **H23** & **H24** & **H25**  
Since the average efficiency varied for virtually every pair of devices, a decision was made not to combine them and to include each of them in the study. After defining the form of independent variables, the impact of each of them on the dependent variable, i.e., efficiency, was checked, but presented in a dichotomous form, as an assessment of whether the level achieved was satisfactory for the company. In line with the expectations of the Management Board, it was assumed that the assessment was positive if the productivity was equal to or above 90%. In other cases, the assessment would be negative. The chi-square test allowed for a statistical and substantive study of the relationship between variables.