# Application of Logistic Regression for Production Machinery Efficiency Evaluation
## 3 Estimation of Markov Logistic Model Parameters
The scale of this change is indicated by the unit odds ratio shown for each parameter in Table 7.  
The odds ratio for the 1 st shift is 0.75, which means that compared to the 2 nd shift, the odds of achieving satisfactory efficiency is 0.75 times lower. In other words, the odds for the proper efficiency is 1.34 times higher in the case of the 2 nd shift. The H14 machine was used as a reference when analyzing the impact on the efficiency of individual devices. Its efficiency is the highest in the test sample, so all the model coefficients obtained are negative, which means less odds of achieving a positive result. The odds ratios are given in column 3 of Table 6. The worst result was obtained for the H2 machine with an odds ratio of 0.289, which means an almost 3.5-fold increase in the odds of achieving satisfactory efficiency when replacing H2 with H14. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Effect** & **Effect Level** & **Odds Ratio** & **95.00\% CI** & **\(-\)95.00\% CI** & **\(p\)** \\ \hline Shift & 1 st & 0.747 & 0.704 & 0.793 & 0.00 \\ Device & H2 & 0.289 & 0.253 & 0.330 & 0.00 \\ Device & H5 & 0.344 & 0.302 & 0.392 & 0.00 \\ Device & H6 & 0.316 & 0.276 & 0.360 & 0.00 \\ Device & H12 & 0.513 & 0.451 & 0.583 & 0.00 \\ Device & H21 & 0.445 & 0.392 & 0.506 & 0.00 \\ Device & H22 & 0.609 & 0.534 & 0.694 & 0.00 \\ Device & H23 & 0.592 & 0.519 & 0.675 & 0.00 \\ Device & H24 & 0.418 & 0.366 & 0.477 & 0.00 \\ Device & H25 & 0.311 & 0.273 & 0.356 & 0.00 \\ Device & H4 & 0.729 & 0.633 & 0.839 & 0.00 \\ No order—yes no & no & 155.489 & 95.002 & 254.487 & 0.00 \\ Failure—yes no & no & 21.7679 & 16.7083 & 28.3595 & 0.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Odds ratios for individual predictors. Figure 5: Logistic regression curve. The results for the other machines, showing how much the efficiency of each machine should be increased in order to obtain the efficiency evaluation as in the case of the H12 machine, are shown in Table 8.  
The last two parameters in Table 7 refer to the absence of an order or failure, as their occurrence has a negative impact on the efficiency. Where there is no downtime, the odds of achieving the expected efficiency are 155 times greater than otherwise. Similarly, the occurrence of a failure has a similar effect, but its absence is not as spectacular. There is a 21-fold increase in the odds if the failure does not occur. The presented model can also be used for predictive purposes, allowing for forecasting the probability of achieving the predicted success (here, the assumed efficiency). It is therefore important to assess the quality of the prediction. For this purpose, it is helpful to determine the so-called cut-off point \(\pi_{0}\). This parameter allows the observed dichotomous values of a dependent variable to be compared with the continuous probability values calculated on the basis of the model. This value falls within the range (0, 1) and is defined as follows [36] when:  
\[\hat{\pi}(x)=\hat{P}(Y=1|x)>\pi_{0}, \tag{14}\]  
it is assumed that an event has occurred (\(\hat{y}=1\)). In the opposite situation, when  
\[\hat{\pi}(x)\leq\pi_{0}, \tag{15}\]  
it is assumed that an event has not occurred (\(\hat{y}=0\)). Prediction ideally occurs when sensitivity and specificity are equal to 1, which means no false positive or negative results. In real life research, the point corresponding to a case where a model best discriminates occurrences is called the optimal cut-off point. It is determined using the [PERSON]'s index (\(J\)), which takes the following form:  
\[J=\textit{sensitivity}+\textit{specificity}-1. \tag{16}\]  
The optimum cut-off point corresponds to the case where the \(J\) value reaches its maximum. For the case under consideration, the proposed cut-off point is shown in Table 9.  
For the proposed cut-off point, the sensitivity is 0.32, and the specificity is 0.86. There are 16,767 well classified cases (2864 true positive and 13,903 true negative) and 8115 badly classified cases (2210 false positive and 5905 false negative cases).