# Application of Logistic Regression for Production Machinery Efficiency Evaluation
## 3 Estimation of Markov Logistic Model Parameters
In all cases, the calculated test statistic did not allow the zero hypothesis on the lack of relationships between variables to be accepted. It was therefore rejected in favor of the alternative hypothesis of the existence of a relationship, the strength of which was measured using [PERSON]'s \(\Phi\) (for binary tables) and [PERSON]'s V coefficient (for tables more complex than \(2\times 2\)). The obtained results are presented in Table 5.  
The observed relationships between variables, although significant, are not strong. This is also confirmed by the graphs of interaction of individual dependent variables with the explained variable (Figure 4). Nevertheless, from the point of view of the analyzed company, the diagnosed bonds should not take place at all. A uniform and efficient operation of all devices is expected, so even minor deviations are undesirable and require further investigation. The calculations carried out (Table 5) and the charts (Figure 4) confirm that the model variables were selected correctly. This allows the parameters of the logistic regression model to be estimated, the values of which are presented in Table 6.  
Figure 3: Graphs of normality of distribution of the efficiency variable grouped by the device variable. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{6}{c}{**Modeled Probability Evaluation of Effectiveness—Positive Result Distribution: Binomial,**} \\ \multicolumn{6}{c}{**Binding Function: LOGIT**} \\ \cline{2-7}
**Effect** & **Effect Level** & **Parameter** & \multicolumn{2}{c}{**Estimated Standard Error**} & \multicolumn{2}{c}{**[PERSON]’s Test Statistics**} & \(p\) & **95.00\% CI** & \multicolumn{2}{c}{**\textasciitilde{\textasciitilde{\textasciitilde{\textasciitilde{\textasciitilde{ \textasciitilde{\textasciitilde{\textasciitilde{\textasciitilde{\textasciitilde{ \textasciitilde{\textasciitilde{\textasciitilde{\textasciitilde{\textasciasciitilde{ \textasciasciasciitilde{\textasciasciasciasciitilde{asci  
All calculated parameters turned out to be statistically significant, which is confirmed by the calculated Wald's statistic value and the associated probability value \(p\), which for each line is lower than the assumed level of significance \(\alpha=0.05\). (Table 6). This means that all the distinguished factors significantly affect the evaluation of production efficiency. This allows the equation of the logistic regression model to be written in the following form:  
\[P\Big{(}efficiency=yes\big{|}X\Big{)}=\frac{e^{a}}{1+e^{a}}, \tag{9}\]  
where  
\[ \begin{array}{c}a=-7.549-0.292\ shift\ l-1.241*H2-1.066*H5-1.153*H6-0.668*\\ H12-0.809*H21-0.496*H22-0.525*H23-0.873*H24-1.167*\\ H25-0.317*H4\ +5.047*no\ order\ +3.08*no\ failure.\end{array}  \tag{10}\]  
The logistic regression curve is shown in Figure 5.  
The logistic regression equation presented above can also take equivalent forms:  
* logistic regression logit function: \[ \begin{array}{c}logit\ P(efficient=1\big{|}X)=\mathrm{In}\frac{P(efficient =1\big{|}X)}{1-P(efficient=1\big{|}X)}\ =\ -7.549-0.292\ 1 st\ shift-\\ 1.241*H2-1.006*H5-1.153*H6-0.668*H12-0.809*H21-0.496*\\ H22-0.525*H23-0.873*H24-1.167*H25-0.317*H4\ +5.047*\\ \ no\ order\ +3.08*no\ failure,\end{array} \] (11)
* in the form of the odds: \[P\frac{P(efficient=1\big{|}X)}{1-P(efficient=1\big{|}X)}=e^{a},\] (12)  
Figure 4: Interaction charts of dependent variable and predictors. where  
\[ \begin{split} a=-7.549-0.292\ shift\ {I-1.241*H2-1.066*H5-1.153*H6-0.668*}\\ H12-0.809*H21-0.496*H22-0.525*H23-0.873*H24-1.167*\\ H25-0.317*H4\ +5.047*no\ order\ +3.08*no\ failure.\end{split}  \tag{13}\]  
An important element of the evaluation of the studied process is the calculation of the odds of an event occurrence (in this case a satisfactory result of efficiency). The sign at the estimated parameter of the logistic regression model indicates whether the analyzed odds are greater (plus) or smaller (minus) in relation to the reference level.