## 2 Material and Methods
### Logistic Regression Model  
Logistic regression is popular with data analysis dealing with describing the relationship between various explanatory input variables and a discrete outcome with two or more levels [27; 28]. What distinguishes LR from linear regression models is that the output variable is binary [29]. LR is a powerful statistical method for analyzing categorical data and is used to predict the probability of a nominal response [30]. Nominal responses are analyzed with a straightforward extension of the logit model. A linear regression model assumes that the expected value of Y for a given x is expressed as \(\mathrm{E}(\mathrm{Y}\mathrm{p})=\beta_{0}+\beta_{1}\mathrm{x}\), for which the specific form of the LR model for calculating the probability of an outcome of interest was:  
\[\pi(\mathrm{Y}=\mathrm{outcome\ of\ interest}\mathrm{X}=\mathrm{x})=\frac{ \mathrm{e}^{\beta_{0}+\beta_{1}\mathrm{x}}}{(1+\mathrm{e}^{\beta_{0}+\beta_{1 }\mathrm{x}})}\]  
Extending the logic of simple regression to multiple input variables (x\({}_{1}\), x\({}_{2}\),..., x\({}_{n}\)), and with a nominal binary response, the logit function of the multiple LR was defined as a linear combination of input variables as in Equation (1) [28], and the LR model was used as in Equation (2). Equation (1): Linear combination of individual predictors. \[\mathrm{z}=\beta_{0}+\beta_{1}\mathrm{x}_{1}+\beta_{2}\mathrm{x}_{2}+\cdots+ \beta_{n}\mathrm{x}_{n} \tag{1}\]  
Equation (1): Probability of outcome of interest. \[\pi(\mathrm{Y}=\mathrm{outcome\ of\ interest}\mathrm{X}=\mathrm{z})=\frac{ \mathrm{e}^{\mathrm{z}}}{(1+\mathrm{e}^{\mathrm{z}})} \tag{2}\]  
In this study, multiple LR was used to estimate the odd ratios of the post-incident state of a non-fatal occupational injury. The input variables used were injured body part group (x\({}_{1}\)), cause of injury group (x\({}_{2}\)), nature of injury group (x\({}_{3}\)), injured worker's occupation (x\({}_{4}\)), agribusiness industry (x\({}_{5}\)), age of the injured worker (x\({}_{6}\)), and years of experience of the injured worker (x\({}_{7}\)). The binary output variable was post-incident state (medical vs disability). The outcome of interest is disability post-incident state. Therefore, the probability of post-incident state of disability is  
\[\mathrm{P}(\mathrm{disability})=\frac{1}{(1+\mathrm{e}^{-\mathrm{Z}})}\]  
and the probability of medical post-incident state is  
\[\mathrm{P}(\mathrm{medical})=\frac{1}{(1+\mathrm{e}^{+\mathrm{Z}})}\] \begin{table}
\begin{tabular}{c c c c c c} \hline
**Predictor** & **Freq.** & **\% Total** & **Predictor** & **Freq.** & **\% Total** \\ \hline Striking against/stepping on & 436 & 3.14 & Grain elevators & 6966 & 50.2 \\ lifting or handling & 228 & 1.64 & Refined fuels & 2460 & 17.7 \\  & & & Feed mills for livestock/pet foods & 1490 & 10.7 \\ Nature of injury group & & & Food distributors & 1156 & 8.33 \\ Specific injury & 13,655 & 98.4 & Fertilizer blending and distribution & 1154 & 8.32 \\ Occupational disease & 158 & 1.13 & Poultry hatchery/grower/processor & 402 & 2.89 \\ Multiple injuries & 54 & 0.389 & Grain milling plants & 239 & 1.72 \\ \hline \end{tabular}
\end{table} 
Table 1: _Cont._To evaluate the effectiveness of the LR model formula in estimation and prediction of the outcome of interest, researchers should consider overall model evaluation statistics, statistical significance of the individual predictors, and validation of the predicted probabilities of the outcome of interest versus the actual outcomes from the data set using a classification table [28]. To examine the effects of the individual predictors on the probability of a disability versus medical post-incident state, the Wald chi-squared values and their correspondent observed significance (\(p\)) values are determined. The Wald chi-squared values are the Wald tests for the hypotheses that each of the parameters is zero and is computed as (estimate per independent variable/SE of estimate)\({}^{2}\).