## 3 Results
### Summary of Latent Class Analysis  
The LCA is conducted to identify statistically distinctive and meaningful risk subgroups of occupational incidents in agribusiness industries based on _injury type_, _class codes_, _injured body part(s)_, _cause_, and _nature of the injury_. In the first step, the latent class analysis is employed as an explorative method for pattern recognition in the data fitting eight models with 3 to 10 latent classes. AIC and BIC are used as the relative fit measures. Lower values for BIC and AIC show a better fit to the data. The fit statistics for models with different numbers of classes are shown in Table 3.  
The changes in BIC and AIC represent the model with three classes as the best fit. Based on values of BIC and AIC, three classes with different injury patterns are found: class one (44.32% of the population), class two (34.31% of the population), and class three (21.37% of the population). In order to decide each row of the data, which includes various levels of input variables, belonging to each latent class (class 1, 2, or 3), the probability of the class membership is calculated for each latent class. By comparing the three probabilities, the one with the highest value determines the latent class to which that specific data row belongs. The statistical details for calculating per class formula is given based on [38]. Let j = 1,..., J represent the observed columns (Y) of input variables. For this study, those Y columns are the input variables of _injury type_, _class codes_, _injured body part(s)_, _cause of injury_, and _nature of injury_. Denote the number of levels for column j by Rj. A multidimensional contingency table of the J variables contains W = R1*...*RJ cells. Each of these cells is defined by its response pattern for the J variables. Therefore, each response pattern is a J-length vector of the form y = y1,..., yj. Define Y to be the W by J array of all the response patterns considered as row vectors. Each element, y\({}_{w}\), in Y has a probability Pr(y\({}_{w}\)). These probabilities sum to 1, as given in Equation (4):  
\[\sum_{w=1}^{W}\Pr(y_{w})=1 \tag{4}\]  
Consider the following notation:  
* C is the number of clusters in the latent class model. * \(\gamma_{c}\) is the probability of membership in cluster c (the \(\gamma_{c}\) are the latent class prevalence). These parameters sum to 1. * \(r_{j,k}\) is the \(k\)th level of the \(j\)th response. * \(\rho_{j,\,k\!c}\) is the probability of observing response rj, k in column j conditional on membership in class c (the \(\rho_{j,\,k\!c}\) are the item-response probabilities). For a given cluster and response variable j, the sum of the \(\rho_{j,\,k\!c}\) is 1. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Number of Classes** & **AIC** & **BIC** & **LL** \\ \hline
3-Class & 26,125 & 29,735 & 12,331 \\
4-Class & 26,139 & 30,954 & 12,094 \\
5-Class & 26,298 & 32,317 & 11,930 \\
6-Class & 26,543 & 33,768 & 11,808 \\
7-Class & 26,811 & 35,240 & 11,698 \\
8-Class & 27,097 & 36,731 & 11,597 \\
9-Class & 27,424 & 38,264 & 11,517 \\
10-Class & 27,812 & 39,856 & 11,467 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Model fit statistics. \(I(y_{j}=r_{j,k})\) is an indicator function that equals 1 when the \(y_{j}\) response is the \(k\)th level of the \(j\)th response, and 0 otherwise. As presented in Equation (5), the probability of observing a specific vector of responses \(y_{w}\) = y1,..., \(y_{j}\) is the sum of the conditional probabilities of observing that vector of responses for each of the C latent classes:  
\[\Pr(y)=\sum_{c=1}^{C}\gamma_{c}\prod_{j=1}^{J}\prod_{k=1}^{R_{j}}\rho_{j,\ k _{C}}^{I(y_{j}=r_{j,k})} \tag{5}\]  
Thus, Equation (5) is the denominator of the probability formula that is saved to the dataset for each row. The final formula for probability per latent class gives \(\Pr(\text{Cluster}=c|y_{w})\), which equals to \(\Pr\) (\(y_{w}\), \(\text{Cluster}=c\))/\(\Pr\) (\(y_{w}\)).