# Explainable Machine Learning for Lung Cancer Screening Models
## 2 Models for Lung Cancer Screening  
According to The National Lung Screening Trial (NLST) [25], it was proved that low-dose computed tomography screening could reduce lung cancer mortality by up to 20% in high-risk patients. Statistical models help to more precisely enroll those who may benefit most, securing, at the same time, efficiently allocated, available financial resources. Additionally, many models that predict the risk of lung cancer or the risk of death from lung cancer have been developed. The authors in [31] developed a comparison of nine screening models. In the study, they demonstrated that four out of nine prominent risk models are the most accurate. It turns out that based on the data from the National Health Interview Survey from 2010 to 2012, there are four models that most accurately predict lung cancer risk: the BACH model [26], PLCO\({}_{m2012}\) model [27], LCRAT model, and LCDRAT models [28]. In our analysis, we decided to present and compare only models assessing lung cancer risk: the BACH model, PLCO\({}_{m2012}\) model, and LCRAT model. As the LCDRAT model predicts the risk of lung cancer death, not the risk of the lung cancer occurrence, we excluded it from our survey. According to [31], the models' performance was compared on two cohorts (AARP and CPS-II). The Area Under the ROC curve's (AUC) values for the chosen three models vary from 0.75 to 0.77 and are higher than for the other screening models. When we compare only AUCs of the analyzed models, the LCRAT model achieves the highest value equal to 0.77, whereas the Bach model the lowest value equal to 0.75. Note that the differences between AUCs of those three models are small. The authors of [31] also calculated the ratio of model predicted cases to the number of observed cases. This provides information on how the model is calibrated. Bach, LCRAT and PLCO\({}_{m2012}\) models are well-calibrated, with the values of Expected/Observed from 0.92 to 1.12 on both cohorts. The lung cancer risk models, such as for example BACH or PLCO\({}_{m2012}\), are developed with different statistical techniques. The BACH and LCRAT models use Cox regression models for survival data. The PLCO\({}_{m2012}\) model is based on logistic regression. Moreover, models are based on different sets of variables that were chosen by the authors of the models. For each model, the lung cancer risks are predicted within different time periods (5, 6, 10 years). See Table 1 for a detailed summary. On the whole, these models are complex, and it is difficult to understand how they work by simply looking at the model coefficients. For example, Equation (1) shows the risk function in the BACH model. It is based on the multi-variable Cox proportional hazards regression. \[risk=\sum_{i=0}^{n}(1-5_{0}^{exp(\beta X_{i})})(S_{1}^{exp(\beta X_{i})}) \prod_{j<i}(S_{0}^{exp(\beta X_{j})})(S_{1}^{exp(\beta X_{j})}) \tag{1}\]  
where:  
* \(S_{0}\) corresponds to the baseline survival without being diagnosed with lung cancer beyond 1-year, its estimate is 0.996229;
* \(S_{1}\) corresponds to the baseline overall survival beyond 1 year, its estimate is 0.9917663;
* \(X\) is the matrix of values of variables for a given person;* \(X_{i}\) are the values of \(i\)-th variable;
* \(\beta\) are coefficients estimated in the model, see Table 2.  
The BACH model estimates the risk of death within 10 years. Nonlinear effects were provided by cubic splines that have been fitted for continuous predictors (for example, age). This precise mathematical formalism shows that even a model with a low number of predictors, based on well-known mathematical formula, could be difficult to understand. Many decimal numbers and the cubic expression for continuous variables make the expression very complex. There is no one common definition of the interpretable model. For example, the author in [32] treats interpretability as \"the degree to which an observer can understand the cause of a decision\".