### Study steps

Figure 1 identifies the main steps of our probabilistic modeling approach for the economic evaluation of silica exposure reduction interventions. In the first step, we identify all variables that impact the net benefit of the silica reduction interventions, such as the number of silica-exposed workers, level of exposure and intervention effectiveness. Variables not dependent on any other variables (called root nodes in BN vocabulary) have a single probability distribution, whereas variables dependent on one or more other variables (called child nodes) have a conditional probability table (dependency of a child variable to its parent’s variables) [19]. In the second step, we identify dependencies between variables via a literature review and expert knowledge. Six researchers with the following backgrounds were involved in all stages of the project meeting: an expert in silica reduction interventions, two occupational health specialists, one economist, and two epidemiologists. Brainstorming sessions and interviews with experts were relatively unstructured. In sessions, participants were all given an opportunity to contribute to the discussion until consensus was reached. Expert feedback also helps us to identify variables and interactions that were overlooked when first developing the model. In the third step, we identify the probability distributions of variables, drawing on several scientific literatures in epidemiology [25, 26], occupational cancer economic burden studies [4, 16], and silica exposure reduction interventions [5,6,7,8,9,10]. Once the distributions of independent variables are determined, we compute the probability distributions of conditional variables according to the knowledge of their parents. The main assumptions about the distribution of each variable are explained in the following paragraphs. To develop the structure of BN model and to compute the probability distributions, we use GeNIe modeller version 2.2.4 (BayesFusion, Pittsburgh University decision system laboratory) [27]. Step four involves establishing the structural validity of the model. We validate the model by setting the variables to extreme values and turn to expert judgment to confirm whether the range of results (e.g. expected lung cancer cases, averted costs, and/or interventions costs) appears reasonable. Sensitivity analysis is also undertaken to quantify how different values of independent variables affect the net benefit of interventions. In the fifth and last step, we select a preferred silica reduction intervention by comparing the expected net benefit of alternatives. Benefits are the expected cost of lung cancer cases averted after implementation of different interventions. We use an incidence cost approach and estimate the societal lifetime cost of lung cancer cases. Then we calculate expected net benefit as the difference between the expected benefit from expected cost of each intervention in a calendar year (i.e., 2020). Costs of the intervention are based on the assumption that there is no use of preventive measures at baseline. Economic evaluation is conducted from societal perspective. A discount rate of 3% was used to obtain the present value. All monetary values are converted to 2017 Canadian dollars.

### Input data

To determine the probability distributions of variables, we combine our model assumptions with secondary data drawn from various sources such as the Occupational Cancer Research Centre (OCRC) [25], CAREX Canada [26], Canadian Life Tables [28], the Labour Force Survey (LFS) [29], the Survey of Labour and Income Dynamics (SLID) [30], Canadian System of National Accounts (CSNA) [31], the General Social Survey (GSS) [32], the Canadian Cancer Risk Management Model (CRMM) [33], the Survey of Employment, Payrolls and Hours (SPEH) [34], Canadian Community Health Survey (CCHS) [35], and various scientific published and grey literature sources.

### BN model

A simplified representation of the model is illustrated in Fig. 2 (the full network is provided in Additional file 1: Part A). With this model we estimate the expected cost of lung cancer cases averted given different silica exposure reduction interventions. The silica reduction intervention decisions in the model include one of three interventions of WM, LEV and PPE, as well as the following combinates: WM-LEV-PPE, WM-LEV, WM-PPE, LEV-PPE, gives rise to seven different silica exposure reduction possibilities (represented by rectangles). These are the main silica reduction interventions in OSHA’s hierarchy of controls, after elimination/ substitution [5]. Although, the elimination/ substitution of silica with materials containing less amount of silica is the most effective way to protect workers, we do not consider it in this study, mainly because of the large dependency of the construction sector to silica-containing supplies. In the BN, to demonstrate the uncertainty related to each domain, we use random variables (represented by ellipses). A random variable can assume more than one value due to chance (e.g. sex of lung cancer cases is a variable with two values, i.e., male and female that each value has a probability of occurrence). In our model, the random variables related to the lung cancer case costs are age, sex, survival rate, direct costs of lung cancer cases, annual wage of workers, and monetary value of a quality-adjusted life-year (QALY). The random variables related to the interventions costs are the number of silica-exposed workers in the construction sector, silica exposure level, intervention’s effectiveness, coverage and unit cost. Implementation of each of these interventions bears on the intervention costs, the exposure reduction experienced by workers, and in the long run, on the total number of lung cancer cases and related costs averted. BN uses utility nodes for estimation of the expected costs and benefits of the decision to be made (represented by hexagons). These two types of nodes (i.e., decision nodes and utility nodes) enhance the BN to decision support tool to determine the decision to make, which gains the highest expected utility, considering the given circumstances [19, 23]. Additional file 1: Part B lists variables definition, distribution and data sources.

#### Number of silica-exposed workers and level of exposure

We estimate the number of the silica-exposed workers in the Ontario, Canada construction sector as about 91 thousand, based on estimates from OCRC Canada [25]. (exposed occupations listed in Additional file 1: Part C). We also identify the level of silica exposure among construction workers into three ranges: low (< 0.0125 mg/m^{3}), medium (0.0125–0.025 mg/m^{3}), and high (> 0.025 mg/m^{3}), with probabilities of 0.47, 0.39, and 0.14, respectively, based on occupational exposure data sources from CAREX Canada [26].

#### Intervention’s effectiveness, coverage and cost

Wide ranges of effectiveness have been reported for silica exposure reduction interventions in the literature [5,6,7,8,9,10]. We identify the lowest reported effectiveness of WM, LEV and PPE at 82% [7], 93% [9], and 90% [7], respectively. However, full effectiveness of interventions is only achieved when they are used under the ideal conditions. For example, WM is fully effective when the system supplies a continuous stream or spray of water at the point of impact, which requires regular filling of the water tank and inspection of hosing and nozzles. Similarly, full effectiveness of PPE is achieved when respirators are used, cleaned and inspected routinely. In construction worksites, the interventions are not always working under ideal conditions. As a conservative assumption, we consider 75% of the reported values, for estimation of interventions effectiveness in the construction projects. For a combined use of each of WM or LEV with PPE (i.e. WM-PPE, LEV-PPE), we consider the additive effects. Level of silica exposure after implementation of intervention is modelled by considering primary silica exposure and the effectiveness of each intervention (Additional file 1: part D).

Some of silica reduction interventions are only applicable to certain occupations in the construction sector. We define intervention coverage, to incorporate this variable into our model. The coverage of WM and LEV are estimated at 60 and 40%, respectively, based on the OSHA [5], which means among all silica-exposed workers in the constructions sector, only these percentages can be protected by each intervention. We assume PPE is applicable to all construction occupations (Additional file 1: part E).

Intervention costs are estimated by using three variables: 1) number of silica-exposed workers that are protected by intervention, 2) intervention unit cost and 3) intervention protection factor, as indicated in expression 1. For estimation of the unit cost of the WM, LEV, and PPE, we use OSHA [5] (Additional file 1: part F). The protection factor represents the number of silica-exposed workers that can be protected by each unit of WM or LEV. Recall, WM and LEV protect a group of workers, so for estimation of the total cost of these interventions, we need to know how many workers are protected by each unit of them. For estimation of the protection factor of both WM and LEV, we drew from Lahiri et al. [7] and estimate their protection factor average at 5 workers, and assume it ranging from 1 to 10 workers with Gaussian distribution. Note that PPE total cost does not depend on the protection factor, as each unit of PPE only protects one silica-exposed worker.

$$ \mathrm{Totalcost}{\mathrm{ofintervention}}_{\left(\mathrm{x}\right)}=\frac{\mathrm{intervention}{\mathrm{unitcost}}_{\left(\mathrm{x}\right)}\times {\mathrm{protectedgroup}}_{\left(\mathrm{x}\right)}}{{\mathrm{protectionfactor}}_{\left(\mathrm{x}\right)}} $$

(1)

#### Lung cancer cases age, sex, survival

We define the age of occupational lung cancer cases in 13 intervals, ranging from 25 to more than 85 years of age [25]. The highest probability of lung cancer is between 70 and 74 years. This older age of onset is due to the long latency of this disease (Additional file 1: part G). Additionally, men have a higher incidence of occupational lung cancer than women (0.7 versus 0.3) because of their higher level of exposure in different male-dominated occupations in the construction sector [25]. We identified the survival probability of lung cancer cases at 0.09 from CRMM [33].

#### Annual wage of workers

To estimate average labour-market earnings of workers for each age and sex group, we used LFS [29], and SLID [30]. Then we add 14% to account for payroll cost paid by employers, based on employer contribution data from the CSNA [31]. We define labour-force participation following treatment of lung cancer cases at 0.77, similar to Earle et al. [36] It is assumed that once they returned to work, their productivity is the same as the productivity of the general population.

#### Monetary value of a quality-adjusted life-year

Given the wide range of monetary values of a QALY in the health economics literature, we consider a range of value in the form of sensitivity analyses. Our baseline value is $150,000 which is reflective of willingness-to-pay values for a QALY identified in recent studies [37]. For sensitivity analyses we use a range from $100,000, which has been used in Canada in the health technology assessment field, to $200,000 which has been extrapolated from increases in health care spending over time and the health gains that have been associated with those increases [38].

#### Lung cancer cases

The number of lung cancer cases expected to arise from different levels of silica exposure, is estimated by using two variables— the number of the silica-exposed workers that are protected by each intervention, and the probability of lung cancer, as described in expression 2. The number of silica-exposed workers that are protected by each intervention depends on the intervention coverage described above. We estimate the probability of lung cancer for different level of silica exposure ranges from low, medium, and high at 9.1E-4, 1.2E-3, and 1.4E-3, respectively, based on OCRC^{.25} (Additional file 1: part H). After the implementation of each intervention, silica exposure is reduced to a lower level, depending on the effectiveness of the intervention (e.g. by using PPE the level of silica exposure shifts from medium to low) and consequently, we expect a lower probability of lung cancer among the protected group of silica-exposed workers. In the expression, x is the silica exposure reduction from the interventions, which is WM, LEV, PPE, or some combination of them.

Lung cancer cases_{(x)} = number of the workers protected_{(x)} × probability of lung cancer_{(x)}(2).

#### Lung cancer direct, indirect and intangible costs

These are three sub-categories of the economic burden of lung cancer cases, which are estimated based on our previous study [16]. We identify the direct cost of lung cancer in three categories: healthcare [33], out-of-pocket costs [39], and informal caregiving costs [40], and assume it follows a Gaussian distribution [41]. We include output/productivity losses and home production losses of lung cancer cases under the indirect cost category, and monetary value of health-related quality of life losses of lung cancer under the intangible cost category. We considered the monetary value of time lost due to poor health or premature death using survival probabilities from the Canadian population [16]. The description of the techniques used to estimate these costs, are presented in Additional file 1: part I.