## 2 Materials and Methods
### Complete Monte Carlo Simulation Procedure
Specifically, for the estimation of the male percentage of total YPLL, first calculate total YPLL for males and females from the simulated ages at death, which are \(\widehat{\text{YPLL}}^{(b)}_{\text{male},js}=\sum_{i=1}^{n_{\text{ male}}}\text{max}\Big{\{}\mathcal{A}-\bar{a}^{(b)}_{i_{m},js},0\Big{\}}\) and \(\widehat{\text{YPLL}}^{(b)}_{\text{female},js}=\sum_{i_{f}=1}^{n_{\text{ sf}}}\text{max}\Big{\{}\mathcal{A}-\bar{a}^{(b)}_{i_{f},js},0\Big{\}}\), respectively. Then, the male percentage of total YPLL, which we denote \(\widehat{\pi}^{(b)}_{\text{male},js}\), is given by: \[\widehat{\pi}^{(b)}_{\text{male},js}=\frac{\widehat{\text{YPLL}}^{(b)}_{ \text{male},js}}{\widehat{\text{YPLL}}^{(b)}_{\text{male},js}+\widehat{ \text{YPLL}}^{(b)}_{\text{female},js}}\times 100\%.\] (4) Similarly, the female percentage of total YPLL, which we denote \(\widehat{\pi}^{(b)}_{\text{female},js}\), is given by: \[\widehat{\pi}^{(b)}_{\text{female},js}=\frac{\widehat{\text{YPLL}}^{(b)}_{ \text{female},js}}{\widehat{\text{YPLL}}^{(b)}_{\text{male},js}+\widehat{ \text{YPLL}}^{(b)}_{\text{female},js}}\times 100\%.\] (5)  
For estimation of the age-adjusted male-to-female YPLL RR, first estimate the age-adjusted male and female YPLL rates using direct age adjustment [58], using the 2019 U.S. Census Bureau age distribution estimate of the overall U.S. population as the standard population. Since the simulated ages at death are continuous and the U.S. Census Bureau age distribution estimates are defined over integer ages from 0 to 84, we aggregate the corresponding simulated YPLL values with respect to the 1-year age intervals implied by these integer ages (i.e., age \(a\in\{0,1,\ldots,84\}\) implies age interval \([a,a+1)\)) to calculate the age-specific YPLL rates, which are subsequently applied to the standard population to obtain the age-adjusted YPLL rate. The male age-adjusted YPLL rate is given by:  
\[\widehat{\text{R}}^{(b)}_{\text{YPLL},\text{male},js}=\frac{\sum_{a<[ \mathcal{A}]}\Bigg{(}n_{a}\times\frac{\widehat{\text{YPLL}}^{(b)}_{\text{ male},js}}{n_{\text{male},a}}\Bigg{)}+\Bigg{(}n_{[\mathcal{A}]}\times\frac{ \widehat{\text{YPLL}}^{(b)}_{\text{male},js,[\mathcal{A}]}}{n_{\text{male}, \mathcal{A}}}\Bigg{)}\times(\mathcal{A}-\lfloor\mathcal{A}\rfloor)}{\sum_{a<[ \mathcal{A}]}n_{a}+n_{[\mathcal{A}]}\times(\mathcal{A}-\lfloor\mathcal{A} \rfloor)}, \tag{6}\]  
where \(\widehat{\text{YPLL}}^{(b)}_{\text{male},ja}=\sum_{i_{m}\atop a^{(b)}_{i_{m },js}\in[a,a+1)}\text{max}\Big{\{}\mathcal{A}-\bar{a}^{(b)}_{i_{m},js},0\Big{\}}\) denotes aggregate male YPLL corresponding to age \(a\in\{0,1,\ldots,84\}\); \(n_{\text{male},sa}\) denotes the 2019 U.S. Census Bureau male population estimate for age \(a\in\{0,1,\ldots,84\}\) in state \(s\); and \(n_{a}\) denotes the 2019 U.S. Census Bureau national population estimate for age \(a\in\{0,1,\ldots,84\}\). Analogously, the female age-adjusted YPLL rate is given by:\[\widehat{\mathrm{R}}^{(b)}_{\mathrm{YPLL,female,js}}=\frac{\sum_{a<\lfloor A \rfloor}\left(n_{a}\times\frac{\widehat{\mathrm{YPLL}}^{(b)}_{\mathrm{female,js}}}{n _{\mathrm{female,js}}}\right)+\left(n_{\lfloor\mathcal{A}\rfloor}\times\frac{ \widehat{\mathrm{YPLL}}^{(b)}_{\mathrm{female,js},\lfloor\mathcal{A}\rfloor} }{n_{\mathrm{female,js},\lfloor\mathcal{A}\rfloor}}\right)\times\left( \mathcal{A}-\lfloor\mathcal{A}\rfloor\right)}{\sum_{a<\lfloor\mathcal{A} \rfloor}n_{a}+n_{\lfloor\mathcal{A}\rfloor}\times\left(\mathcal{A}-\lfloor \mathcal{A}\rfloor\right)}, \tag{7}\]  
\[\text{where }\widehat{\mathrm{YPLL}}^{(b)}_{\mathrm{female,js}}=\sum_{ \begin{subarray}{c}i_{f}\\ \hat{a}^{(b)}_{i_{f},j}\in[a,\mathcal{A}+1)\end{subarray} }\max\Bigl{\{} \mathcal{A}-\hat{a}^{(b)}_{i_{f},j},0\Bigr{\}}\text{ denotes aggregate female YPLL}\]  
corresponding to age \(a\in\{0,1,\ldots,84\}\); and \(n_{\mathrm{female,}sa}\) denotes the 2019 U.S. Census Bureau female population estimate for age \(a\in\{0,1,\ldots,84\}\) in state \(s\).