## 2 Materials and Methods
### Monte Carlo Simulation Procedure for Estimation of Age-Adjusted Mortality Rates and Rate Ratios  
We also considered estimation of age-adjusted mortality rates by sex and age-adjusted male-to-female mortality RRs in the U.S. and in each state for the purpose of comparing them to our estimates of age-adjusted YPLL rates by sex and age-adjusted male-to-female YPLL RRs, respectively, to examine potential differences in the characterization of the disparities when measuring mortality burden in terms of YPLL compared to death counts. Adopting the same methodological motivation as [PERSON] et al. (2021b) in their estimation of age-adjusted mortality rates by race/ethnicity and their associated RRs, we wanted our male and female mortality rates to be standardized to the 2019 U.S. Census Bureau age distribution estimate of the overall U.S. population without combining age intervals in the U.S. Census Bureau data to align them with the NCHS data age intervals so that estimated mortality and YPLL rates in our analysis were age-standardized to as identical as possible standard populations in terms of age interval granularity. To this end, we performed a MC simulation procedure to obtain conservative \((1-\alpha)\times 100\%\) interval estimates of male and female age-adjusted mortality rates and age-adjusted male-to-female mortality RRs that largely mirrors the MC simulation procedure for the YPLL-based estimands of interest described in Section 2.6. The key differences were that ages at death were simulated for all individuals in non-85+ age groups, and in the direct age adjustment procedure, we summed the number of simulated ages at death falling within the 1-year intervals implied by integer ages 0 to 84 to calculate the age-specific mortality rates for ages 0 to 84, as well as calculate an age 85+ mortality rate, which were subsequently applied to the standard population to obtain the male and female age-adjusted mortality rates, which we denote as \(\widehat{\mathrm{R}}_{\mathrm{mort,male,js}}^{(b)}\) and \(\widehat{\mathrm{R}}_{\mathrm{mort,female,js}}^{(b)}\), respectively.  
Mathematically, the male age-adjusted mortality rate is given by:  
\[\widehat{\mathrm{R}}_{\mathrm{mort,male,js}}^{(b)}=\frac{\sum_{a=0}^{84} \left(n_{a}\times\frac{\hat{d}_{\mathrm{male,js}}^{(b)}}{n_{\mathrm{male,js}} }\right)+\left(n_{85+}\times\frac{d_{\mathrm{male,js,85+}}}{n_{\mathrm{male,js,85+}}}\right)}{\sum_{a=0}^{84}n_{a}+n_{85+}}, \tag{9}\]  
where \(\hat{d}_{\mathrm{male,js}}^{(b)}\ =\ \sum\limits_{\begin{subarray}{c}i_{\mathrm{ j}}\\ \hat{a}_{i_{\mathrm{j}},\mathrm{s}}\in[a+1)\end{subarray}}1\) is the number of male simulated ages at death equal to age \(a\in\{0,1,\ldots,84\}\), \(\hat{d}_{\mathrm{male,js,85+}}\) denotes the number of male deaths in the 85+ age group, and \(n_{85+}\) denotes the 2019 U.S. Census Bureau national population estimate for age group 85+.  
Similarly, the female age-adjusted mortality rate is given by:  
\[\widehat{\mathrm{R}}_{\mathrm{mort,female,js}}^{(b)}=\frac{\sum_{a=0}^{84} \left(n_{a}\times\frac{\hat{d}_{\mathrm{female,js}}^{(b)}}{n_{\mathrm{ female,js}}}\right)+\left(n_{85+}\times\frac{\hat{d}_{\mathrm{female,js,85+}}}{n_{ \mathrm{female,js,85+}}}\right)}{\sum_{a=0}^{84}n_{a}+n_{85+}}, \tag{10}\]  
where \(\hat{d}_{\mathrm{female,js}}^{(b)}=\sum\limits_{\begin{subarray}{c}i_{ \mathrm{ j}}\\ \hat{a}_{i_{\mathrm{j}},\mathrm{s}}\in[a,a+1)\end{subarray}}1\) is the number of female simulated ages at death equal to age \(a\in\{0,1,\ldots,84\}\), and \(\hat{d}_{\mathrm{female,js,85+}}\) denotes the number of female deaths in the 85+ age group.  
The age-adjusted male-to-female mortality RR is defined as the quotient of \(\widehat{\mathrm{R}}_{\mathrm{mort,male,js}}^{(b)}\) and \(\widehat{\mathrm{R}}_{\mathrm{mort,female,js}}^{(b)}\):  
\[\widehat{\mathrm{RR}}_{\mathrm{mort,male,female,js}}^{(b)}=\frac{\widehat{ \mathrm{R}}_{\mathrm{mort,male,js}}^{(b)}}{\widehat{\mathrm{R}}_{\mathrm{ mort,female,js}}^{(b)}}. \tag{11}\]