Calculating cumulative excess risks

Installation

library(CanEpiRisk)

1. Overview

With an exposure scenario, baseline reference data, and a risk model, CanEpiRisk computes:

• Cumulative Excess Risk (CER) via CER()
  
• Years of Life Lost (YLL) via YLL()
  
• Population-averaged versions via population_LAR() and population_YLL()

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2. Inputs at a Glance

1. Exposure (exposure; list): attained-age grid is implicit (ages 1–100). Key fields
   • agex: age(s) at exposure (scalar or vector)
   • doseGy: dose(s) in Gy (same length as agex if vectorized)
   • sex: typically 1 = male, 2 = female
2. Reference (reference; list): region/site-specific tables
   • baseline: site-specific baseline incidence or mortality (per person-year) on ages 1–100
   • mortality: all-cause mortality (per person-year) on ages 1–100
3. Risk model (riskmodel; list): e.g., LSS/INWORKS style objects
   • For packaged LSS models: LSS_mortality$<site>$L, LSS_incidence$<site>$LQ, etc.
   • For user models: see §5
4. Options (option; list):
   • maxage: upper bound of attained age for accumulation (e.g., 100)
   • err_wgt: blend ERR vs EAR (1 = pure ERR; 0 = pure EAR)
   • n_mcsamp: Monte Carlo sample size for uncertainty (e.g., 10000)

Units & alignment. Keep dose in Gy; ensure baseline and mortality come from the same region/population and are age-aligned (ages 1–100).

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3. Quick Start: CER and YLL

3.1 All solid cancer mortality (ERR model; LSS)

set.seed(123)

# Exposure scenario: 0.1 Gy at age 15, female, track to age 100
exp1 <- list(agex = 15, doseGy = 0.1, sex = 2)

# Region 1 reference (example)
ref1 <- list(
  baseline  = Mortality[[1]]$allsolid,  # site-specific baseline mortality
  mortality = Mortality[[1]]$allcause   # all-cause mortality
)

# LSS linear ERR model for all-solid mortality
mod1 <- LSS_mortality$allsolid$L

# Tuning options
opt1 <- list(maxage = 100, err_wgt = 1, n_mcsamp = 10000)

# CER (per 10,000 persons)
cer1 <- CER(exposure = exp1, reference = ref1, riskmodel = mod1, option = opt1)
cer1 * 10000
#>         mle        mean      median  ci_lo.2.5% ci_up.97.5% 
#>    156.0667    158.4113    155.9330    115.0978    214.6693

3.2 Leukaemia incidence (EAR model; LSS LQ)

# Dose delivered evenly across ages 30–45 (15 yearly bins), male
exp2 <- list(agex = 30:44 + 0.5, doseGy = rep(0.1/15, 15), sex = 1)

ref2 <- list(
  baseline  = Incidence[[4]]$leukaemia,  # site-specific baseline incidence
  mortality = Mortality[[4]]$allcause    # all-cause mortality
)

mod2 <- LSS_incidence$leukaemia$LQ
opt2 <- list(maxage = 60, err_wgt = 0, n_mcsamp = 10000)

cer2 <- CER(exposure = exp2, reference = ref2, riskmodel = mod2, option = opt2)
cer2 * 10000
#>         mle        mean      median  ci_lo.2.5% ci_up.97.5% 
#>  3.68374227  3.66876471  3.63603307  0.05966926  7.39791682

3.3 Years of Life Lost (YLL)

# Same inputs; YLL returns expected years-of-life lost attributable to exposure
yll1 <- YLL(exposure = exp1, reference = ref1, riskmodel = mod1, option = opt1)
yll1
#>         mle        mean      median  ci_lo.2.5% ci_up.97.5% 
#>   0.2408580   0.2439038   0.2421660   0.1883518   0.3081081

Interpreting outputs. Functions typically return point summaries (MLE/mean/median) and percentile intervals when simulation is used. Report the scale (e.g., per person vs per 10,000) and follow-up window (maxage).

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4. Population-Averaged Metrics

When you have population age/sex distributions and exposure distributions, use:

# LAR (lifetime attributable risk) and YLL at the population level
# Example 1: allsolid mortality, Region-1, exposed to 0.1 Gy, followed up to age 100, LSS linear ERR
 ref1 <- list(  baseline=Mortality[[1]]$allsolid,     # baseline rates
               mortality=Mortality[[1]]$allcause,     # allcause mortality
                 agedist=agedist_rgn[[1]] )           # age distribution
 mod1 <- LSS_mortality$allsolid$L                     # risk model
 population_LAR( dsGy=0.1, reference=ref1, riskmodel=mod1 )    # CER cases per 10,000

 # Example 2: leukaemia incidence, Region-4, exposed to 0.1 Gy, followed up to age 100, LSS LQ ERR
 ref2 <- list(  baseline=Incidence[[4]]$leukaemia,    # baseline rates
               mortality=Mortality[[4]]$allcause,     # all-cause mortality
                 agedist=agedist_rgn[[4]] )           # age distribution
 mod2 <- LSS_incidence$leukaemia$LQ                   # risk model
 population_LAR( dsGy=0.1, reference=ref2, riskmodel=mod2 )    # CER cases per 10,000

Where - agedist_rgn encodes the population structure of WHO global regions.

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5. Using Your Own Risk Models

A risk model object for an endpoint (site-specific incidence or mortality) is a list with sublists for ERR/EAR, each containing:

• para: vector of parameter estimates
  
• var: variance–covariance matrix or (for 1-parameter models) ci: 95% confidence bounds
  
• f: function implementing the risk as a function of parameters and data, e.g. function(beta, data, lag = 10) { beta[1] * data$dose * (data$age - data$agex >= lag) }

Example skeleton (ERR, single parameter with CI bounds):

MyRiskmodel <- list()
MyRiskmodel$allsolid <- list()
MyRiskmodel$allsolid$L <- list(
  err = list(
    para = c(0.47),
    ci   = c(0.14, 0.85),
    f    = function(beta, data, lag = 10) {
      beta[1] * data$dose * (data$age - data$agex >= lag)
    }
  )
)

When a risk model has one parameter and includes ci, CER() uses the CI directly to form uncertainty intervals; otherwise it uses var to sample. The following example shows how to specify and using a one-parameter model derived from the INWORKS cohort for mortality from all solid cancer (Richardson et al., 2015) and from leukaemia (Leuraud et al., 2015).

INWORKS_mortality <- NULL
INWORKS_mortality$allsolid <- NULL

INWORKS_mortality$allsolid$L <- list(
     err=list(
       para=c(0.47),                  # ERR/Gy=0.47 (90% CI: 0.18,0.79)  
       ci= c(0.1392403, 0.8521128),   #  95% CI coverted from 90%CI by Weibull approx.
       f=function (beta, data, lag=10) {
           beta[1] * data$dose  * (data$age - data$agex >= lag )
       }
       ),
      ear=list(    # dummry object
       para=c(4.8/10000),
       ci= c(0.1068428, 12.5703871)/10000,
       f=function (beta, data, lag=10) {
           beta[1] * data$dose  * (data$age - data$agex >= lag )
       } 
       )
 
  )

INWORKS_mortality$leukaemia$L <- list(
     err=list(
       para=c(2.96),                         # ERR/Gy=2.96 (90% CI: 1.17, 5.21) 
       ci= c(0.8664, 5.6940),
       f=function (beta, data, lag=2) {
           beta[1] * data$dose  * (data$age - data$agex >= lag )
       }
       ),
      ear=list(    # dummry object
       para=c(2.25/10000),
       ci=c(0.5064054, 4.4914628)/10000,
       f=function (beta, data, lag=2) {
           beta[1] * data$dose  * (data$age - data$agex >= lag )
       } 
       )
  )

# Cumulative excess risk (CER) calculations 
# (1) All solid cancer mortality, Region-1, female, 0.1Gy at age 15, followed up to age 100, INWORKS linear ERR
exp1 <- list( agex=5, doseGy=0.1, sex=2 )   # exposure scenario
ref1 <- list( baseline=Mortality[[1]]$allsolid,        # baseline rates
             mortality=Mortality[[1]]$allcause )       # all-cause mortality
mod1 <- INWORKS_mortality$allsolid$L                       # risk model
opt1 <- list( maxage=100, err_wgt=1, n_mcsamp=10000 )  # option
CER(  exposure=exp1, reference=ref1, riskmodel=mod1, option=opt1 ) * 10000 # cases per 10,000
#>       mle      mean    median     ci_lo     ci_up 
#>  78.57894  78.57894  78.57894  23.27948 142.46409

# (2) Leukaemia incidence, Region-4, male, 6.7(100/15)mGy at ages 30-45, followed up to age 60, INWORKS EAR
exp2 <- list( agex=30:44+0.5, doseGy=rep(0.1/15,15), sex=1 )
ref2 <- list( baseline=Incidence[[4]]$leukaemia,       # baseline rates
             mortality=Mortality[[4]]$allcause )       # all-cause mortality rates
mod2 <- INWORKS_mortality$leukaemia$L                            # risk model
opt2 <- list( maxage=60, err_wgt=0, n_mcsamp=10000)    # option
CER(  exposure=exp2, reference=ref2, riskmodel=mod2, option=opt2 ) * 10000 # cases per 10,000
#>      mle     mean   median    ci_lo    ci_up 
#> 4.445936 4.445936 4.445936 1.000643 8.875002

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6. Notes

• maxage: Ensure max(exposure$agex) < maxage.
• err_wgt: Use 1 for pure ERR, 0 for pure EAR, or intermediate values if a combined measure is justified.
• n_mcsamp: Increase for smoother interval estimates; reduce for quick prototyping.
• lag: Implemented within your model’s f(); typical values differ by site (e.g., 5 years for solid cancers vs 2 years for leukaemia).
  
• Scaling: Multiply by 1e4 (or 1e5) to report per 10,000 (or per 100,000).
• doseGy >= 0, consistent vector lengths for agex/doseGy when vectorized, sex ∈ {1,2}.
• Model–endpoint match: Do not mix incidence models with mortality baselines (and vice versa).

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References

Richardson, D.B., E. Cardis, R.D. Daniels et al. Risk of cancer from occupational exposure to ionising radiation: retrospective cohort study of workers in France, the United Kingdom, and the United States (INWORKS). BMJ 351: h5359 (2015).

Richardson, D.B., K. Leuraud, D. Laurier et al. Cancer mortality after low dose exposure to ionising radiation in workers in France, the United Kingdom, and the United States (INWORKS): cohort study. BMJ 382: e074520 (2023).

Leuraud, K., D.B. Richardson, E. Cardis et al. Ionising radiation and risk of death from leukaemia and lymphoma in radiation-monitored workers (INWORKS): an international cohort study. Lancet Haematol 2(7): e276-281 (2015).