# Background Mortality Adjustments

## Why Background Mortality Matters

Health economic models track cohorts over long time horizons (often lifetime). Patients can die from the disease being modelled, but they also face background (all-cause) mortality from other causes. Census life tables provide general population mortality rates, but these must be adjusted for your specific patient population.

ParCC provides four methods for handling background mortality.

## Method 1: SMR Adjustment

### The Scenario – Heart Failure Cohort

You are modelling a cohort of 65-year-old heart failure patients. The general population mortality rate for age 65 (from the Indian Census life table) is 0.013 per year. Published literature reports that heart failure patients have a Standardized Mortality Ratio (SMR) of 2.5 compared to the general population (Meta-analysis by Jhund et al., Eur Heart J 2009).

### The Formula

\[r_{disease} = r_{population} \times SMR\]

\[p_{disease} = 1 - e^{-r_{disease} \times t}\]

### Worked Example

    r_pop <- 0.013   # General population rate, age 65
    smr <- 2.5        # From published meta-analysis

    r_disease <- r_pop * smr
    p_disease <- 1 - exp(-r_disease)

    cat("Population rate:", r_pop, "\n")
    #> Population rate: 0.013
    cat("SMR:", smr, "\n")
    #> SMR: 2.5
    cat("Disease-adjusted rate:", round(r_disease, 5), "\n")
    #> Disease-adjusted rate: 0.0325
    cat("Annual mortality probability:", round(p_disease, 5), "\n")
    #> Annual mortality probability: 0.03198

## Method 2: DEALE (Declining Exponential Approximation of Life Expectancy)

### The Scenario – Rare Disease with Known Life Expectancy

For Duchenne Muscular Dystrophy, the observed mean life expectancy is 25 years (from a registry study). The background life expectancy for the same age-sex group is 65 years.

### The Formula

\[r_{disease} = \frac{1}{LE_{observed}} - \frac{1}{LE_{background}}\]

### Worked Example

    le_observed <- 25    # Disease-specific life expectancy
    le_background <- 65  # General population

    r_excess <- (1 / le_observed) - (1 / le_background)
    cat("Excess mortality rate:", round(r_excess, 5), "\n")
    #> Excess mortality rate: 0.02462
    cat("This means patients face an additional", round(r_excess * 1000, 2),
        "deaths per 1,000 person-years beyond background\n")
    #> This means patients face an additional 24.62 deaths per 1,000 person-years beyond background

## Method 3: Gompertz Parameterization

### When to Use

The Gompertz law describes how mortality increases exponentially with age, which closely matches human mortality patterns above age 30:

\[r(age) = \alpha \times e^{\beta \times age}\]

This is useful when you need age-specific mortality rates and only have a few data points from the life table.

### Worked Example

    # Fit Gompertz from two life table points
    # Age 60: rate = 0.008
    # Age 80: rate = 0.065

    r1 <- 0.008; age1 <- 60
    r2 <- 0.065; age2 <- 80

    beta <- log(r2 / r1) / (age2 - age1)
    alpha <- r1 / exp(beta * age1)

    cat("Gompertz alpha:", format(alpha, scientific = TRUE), "\n")
    #> Gompertz alpha: 1.491488e-05
    cat("Gompertz beta:", round(beta, 5), "\n\n")
    #> Gompertz beta: 0.10475

    # Generate age-specific rates
    ages <- seq(50, 90, by = 5)
    rates <- alpha * exp(beta * ages)
    probs <- 1 - exp(-rates)

    data.frame(
      Age = ages,
      Rate = round(rates, 5),
      Annual_Prob = round(probs, 5)
    )
    #>   Age    Rate Annual_Prob
    #> 1  50 0.00281     0.00280
    #> 2  55 0.00474     0.00473
    #> 3  60 0.00800     0.00797
    #> 4  65 0.01351     0.01342
    #> 5  70 0.02280     0.02255
    #> 6  75 0.03850     0.03777
    #> 7  80 0.06500     0.06293
    #> 8  85 0.10974     0.10393
    #> 9  90 0.18528     0.16913

## Method 4: Linear Interpolation

For ages between available life table entries, linear interpolation provides a simple estimate:

\[r(age) = r_1 + (age - age_1) \times \frac{r_2 - r_1}{age_2 - age_1}\]

## References

- Beck JR, et al. A convenient approximation of life expectancy (the “DEALE”). Am J Med. 1982;73(6):883-888.
- Gompertz B. On the nature of the function expressive of the law of human mortality. Phil Trans Royal Soc. 1825;115:513-583.
- Jhund PS, et al. Long-term trends in first hospitalization for heart failure and subsequent survival. Eur Heart J. 2009;30(4):1807-1812.