PAPP103 - S09: Models of populations with variable growth rates

Application 4:  Case-fatality ratios

As well as being used to estimate single-decrement life tables, variable growth rate methods can be used to estimate certain measures in multiple-decrement life tables. The case-fatality ratio or proportion is the proportion of those contracting a disease that die from it. By adjusting the numbers of deaths from a disease for age-specific growth, one can obtain period life table estimates of fatalities from the disease. No information is required about mortality from other causes of death.

Multiplying the variable r equation for the population with some disease, i, according to duration since diagnosis, x, by the duration-specific death rates from that disease, mⁱ_x, yields:

$$D_x^i=P_x{m}_x^i=B_0{e}^{-{\displaystyle \sum _{a=0}^{x-1}{r}_a-0.5r_x}}\frac{L_x}{l_0}{m}_x^i$$

In this equation, L_xmⁱ_x represents the life table deaths from cause i at duration x. The ‘births’ entering the population with a disease are the current year’s incident cases of that disease (B₀ = I). Summing life table deaths from a specific cause over all durations x, one gets the total proportion of those developing the disease that die from it, ∑dⁱ_x=lⁱ₀. So, rearranging the equation to place L_xmⁱ_x on its LHS and summing over x:

$$\frac{l_0^i}{l_0}=\frac{{\displaystyle \sum _{x=0}^w{D}_x^i{e}^{{\displaystyle \sum _{a=0}^{x-1}{r}_a+0.5r_x}}}}{I}$$

This equation demonstrates that dividing current deaths by current diagnoses only gives an unbiased measure of case-fatality if both the incidence of and deaths from the condition are unchanging. If the numbers of deaths from a condition are growing at all durations, the case-fatality ratio is biased downward. If the number of deaths is falling at all durations, the case-fatality ratio will be biased up. However, if one has two sets of data on cases or deaths from the disease from which one can calculate growth rates, one can adjust for growth and divide the fatalities by the incident cases to obtain an unbiased estimate of the case-fatality ratio.

The same logic can be applied to other processes. One demographic application would be to use the method to calculate the probability that a marriage will end in divorce from data on the married population tabulated by marital duration at two dates and the number of marriages occurring in the intervening period.