How does noise protection affect the accuracy of life expectancy and other demographic indicators?

New and efficient methods based on noise addition to protect the confidentiality in population statistics have been developed, tested and applied in census production by various members of the European Statistical System over the past years. Basic demographic statistics - such as population stocks, live births and deaths by age, sex and region - may be protected in a similar way, but also form the raw input to calculate various demographic indicators. This paper analyses the impact on the accuracy of some selected indicators, namely fertility and mortality rates and life expectancies, under the assumption that the raw input counts are protected with a generic noise method with fixed variance parameter, by comparing the size of noise uncertainties with intrinsic statistical uncertainties using a Poisson model. As a by-product, we derive and validate numerically a closed analytical expression for the variance of life expectancies in a certain class of calculation models as a function of the variance of input mortality data. This expression also allows to calculate analytically the statistical uncertainty of life expectancies using the mentioned Poisson model for the input death counts.

💡 Research Summary

This paper investigates how the addition of statistical disclosure control (SDC) noise—specifically the Cell‑Key Method (CKM) used by the European Statistical System—affects the accuracy of key demographic indicators: age‑specific fertility rates, age‑specific mortality rates, and life expectancy. CKM adds integer‑valued noise to each cell of a tabulation; the variance of the added noise is a fixed parameter V, giving a standard deviation Δ = √V that is identical for all input counts. Using the classical error‑propagation formula Δf = √∑(∂f/∂xᵢ)² Δ², the authors derive simple propagation rules for sums (absolute errors add in quadrature) and for products or ratios (relative errors add in quadrature).
For fertility, the age‑specific rate fₓ = bₓ/wₓ simplifies to a relative error δfₓ ≈ Δ/bₓ because births bₓ are typically far smaller than the female population wₓ. With V = 1 (Δ = 1) and 2023 NUTS‑2 data, the median relative error across all regions and ages is 0.37 %; however, when bₓ ≤ 10 the error can exceed 1 % and reach up to 100 % for single births. The total fertility rate aggregates these errors according to Eq. (7).
For mortality, the age‑specific crude death rate Mₓ = Dₓ/Bₓ yields δMₓ ≈ Δ/Dₓ. Because deaths Dₓ are often very small at young ages, the median relative error for the total population is about 1.4 %, with 23 % of region‑age cells showing errors above 10 %.
Life expectancy requires a more elaborate treatment. The authors use the standard life‑table quantities ℓₓ (survivorship), Lₓ (person‑years lived), and Tₓ (total person‑years above age x). Assuming the simplest mortality model Mₖ = Dₖ/Bₖ, they differentiate the life‑expectancy formula (Eq. 10‑13) with respect to each death count Dₖ, apply the chain rule, and obtain a closed‑form expression for the relative error δEₓ (Eq. 16). This formula depends only on Δ, the survivorship values ℓ, the population B, and the terminal death count D. Applying V = 1 to the 2023 NUTS‑2 dataset gives a median δEₓ of 0.018 % for the total population; only two regions (Mayotte, France and Åland, Finland) exceed a 1 % error, illustrating the robustness of life expectancy to CKM noise when death counts are sufficiently large.
To place CKM‑induced errors in context, the paper adopts a Poisson model for intrinsic statistical variation: for any count xᵢ, the variance equals the count (λ ≈ xᵢ), giving a statistical standard deviation √xᵢ. By combining this with CKM noise (Eq. 18‑19), the authors provide formulas for the total uncertainty when both sources are present. Setting Δ = 0 recovers the pure Poisson uncertainty; setting the Poisson term to zero isolates the pure CKM effect.
Empirical results show that (1) fertility and mortality rates are sensitive to noise in cells with low counts, especially at fine geographic or age granularity; (2) life expectancy is remarkably insensitive to the same noise because it aggregates over many ages and large death counts; (3) the magnitude of CKM noise can be directly controlled via the variance parameter V, allowing statistical offices to balance confidentiality against data utility. The authors suggest that for high‑resolution demographic releases, pre‑release simulations using the derived formulas are essential to decide on an appropriate V. They also note that the analytical framework can be extended to other demographic measures and to other countries, providing a practical tool for policymakers and data producers navigating the trade‑off between privacy protection and analytical accuracy.