# F6-dev-01 — The Preston curve in 2023

# Question

Across countries/economies, how strongly is life expectancy at birth (LE, years) linearly associated with the natural log of GDP per capita (PPP, constant 2021 international $) in the most recent year with usable World Bank data? Estimand: the slope β in LE_i = α + β·ln(GDPpc_i) + ε_i, one equally weighted observation per country. The relationship is descriptive and associational only. No causal claim is made.

# Data and exclusions

• Inputs: data/wb_indicators.csv (2,349 entity-year rows, 2015–2023) and data/wb_entities.csv (296 entities), both World Bank WDI (see data/README.md). The CSVs were read with only empty cells treated as missing, so a literal "NA" in the data cannot be mistaken for a missing value.
• Population: all entities whose region is Aggregates were removed, leaving 217 countries/economies. Every iso3 code in the indicator file matched the metadata, and there were no duplicate iso3-year rows.
• Analysis year: in each year 2015–2022, 199 countries had both indicators, and 197 did in 2023. 2023 is therefore the most recent year with at least 150 countries.
• In 2023, 20 countries/economies are excluded for missing data (n_excluded_missing). In every case GDP pc PPP is the missing indicator: ASM, CHI, CUB, ERI, GIB, GUM, IMN, LIE, MAF, MCO, MNP, NCL, PRK, PYF, SSD, SYR, VEN, VGB, VIR, YEM. Nothing was imputed. n = 197 countries are analysed.
• Anomaly found: the Central African Republic has implausible LE values in years outside the analysis year (31.5 in 2019, 18.8 in 2022). They do not enter the main estimate and were left unchanged, but they inflate the robust SE of the 2022 slope in the by-year sensitivity check. All data issues are listed in results.json → data_issues.

# Method

• Baseline (no information): predict every country's LE with the sample mean. Its RMSE equals the population SD of LE.
• Model: OLS of LE on ln(GDPpc) (statsmodels), with the HC3 heteroskedasticity-robust standard error. HC3 was chosen because it is the most conservative of HC0–HC3 in small samples with leverage points. The 95% CI is β ± 1.96·SE_HC3, a normal approximation, which is what statsmodels uses with robust covariance.
• Sensitivity checks (written to out/sensitivity.json by analysis.py, not required keys):
  • HC0/HC1/HC2 and classical SEs;
  • the same regression for every year 2015–2023;
  • re-estimation after dropping high-influence points (Cook's D > 4/n);
  • a quadratic-in-log specification.
• Outputs: Figure 1 (figure1.png) is a scatter plot with the OLS line, coloured by income level. Table 1 (table1.csv) gives means by World Bank income level. No randomness is used (random_seed = null).

# Results

The Preston curve holds clearly in 2023. A 1-log-point difference in GDP per capita (about 2.7× income) goes with about 5.3 more years of life expectancy. Doubling income goes with about 5.27·ln 2 ≈ 3.65 years. Income alone cuts the prediction error roughly in half, from 7.08 to 3.59 years RMSE.

Table 1 (from table1.csv):

Conclusion code: positive_effect. The interval lies entirely above zero.

# Uncertainty

• Main result: HC3 SE = 0.211, 95% CI [4.85, 5.68].
• Other standard-error estimators: SEs are 0.207 (HC0), 0.208 (HC1), 0.209 (HC2) and 0.222 (classical), so the inference does not depend on the choice.
• Slope in other years: the slope ranges from 5.27 (2023) to 5.92 (2019) across 2015–2023, with all intervals far above zero. The 2023 slope is the lowest in the window, and slopes are somewhat lower after 2019. This could reflect post-COVID LE recovery patterns or data revisions; it was not investigated.
• Influential points: dropping the 9 points with Cook's D > 4/n gives β = 5.42 (SE 0.17). The most influential are Nigeria, Greenland, Chad, Solomon Islands and Guyana.
• Curvature: the quadratic term in ln GDPpc is 0.17 (SE 0.18), which is not significant. There is no clear evidence of curvature in log-income at this sample size.
• Not covered by the CI: the interval only reflects sampling-style variation given these 197 countries. It does not cover measurement error in either indicator, or the fact that the countries are close to a whole population rather than a random sample.

# Limitations and open issues

• What would invalidate the result:
  • Non-random missingness. The 20 excluded economies include conflict or sanctioned states (SSD, SYR, YEM, PRK, VEN, ERI, CUB) and wealthy microstates or territories (MCO, LIE, IMN, CHI, GIB). Several of these probably sit far from the fitted line. For example, Cuba plausibly has high LE at moderate income, and some conflict states have low income. Their omission could bias β in either direction.
  • Systematic errors in the 2023 data. Many 2023 LE values are WDI/UN WPP modelled estimates and projections, not vital-registration measurements, and they may be revised. The CAF series shows WDI can contain implausible values. Only the most extreme ones (LE < 35) were screened.
  • Mismeasured GDP. GDP pc PPP is poorly measured for some economies and distorted by profit shifting (e.g. Ireland, Luxembourg) or resource rents (e.g. Guyana, Equatorial Guinea). This creates errors-in-variables, which would tend to attenuate β.
• What was not checked:
  • population weighting (the estimand is deliberately unweighted);
  • alternative functional forms beyond a quadratic in log;
  • spatial or regional dependence of residuals: HC SEs assume independent countries, and regional clustering could widen the CI;
  • the time-varying vintage of income classifications. Table 1 uses the current World Bank income level from metadata, not the 2023 classification;
  • why the 2023 slope is the lowest in the window.
• Main threats:
  • Confounding: education, health systems, disease environment and institutions all move with income. β is an association, not the effect of raising income.
  • Ecological interpretation: the result says nothing about individuals.
  • A single cross-section: it is sensitive to year-specific shocks such as COVID-19 mortality in 2020–2022, which is partly why the most recent year was required.