Bootstrap inference, information theory, non-parametric tests, power-law estimation, Bayesian posteriors, and health equity measures — all from 22,110 trials across 54 nations.
Non-parametric bootstrap resampling (10,000 iterations) provides distribution-free confidence intervals. The Gini CI excludes 0.80, confirming extreme inequality is not a sampling artefact. The narrow HHI interval confirms high market-like concentration.
Unlike Gini, Theil indices are additively decomposable into between-group and within-group components. Theil T is sensitive to changes at the top of the distribution (Egypt), while Theil L is sensitive to changes at the bottom (zero-trial countries). Both confirm extreme inequality from different perspectives.
The Atkinson index incorporates a normative parameter ε reflecting society's aversion to inequality. At ε=2.0 (high aversion to inequality), 100% of total trials would need to be redistributed to achieve equality. This means that 22,110 of 22,110 trials are "wasted" from an equity perspective.
KL divergence measures how many extra bits are needed to encode the observed distribution using an optimal code for the uniform distribution — 2.9 bits of "surprise." Jensen-Shannon divergence (symmetric, bounded) of 0.361 between trial distribution and population distribution confirms that trials are not allocated proportionally to population need.
Spearman ρ = 0.7235 indicates strong monotonic association between population and trials. But ρ = -0.0112 for per-capita rates suggests no relationship — large countries do not necessarily have higher per-capita trial access.
Non-parametric comparison of the top-10 versus bottom-44 African nations confirms that the trial volume difference is statistically significant. The distribution is bimodal: a small cluster of research-active nations and a large cluster of research-desert nations.
The distribution is consistent with a log-normal model, as expected for hierarchical socioeconomic phenomena.
A power-law exponent α = 1.40 estimated via maximum likelihood (Clauset et al., 2009) indicates an extremely steep distribution where the largest values dominate — steeper than most natural phenomena (earthquakes α ~ 2.0, city sizes α ~ 2.1).
Using a Beta-Binomial model with uniform prior, Africa's posterior probability of hosting a randomly selected global trial is 5.5%. The 95% credible interval [5.4%, 5.5%] is extremely narrow, confirming high precision in this estimate of Africa's marginal global share.
Jackknife and bootstrap standard errors agree closely, providing cross-validation of uncertainty estimates. The Gini is estimated with high precision (SE ~ 0.1097), meaning even small changes to the distribution would not alter the conclusion of extreme inequality.
A permutation test confirms that the trial volume difference between North African nations and the rest of the continent is statistically significant (p < 0.05). This tests whether the North-South African divide is a genuine structural pattern or could arise by chance.
Africa's trial growth is better described by a quadratic (accelerating) model. The slope of 2729 additional trials per epoch indicates strong absolute growth, but the US slope of 7838 means the absolute gap widens each epoch.
The concentration index of 0.741, borrowed from health economics (Wagstaff et al.), measures whether trials are concentrated among high-population countries. A positive CI means that large-population nations host disproportionately more trials, but this does not imply equitable per-capita access.
| # | Method | Family | Key Result |
|---|---|---|---|
| 1 | Bootstrap CI | Resampling | Gini 0.809 [0.606, 0.897] |
| 2 | Theil T / L | Generalised Entropy | T=2.0283, L=2.069 |
| 3 | Atkinson | Welfare Economics | A(2.0)=1.0 |
| 4 | KL Divergence | Information Theory | 2.93 bits from uniform |
| 5 | JS Divergence | Information Theory | 0.361 bits from population |
| 6 | Spearman ρ | Rank Correlation | ρ=0.7235 |
| 7 | Kendall τ | Rank Correlation | τ=0.5332 |
| 8 | Mann-Whitney U | Non-parametric Test | z=-4.899 |
| 9 | KS Test | Distribution Fit | D=0.0623 vs crit=0.1868 |
| 10 | Power-Law α | MLE | α=1.402 |
| 11 | Bayesian Posterior | Beta-Binomial | P=0.0546 |
| 12 | Jackknife SE | Resampling | Gini SE=0.1097 |
| 13 | Permutation Test | Exact Test | p=0.0372 |
| 14 | Trend Decomposition | Time Series | R²=0.8892 |
| 15 | Concentration Index | Health Equity | CI=0.7405 |