# WP35 — Measurement report (Phase 3)

All numbers come from `results/empirical/measurement_diagnostics.json`, `measurement_diagnostics_modern.json`,
`measurement_irt_item_parameters.csv`, `power_h2a_*.csv`, produced by `python -m wp35.measurement.build_indices`
(2026-11-01). **No outcome variable was read by that module.** Definitions follow `research/preregistration.md` §3.

## 1. Seshat (polity level)

### 1.1 Coordination demand D
Four logged scale variables (polity population, territory, largest-settlement population; settlement hierarchy levels
unlogged). First principal component of their pairwise-complete correlation matrix explains **82.1%** of variance with
near-equal loadings (log pop 0.26, log territory 0.24, log capital 0.26, settlement hierarchy 0.24). Score = loading-
weighted mean of available standardized variables, requiring ≥ 2 of 4; **442 polities** have `D_pc1`; 299 have
population alone (`log_pop`). Range bounds (`value_from`/`value_to`) propagate to `D_pc1_lo/hi`: the median half-width
is 0.017 SD, the 90th percentile 0.093 SD, so range uncertainty in the scale variables is small relative to
between-polity variation (the larger uncertainty is *which* polities have any estimate at all).

### 1.2 Coordination capacity K
Four blocks of presence indicators, each the share coded present among coded items (≥ 50% coded required):
administration (5 items incl. rank-scaled administrative levels), standardized rules and money (10), communication
infrastructure (7), record-keeping (9). `K_mean` = mean of blocks with ≥ 3 blocks: **344 polities**. Inter-block
correlations 0.75–0.79; first PC of the blocks explains 82.4% (loadings 0.25 each), and `K_pc1` correlates 0.999 with
`K_mean`. Cronbach α of the four blocks by macro-region ranges 0.83 (Europe) to 0.97 (South Asia); mean off-diagonal
block correlation by era 0.67 (1001–1500) to 0.81 (pre-1000 BCE) — the four capacity blocks hang together
in every region and era.
Latent alternative: 2PL IRT (MAP, log-normal prior on discrimination, EM converged in 135 iterations) on the 30 binary
items (disputed 0.5 → missing; administrative levels excluded) gives `K_irt` for **579 polities** (≥ 3 items). It
correlates **0.96** with `K_mean`. Item difficulties order sensibly (tokens, non-written records, roads, couriers easy;
paper currency, general postal service, examination system, merit promotion hard; `article` is nearly universal and
uninformative). Five record-keeping items (written record, script, lists/tables, calendar, practical literature) hit the
discrimination ceiling (8): they are coded almost jointly in Seshat, i.e. locally dependent; the latent score is
therefore dominated by the "writing cluster", which is why we keep the transparent block mean as primary. Posterior SD of
`K_irt`: median 0.34, 90th percentile 0.61 (on a unit-variance scale) — individual-polity capacity scores carry
substantial uncertainty; this is carried into the econometrics through the IRT variant and the bound runs.
Cross-region invariance: refitting the IRT separately on the Old World (n = 532) and on the Americas + Southeast
Asia/Oceania (n = 94) gives item difficulties correlated **0.88** across the two fits (mean absolute difference 1.16
logits, inflated by boundary estimates on the small New-World sample). Comparability is acceptable for the ordering
of items but the New-World sample is too small for a strict invariance test; leave-region-out results (H5) are the
operational safeguard.
Leave-one-block-out: dropping any block leaves `K` correlated ≥ 0.98 with the full index and `P` correlated ≥ 0.95
with `P_main`.

### 1.3 Coordination pressure P and the collinearity problem
`P_main = z(D_pc1) − z(K_mean)` is defined for **317 polities** and, in pre-crisis form, for **127 of the 166 CrisisDB
records**. As anticipated from Turchin et al. (2018) and Shin et al. (2020), scale and capacity are strongly
collinear: corr(`D_pc1`, `K_mean`) = **0.84** (0.80 with `K_irt`). Consequently `P_main` retains only **16.6%** of the
variance of z(D)+z(K); it correlates 0.39 with D and −0.18 with K, and −0.04 with the Turchin-style single complexity
axis (`complexity_pc1`, first PC of 57 social-complexity indicators, 49.7% of variance). The mismatch index is thus
nearly orthogonal to overall complexity, which is exactly what makes the H2 test informative *if* P carries signal, and
what makes it fragile: P is a residual, so measurement error in either D or K leaks into it. The pre-registered
baselines (D-only, K-only, complexity-PC1-only) address this directly.

### 1.4 A-priori power (pre-registration §5.8; random outcomes, actual P distribution, n = 127)
| Outcome | β (per SD of P, logit) | power |
|---|---|---|
| Y1b binary (base rate 0.77) | 0.4 / 0.5 / 0.6 / 0.8 | 0.47 / 0.63 / 0.79 / 0.95 |
| Y1 fractional (14 items, base 0.29) | 0.10 / 0.15 / 0.20 | 0.49 / 0.81 / 0.96 |
**Decision (pre-registered rule):** the binary severe-outcome test Y1b is *underpowered a priori* (MDE ≈ 0.6 > 0.5);
the fractional severity Y1 is adequately powered for effects of ≈ 0.15 log-odds per SD. Y1 remains the primary
Seshat H2 outcome; Y1b is reported as secondary.

## 2. Modern panel
`K_modern` = mean of six standardized V-Dem items (≥ 4 coded): rigorous/impartial administration, meritocratic
appointment, transparent laws, fiscal capacity source, territorial control, EMB capacity; PC1 explains 60.5% with
loadings 0.13–0.19. `D_modern` = log Maddison population (1820+); `P_modern` defined for **15,697 country-years**.
corr(D, K) = 0.06 (no collinearity problem in the modern panel). **Caveat fixed before analysis:** corr(`K_modern`,
`v2x_polyarchy`) = **0.75** — the V-Dem capacity items are strongly associated with electoral democracy even though no
democracy index enters K; all modern specifications therefore control for polyarchy as pre-registered, and the K-only
baseline is reported.

## 3. Ethnographic Atlas
`log_community_size` from EA031 category midpoints (611 societies), `log_population` from continuous EA202 (953),
`jurisdictional_hierarchy_beyond` = EA033 − 1 (0–4 levels; 1,155). No outcome; used for H1 only, with Glottolog
family (173 families) and region controls.

## 4. What the measurement model cannot do
- Seshat K items are presence codes: they measure *existence*, not reach or quality, of institutions.
- Most facts are undated, so "pre-crisis" K is mostly whole-polity K (see pre-registration §3.6).
- No direct indicator of division of labour or dispersion in Polaris; D is scale only.
- IRT local dependence among writing items; invariance only coarsely checked.