Correlates of War, participant-engagement series (A_COW_P), 1816–2007 · epoch-folding χ² contrast using a fixed window T = 35.1 years · the window is specified, not an estimated cycle length · H2 concerns the structure of engagement; H1 is tracked separately
Each bar is the mean deviation from trend for years assigned to that phase bin in a fixed 35.1-year folding window (10 bins, about 3.5 years each). The chart shows unnormalized phase-bin means. Panel 2 shows M2, the difference between the epochs' variance-normalized χ² statistics; bar heights alone do not determine that contrast.
| bin | Epoch 1 mean | Epoch 1 n | Epoch 2 mean | Epoch 2 n |
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k is the number of years with the highest values in the detrended Epoch 2 series removed before recomputing M2 (χ² of Epoch 2 minus χ² of Epoch 1). Blue: observed M2. Orange: removal of k random years (mean ± 1 SD, 500 draws). This is a partial comparator: random removal does not reproduce the variance reduction caused by removing the highest values. k=0 is the before-removal result on the detrended series; k=5 is the primary robustness decision point specified before the run. The orange band is not a confidence interval.
The contrast remains significant at k=5 (M2=24.91, p=0.0085) and k=10 (M2=30.23, p=0.0020). The higher normalized statistic does not indicate a stronger underlying phase pattern: the unnormalized discrepancy decreases while the normalizing variance falls faster. Separately, excluding all twelve world-war years gives M2=14.65, p=0.096, which does not meet the 0.05 significance threshold.
| k | n (Epoch 2) | M2 observed | p | random-removal mean | random-removal SD |
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