============================================================================== REGRESSION TO THE MEAN: HOW MUCH IMPROVEMENT COMES FROM NOTHING Science Journaling Club, Volume 2 Issue 2, Winter 2026 ============================================================================== master seed : 20260211 generator : numpy PCG64 via SeedSequence.spawn, one stream per cell python : 3.12.3 numpy : 2.4.2 individuals/rep : 100,000 replicates/cell : 60 individuals/cell : 6,000,000 grid : 6 selection thresholds x 7 noise ratios = 42 cells Model: T ~ N(0, r), e ~ N(0, 1-r), X1 = T + e1, X2 = T + e2. Total variance of one measurement is 1, so every number below is in units of the population SD of a single measurement. r = 1/(1+lam^2) is the correlation between the two measurements. Selection takes the bottom p on X1. NO TREATMENT IS APPLIED in Part A. All standard errors are computed from the spread across the 25 independent replicates of each cell (batched means), not from an assumed formula. ============================================================================== PART A.1 VALIDATION OF THE SHRINKAGE LAW E[X2|sel] = r * E[X1|sel] ============================================================================== For a bivariate normal with correlation r, the mean of a group selected on the first measurement is pulled toward the population mean by exactly r, whatever the selection rule, as long as it uses X1 alone. p lam r (exact) shrink sim SE difference z corr(X1,X2) ------------------------------------------------------------------------------ 0.50 0.00 1.000000 1.000000 0.000000 +0.000000 exact 1.000000 0.50 0.25 0.941176 0.941037 0.000243 -0.000139 -0.57 0.941167 0.50 0.50 0.800000 0.799888 0.000371 -0.000112 -0.30 0.800246 0.50 0.75 0.640000 0.639357 0.000486 -0.000643 -1.32 0.640097 0.50 1.00 0.500000 0.500216 0.000651 +0.000216 0.33 0.499847 0.50 1.50 0.307692 0.306458 0.000764 -0.001234 -1.62 0.307726 0.50 2.00 0.200000 0.201599 0.000631 +0.001599 2.53 0.200590 .............................................................................. 0.25 0.00 1.000000 1.000000 0.000000 +0.000000 exact 1.000000 0.25 0.25 0.941176 0.941216 0.000164 +0.000039 0.24 0.941141 0.25 0.50 0.800000 0.800015 0.000421 +0.000015 0.03 0.800110 0.25 0.75 0.640000 0.639732 0.000484 -0.000268 -0.55 0.639910 0.25 1.00 0.500000 0.500691 0.000524 +0.000691 1.32 0.500151 0.25 1.50 0.307692 0.308148 0.000645 +0.000456 0.71 0.307547 0.25 2.00 0.200000 0.199595 0.000687 -0.000405 -0.59 0.200392 .............................................................................. 0.10 0.00 1.000000 1.000000 0.000000 +0.000000 exact 1.000000 0.10 0.25 0.941176 0.940935 0.000248 -0.000242 -0.98 0.941188 0.10 0.50 0.800000 0.800450 0.000417 +0.000450 1.08 0.799803 0.10 0.75 0.640000 0.639593 0.000542 -0.000407 -0.75 0.639829 0.10 1.00 0.500000 0.500825 0.000694 +0.000825 1.19 0.499742 0.10 1.50 0.307692 0.307786 0.000759 +0.000094 0.12 0.307373 0.10 2.00 0.200000 0.200895 0.000676 +0.000895 1.32 0.199821 .............................................................................. 0.05 0.00 1.000000 1.000000 0.000000 +0.000000 exact 1.000000 0.05 0.25 0.941176 0.941419 0.000256 +0.000243 0.95 0.941108 0.05 0.50 0.800000 0.799598 0.000518 -0.000402 -0.78 0.800102 0.05 0.75 0.640000 0.640655 0.000891 +0.000655 0.73 0.639949 0.05 1.00 0.500000 0.499248 0.000843 -0.000752 -0.89 0.499867 0.05 1.50 0.307692 0.306997 0.000859 -0.000695 -0.81 0.307719 0.05 2.00 0.200000 0.200563 0.000855 +0.000563 0.66 0.199970 .............................................................................. 0.02 0.00 1.000000 1.000000 0.000000 +0.000000 exact 1.000000 0.02 0.25 0.941176 0.940739 0.000354 -0.000438 -1.23 0.941216 0.02 0.50 0.800000 0.800017 0.000653 +0.000017 0.03 0.800081 0.02 0.75 0.640000 0.639903 0.000907 -0.000097 -0.11 0.639972 0.02 1.00 0.500000 0.500582 0.000994 +0.000582 0.59 0.499913 0.02 1.50 0.307692 0.308336 0.001096 +0.000644 0.59 0.307417 0.02 2.00 0.200000 0.198993 0.001073 -0.001007 -0.94 0.199996 .............................................................................. 0.01 0.00 1.000000 1.000000 0.000000 +0.000000 exact 1.000000 0.01 0.25 0.941176 0.940421 0.000616 -0.000756 -1.23 0.941149 0.01 0.50 0.800000 0.801215 0.001046 +0.001215 1.16 0.799937 0.01 0.75 0.640000 0.639805 0.001311 -0.000195 -0.15 0.640035 0.01 1.00 0.500000 0.500375 0.001439 +0.000375 0.26 0.500512 0.01 1.50 0.307692 0.305843 0.001534 -0.001850 -1.21 0.307184 0.01 2.00 0.200000 0.199447 0.001489 -0.000553 -0.37 0.200717 .............................................................................. Largest |z| on the shrinkage factor across all 42 cells: 2.53 worst cell p=0.5, lam=2.0: simulated 0.201599 vs exact 0.200000, SE 0.000631, z = +2.53 Bonferroni threshold for 42 cells at family-wise 5%, Student t with 59 df: 3.41 ============================================================================== PART A.2 THE SPURIOUS IMPROVEMENT, NO TREATMENT AT ALL ============================================================================== Delta = mean(X2 - X1) over the selected group, in population SDs. Analytic value: Delta = (1 - r) * phi(z_p) / p. p lam r pre-mean post-mean Delta sim SE Delta exact diff z d_within -------------------------------------------------------------------------------------------------------------------- 0.50 0.00 1.0000 -0.7983 -0.7983 0.00000 0.00000 0.00000 +0.00000 exact 0.0000 0.50 0.25 0.9412 -0.7974 -0.7504 0.04702 0.00020 0.04693 +0.00008 0.42 0.0780 0.50 0.50 0.8000 -0.7982 -0.6385 0.15973 0.00031 0.15958 +0.00015 0.48 0.2648 0.50 0.75 0.6400 -0.7981 -0.5103 0.28784 0.00040 0.28724 +0.00060 1.49 0.4775 0.50 1.00 0.5000 -0.7975 -0.3989 0.39856 0.00056 0.39894 -0.00039 -0.69 0.6606 0.50 1.50 0.3077 -0.7983 -0.2447 0.55367 0.00069 0.55238 +0.00129 1.86 0.9177 0.50 2.00 0.2000 -0.7979 -0.1609 0.63708 0.00057 0.63831 -0.00123 -2.14 1.0571 .............................................................................. 0.25 0.00 1.0000 -1.2703 -1.2703 0.00000 0.00000 0.00000 +0.00000 exact 0.0000 0.25 0.25 0.9412 -1.2705 -1.1958 0.07468 0.00021 0.07477 -0.00009 -0.41 0.1520 0.25 0.50 0.8000 -1.2715 -1.0172 0.25429 0.00057 0.25422 +0.00007 0.12 0.5175 0.25 0.75 0.6400 -1.2716 -0.8135 0.45813 0.00064 0.45760 +0.00053 0.83 0.9313 0.25 1.00 0.5000 -1.2711 -0.6364 0.63465 0.00075 0.63555 -0.00090 -1.21 1.2920 0.25 1.50 0.3077 -1.2713 -0.3917 0.87954 0.00090 0.88000 -0.00046 -0.51 1.7886 0.25 2.00 0.2000 -1.2708 -0.2536 1.01714 0.00101 1.01689 +0.00025 0.25 2.0673 .............................................................................. 0.10 0.00 1.0000 -1.7556 -1.7556 0.00000 0.00000 0.00000 +0.00000 exact 0.0000 0.10 0.25 0.9412 -1.7556 -1.6519 0.10369 0.00043 0.10323 +0.00046 1.06 0.2518 0.10 0.50 0.8000 -1.7539 -1.4039 0.34999 0.00075 0.35100 -0.00101 -1.35 0.8513 0.10 0.75 0.6400 -1.7559 -1.1231 0.63285 0.00100 0.63179 +0.00106 1.06 1.5428 0.10 1.00 0.5000 -1.7549 -0.8789 0.87598 0.00131 0.87749 -0.00151 -1.15 2.1263 0.10 1.50 0.3077 -1.7553 -0.5403 1.21506 0.00138 1.21499 +0.00007 0.05 2.9531 0.10 2.00 0.2000 -1.7548 -0.3525 1.40228 0.00132 1.40399 -0.00171 -1.29 3.4055 .............................................................................. 0.05 0.00 1.0000 -2.0609 -2.0609 0.00000 0.00000 0.00000 +0.00000 exact 0.0000 0.05 0.25 0.9412 -2.0615 -1.9407 0.12077 0.00054 0.12134 -0.00057 -1.06 0.3263 0.05 0.50 0.8000 -2.0627 -1.6493 0.41337 0.00109 0.41254 +0.00083 0.77 1.1121 0.05 0.75 0.6400 -2.0630 -1.3217 0.74131 0.00185 0.74258 -0.00126 -0.68 1.9850 0.05 1.00 0.5000 -2.0631 -1.0300 1.03310 0.00183 1.03136 +0.00175 0.95 2.7736 0.05 1.50 0.3077 -2.0634 -0.6335 1.42995 0.00187 1.42803 +0.00192 1.03 3.8377 0.05 2.00 0.2000 -2.0619 -0.4135 1.64836 0.00194 1.65017 -0.00181 -0.93 4.4336 .............................................................................. 0.02 0.00 1.0000 -2.4229 -2.4229 0.00000 0.00000 0.00000 +0.00000 exact 0.0000 0.02 0.25 0.9412 -2.4189 -2.2756 0.14335 0.00086 0.14241 +0.00094 1.09 0.4300 0.02 0.50 0.8000 -2.4215 -1.9373 0.48426 0.00158 0.48418 +0.00008 0.05 1.4555 0.02 0.75 0.6400 -2.4212 -1.5494 0.87188 0.00221 0.87153 +0.00035 0.16 2.6242 0.02 1.00 0.5000 -2.4182 -1.2105 1.20770 0.00255 1.21045 -0.00275 -1.08 3.6293 0.02 1.50 0.3077 -2.4205 -0.7463 1.67413 0.00279 1.67601 -0.00189 -0.67 5.0147 0.02 2.00 0.2000 -2.4214 -0.4818 1.93955 0.00303 1.93673 +0.00282 0.93 5.7811 .............................................................................. 0.01 0.00 1.0000 -2.6624 -2.6624 0.00000 0.00000 0.00000 +0.00000 exact 0.0000 0.01 0.25 0.9412 -2.6642 -2.5055 0.15874 0.00166 0.15678 +0.00196 1.19 0.5085 0.01 0.50 0.8000 -2.6662 -2.1362 0.53000 0.00280 0.53304 -0.00305 -1.09 1.7072 0.01 0.75 0.6400 -2.6646 -1.7048 0.95978 0.00356 0.95948 +0.00030 0.08 3.0880 0.01 1.00 0.5000 -2.6628 -1.3324 1.33044 0.00402 1.33261 -0.00217 -0.54 4.2966 0.01 1.50 0.3077 -2.6689 -0.8163 1.85265 0.00414 1.84515 +0.00750 1.81 5.9384 0.01 2.00 0.2000 -2.6669 -0.5319 2.13499 0.00418 2.13217 +0.00282 0.68 6.8722 .............................................................................. d_within is the spurious improvement divided by the SD of the selected group's own first measurement, which is the effect size a single-arm paper would usually report. Largest |z| on Delta across all 42 cells: 2.14 worst cell p=0.5, lam=2.0: simulated 0.63708 vs exact 0.63831, SE 0.00057, z = -2.14 Bonferroni threshold for 42 cells at family-wise 5%: 3.41 Pooled goodness of fit over the 36 cells that carry a standard error: sum of z^2 = 36.20 on 36 degrees of freedom, p = 0.459 mean z = +0.043, SD of z = 1.016 (a correct implementation should give sum z^2 near the degrees of freedom, mean z near 0 and SD of z near 1) RE-RUN RULE, fixed before the run: any cell with |z| > 3.0 on Delta is simulated again from a completely independent stream. A real disagreement repeats; a fluctuation does not. no cell exceeded |z| = 3.0. Nothing to re-run. ============================================================================== PART A.3 THE ZERO-NOISE CONTROL (lam = 0, r = 1) ============================================================================== With no measurement noise the second measurement equals the first exactly, so no spurious improvement can appear. Every p is checked. p r Delta sim SE Delta exact verdict ------------------------------------------------------------------------------ 0.50 1.0 0.0000000000 0.000000 0.0000000000 exactly 0 0.25 1.0 0.0000000000 0.000000 0.0000000000 exactly 0 0.10 1.0 0.0000000000 0.000000 0.0000000000 exactly 0 0.05 1.0 0.0000000000 0.000000 0.0000000000 exactly 0 0.02 1.0 0.0000000000 0.000000 0.0000000000 exactly 0 0.01 1.0 0.0000000000 0.000000 0.0000000000 exactly 0 zero-noise control: PASSED, all Delta identically zero ============================================================================== PART A.4 THE HEADLINE CELL IN FULL ============================================================================== Selection: bottom 10% on the first measurement. Noise-to-signal ratio lam = 1.00, so reliability r = 0.5000. Individuals simulated: 6,000,000 in 60 replicates. Selected per replicate: 10,000 total selected: 600,000 mean first measurement, selected : -1.7549 +/- 0.0008 (exact -1.7550) mean second measurement, selected : -0.8789 +/- 0.0012 (exact -0.8775) apparent improvement Delta : +0.8760 +/- 0.0013 (exact +0.8775) shrinkage m2/m1 : 0.500825 +/- 0.000694 (exact r 0.500000) SD of selected group on X1 : 0.4120 (exact 0.4113) improvement / that SD : 2.1263 percent of the selected group's gap to the mean closed by doing nothing: 50.0% (exactly 1 - r) ============================================================================== PART B A REAL TREATMENT EFFECT, TWO DESIGNS, SAME SIMULATED PEOPLE ============================================================================== tau is the true effect in population SDs, added to the second measurement of anyone treated. Single-arm treats everyone selected and reports mean(X2 - X1). Randomised splits the selected group 50/50 and reports mean(X2 | treated) - mean(X2 | control). -------------------------------------------------------------------------------------------------------------- SCENARIO: tight cut, clean measure p = 0.05, lam = 0.50, r = 0.8000, regression bias = 0.4125 SD -------------------------------------------------------------------------------------------------------------- tau single-arm SE expected bias overstate RCT est SE z(RCT) DiD est sig% 0.00 0.41187 0.00097 0.41254 0.4125 inf 0.00002 0.00247 0.01 0.00029 100% 0.10 0.51460 0.00100 0.51254 0.4125 5.15x 0.09888 0.00218 -0.51 0.09929 100% 0.20 0.61445 0.00111 0.61254 0.4125 3.07x 0.20162 0.00265 0.61 0.19738 100% 0.30 0.71178 0.00092 0.71254 0.4125 2.37x 0.29854 0.00223 -0.66 0.30124 100% 0.50 0.91139 0.00088 0.91254 0.4125 1.82x 0.50076 0.00250 0.31 0.50073 100% 1.00 1.41275 0.00101 1.41254 0.4125 1.41x 1.00023 0.00251 0.09 0.99845 100% -------------------------------------------------------------------------------------------------------------- SCENARIO: headline cell p = 0.10, lam = 1.00, r = 0.5000, regression bias = 0.8775 SD -------------------------------------------------------------------------------------------------------------- tau single-arm SE expected bias overstate RCT est SE z(RCT) DiD est sig% 0.00 0.87676 0.00130 0.87749 0.8775 inf -0.00057 0.00230 -0.25 -0.00133 100% 0.10 0.97747 0.00096 0.97749 0.8775 9.77x 0.10051 0.00230 0.22 0.09838 100% 0.20 1.07694 0.00103 1.07749 0.8775 5.38x 0.19819 0.00210 -0.87 0.19821 100% 0.30 1.17708 0.00139 1.17749 0.8775 3.92x 0.29886 0.00214 -0.53 0.29838 100% 0.50 1.37986 0.00117 1.37749 0.8775 2.76x 0.50290 0.00235 1.23 0.50244 100% 1.00 1.87742 0.00103 1.87749 0.8775 1.88x 1.00346 0.00232 1.49 1.00483 100% -------------------------------------------------------------------------------------------------------------- SCENARIO: loose cut, noisy measure p = 0.25, lam = 1.50, r = 0.3077, regression bias = 0.8800 SD -------------------------------------------------------------------------------------------------------------- tau single-arm SE expected bias overstate RCT est SE z(RCT) DiD est sig% 0.00 0.87965 0.00081 0.88000 0.8800 inf 0.00019 0.00168 0.11 0.00070 100% 0.10 0.98131 0.00080 0.98000 0.8800 9.81x 0.09939 0.00150 -0.41 0.09983 100% 0.20 1.07879 0.00097 1.08000 0.8800 5.39x 0.19991 0.00164 -0.05 0.19972 100% 0.30 1.18008 0.00095 1.18000 0.8800 3.93x 0.30053 0.00149 0.35 0.30087 100% 0.50 1.38014 0.00080 1.38000 0.8800 2.76x 0.50042 0.00172 0.24 0.49901 100% 1.00 1.88177 0.00088 1.88000 0.8800 1.88x 0.99690 0.00176 -1.77 0.99696 100% Largest |z| on the randomised estimate against the true tau, across all 18 trial cells: 1.77 worst: p=0.25, lam=1.5, tau=1.0: RCT 0.99690 vs true 1.00000, SE 0.00176, z = -1.77 sig% is the share of replicates in which the single-arm paired t statistic exceeded 1.96 in absolute value. At tau = 0 it is the false-positive rate of a before-and-after design applied to a selected group, and it is not 5%. ============================================================================== PART B.2 THE SHARE OF THE APPARENT EFFECT THAT IS ARTEFACT ============================================================================== scenario r tau single-arm artefact share inflation ------------------------------------------------------------------------------------ tight cut, clean measure 0.8000 0.10 0.51460 80.2% 5.15x tight cut, clean measure 0.8000 0.20 0.61445 67.1% 3.07x tight cut, clean measure 0.8000 0.30 0.71178 58.0% 2.37x tight cut, clean measure 0.8000 0.50 0.91139 45.3% 1.82x tight cut, clean measure 0.8000 1.00 1.41275 29.2% 1.41x .............................................................................. headline cell 0.5000 0.10 0.97747 89.8% 9.77x headline cell 0.5000 0.20 1.07694 81.5% 5.38x headline cell 0.5000 0.30 1.17708 74.5% 3.92x headline cell 0.5000 0.50 1.37986 63.6% 2.76x headline cell 0.5000 1.00 1.87742 46.7% 1.88x .............................................................................. loose cut, noisy measure 0.3077 0.10 0.98131 89.7% 9.81x loose cut, noisy measure 0.3077 0.20 1.07879 81.6% 5.39x loose cut, noisy measure 0.3077 0.30 1.18008 74.6% 3.93x loose cut, noisy measure 0.3077 0.50 1.38014 63.8% 2.76x loose cut, noisy measure 0.3077 1.00 1.88177 46.8% 1.88x .............................................................................. ============================================================================== PART C THE CLUB'S OWN HOMEWORK, SIMULATED ============================================================================== A quiz taken twice by a class of 30. This is a SIMULATION with class-sized numbers, not our actual marks; the club has no such dataset and did not collect one. Reliability of a short classroom quiz is taken as r = 0.70, which is a typical published value for a short classroom test. The bottom 8 of 30 are picked out for 'extra help' that in this model does nothing. classes simulated : 200,000 class size : 30, bottom 8 selected (p = 0.2667) quiz reliability r : 0.70 mean gain of the bottom 8 : +0.3597 SD +/- 0.00059 continuous-population exact : +0.3697 SD (a finite class of 30 differs slightly; order statistics, not a fixed cut) mean gain of the whole class : +0.000748 SD +/- 0.000316 mean gain of the other 22 : -0.1298 SD +/- 0.00036 mean gain of the TOP 8 : -0.3578 SD +/- 0.00059 (the same effect, running downhill) on a 100-point scale, mean 68 and SD 12: bottom 8 first attempt : 53.7 marks bottom 8 second attempt : 58.0 marks apparent improvement : +4.3 marks whole class moved : +0.01 marks the other 22 moved : -1.56 marks the top 8 moved : -4.29 marks share of classes in which the bottom 8 improved : 91.4% share of classes in which the top 8 got worse : 91.3% Both halves of that are the same arithmetic. Help the bottom 8 and they improve. Leave the top 8 alone and they decline. Neither group was touched. ============================================================================== PART E SET AGAINST A PUBLISHED NUMBER ============================================================================== Krogsboll, Hrobjartsson and Gotzsche (2009) pooled 37 three-armed trials in 2,900 patients across 8 clinical conditions, in which one arm received no treatment at all. Change from baseline in those untreated arms had a pooled standardised mean difference of -0.24. Their active-treatment arms gave -1.01 and their placebo arms -0.44, so the untreated arms accounted for about 24% of the movement seen under active treatment. Their -0.24 is real patients and mixes several causes: natural history of the illness, regression to the mean, and anything else that changes between two visits. Our model produces only the regression part. So the fair question is: what reliability would a trial's entry measurement need for regression alone to account for the whole -0.24? entry cut p phi(z_p)/p r needed lam needed our Delta at r=0.5 our Delta at r=0.8 -------------------------------------------------------------------------------------------- 0.50 0.7979 0.6992 0.6559 0.3989 0.1596 0.25 1.2711 0.8112 0.4825 0.6356 0.2542 0.10 1.7550 0.8632 0.3980 0.8775 0.3510 0.05 2.0627 0.8836 0.3629 1.0314 0.4125 0.02 2.4209 0.9009 0.3317 1.2105 0.4842 0.01 2.6652 0.9100 0.3146 1.3326 0.5330 Read the table this way. If a trial enrols the worst 10% on a single entry measurement, regression alone matches the published -0.24 as soon as that measurement's test-retest reliability is 0.863 or worse. Reliabilities below that are ordinary for single-occasion clinical and behavioural measurements. The published number therefore sets no lower bound on how much of it is natural history: regression can supply all of it on plausible numbers, and our model cannot tell you how much it actually did supply in any particular trial. What it can say is that a single-arm design has no way of separating the two, and a randomised design does not need to. The opposite reading matters as much. If entry is loose, say the worst half, then phi(z_p)/p is only 0.7979 and regression can reach -0.24 only if reliability falls to 0.699. Tight entry criteria are what make the artefact large. That is a design choice, not a property of the disease. ============================================================================== PART D CONVERGENCE OF THE HEADLINE CELL ============================================================================== The running estimate of Delta as replicates accumulate, for the headline cell p = 0.10, lam = 1.00. Each replicate adds 10,000 selected individuals. selected so far running Delta exact diff ------------------------------------------------------------------------------ 10,000 0.851806 0.877492 -0.025686 20,000 0.861152 0.877492 -0.016339 30,000 0.867103 0.877492 -0.010389 40,000 0.871690 0.877492 -0.005802 50,000 0.876656 0.877492 -0.000836 60,000 0.874730 0.877492 -0.002762 70,000 0.876963 0.877492 -0.000529 80,000 0.877236 0.877492 -0.000256 90,000 0.876753 0.877492 -0.000739 100,000 0.876408 0.877492 -0.001084 110,000 0.876766 0.877492 -0.000726 120,000 0.877610 0.877492 +0.000118 130,000 0.877593 0.877492 +0.000102 140,000 0.878246 0.877492 +0.000755 150,000 0.878967 0.877492 +0.001475 160,000 0.879077 0.877492 +0.001585 170,000 0.878908 0.877492 +0.001416 180,000 0.878853 0.877492 +0.001361 190,000 0.879110 0.877492 +0.001618 200,000 0.879196 0.877492 +0.001705 210,000 0.878882 0.877492 +0.001390 220,000 0.878918 0.877492 +0.001427 230,000 0.878805 0.877492 +0.001314 240,000 0.878931 0.877492 +0.001439 250,000 0.878538 0.877492 +0.001047 260,000 0.878710 0.877492 +0.001218 270,000 0.878511 0.877492 +0.001020 280,000 0.878787 0.877492 +0.001296 290,000 0.878603 0.877492 +0.001111 300,000 0.878332 0.877492 +0.000840 310,000 0.878515 0.877492 +0.001023 320,000 0.878079 0.877492 +0.000587 330,000 0.877586 0.877492 +0.000094 340,000 0.878209 0.877492 +0.000718 350,000 0.878297 0.877492 +0.000805 360,000 0.878065 0.877492 +0.000573 370,000 0.877749 0.877492 +0.000257 380,000 0.877593 0.877492 +0.000102 390,000 0.877426 0.877492 -0.000066 400,000 0.877670 0.877492 +0.000179 410,000 0.877480 0.877492 -0.000012 420,000 0.877466 0.877492 -0.000025 430,000 0.877624 0.877492 +0.000132 440,000 0.877353 0.877492 -0.000139 450,000 0.877200 0.877492 -0.000291 460,000 0.877162 0.877492 -0.000330 470,000 0.877062 0.877492 -0.000430 480,000 0.876787 0.877492 -0.000704 490,000 0.876591 0.877492 -0.000901 500,000 0.876380 0.877492 -0.001112 510,000 0.876347 0.877492 -0.001145 520,000 0.876393 0.877492 -0.001099 530,000 0.876713 0.877492 -0.000778 540,000 0.876727 0.877492 -0.000764 550,000 0.876883 0.877492 -0.000608 560,000 0.877216 0.877492 -0.000276 570,000 0.877208 0.877492 -0.000284 580,000 0.876915 0.877492 -0.000576 590,000 0.876944 0.877492 -0.000548 600,000 0.876885 0.877492 -0.000607 final running estimate : 0.876885 exact value : 0.877492 replicate-SE : 0.001205 final z : -0.50 ============================================================================== FIGURE DATA ============================================================================== FIG1 scatter of first against second measurement, headline cell 600 individuals drawn at random, x1 then x2, three decimals selection cut at z_p = -1.2816 -0.279 0.651 0.556 1.638 -0.967 0.259 -0.771 0.158 -0.605 0.106 0.295 0.092 -1.247 -0.670 0.276 -0.629 2.210 0.168 -0.150 -3.230 -0.100 -0.701 -0.406 -0.129 -0.509 0.313 -0.540 -0.381 0.375 -0.349 -0.800 -0.087 -0.702 -2.124 -1.000 0.123 -1.043 -0.831 -0.037 0.603 -1.856 -0.357 -0.816 -1.022 1.818 0.004 0.242 0.800 -2.481 -1.571 3.505 0.209 -0.874 -0.665 -0.322 -0.494 0.510 0.398 -0.720 -0.691 0.479 0.690 1.506 1.127 -1.140 -1.561 0.787 -0.712 -1.032 -0.745 -2.687 -2.088 0.104 0.566 -1.411 -0.000 -0.308 0.033 0.513 -0.153 -0.144 0.427 0.591 -0.627 0.874 0.112 0.118 -2.420 -0.568 -0.432 0.597 0.786 0.076 0.348 -0.094 -1.486 2.243 1.141 -0.190 -1.518 -0.767 0.559 0.170 -2.095 0.867 1.729 0.652 1.458 3.133 1.290 -1.588 -2.422 1.202 0.995 1.594 -0.074 -0.480 0.033 -0.428 -0.001 -1.095 -0.434 0.601 -0.595 0.447 0.005 0.211 0.835 1.217 1.822 -0.029 0.503 -0.609 -0.171 -0.048 -0.177 -0.672 -0.572 0.221 1.222 -0.008 0.731 0.811 -0.173 -0.069 0.636 0.582 0.748 0.423 1.448 -1.566 0.150 -1.015 -1.443 0.260 -0.493 -0.113 0.903 1.796 1.912 0.664 -0.623 -0.783 -1.365 -1.404 -0.046 -2.109 -0.077 1.112 0.395 0.346 0.413 0.416 0.501 -0.710 -0.262 0.822 2.703 0.251 1.667 -1.274 -1.569 0.253 0.331 0.138 0.124 -0.805 -0.031 -0.711 -1.791 -0.333 -0.663 1.111 -0.201 0.870 -0.246 0.468 -0.497 0.032 2.179 -0.501 -0.701 -0.437 -1.221 -0.886 -0.703 0.209 -0.913 0.840 1.839 1.142 1.205 -2.363 -1.595 0.603 1.964 -1.294 -0.080 -0.515 0.270 0.898 1.116 0.566 0.553 -1.047 -1.595 0.663 0.200 1.646 2.387 -1.308 -1.223 0.669 2.161 -2.028 -1.718 1.048 0.900 -0.742 -1.148 2.202 2.122 -1.140 -0.466 0.885 1.058 1.132 -0.671 1.424 0.951 0.366 -0.607 -0.593 -0.096 0.661 -0.443 0.025 -0.335 -0.363 0.524 0.976 0.118 -1.177 -0.411 1.186 0.237 -0.761 0.351 0.255 0.557 -0.772 -1.944 -0.960 -0.015 0.024 0.206 2.055 1.878 -0.133 0.795 -2.062 -1.100 -0.587 -0.596 -0.692 -0.573 -0.294 -0.325 0.645 0.411 0.436 -1.131 0.569 1.602 -1.137 -0.797 -0.450 1.019 -0.149 -0.426 -0.150 -0.061 0.871 -0.809 -0.120 -0.094 -1.218 -2.247 -1.142 -1.022 -0.821 -0.282 0.805 1.055 0.235 0.192 1.290 0.930 -0.527 -2.354 0.777 0.977 0.285 0.363 1.236 0.608 -0.751 -0.059 -0.686 0.523 0.586 1.384 -1.613 -2.008 -1.644 -1.564 -0.654 -1.604 -0.307 -1.200 -0.295 -0.548 0.414 -0.355 -0.765 0.591 2.883 1.355 1.555 1.369 -0.714 1.010 0.929 0.832 1.476 1.097 0.150 0.498 0.512 0.463 -1.496 1.187 -0.484 -1.280 -0.517 -1.566 0.993 0.292 0.147 -1.540 1.351 0.653 -0.229 0.112 -0.016 1.498 0.074 0.478 -0.241 0.421 2.296 3.130 -1.811 -0.349 -0.383 -0.964 -0.332 -0.946 -0.457 0.260 -1.419 -0.395 -0.424 -0.465 1.273 2.454 0.358 0.213 0.604 0.732 0.688 0.301 -1.674 0.426 0.240 1.303 -0.193 -0.215 0.775 0.519 0.340 0.730 -0.654 -1.304 0.700 1.571 -1.053 -1.177 1.699 0.809 -0.500 -1.068 0.945 0.131 -0.961 -0.816 -0.498 1.261 -0.550 -0.991 -1.755 -1.322 -1.884 -0.425 0.633 -1.575 -0.556 -0.346 -0.939 -1.131 -0.504 -0.620 0.370 -0.295 -0.049 -0.771 -0.654 1.056 -0.138 0.773 0.475 1.974 -0.104 0.434 -1.010 -1.411 0.533 -1.084 0.447 0.537 -1.858 0.068 -1.566 -2.101 -1.361 -0.868 -1.110 -0.562 0.520 -1.980 1.650 1.719 -1.153 -1.049 -1.316 -0.030 -0.659 -0.881 -0.711 0.801 -1.246 -1.373 0.085 -0.189 -0.558 -0.834 1.332 0.362 0.884 -0.972 0.241 0.484 0.634 -0.549 1.337 -0.185 1.117 1.080 0.370 -0.312 -0.830 -0.806 0.639 0.937 1.039 0.981 -0.090 1.526 -0.214 0.522 1.267 0.644 -0.133 -0.307 -0.765 -0.374 -0.706 -0.039 0.336 1.924 -1.975 -0.271 1.993 1.093 -1.022 -2.235 1.366 1.528 -2.505 -1.239 -0.973 -0.765 1.035 -0.752 -0.829 1.234 0.754 1.020 0.380 -0.185 0.337 0.147 -0.223 0.195 0.165 0.685 0.558 -0.249 0.673 0.338 -0.999 -0.922 -1.751 -0.450 0.322 0.172 -0.177 -1.189 1.520 0.369 -1.035 -1.171 -0.560 -0.012 -0.062 0.187 2.230 2.154 1.522 0.479 -0.445 0.026 0.186 -0.775 -0.611 0.096 1.459 1.589 1.314 0.427 -1.216 -1.488 0.595 -1.179 -1.487 0.360 0.409 0.846 -0.090 1.622 0.612 0.477 1.360 1.514 -0.754 -0.105 -0.997 0.818 -0.879 -0.305 -0.406 -0.507 0.624 0.520 0.597 -1.325 -0.026 -0.686 0.377 -1.401 -0.873 -0.962 0.109 -1.182 -0.426 -0.049 0.244 -0.861 -1.322 -1.166 -0.307 -0.906 -0.923 0.798 -0.321 -0.297 2.187 0.951 -0.258 -0.414 0.067 1.428 -0.643 -0.106 -1.133 -1.570 0.417 -1.146 -0.083 0.313 -0.876 -1.258 1.594 0.151 0.525 0.873 0.017 -1.102 1.224 -0.434 0.347 -1.017 -1.466 -0.060 -0.044 0.038 0.386 -0.148 0.624 1.438 0.652 0.611 0.816 0.491 0.778 -0.547 -0.280 -0.188 -0.741 -0.098 0.897 1.261 1.529 0.826 1.448 0.660 0.104 0.016 0.778 0.861 -1.122 -0.425 -1.042 0.320 -1.738 -1.080 0.296 1.013 -0.812 -0.351 1.273 1.040 -0.107 -1.618 -0.681 -0.216 -0.022 -0.981 -1.752 -1.538 0.029 -0.065 -1.579 -0.676 0.297 0.645 0.958 1.360 -1.771 -0.567 -0.700 -1.102 -0.130 -0.570 1.494 1.179 -1.644 0.045 -0.386 -0.738 0.152 0.384 -1.164 -0.591 0.179 -0.150 0.831 -0.271 -0.955 -0.281 -0.366 0.199 0.362 1.304 -0.890 -0.697 -1.744 -0.171 0.597 0.323 1.260 -0.881 1.214 -0.151 -1.580 -1.187 0.268 -1.049 0.640 -0.143 -0.389 2.136 0.712 0.822 0.976 0.097 -0.867 -0.021 0.056 -1.623 -1.133 0.271 -0.235 -0.221 0.083 0.565 0.456 0.170 -1.335 -0.432 0.958 0.023 0.871 0.148 -0.408 1.031 0.398 0.399 1.240 1.469 -0.464 -1.256 0.342 0.395 0.023 0.385 -1.079 -1.078 0.658 0.440 2.620 1.188 -1.514 -1.388 -0.806 -0.303 -1.489 1.111 -0.685 0.143 0.251 -0.295 -0.746 2.190 1.112 1.278 -2.915 0.914 0.511 -1.134 0.187 0.506 -0.643 -0.699 0.479 -0.224 1.130 0.516 2.246 0.931 0.129 -0.632 -1.636 -0.476 -1.971 -0.539 -0.141 0.769 -2.359 -1.717 1.787 1.204 -0.822 -1.038 -0.532 -0.369 0.291 0.637 0.012 0.020 2.077 -0.594 -0.987 -2.648 -0.971 -0.629 0.376 0.034 -0.923 -0.172 -1.927 -1.147 0.378 1.061 1.818 0.977 0.376 1.593 -1.607 -1.425 2.203 1.104 0.775 0.995 0.505 1.517 -0.270 -0.421 0.547 0.970 0.136 -0.455 1.363 2.323 -0.281 0.874 -0.026 0.343 -0.241 -2.566 1.343 -0.517 -0.705 -0.637 0.684 1.977 -0.729 0.930 0.105 -1.334 0.523 0.051 -0.952 -0.583 -1.174 0.078 -2.393 -0.009 0.246 0.146 -0.334 -1.349 -1.488 -0.645 -0.483 -0.815 -1.683 0.024 -0.978 1.052 0.015 -0.894 -0.706 -0.786 -0.926 0.111 -0.257 -1.348 1.198 0.760 0.826 0.191 -0.336 -2.036 -0.561 -0.664 -0.598 -1.350 0.323 0.227 0.962 0.928 -0.927 -0.410 1.037 -0.405 -0.238 0.070 -1.357 -1.128 -1.689 -1.692 1.175 1.889 0.838 0.552 0.279 -0.437 -1.574 0.449 0.892 1.469 0.935 1.611 -0.855 -1.197 1.322 0.902 0.354 -1.097 -0.661 -1.708 0.757 0.803 1.190 -0.226 1.881 0.462 0.393 0.288 0.844 -1.964 1.057 1.067 -0.843 -0.956 -1.258 -1.045 0.889 0.751 -0.139 -0.417 -0.534 -0.676 -0.179 -0.726 -0.560 -0.911 0.282 0.555 -1.074 -0.883 0.052 0.613 -0.605 -0.298 0.582 1.572 -1.337 -0.240 0.215 0.274 -0.029 -0.064 0.396 -0.683 2.270 1.933 1.460 2.169 0.816 0.442 0.742 -0.388 1.850 -0.578 1.913 2.661 -1.050 0.991 -0.843 0.348 -1.211 0.471 -0.020 0.438 -0.889 1.356 -0.083 -0.334 1.203 -1.698 -0.182 -0.447 1.482 0.915 0.256 0.024 0.878 1.762 -0.467 -0.470 0.643 1.840 -0.921 -0.969 -1.278 1.226 0.447 -0.520 -1.549 -0.267 0.274 -0.153 1.886 1.977 -1.044 -0.414 -0.632 0.564 -0.201 -0.755 0.775 -0.200 0.014 -0.361 0.477 -0.493 -2.028 0.563 1.500 1.119 -0.545 -0.593 0.511 0.443 -1.713 -0.021 -0.451 0.386 0.354 1.790 -1.537 -1.093 -2.216 -0.949 -2.512 -0.455 0.590 -1.971 0.487 0.077 0.509 2.565 -0.288 0.240 -0.813 -0.190 -0.676 -0.273 -1.436 -1.011 0.881 0.880 -0.401 -0.136 -0.527 0.515 1.337 0.412 -1.013 -0.716 1.203 1.220 0.025 0.928 -0.125 -2.182 0.736 -1.361 0.508 -0.409 -0.628 -1.286 0.721 0.452 0.617 -1.125 -0.953 -2.126 -0.265 -1.746 1.490 0.262 0.592 -0.363 -0.901 0.153 0.129 -1.199 1.353 0.252 -0.133 0.051 1.720 1.290 -0.467 -0.592 0.479 1.094 0.642 1.078 1.369 -0.277 -1.614 -1.486 1.379 0.788 0.537 0.813 -0.719 0.335 1.935 1.231 0.785 0.465 -0.239 1.220 -1.961 -2.125 -1.101 -0.869 0.915 0.501 -0.890 0.188 -0.876 0.420 1.791 1.065 -0.700 -0.578 -0.859 -0.658 -2.314 -0.982 -0.197 -1.059 -2.724 -2.060 1.647 1.160 -0.557 -0.097 0.448 0.741 -0.924 0.149 1.411 -0.264 FIG2 Delta against selection threshold p, one series per lam lam r p Delta sim SE Delta exact 0.00 1.0000 0.500 0.00000 0.00000 0.00000 0.00 1.0000 0.250 0.00000 0.00000 0.00000 0.00 1.0000 0.100 0.00000 0.00000 0.00000 0.00 1.0000 0.050 0.00000 0.00000 0.00000 0.00 1.0000 0.020 0.00000 0.00000 0.00000 0.00 1.0000 0.010 0.00000 0.00000 0.00000 0.25 0.9412 0.500 0.04702 0.00020 0.04693 0.25 0.9412 0.250 0.07468 0.00021 0.07477 0.25 0.9412 0.100 0.10369 0.00043 0.10323 0.25 0.9412 0.050 0.12077 0.00054 0.12134 0.25 0.9412 0.020 0.14335 0.00086 0.14241 0.25 0.9412 0.010 0.15874 0.00166 0.15678 0.50 0.8000 0.500 0.15973 0.00031 0.15958 0.50 0.8000 0.250 0.25429 0.00057 0.25422 0.50 0.8000 0.100 0.34999 0.00075 0.35100 0.50 0.8000 0.050 0.41337 0.00109 0.41254 0.50 0.8000 0.020 0.48426 0.00158 0.48418 0.50 0.8000 0.010 0.53000 0.00280 0.53304 0.75 0.6400 0.500 0.28784 0.00040 0.28724 0.75 0.6400 0.250 0.45813 0.00064 0.45760 0.75 0.6400 0.100 0.63285 0.00100 0.63179 0.75 0.6400 0.050 0.74131 0.00185 0.74258 0.75 0.6400 0.020 0.87188 0.00221 0.87153 0.75 0.6400 0.010 0.95978 0.00356 0.95948 1.00 0.5000 0.500 0.39856 0.00056 0.39894 1.00 0.5000 0.250 0.63465 0.00075 0.63555 1.00 0.5000 0.100 0.87598 0.00131 0.87749 1.00 0.5000 0.050 1.03310 0.00183 1.03136 1.00 0.5000 0.020 1.20770 0.00255 1.21045 1.00 0.5000 0.010 1.33044 0.00402 1.33261 1.50 0.3077 0.500 0.55367 0.00069 0.55238 1.50 0.3077 0.250 0.87954 0.00090 0.88000 1.50 0.3077 0.100 1.21506 0.00138 1.21499 1.50 0.3077 0.050 1.42995 0.00187 1.42803 1.50 0.3077 0.020 1.67413 0.00279 1.67601 1.50 0.3077 0.010 1.85265 0.00414 1.84515 2.00 0.2000 0.500 0.63708 0.00057 0.63831 2.00 0.2000 0.250 1.01714 0.00101 1.01689 2.00 0.2000 0.100 1.40228 0.00132 1.40399 2.00 0.2000 0.050 1.64836 0.00194 1.65017 2.00 0.2000 0.020 1.93955 0.00303 1.93673 2.00 0.2000 0.010 2.13499 0.00418 2.13217 FIG2b analytic Delta curve, dense p, for drawing smooth lines lam=0.25 0.0100:0.15678 0.0107:0.15539 0.0115:0.15399 0.0123:0.15258 0.0132:0.15116 0.0141:0.14973 0.0152:0.14828 0.0163:0.14683 0.0174:0.14536 0.0187:0.14388 0.0200:0.14239 0.0215:0.14088 0.0230:0.13936 0.0246:0.13783 0.0264:0.13629 0.0283:0.13472 0.0304:0.13315 0.0325:0.13156 0.0349:0.12995 0.0374:0.12833 0.0401:0.12669 0.0429:0.12503 0.0460:0.12336 0.0493:0.12166 0.0529:0.11995 0.0567:0.11822 0.0608:0.11647 0.0651:0.11470 0.0698:0.11290 0.0748:0.11109 0.0802:0.10925 0.0860:0.10739 0.0921:0.10550 0.0988:0.10358 0.1059:0.10164 0.1135:0.09967 0.1216:0.09767 0.1304:0.09564 0.1397:0.09358 0.1498:0.09148 0.1605:0.08935 0.1721:0.08718 0.1844:0.08498 0.1977:0.08273 0.2119:0.08043 0.2271:0.07809 0.2434:0.07570 0.2609:0.07326 0.2797:0.07076 0.2998:0.06821 0.3213:0.06558 0.3444:0.06289 0.3691:0.06012 0.3957:0.05727 0.4241:0.05433 0.4546:0.05129 0.4872:0.04814 0.5222:0.04487 0.5598:0.04145 0.6000:0.03788 lam=0.50 0.0100:0.53304 0.0107:0.52832 0.0115:0.52357 0.0123:0.51877 0.0132:0.51394 0.0141:0.50907 0.0152:0.50416 0.0163:0.49922 0.0174:0.49423 0.0187:0.48920 0.0200:0.48412 0.0215:0.47900 0.0230:0.47384 0.0246:0.46863 0.0264:0.46337 0.0283:0.45806 0.0304:0.45270 0.0325:0.44729 0.0349:0.44183 0.0374:0.43631 0.0401:0.43074 0.0429:0.42510 0.0460:0.41941 0.0493:0.41366 0.0529:0.40784 0.0567:0.40195 0.0608:0.39600 0.0651:0.38997 0.0698:0.38387 0.0748:0.37770 0.0802:0.37145 0.0860:0.36511 0.0921:0.35869 0.0988:0.35218 0.1059:0.34558 0.1135:0.33888 0.1216:0.33209 0.1304:0.32518 0.1397:0.31817 0.1498:0.31105 0.1605:0.30380 0.1721:0.29642 0.1844:0.28892 0.1977:0.28127 0.2119:0.27347 0.2271:0.26551 0.2434:0.25739 0.2609:0.24909 0.2797:0.24060 0.2998:0.23190 0.3213:0.22298 0.3444:0.21383 0.3691:0.20441 0.3957:0.19472 0.4241:0.18472 0.4546:0.17439 0.4872:0.16367 0.5222:0.15254 0.5598:0.14093 0.6000:0.12878 lam=0.75 0.0100:0.95948 0.0107:0.95098 0.0115:0.94242 0.0123:0.93379 0.0132:0.92510 0.0141:0.91633 0.0152:0.90750 0.0163:0.89859 0.0174:0.88961 0.0187:0.88055 0.0200:0.87142 0.0215:0.86221 0.0230:0.85291 0.0246:0.84353 0.0264:0.83407 0.0283:0.82451 0.0304:0.81487 0.0325:0.80513 0.0349:0.79530 0.0374:0.78536 0.0401:0.77533 0.0429:0.76519 0.0460:0.75494 0.0493:0.74458 0.0529:0.73411 0.0567:0.72351 0.0608:0.71279 0.0651:0.70195 0.0698:0.69097 0.0748:0.67986 0.0802:0.66860 0.0860:0.65720 0.0921:0.64564 0.0988:0.63393 0.1059:0.62204 0.1135:0.60999 0.1216:0.59776 0.1304:0.58533 0.1397:0.57271 0.1498:0.55988 0.1605:0.54684 0.1721:0.53356 0.1844:0.52005 0.1977:0.50628 0.2119:0.49225 0.2271:0.47793 0.2434:0.46330 0.2609:0.44836 0.2797:0.43307 0.2998:0.41742 0.3213:0.40136 0.3444:0.38489 0.3691:0.36795 0.3957:0.35050 0.4241:0.33250 0.4546:0.31390 0.4872:0.29461 0.5222:0.27458 0.5598:0.25368 0.6000:0.23181 lam=1.00 0.0100:1.33261 0.0107:1.32081 0.0115:1.30891 0.0123:1.29693 0.0132:1.28485 0.0141:1.27268 0.0152:1.26041 0.0163:1.24804 0.0174:1.23557 0.0187:1.22299 0.0200:1.21030 0.0215:1.19751 0.0230:1.18460 0.0246:1.17157 0.0264:1.15843 0.0283:1.14516 0.0304:1.13176 0.0325:1.11824 0.0349:1.10458 0.0374:1.09078 0.0401:1.07685 0.0429:1.06276 0.0460:1.04853 0.0493:1.03414 0.0529:1.01959 0.0567:1.00488 0.0608:0.98999 0.0651:0.97493 0.0698:0.95968 0.0748:0.94425 0.0802:0.92861 0.0860:0.91278 0.0921:0.89673 0.0988:0.88045 0.1059:0.86395 0.1135:0.84721 0.1216:0.83022 0.1304:0.81296 0.1397:0.79543 0.1498:0.77762 0.1605:0.75950 0.1721:0.74106 0.1844:0.72229 0.1977:0.70317 0.2119:0.68368 0.2271:0.66379 0.2434:0.64348 0.2609:0.62272 0.2797:0.60149 0.2998:0.57974 0.3213:0.55745 0.3444:0.53456 0.3691:0.51103 0.3957:0.48680 0.4241:0.46181 0.4546:0.43597 0.4872:0.40919 0.5222:0.38135 0.5598:0.35233 0.6000:0.32195 lam=1.50 0.0100:1.84515 0.0107:1.82881 0.0115:1.81234 0.0123:1.79575 0.0132:1.77903 0.0141:1.76218 0.0152:1.74519 0.0163:1.72806 0.0174:1.71079 0.0187:1.69337 0.0200:1.67581 0.0215:1.65809 0.0230:1.64021 0.0246:1.62218 0.0264:1.60397 0.0283:1.58560 0.0304:1.56705 0.0325:1.54833 0.0349:1.52942 0.0374:1.51031 0.0401:1.49102 0.0429:1.47152 0.0460:1.45181 0.0493:1.43189 0.0529:1.41174 0.0567:1.39137 0.0608:1.37076 0.0651:1.34990 0.0698:1.32879 0.0748:1.30742 0.0802:1.28577 0.0860:1.26384 0.0921:1.24162 0.0988:1.21909 0.1059:1.19624 0.1135:1.17306 0.1216:1.14953 0.1304:1.12564 0.1397:1.10137 0.1498:1.07670 0.1605:1.05161 0.1721:1.02609 0.1844:1.00010 0.1977:0.97362 0.2119:0.94663 0.2271:0.91909 0.2434:0.89097 0.2609:0.86223 0.2797:0.83283 0.2998:0.80272 0.3213:0.77186 0.3444:0.74017 0.3691:0.70759 0.3957:0.67404 0.4241:0.63943 0.4546:0.60365 0.4872:0.56657 0.5222:0.52803 0.5598:0.48785 0.6000:0.44578 lam=2.00 0.0100:2.13217 0.0107:2.11329 0.0115:2.09426 0.0123:2.07509 0.0132:2.05577 0.0141:2.03629 0.0152:2.01666 0.0163:1.99687 0.0174:1.97691 0.0187:1.95679 0.0200:1.93649 0.0215:1.91601 0.0230:1.89536 0.0246:1.87451 0.0264:1.85348 0.0283:1.83225 0.0304:1.81082 0.0325:1.78918 0.0349:1.76733 0.0374:1.74525 0.0401:1.72295 0.0429:1.70042 0.0460:1.67765 0.0493:1.65462 0.0529:1.63135 0.0567:1.60780 0.0608:1.58399 0.0651:1.55989 0.0698:1.53549 0.0748:1.51079 0.0802:1.48578 0.0860:1.46044 0.0921:1.43476 0.0988:1.40873 0.1059:1.38232 0.1135:1.35553 0.1216:1.32835 0.1304:1.30074 0.1397:1.27269 0.1498:1.24418 0.1605:1.21519 0.1721:1.18570 0.1844:1.15567 0.1977:1.12507 0.2119:1.09388 0.2271:1.06206 0.2434:1.02956 0.2609:0.99635 0.2797:0.96238 0.2998:0.92759 0.3213:0.89192 0.3444:0.85530 0.3691:0.81766 0.3957:0.77889 0.4241:0.73889 0.4546:0.69755 0.4872:0.65470 0.5222:0.61017 0.5598:0.56374 0.6000:0.51512 FIG3 validation, measured shrinkage against exact r, all cells p lam r exact shrink sim SE z 0.50 0.00 1.000000 1.000000 0.000000 nan 0.50 0.25 0.941176 0.941037 0.000243 -0.573 0.50 0.50 0.800000 0.799888 0.000371 -0.301 0.50 0.75 0.640000 0.639357 0.000486 -1.323 0.50 1.00 0.500000 0.500216 0.000651 0.331 0.50 1.50 0.307692 0.306458 0.000764 -1.616 0.50 2.00 0.200000 0.201599 0.000631 2.535 0.25 0.00 1.000000 1.000000 0.000000 nan 0.25 0.25 0.941176 0.941216 0.000164 0.240 0.25 0.50 0.800000 0.800015 0.000421 0.035 0.25 0.75 0.640000 0.639732 0.000484 -0.554 0.25 1.00 0.500000 0.500691 0.000524 1.320 0.25 1.50 0.307692 0.308148 0.000645 0.706 0.25 2.00 0.200000 0.199595 0.000687 -0.589 0.10 0.00 1.000000 1.000000 0.000000 nan 0.10 0.25 0.941176 0.940935 0.000248 -0.977 0.10 0.50 0.800000 0.800450 0.000417 1.079 0.10 0.75 0.640000 0.639593 0.000542 -0.752 0.10 1.00 0.500000 0.500825 0.000694 1.189 0.10 1.50 0.307692 0.307786 0.000759 0.124 0.10 2.00 0.200000 0.200895 0.000676 1.325 0.05 0.00 1.000000 1.000000 0.000000 nan 0.05 0.25 0.941176 0.941419 0.000256 0.947 0.05 0.50 0.800000 0.799598 0.000518 -0.777 0.05 0.75 0.640000 0.640655 0.000891 0.735 0.05 1.00 0.500000 0.499248 0.000843 -0.892 0.05 1.50 0.307692 0.306997 0.000859 -0.809 0.05 2.00 0.200000 0.200563 0.000855 0.658 0.02 0.00 1.000000 1.000000 0.000000 nan 0.02 0.25 0.941176 0.940739 0.000354 -1.234 0.02 0.50 0.800000 0.800017 0.000653 0.026 0.02 0.75 0.640000 0.639903 0.000907 -0.107 0.02 1.00 0.500000 0.500582 0.000994 0.585 0.02 1.50 0.307692 0.308336 0.001096 0.588 0.02 2.00 0.200000 0.198993 0.001073 -0.939 0.01 0.00 1.000000 1.000000 0.000000 nan 0.01 0.25 0.941176 0.940421 0.000616 -1.228 0.01 0.50 0.800000 0.801215 0.001046 1.161 0.01 0.75 0.640000 0.639805 0.001311 -0.148 0.01 1.00 0.500000 0.500375 0.001439 0.260 0.01 1.50 0.307692 0.305843 0.001534 -1.206 0.01 2.00 0.200000 0.199447 0.001489 -0.371 FIG4 convergence trace, headline cell (n selected, running Delta) 10000 0.851806 20000 0.861152 30000 0.867103 40000 0.871690 50000 0.876656 60000 0.874730 70000 0.876963 80000 0.877236 90000 0.876753 100000 0.876408 110000 0.876766 120000 0.877610 130000 0.877593 140000 0.878246 150000 0.878967 160000 0.879077 170000 0.878908 180000 0.878853 190000 0.879110 200000 0.879196 210000 0.878882 220000 0.878918 230000 0.878805 240000 0.878931 250000 0.878538 260000 0.878710 270000 0.878511 280000 0.878787 290000 0.878603 300000 0.878332 310000 0.878515 320000 0.878079 330000 0.877586 340000 0.878209 350000 0.878297 360000 0.878065 370000 0.877749 380000 0.877593 390000 0.877426 400000 0.877670 410000 0.877480 420000 0.877466 430000 0.877624 440000 0.877353 450000 0.877200 460000 0.877162 470000 0.877062 480000 0.876787 490000 0.876591 500000 0.876380 510000 0.876347 520000 0.876393 530000 0.876713 540000 0.876727 550000 0.876883 560000 0.877216 570000 0.877208 580000 0.876915 590000 0.876944 600000 0.876885 exact 0.877492 rep_se 0.001205 FIG5 single-arm against randomised estimate, by true tau scenario=tight cut, clean measure p=0.05 lam=0.5 r=0.8000 bias=0.41254 tau=0.00 single=0.41187 single_se=0.00097 rct=0.00002 rct_se=0.00247 tau=0.10 single=0.51460 single_se=0.00100 rct=0.09888 rct_se=0.00218 tau=0.20 single=0.61445 single_se=0.00111 rct=0.20162 rct_se=0.00265 tau=0.30 single=0.71178 single_se=0.00092 rct=0.29854 rct_se=0.00223 tau=0.50 single=0.91139 single_se=0.00088 rct=0.50076 rct_se=0.00250 tau=1.00 single=1.41275 single_se=0.00101 rct=1.00023 rct_se=0.00251 scenario=headline cell p=0.1 lam=1.0 r=0.5000 bias=0.87749 tau=0.00 single=0.87676 single_se=0.00130 rct=-0.00057 rct_se=0.00230 tau=0.10 single=0.97747 single_se=0.00096 rct=0.10051 rct_se=0.00230 tau=0.20 single=1.07694 single_se=0.00103 rct=0.19819 rct_se=0.00210 tau=0.30 single=1.17708 single_se=0.00139 rct=0.29886 rct_se=0.00214 tau=0.50 single=1.37986 single_se=0.00117 rct=0.50290 rct_se=0.00235 tau=1.00 single=1.87742 single_se=0.00103 rct=1.00346 rct_se=0.00232 scenario=loose cut, noisy measure p=0.25 lam=1.5 r=0.3077 bias=0.88000 tau=0.00 single=0.87965 single_se=0.00081 rct=0.00019 rct_se=0.00168 tau=0.10 single=0.98131 single_se=0.00080 rct=0.09939 rct_se=0.00150 tau=0.20 single=1.07879 single_se=0.00097 rct=0.19991 rct_se=0.00164 tau=0.30 single=1.18008 single_se=0.00095 rct=0.30053 rct_se=0.00149 tau=0.50 single=1.38014 single_se=0.00080 rct=0.50042 rct_se=0.00172 tau=1.00 single=1.88177 single_se=0.00088 rct=0.99690 rct_se=0.00176 ============================================================================== SUMMARY OF CHECKS ============================================================================== 1. Shrinkage law E[X2|sel] = r E[X1|sel], 42 cells, largest |z| = 2.53 2. Spurious improvement against (1-r) phi(z_p)/p, 42 cells, largest |z| = 2.14 3. Zero-noise control at r = 1: PASSED, Delta identically 0 at every p 4. Randomised design recovers the true tau, 18 cells, largest |z| = 1.77 5. Pooled goodness of fit on Delta: sum z^2 = 36.20 on 36 df, p = 0.459 Largest |z| anywhere in the study: 2.53 Checks 1 and 2 are not independent of one another: the shrinkage factor and Delta are the same measurement written two ways, so a cell that wanders in one wanders in the other. Counting them as one family of 42 and the trial cells as a second family of 18, the Bonferroni threshold at family-wise 5% is |t| = 3.52 with 59 df. No disagreement between simulation and closed form survives. headline number: selecting the bottom 10% of a population on a measurement with reliability r = 0.50 and applying no treatment produces an apparent improvement of 0.8760 +/- 0.0013 population SDs (exact value 0.8775). wall clock: 36.0 s ==============================================================================