=== 1. Chance alarms per year, England and Wales [article: Step one, Table 1] === CUSUM h: NICU E10=2.92 E50=4.45 CUSUM h: LNU E10=2.19 E50=3.57 CUSUM h: SCU E10=1.39 E50=2.51 A Annual check, 2-sigma alarm unit-yrs 11.3 episodes 10.8 share of unit-years with an alarm: NICU 0.114 LNU 0.058 SCU 0.028 B Annual check, 3-sigma alarm unit-yrs 3.1 episodes 3.0 share of unit-years with an alarm: NICU 0.043 LNU 0.012 SCU 0.003 C Rolling 12m, monthly, 2-sigma alarm unit-yrs 31.5 episodes 30.2 share of unit-years with an alarm: NICU 0.278 LNU 0.177 SCU 0.088 D Rolling 12m, monthly, doubling heuristic alarm unit-yrs 34.0 episodes 33.5 share of unit-years with an alarm: NICU 0.071 LNU 0.328 SCU 0.067 E10 Risk-adjusted CUSUM, 1 false alarm per 10 unit-years alarm unit-yrs 17.1 episodes 17.5 share of unit-years with an alarm: NICU 0.095 LNU 0.100 SCU 0.097 E50 Risk-adjusted CUSUM, 1 false alarm per 50 unit-years alarm unit-yrs 3.5 episodes 3.5 share of unit-years with an alarm: NICU 0.018 LNU 0.021 SCU 0.020 === 2. Rota review of chance alarms, system-weighted shares [article: Step two, Table 2] === avg-exp = average-exposure test; own-exp = own-exposure test; max-adj = maximum-adjusted test (see MODEL.md) A avg-exp 0.79 own-exp 0.58 max-adj 0.024 | flag episodes/yr 6.2 distinct nurses/yr 6.2 NICU: avg-exp 1.00 own-exp 0.68 max-adj 0.035 | median deaths reviewed 30, median top-nurse attendance 13 LNU: avg-exp 0.71 own-exp 0.53 max-adj 0.013 | median deaths reviewed 8, median top-nurse attendance 4 SCU: avg-exp 0.31 own-exp 0.32 max-adj 0.018 | median deaths reviewed 4, median top-nurse attendance 2 B avg-exp 0.87 own-exp 0.60 max-adj 0.031 | flag episodes/yr 1.8 distinct nurses/yr 1.8 NICU: avg-exp 1.00 own-exp 0.69 max-adj 0.028 | median deaths reviewed 34, median top-nurse attendance 14 LNU: avg-exp 0.69 own-exp 0.46 max-adj 0.037 | median deaths reviewed 10, median top-nurse attendance 5 SCU: avg-exp 0.40 own-exp 0.32 max-adj 0.040 | median deaths reviewed 5, median top-nurse attendance 3 C avg-exp 0.76 own-exp 0.55 max-adj 0.023 | flag episodes/yr 16.7 distinct nurses/yr 16.4 NICU: avg-exp 1.00 own-exp 0.69 max-adj 0.031 | median deaths reviewed 28, median top-nurse attendance 12 LNU: avg-exp 0.68 own-exp 0.51 max-adj 0.018 | median deaths reviewed 8, median top-nurse attendance 4 SCU: avg-exp 0.26 own-exp 0.26 max-adj 0.014 | median deaths reviewed 3, median top-nurse attendance 2 D avg-exp 0.66 own-exp 0.49 max-adj 0.015 | flag episodes/yr 16.6 distinct nurses/yr 16.2 NICU: avg-exp 1.00 own-exp 0.69 max-adj 0.029 | median deaths reviewed 31, median top-nurse attendance 13 LNU: avg-exp 0.64 own-exp 0.49 max-adj 0.011 | median deaths reviewed 6, median top-nurse attendance 4 SCU: avg-exp 0.46 own-exp 0.32 max-adj 0.041 | median deaths reviewed 4, median top-nurse attendance 2 E10 avg-exp 0.63 own-exp 0.50 max-adj 0.018 | flag episodes/yr 8.7 distinct nurses/yr 8.6 NICU: avg-exp 0.95 own-exp 0.73 max-adj 0.026 | median deaths reviewed 12, median top-nurse attendance 7 LNU: avg-exp 0.64 own-exp 0.49 max-adj 0.015 | median deaths reviewed 7, median top-nurse attendance 4 SCU: avg-exp 0.27 own-exp 0.28 max-adj 0.015 | median deaths reviewed 3, median top-nurse attendance 2 E50 avg-exp 0.75 own-exp 0.52 max-adj 0.021 | flag episodes/yr 1.8 distinct nurses/yr 1.8 NICU: avg-exp 0.98 own-exp 0.70 max-adj 0.024 | median deaths reviewed 20, median top-nurse attendance 9 LNU: avg-exp 0.79 own-exp 0.53 max-adj 0.014 | median deaths reviewed 12, median top-nurse attendance 6 SCU: avg-exp 0.46 own-exp 0.33 max-adj 0.030 | median deaths reviewed 7, median top-nurse attendance 3 === 3. Offender adds expected deaths for 12 months; caught within 12 months [bg = no offender] [article: Step three, Table 3 and background result] === +4 A NICU: 0.24 [bg 0.12] (offender identified 0.14) LNU: 0.40 [bg 0.06] (offender identified 0.65) SCU: 0.66 [bg 0.03] (offender identified 0.86) +4 B NICU: 0.10 [bg 0.04] (offender identified 0.14) LNU: 0.17 [bg 0.01] (offender identified 0.67) SCU: 0.38 [bg 0.00] (offender identified 0.91) +4 C NICU: 0.40 [bg 0.27] (offender identified 0.06) LNU: 0.55 [bg 0.18] (offender identified 0.34) SCU: 0.73 [bg 0.08] (offender identified 0.58) +4 D NICU: 0.11 [bg 0.07] (offender identified 0.09) LNU: 0.67 [bg 0.33] (offender identified 0.27) SCU: 0.77 [bg 0.07] (offender identified 0.61) +4 E10 NICU: 0.25 [bg 0.08] (offender identified 0.10) LNU: 0.55 [bg 0.07] (offender identified 0.46) SCU: 0.83 [bg 0.06] (offender identified 0.61) +4 E50 NICU: 0.08 [bg 0.01] (offender identified 0.16) LNU: 0.23 [bg 0.01] (offender identified 0.63) SCU: 0.53 [bg 0.00] (offender identified 0.82) +7 A NICU: 0.36 [bg 0.12] (offender identified 0.42) LNU: 0.71 [bg 0.06] (offender identified 0.89) SCU: 0.92 [bg 0.03] (offender identified 0.98) +7 B NICU: 0.17 [bg 0.04] (offender identified 0.39) LNU: 0.44 [bg 0.01] (offender identified 0.92) SCU: 0.77 [bg 0.00] (offender identified 0.98) +7 C NICU: 0.52 [bg 0.27] (offender identified 0.17) LNU: 0.79 [bg 0.18] (offender identified 0.53) SCU: 0.94 [bg 0.09] (offender identified 0.68) +7 D NICU: 0.17 [bg 0.07] (offender identified 0.21) LNU: 0.86 [bg 0.33] (offender identified 0.44) SCU: 0.96 [bg 0.07] (offender identified 0.71) +7 E10 NICU: 0.43 [bg 0.08] (offender identified 0.26) LNU: 0.84 [bg 0.07] (offender identified 0.65) SCU: 0.98 [bg 0.06] (offender identified 0.74) +7 E50 NICU: 0.19 [bg 0.02] (offender identified 0.35) LNU: 0.60 [bg 0.01] (offender identified 0.83) SCU: 0.88 [bg 0.00] (offender identified 0.90) === 4. Falsely flagged unit-years per detected offender-year (+4); national years per detection [article: Step three, Table 4] === base rates per 10,000 unit-years as rates per nurse-year: p=0.1: 1 in 5,028,571 p=1: 1 in 502,857 p=3: 1 in 167,619 p=10: 1 in 50,286 p=30: 1 in 16,762 main case, risk in proportion to staff: A p=0.1: 18716:1 ( 1658y) p=1: 1871:1 ( 166y) p=3: 624:1 ( 55y) p=10: 187:1 ( 17y) p=30: 62:1 ( 6y) B p=0.1: 11278:1 ( 3680y) p=1: 1128:1 ( 368y) p=3: 376:1 ( 123y) p=10: 113:1 ( 37y) p=30: 37:1 ( 12y) C p=0.1: 36488:1 ( 1158y) p=1: 3648:1 ( 116y) p=3: 1216:1 ( 39y) p=10: 364:1 ( 12y) p=30: 121:1 ( 4y) D p=0.1: 49278:1 ( 1448y) p=1: 4927:1 ( 145y) p=3: 1642:1 ( 48y) p=10: 492:1 ( 14y) p=30: 164:1 ( 5y) E10 p=0.1: 22940:1 ( 1342y) p=1: 2294:1 ( 134y) p=3: 764:1 ( 45y) p=10: 229:1 ( 13y) p=30: 76:1 ( 4y) E50 p=0.1: 10865:1 ( 3115y) p=1: 1086:1 ( 312y) p=3: 362:1 ( 104y) p=10: 109:1 ( 31y) p=30: 36:1 ( 10y) sensitivity, equal risk per unit: A p=0.1: 15136:1 ( 1341y) p=1: 1513:1 ( 134y) p=3: 504:1 ( 45y) p=10: 151:1 ( 13y) p=30: 50:1 ( 4y) B p=0.1: 8536:1 ( 2785y) p=1: 854:1 ( 279y) p=3: 284:1 ( 93y) p=10: 85:1 ( 28y) p=30: 28:1 ( 9y) C p=0.1: 32237:1 ( 1023y) p=1: 3223:1 ( 102y) p=3: 1074:1 ( 34y) p=10: 322:1 ( 10y) p=30: 107:1 ( 3y) D p=0.1: 35255:1 ( 1036y) p=1: 3525:1 ( 104y) p=3: 1175:1 ( 35y) p=10: 352:1 ( 10y) p=30: 117:1 ( 3y) E10 p=0.1: 17941:1 ( 1050y) p=1: 1794:1 ( 105y) p=3: 598:1 ( 35y) p=10: 179:1 ( 10y) p=30: 60:1 ( 3y) E50 p=0.1: 7430:1 ( 2130y) p=1: 743:1 ( 213y) p=3: 248:1 ( 71y) p=10: 74:1 ( 21y) p=30: 25:1 ( 7y) === 5. P(flagged nurse is the offender | alarm and own-exposure flag), +4 [article: Step three, closing paragraph] === main case, risk in proportion to staff: A p=0.1: 0.005% p=1: 0.049% p=3: 0.148% p=10: 0.491% p=30: 1.455% B p=0.1: 0.008% p=1: 0.084% p=3: 0.250% p=10: 0.827% p=30: 2.428% C p=0.1: 0.001% p=1: 0.013% p=3: 0.040% p=10: 0.134% p=30: 0.400% D p=0.1: 0.001% p=1: 0.013% p=3: 0.039% p=10: 0.130% p=30: 0.388% E10 p=0.1: 0.003% p=1: 0.033% p=3: 0.099% p=10: 0.327% p=30: 0.972% E50 p=0.1: 0.010% p=1: 0.103% p=3: 0.308% p=10: 1.017% p=30: 2.968% sensitivity, equal risk per unit: A p=0.1: 0.008% p=1: 0.079% p=3: 0.236% p=10: 0.782% p=30: 2.304% B p=0.1: 0.014% p=1: 0.143% p=3: 0.426% p=10: 1.404% p=30: 4.078% C p=0.1: 0.002% p=1: 0.022% p=3: 0.065% p=10: 0.215% p=30: 0.642% D p=0.1: 0.002% p=1: 0.022% p=3: 0.067% p=10: 0.222% p=30: 0.663% E10 p=0.1: 0.005% p=1: 0.052% p=3: 0.157% p=10: 0.521% p=30: 1.542% E50 p=0.1: 0.018% p=1: 0.179% p=3: 0.533% p=10: 1.752% p=30: 5.044% === 6. Nurse-flagging episodes per year (distinct nurses) by q [article: Step three, Table 5] === A q=1: 6.2 (6.2) q=0.5: 3.1 (3.1) q=0.25: 1.6 (1.6) q=0.1: 0.6 (0.6) B q=1: 1.8 (1.8) q=0.5: 0.9 (0.9) q=0.25: 0.4 (0.4) q=0.1: 0.2 (0.2) C q=1: 16.7 (16.4) q=0.5: 8.4 (8.2) q=0.25: 4.2 (4.1) q=0.1: 1.7 (1.6) D q=1: 16.6 (16.2) q=0.5: 8.3 (8.1) q=0.25: 4.1 (4.0) q=0.1: 1.7 (1.6) E10 q=1: 8.7 (8.6) q=0.5: 4.3 (4.3) q=0.25: 2.2 (2.2) q=0.1: 0.9 (0.9) E50 q=1: 1.8 (1.8) q=0.5: 0.9 (0.9) q=0.25: 0.5 (0.5) q=0.1: 0.2 (0.2) === 7. Robustness: pure Poisson, true baseline known [article: Step one, robustness check] === C: 13.4 alarm unit-years per year D: 16.4 alarm unit-years per year === 8. Exactly 4 offender deaths, not an expected 4, p = 1/10k, risk in proportion to staff [article: Step three, exact-count result] === A: 1845:1 (expected-4 model 1871:1) B: 1318:1 (expected-4 model 1128:1) C: 3580:1 (expected-4 model 3648:1) D: 4559:1 (expected-4 model 4927:1) E10: 2228:1 (expected-4 model 2294:1) E50: 1293:1 (expected-4 model 1086:1) === 9. Idealised CUSUM threshold sweep, fixed paths, +4 expected deaths, risk in proportion to staff [article: Step three, Figure 2] === false alarms per unit-year | national false alarms/yr | caught within 12m NICU LNU SCU | LNU with no offender | p=0.1: false:true, years per detection | p=1: false:true, years per detection 1 in 5 | 35.00 | 0.388 0.697 0.910 | 0.155 | 35664:1 1019 | 3566:1 102 1 in 10 | 17.50 | 0.247 0.545 0.832 | 0.068 | 23695:1 1354 | 2370:1 135 1 in 20 | 8.75 | 0.152 0.399 0.716 | 0.027 | 16375:1 1871 | 1637:1 187 1 in 50 | 3.50 | 0.077 0.235 0.529 | 0.007 | 10857:1 3102 | 1086:1 310 1 in 100 | 1.75 | 0.045 0.147 0.390 | 0.002 | 8362:1 4778 | 836:1 478 1 in 200 | 0.88 | 0.025 0.084 0.270 | 0.000 | 6849:1 7828 | 685:1 783 1 in 500 | 0.35 | 0.012 0.036 0.153 | 0.000 | 5612:1 16035 | 561:1 1603 1 in 1000 | 0.17 | 0.006 0.019 0.098 | 0.000 | 4858:1 27757 | 486:1 2776 1 in 2000 | 0.09 | 0.003 0.009 0.058 | 0.000 | 4497:1 51390 | 450:1 5139 1 in 5000 | 0.04 | 0.002 0.003 0.027 | 0.000 | 4003:1 114385 | 400:1 11438 1 in 10000 | 0.02 | 0.001 0.002 0.014 | 0.000 | 4014:1 229385 | 401:1 22938 Total elapsed wall time: 200.90 s (Numba)