======================================================================== PART A. Body geometry across the observed length range ======================================================================== Idealised body: triaxial ellipsoid, SL = 0.82*TL, depth = 0.45*SL, width = 0.18*SL, tissue density 1.05 g/cm^3. Shape held constant, so SA/V is exactly proportional to 1/TL and mass to TL^3. TL (mm) mass (g) area (mm2) SA/V (1/mm) rel. SA/V ---------------------------------------------------------- 27.00 0.483 410.5 0.89180 2.500 35.00 1.053 689.7 0.68796 1.929 45.00 2.237 1140.2 0.53508 1.500 52.60 3.573 1557.9 0.45777 1.283 60.00 5.304 2027.0 0.40131 1.125 67.50 7.551 2565.4 0.35672 1.000 75.00 10.359 3167.2 0.32105 0.900 83.50 14.295 3925.8 0.28837 0.808 Mass range over the observed lengths: 0.483 g at 27.00 mm to 14.29 g at 83.50 mm, a factor of 29.6. SA/V at the rank-1 mean (67.50 mm): 0.35672 /mm SA/V at the rank-2 mean (52.60 mm): 0.45777 /mm (28.3% higher) Effect of a shrink on the rank-1 mean fish: shrink %TL new TL new mass mass lost % SA/V gain % -------------------------------------------------------- 0.5 67.16 7.439 1.49 0.50 1.0 66.83 7.327 2.97 1.01 1.4 66.55 7.239 4.14 1.42 2.0 66.15 7.107 5.88 2.04 2.5 65.81 6.999 7.31 2.56 3.0 65.47 6.892 8.73 3.09 4.0 64.80 6.681 11.53 4.17 5.0 64.12 6.474 14.26 5.26 ======================================================================== PART B1. Oxygen solubility falls while demand rises ======================================================================== Salinity 35, air saturation, 1 atm. Reference state: TL 67.50 mm at 28.5 degC. T (degC) O2 (umol/kg) O2 (mg/L) supply rel demand rel S/D ratio ---------------------------------------------------------------------- 26.0 203.38 6.658 1.0413 0.8409 1.2384 27.0 200.08 6.550 1.0244 0.9013 1.1367 28.5 195.31 6.393 1.0000 1.0000 1.0000 29.5 192.24 6.293 0.9843 1.0718 0.9184 30.5 189.26 6.195 0.9690 1.1487 0.8436 31.5 186.35 6.100 0.9541 1.2311 0.7750 32.5 183.53 6.008 0.9397 1.3195 0.7121 33.5 180.77 5.917 0.9255 1.4142 0.6545 34.0 179.41 5.873 0.9186 1.4641 0.6274 Over the 4.0 degC anomaly, from 28.5 to 32.5 degC: dissolved oxygen falls 6.03% (195.31 to 183.53 umol/kg, 6.393 to 6.008 mg/L) metabolic demand rises 31.95% at Q10 = 2.0 supply-to-demand ratio for a fish that does not change size: 0.7121 that is a squeeze of 28.79% on the oxygen budget. ======================================================================== PART B2. How much warming does shrinking actually offset? ======================================================================== Two currencies. RATIO asks how much warming is cancelled if the fish is limited by supply per unit demand. ABSOLUTE asks the same question if the fish is limited by its total oxygen and food bill. Same model, same shrink, answers an order of magnitude apart. shrink %TL ratio gain % offset degC demand cut % offset degC (d_g=0.70) (ratio) (M^0.75) (absolute) ----------------------------------------------------------------- 0.5 0.075 0.0088 1.12 0.1627 1.0 0.151 0.0177 2.24 0.3262 1.4 0.212 0.0248 3.12 0.4577 2.0 0.304 0.0355 4.44 0.6558 2.5 0.380 0.0445 5.54 0.8218 3.0 0.458 0.0536 6.62 0.9887 4.0 0.614 0.0718 8.78 1.3251 5.0 0.772 0.0903 10.90 1.6650 A typical 1.4% shrink offsets 0.0248 degC in the ratio currency and 0.4577 degC in the absolute currency. The absolute answer is 18.4 times larger. Against the 4.0 degC anomaly, the ratio currency recovers 0.62% of the heat; the absolute currency recovers 11.44%. To cancel the full 4.0 degC in the RATIO currency, a fish would have to shrink to 10.40% of its length: 67.50 mm down to 7.02 mm. That is a 89.60% reduction, which is not a fish. To cancel it in the ABSOLUTE currency: shrink to 88.41% of length, 59.67 mm. That is a 11.59% reduction. Still far beyond anything observed, but within the same order of magnitude as a real animal. ======================================================================== PART B3. Sensitivity to the gill-area exponent ======================================================================== d_g is the exponent in gill surface area proportional to mass^d_g. The whole geometric argument lives or dies on whether it is below 0.75. d_g ratio exponent gain at 1.4% offset degC verdict -------------------------------------------------------------------------- 0.60 -0.450 0.636 0.0744 shrinking helps 0.65 -0.300 0.424 0.0496 shrinking helps 0.67 -0.240 0.339 0.0397 shrinking helps 0.70 -0.150 0.212 0.0248 shrinking helps 0.75 0.000 0.000 -0.0000 shrinking is neutral 0.80 0.150 -0.211 -0.0248 shrinking hurts 0.85 0.300 -0.422 -0.0496 shrinking hurts 0.90 0.450 -0.632 -0.0744 shrinking hurts At d_g = 0.75 the two exponents cancel and body size drops out of the ratio entirely. Above it, the sign flips and a smaller fish is worse off. Published intraspecific gill exponents straddle 0.75. This is the single softest joint in the whole calculation. ======================================================================== PART B4. The length that restores balance, temperature by temperature ======================================================================== T (degC) S/D at 67.50 restoring TL shrink needed % ------------------------------------------------------- 29.0 0.9583 50.8116 24.72 29.5 0.9184 38.2618 43.32 30.0 0.8802 28.8207 57.30 30.5 0.8436 21.7160 67.83 31.0 0.8085 16.3676 75.75 31.5 0.7750 12.3400 81.72 32.0 0.7429 9.3061 86.21 32.5 0.7121 7.0200 89.60 Read the last column as a reductio. Half a degree of warming already asks for a shrink of tens of percent under these assumptions. The fish shrank by one or two. Whatever shrinking is doing, the gill-geometry ratio on its own is not a sufficient explanation. ======================================================================== PART C1. Building the simulated population ======================================================================== pairs: 3000, individuals: 6000 shrank at least once: 4497 (74.95%) rank 1: 2122 (70.73%), target 71% rank 2: 2375 (79.17%), target 79% pairs where both shrank (tandem): 1683 (56.10% of pairs) shrink magnitude among shrinkers: mean 1.396% TL, median 1.209%, max 7.124% in mm on a 67.50 mm fish: mean 0.943 mm, max 4.809 mm generator coefficients (logit scale): b0 intercept = +0.944462 (non-shrinker survival 0.72) b1 per 1% TL shrink = +0.191896 (solved so the population odds ratio is exactly 1.78) b2 tandem pair = +0.438255 (odds ratio 1.55) population OR achieved = 1.780000 drawn outcomes: 4744 survived of 6000 (79.07%) ======================================================================== PART C2. Fitted logistic regression ======================================================================== Model 1: survival ~ shrank (binary). The headline comparison. intercept +0.93406 (SE 0.05732) shrank +0.54869 (SE 0.06899) ODDS RATIO 1.7310 95% CI 1.5121 to 1.9816 Wald z 7.953, p = 1.822e-15 paper's reported effect: 78% increase, i.e. OR 1.78 recovery error: -2.75% Model 2: survival ~ shrink magnitude + tandem. The dose-response. intercept +0.92423 (SE 0.04933) -> survival 71.59% at zero shrink per 1% TL +0.16881 (SE 0.04070) -> OR 1.1839 per percentage point tandem +0.46211 (SE 0.07168) -> OR 1.5874 generator truth: b1 +0.19190, b2 +0.43825 Fitted survival probability against shrink magnitude: shrink %TL solo tandem gap (pp) ------------------------------------------ 0.0 71.59% 80.00% 8.41 0.5 73.28% 81.32% 8.04 1.0 74.90% 82.57% 7.67 1.4 76.14% 83.52% 7.37 2.0 77.93% 84.86% 6.93 2.5 79.35% 85.92% 6.56 3.0 80.70% 86.91% 6.21 4.0 83.19% 88.71% 5.52 5.0 85.42% 90.29% 4.87 HEADLINE NUMBERS at the club's default settings: non-shrinker survival 71.59% 1.4% solo shrinker survival 76.14% 1.4% tandem shrinker 83.52% fitted binary odds ratio 1.7310 ======================================================================== PART C3. Does tandem shrinking beat solo shrinking? ======================================================================== Shrinkers only, so the comparison is tandem against solo with shrink magnitude held in the model. n = 4497 shrinkers, of whom 3366 were in tandem pairs tandem coefficient +0.47676 (SE 0.08359) ODDS RATIO 1.6109 95% CI 1.3674 to 1.8976 Wald z 5.703, p = 1.175e-08 verdict: tandem beats solo at the 5% level Raw survival rates, no model: did not shrink 71.79% (n = 1503) shrank, solo 75.77% (n = 1131) shrank, tandem 83.42% (n = 3366) tandem minus solo: +7.65 percentage points ======================================================================== PART C4. Could 134 fish have seen this with a binary endpoint? ======================================================================== 4000 simulated studies of 134 fish, non-shrinker survival 0.87, true OR 1.78. mean deaths per study: 12.18 (9.09% mortality; the paper observed 11, 8.21%) usable 2x2 tables: 3951 of 4000 median estimated OR: 1.759 studies whose 95% CI excluded 1: 611 (15.28% power) With a plain survived-or-not endpoint, a study this size would almost never reach significance. The paper did not use a binary endpoint. It used repeated measures and time-to-event models, which carry far more information per fish. Read this number as a floor on the design's power, not as a criticism of it. ======================================================================== SUMMARY OF QUOTED FIGURES ======================================================================== A1 mass at TL 27.00 mm 0.483 g A2 mass at TL 83.50 mm 14.29 g A3 SA/V at TL 67.50 mm 0.35672 /mm A4 SA/V gain from a 1.4% shrink 1.420% A5 mass lost in a 1.4% shrink 4.14% B1 O2 at 28.5 degC 195.31 umol/kg (6.393 mg/L) B2 O2 at 32.5 degC 183.53 umol/kg (6.008 mg/L) B3 solubility drop over 4 degC 6.03% B4 demand rise over 4 degC (Q10 2.0) 31.95% B5 supply/demand ratio at 32.5 degC 0.7121 B6 offset of a 1.4% shrink, ratio 0.0248 degC B7 offset of a 1.4% shrink, absolute 0.4577 degC B8 absolute demand cut, 1.4% shrink 3.12% B9 shrink needed to cancel 4 degC, ratio 89.60% (to 7.02 mm) B10 gill exponent where sign flips 0.75 C1 fitted binary odds ratio 1.7310 [1.5121, 1.9816] C2 non-shrinker survival, fitted 71.59% C3 1.4% solo shrinker survival 76.14% C4 1.4% tandem shrinker survival 83.52% C5 tandem odds ratio among shrinkers 1.6109, p = 1.175e-08 C6 raw tandem minus solo +7.65 pp C7 power of a 134-fish binary study 15.28% seed = 20250521; numpy 2.4.2