============================================================================== SECTION 1. VALIDATION AGAINST CLOSED FORM ============================================================================== Nothing below this line is worth reading unless the two regression routines are right. So they are checked first, on a five-point dataset small enough to do by hand, against closed-form answers computed independently with exact rational arithmetic (Python's fractions module, no floating point at all). Hand dataset: x = [0, 1, 2, 3, 4] y = [1, 3, 2, 5, 4] Exact sums (rational): Sxx = 10 Syy = 10 Sxy = 8 Exact OLS slope b = Sxy/Sxx = 4/5 = 0.800000000000 Exact OLS intercept a = ybar - b*xbar = 7/5 = 1.400000000000 Exact SSE = Syy - b*Sxy = 18/5 = 3.600000000000 Exact r^2 = Sxy^2/(Sxx*Syy) = 16/25 = 0.640000000000 Exact RMA slope = sqrt(Syy/Sxx) = sqrt(1) = 1.000000000000 Exact RMA via identity b_ols/|r| = 1.000000000000 quantity club code closed form difference -------------------------------------------------------------------------------------- OLS slope b 0.800000000000 0.800000000000 0.00e+00 OLS intercept a 1.400000000000 1.400000000000 0.00e+00 residual sum of squares 3.600000000000 3.600000000000 -1.78e-15 r^2 0.640000000000 0.640000000000 1.11e-16 SE(b) 0.346410161514 0.346410161514 -5.55e-17 RMA slope 1.000000000000 1.000000000000 0.00e+00 RMA slope via b_ols/|r| 1.000000000000 1.000000000000 0.00e+00 -------------------------------------------------------------------------------------- largest absolute discrepancy: 1.776e-15 Residuals, printed in full, club code against exact rational values: i x y club resid exact resid difference 0 0.0000 1.0000 -0.400000000000 -0.400000000000 1.11e-16 1 1.0000 3.0000 0.800000000000 0.800000000000 -2.22e-16 2 2.0000 2.0000 -1.000000000000 -1.000000000000 0.00e+00 3 3.0000 5.0000 1.200000000000 1.200000000000 -2.22e-16 4 4.0000 4.0000 -0.600000000000 -0.600000000000 3.33e-16 sum of club residuals = 0.000e+00 (must be 0 for a fit with an intercept) sum of x*resid = 4.441e-16 (must be 0: the normal equations) numpy.polyfit slope 0.800000000000 intercept 1.400000000000 numpy.linalg.lstsq slope 0.800000000000 intercept 1.400000000000 club code slope 0.800000000000 intercept 1.400000000000 max difference vs numpy: 1.110e-15 Perfectly collinear check, y = 3 + 0.75x exactly: OLS slope 0.750000000000000 RMA slope 0.750000000000000 difference 0.000e+00 r^2 = 1.000000000000000, so the OLS/RMA gap 1/|r| - 1 = 0.000e+00 Student t critical values, two-sided 95%, club code against published tables (Fisher & Yates / any standard statistical table): df club code table value difference 1 12.706205 12.706205 -2.64e-07 2 4.302653 4.302653 -2.70e-07 3 3.182446 3.182446 3.05e-07 5 2.570582 2.570582 -1.64e-07 10 2.228139 2.228139 -1.48e-07 20 2.085963 2.085963 4.47e-07 30 2.042272 2.042272 4.56e-07 60 2.000298 2.000298 -1.78e-07 120 1.979930 1.979930 4.05e-07 largest discrepancy: 4.56e-07 (table values are quoted to six decimal places, so a few units in the last place is the closest agreement the comparison can show) df = 1000 club t = 1.96233908 z(1+(z^2+1)/4v) = 1.96233626 diff 2.83e-06 df = 10000 club t = 1.96020124 z(1+(z^2+1)/4v) = 1.96020121 diff 2.82e-08 df = 100000 club t = 1.95998771 z(1+(z^2+1)/4v) = 1.95998771 diff 2.74e-10 the club value converges on the normal quantile 1.959964 from above, which is the behaviour a correct t quantile must show. F distribution check: for df1 = 1, F = t^2 exactly, so the upper tail of F must equal the two-sided tail of t. t = 2.000000 df = 10 2-sided t tail 0.0733880348 F=4.000000 tail 0.0733880348 diff 0.00e+00 t = 3.182446 df = 3 2-sided t tail 0.0500000117 F=10.127963 tail 0.0500000117 diff 0.00e+00 t = 1.500000 df = 420 2-sided t tail 0.1343657104 F=2.250000 tail 0.1343657104 diff 0.00e+00 I_0.5(1,1) = 0.500000000000000 (must be exactly 0.5) I_0.25(2,3) = 0.261718750000 (closed form 1 - (1-x)^3(1+3x) at x=0.25 = 0.261718750000) Internal consistency: the RMA confidence interval and the RMA point hypothesis test are computed by two different routes, so testing the slope value sitting exactly on the interval endpoint must return p = 0.05 to within rounding. lower endpoint b = 0.820605 p = 0.050000 |p - 0.05| = 2.78e-17 upper endpoint b = 1.018764 p = 0.050000 |p - 0.05| = 5.62e-16 VALIDATION PASSED. Both estimators reproduce closed-form answers to better than 1e-12 (largest 1.78e-15), and the t quantiles match published tables to better than 1e-5 (largest 4.56e-07). ============================================================================== SECTION 2. THE DATA ============================================================================== source : AnAge, Animal Ageing and Longevity Database, Build 15 release date : 3 July 2023 url : https://genomics.senescence.info/species/dataset.zip retrieved : 2026-09-14 cached archive: analysis/data/anage_dataset.zip sha256 : e3ddb66e32e973a79932859ba53013e8f60d957c6ec01c6eb573e3ea3018d630 cached table : analysis/data/anage_data.txt sha256 : 98867969fbd4d0bed6bab415c2715bb19079dbd7f92bdc26e5961856aa1c1519 citation : Tacutu et al. (2018) Nucleic Acids Research 46, D1083 rows in AnAge build 15 : 4645 rows carrying BOTH mass and BMR > 0 : 627 distinct species names : 627 duplicate species names : 0 mass range : 0.76 g to 3672000 g : 6.684 decades (a factor of 4.83e+06) BMR range : 9.9e-05 W to 2336 W (7.373 decades) breakdown by class: class n decades median mass g share Mammalia 422 6.185 240.0 67.3% Aves 171 4.523 37.5 27.3% Reptilia 16 3.309 415.0 2.6% Amphibia 18 2.673 16.6 2.9% TOTAL 627 6.684 131.3 100.0% lightest five and heaviest five, as a sanity check that the units are what we think they are: species class mass (g) BMR (W) Plethodon cinereus Amphibia 0.76 9.9e-05 Pseudacris crucifer Amphibia 0.9 0.00105 Suncus etruscus Mammalia 2.4 0.063 Stellula calliope Aves 3 0.06853 Archilochus colubris Aves 3.2 0.07832 Alces alces Mammalia 3.25e+05 286.85 Bos taurus Mammalia 3.47e+05 306.77 Camelus dromedarius Mammalia 4.07e+05 224.78 Trichechus manatus Mammalia 4.5e+05 70.006 Elephas maximus Mammalia 3.672e+06 2336.5 A human check: our mass-specific BMR for Homo sapiens should land near the textbook 1.2 W/kg resting. Homo sapiens: mass 70000 g, BMR 82.78 W, that is 1.18 W/kg and 1709 kcal/day ============================================================================== SECTION 3. EXPONENTS BY BOTH METHODS, WITH INTERVALS ============================================================================== group n dec r^2 method b 95% CI SE -------------------------------------------------------------------------------------------- Mammalia 422 6.18 0.9505 OLS 0.7091 [0.6936, 0.7246] 0.0079 RMA 0.7273 [0.7120, 0.7430] - Aves 171 4.52 0.9646 OLS 0.6549 [0.6358, 0.6739] 0.0097 RMA 0.6668 [0.6480, 0.6861] - Reptilia 16 3.31 0.9022 OLS 0.7272 [0.5900, 0.8644] 0.0640 RMA 0.7656 [0.6406, 0.9150] - Amphibia 18 2.67 0.7858 OLS 0.6933 [0.5015, 0.8851] 0.0905 RMA 0.7821 [0.6135, 0.9971] - Endotherms (M+A) 593 6.18 0.9426 OLS 0.6778 [0.6643, 0.6913] 0.0069 RMA 0.6982 [0.6848, 0.7118] - Ectotherms (R+A) 34 4.62 0.9134 OLS 0.7920 [0.7042, 0.8798] 0.0431 RMA 0.8287 [0.7455, 0.9212] - All four classes 627 6.68 0.8187 OLS 0.7132 [0.6869, 0.7396] 0.0134 RMA 0.7883 [0.7623, 0.8151] - -------------------------------------------------------------------------------------------- The raw sums behind the headline mammal fit, printed so the division can be done on paper: n = 422 xbar = 2.562455 (log10 grams) ybar = 0.144631 (log10 watts) Sxx = 559.2344 Sxy = 396.5571 Syy = 295.8555 b = Sxy/Sxx = 396.5571 / 559.2344 = 0.709107 r = Sxy/sqrt(Sxx*Syy) = 0.974920 b_rma = sqrt(Syy/Sxx) = sqrt(295.8555/559.2344) = 0.727349 SSE = Syy - b*Sxy = 14.6539 SE(b) = sqrt( SSE/(n-2) / Sxx ) = sqrt(0.034890/559.2344) = 0.007899 t(3/4)= (0.709107 - 0.75)/0.007899 = -5.1772 on 420 df Intercepts (log10 a, with B in watts and M in grams), and the predicted BMR of a 1 kg animal, which is what the intercept means: group OLS log a RMA log a BMR at 1 kg, W Mammalia -1.6724 -1.7192 2.8503 Aves -1.3788 -1.4009 3.8529 Reptilia -2.9380 -3.0372 0.1752 Amphibia -3.1469 -3.2537 0.0857 Endotherms (M+A) -1.5430 -1.5909 3.0937 Ectotherms (R+A) -3.1902 -3.2582 0.1534 All four classes -1.7034 -1.8783 2.7311 ============================================================================== SECTION 4. DOES THE INTERVAL CONTAIN 3/4? DOES IT CONTAIN 2/3? ============================================================================== 3/4 = 0.750000 2/3 = 0.666667 difference = 0.083333 OLS, t = (b - beta0)/SE(b), df = n-2: group b SE t(3/4) p(3/4) t(2/3) p(2/3) verdict ---------------------------------------------------------------------------------------------------- Mammalia 0.7091 0.0079 -5.18 3.5e-07 5.37 1.29e-07 excludes BOTH 3/4 and 2/3 Aves 0.6549 0.0097 -9.85 2.2e-18 -1.22 0.224 excludes 3/4, contains 2/3 Reptilia 0.7272 0.0640 -0.36 0.727 0.95 0.36 excludes NEITHER Amphibia 0.6933 0.0905 -0.63 0.54 0.29 0.772 excludes NEITHER Endotherms (M+A) 0.6778 0.0069 -10.49 9.85e-24 1.62 0.105 excludes 3/4, contains 2/3 Ectotherms (R+A) 0.7920 0.0431 0.97 0.337 2.91 0.00658 excludes 2/3, contains 3/4 All four classes 0.7132 0.0134 -2.74 0.00636 3.47 0.000558 excludes BOTH 3/4 and 2/3 RMA, exact zero-correlation test on the rescaled axis scores: group b t(3/4) p(3/4) t(2/3) p(2/3) verdict ---------------------------------------------------------------------------------------------------- Mammalia 0.7273 -2.82 0.00496 8.03 9.75e-15 excludes BOTH 3/4 and 2/3 Aves 0.6668 -8.14 8.21e-14 0.01 0.99 excludes 3/4, contains 2/3 Reptilia 0.7656 0.25 0.809 1.66 0.119 excludes NEITHER Amphibia 0.7821 0.36 0.722 1.39 0.185 excludes NEITHER Endotherms (M+A) 0.6982 -7.28 1.09e-12 4.69 3.47e-06 excludes BOTH 3/4 and 2/3 Ectotherms (R+A) 0.8287 1.92 0.0636 4.21 0.000191 excludes 2/3, contains 3/4 All four classes 0.7883 2.92 0.00359 9.88 1.68e-21 excludes BOTH 3/4 and 2/3 Distance of each fitted exponent from the two candidates, in standard errors of that fit (OLS only, since OLS is the one with a closed-form SE): group SEs from 3/4 SEs from 2/3 Mammalia -5.18 5.37 Aves -9.85 -1.22 Reptilia -0.36 0.95 Amphibia -0.63 0.29 Endotherms (M+A) -10.49 1.62 Ectotherms (R+A) 0.97 2.91 All four classes -2.74 3.47 ============================================================================== SECTION 5. HOW FAR APART THE TWO METHODS SIT, ON THE SAME DATA ============================================================================== Algebraically b_rma = b_ols / |r| exactly, so the gap depends only on the scatter. Confirm the identity numerically, then read the size off. group r^2 b_ols b_rma b_ols/|r| gap gap % ------------------------------------------------------------------------------------ Mammalia 0.9505 0.7091 0.7273 0.7273 0.0182 2.57% Aves 0.9646 0.6549 0.6668 0.6668 0.0119 1.82% Reptilia 0.9022 0.7272 0.7656 0.7656 0.0384 5.28% Amphibia 0.7858 0.6933 0.7821 0.7821 0.0888 12.81% Endotherms (M+A) 0.9426 0.6778 0.6982 0.6982 0.0203 3.00% Ectotherms (R+A) 0.9134 0.7920 0.8287 0.8287 0.0367 4.63% All four classes 0.8187 0.7132 0.7883 0.7883 0.0750 10.52% ------------------------------------------------------------------------------------ largest deviation from the identity b_rma = b_ols/|r| : 3.331e-16 Read that column again. The OLS/RMA gap for mammals is 0.0182, which is 21.9% of the whole distance between 3/4 and 2/3 (0.0833). For all four classes pooled it is 0.0750, which is 90% of that distance: bigger than the entire quantity under dispute. What r^2 would a dataset need for the two methods to agree to within a given tolerance, at b_ols = 0.70? tolerance required |r| required r^2 0.0833 0.89366 0.79862 0.0200 0.97222 0.94522 0.0100 0.98592 0.97203 0.0050 0.99291 0.98587 0.0010 0.99857 0.99715 ============================================================================== SECTION 6. BOOTSTRAP INTERVALS (seed 20260614, 20000 resamples) ============================================================================== Non-parametric bootstrap over species, 20000 resamples per group. If the analytic intervals are right, these should agree closely. They do NOT fix the phylogenetic non-independence: resampling species keeps the same shared ancestry, so both intervals are too narrow together. group method analytic 95% CI bootstrap 95% CI boot SE ------------------------------------------------------------------------------------ Mammalia OLS [0.6936, 0.7246] [0.6890, 0.7285] 0.0102 RMA [0.7120, 0.7430] [0.7085, 0.7456] 0.0095 Aves OLS [0.6358, 0.6739] [0.6360, 0.6740] 0.0096 RMA [0.6480, 0.6861] [0.6481, 0.6863] 0.0097 Reptilia OLS [0.5900, 0.8644] [0.6154, 0.8263] 0.0539 RMA [0.6406, 0.9150] [0.6608, 0.8715] 0.0546 Amphibia OLS [0.5015, 0.8851] [0.4348, 1.0067] 0.1523 RMA [0.6135, 0.9971] [0.4920, 1.0775] 0.1639 Endotherms (M+A) OLS [0.6643, 0.6913] [0.6609, 0.6942] 0.0085 RMA [0.6848, 0.7118] [0.6821, 0.7137] 0.0081 Ectotherms (R+A) OLS [0.7042, 0.8798] [0.7031, 0.8793] 0.0450 RMA [0.7455, 0.9212] [0.7499, 0.9228] 0.0456 All four classes OLS [0.6869, 0.7396] [0.6839, 0.7436] 0.0153 RMA [0.7623, 0.8151] [0.7508, 0.8272] 0.0195 Bootstrap probability that the true exponent exceeds each candidate, read straight off the resample distribution: group method P(b > 3/4) P(b > 2/3) Mammalia OLS 0.0000 1.0000 RMA 0.0072 1.0000 Aves OLS 0.0000 0.1113 RMA 0.0000 0.5073 Endotherms (M+A) OLS 0.0000 0.9041 RMA 0.0000 0.9998 All four classes OLS 0.0097 0.9991 RMA 0.9769 1.0000 ============================================================================== SECTION 7. TAXONOMIC HETEROGENEITY ============================================================================== All four classes, separate slopes against one common slope: classes : Amphibia, Aves, Mammalia, Reptilia common slope : 0.7000 (SE 0.0066) SSE separate slopes: 18.5905 on 619 df SSE common slope : 18.8942 on 622 df F(3, 619) = 3.371, p = 0.01823 verdict : slopes differ between classes Mammals against birds only (the two well-sampled classes): common slope : 0.6995 (SE 0.0065) F(1, 589) = 10.378, p = 0.001346 Pairwise slope contrasts, OLS, Welch-style on the two standard errors: pair b1 b2 b1-b2 t p -------------------------------------------------------------------------- Mammalia vs Aves 0.7091 0.6549 0.0542 4.35 1.75e-05 Mammalia vs Reptilia 0.7091 0.7272 -0.0181 -0.28 0.783 Aves vs Reptilia 0.6549 0.7272 -0.0723 -1.12 0.282 Mammalia vs Amphibia 0.7091 0.6933 0.0158 0.17 0.864 Reptilia vs Amphibia 0.7272 0.6933 0.0339 0.31 0.762 Mammalian orders with at least 8 species, fitted separately. This is the same question asked one level down. order n decades b (OLS) 95% CI r^2 ---------------------------------------------------------------------------- Artiodactyla 20 2.40 0.7533 [0.6884, 0.8182] 0.9706 Carnivora 52 3.04 0.7791 [0.7087, 0.8495] 0.9081 Chiroptera 33 2.44 0.7658 [0.6900, 0.8417] 0.9318 Cingulata 9 1.61 0.6691 [0.5130, 0.8253] 0.9362 Dasyuromorphia 20 2.94 0.7401 [0.6962, 0.7841] 0.9858 Didelphimorphia 10 1.49 0.8100 [0.7270, 0.8931] 0.9844 Diprotodontia 22 3.47 0.7064 [0.6605, 0.7522] 0.9810 Lagomorpha 8 1.44 0.7029 [0.6037, 0.8021] 0.9804 Primates 26 3.05 0.7810 [0.7006, 0.8614] 0.9437 Rodentia 155 3.56 0.6666 [0.6347, 0.6985] 0.9175 Soricomorpha 19 1.31 0.4772 [0.2657, 0.6887] 0.5714 ---------------------------------------------------------------------------- within-order slopes: n = 11 orders, mean 0.7138, sd 0.0913, range 0.4772 to 0.8100 how many of those 11 order-level intervals contain 3/4: 9 how many contain 2/3 : 5 how many contain both : 3 Order-level means (one point per mammalian order, n >= 3 species). This is the crudest possible answer to the pseudoreplication problem: it throws away nearly all the data to buy near-independence. orders used : 18 (Afrosoricida n=7, Artiodactyla n=20, Carnivora n=52, Chiroptera n=33, Cingulata n=9, Dasyuromorphia n=20, Didelphimorphia n=10, Diprotodontia n=22, Erinaceomorpha n=6, Lagomorpha n=8, Macroscelidea n=7, Monotremata n=3, Peramelemorphia n=7, Pilosa n=5, Primates n=26, Rodentia n=155, Scandentia n=3, Soricomorpha n=19) OLS slope : 0.6838 95% CI [0.5880, 0.7796] excludes NEITHER RMA slope : 0.7073 95% CI [0.6179, 0.8095] excludes NEITHER r^2 = 0.9347, n = 18, so the interval is 6.2 times wider than the species-level one. Losing independence costs precision, and that is the honest trade. ============================================================================== SECTION 8. WHAT HAPPENS WHEN YOU NARROW THE MASS WINDOW ============================================================================== The core statistical criticism of Kleiber's law is that the exponent is estimated across an enormous mass range, and that a line fitted across six decades will look straight whatever the data do locally. So: fit the same mammal data inside sliding windows and watch. Sliding window of width 2.0 decades, step 0.25 decades, mammals only. win lo win hi n b (OLS) 95% CI b (RMA) r^2 verdict -------------------------------------------------------------------------------------------------------- 0.25 2.25 194 0.5911 [0.5308, 0.6514] 0.7272 0.6607 excludes BOTH 3/4 and 2/3 0.50 2.50 223 0.6083 [0.5600, 0.6566] 0.7091 0.7360 excludes BOTH 3/4 and 2/3 0.75 2.75 239 0.6107 [0.5690, 0.6525] 0.6925 0.7779 excludes BOTH 3/4 and 2/3 1.00 3.00 254 0.6472 [0.6128, 0.6816] 0.7041 0.8449 excludes 3/4, contains 2/3 1.25 3.25 261 0.6509 [0.6195, 0.6823] 0.6997 0.8653 excludes 3/4, contains 2/3 1.50 3.50 249 0.6841 [0.6531, 0.7151] 0.7273 0.8846 excludes 3/4, contains 2/3 1.75 3.75 235 0.6899 [0.6588, 0.7210] 0.7307 0.8913 excludes 3/4, contains 2/3 2.00 4.00 217 0.7245 [0.6902, 0.7587] 0.7680 0.8899 excludes 2/3, contains 3/4 2.25 4.25 189 0.7318 [0.6895, 0.7740] 0.7881 0.8622 excludes 2/3, contains 3/4 2.50 4.50 169 0.7880 [0.7369, 0.8391] 0.8562 0.8471 excludes 2/3, contains 3/4 2.75 4.75 157 0.7873 [0.7346, 0.8400] 0.8545 0.8489 excludes 2/3, contains 3/4 3.00 5.00 130 0.8426 [0.7826, 0.9025] 0.9096 0.8580 excludes BOTH 3/4 and 2/3 3.25 5.25 109 0.8511 [0.7849, 0.9173] 0.9185 0.8586 excludes BOTH 3/4 and 2/3 3.50 5.50 86 0.8701 [0.7869, 0.9533] 0.9509 0.8373 excludes BOTH 3/4 and 2/3 3.75 5.75 64 0.7749 [0.6614, 0.8884] 0.8946 0.7502 excludes NEITHER 4.00 6.00 49 0.7554 [0.6002, 0.9106] 0.9221 0.6710 excludes NEITHER 4.25 6.25 38 0.6386 [0.4265, 0.8507] 0.8953 0.5088 excludes NEITHER 4.50 6.50 28 0.6289 [0.3713, 0.8866] 0.8967 0.4919 excludes NEITHER 4.75 6.75 20 0.6656 [0.4064, 0.9248] 0.8467 0.6179 excludes NEITHER -------------------------------------------------------------------------------------------------------- window slopes: n = 19 windows, min 0.5911, max 0.8701, range 0.2790, sd 0.0877 that range is 3.35 times the whole 3/4-minus-2/3 gap of 0.0833 windows whose interval excludes 3/4 : 10 of 19 windows whose interval excludes 2/3 : 10 of 19 windows whose interval excludes both: 6 of 19 windows whose interval excludes neither: 5 of 19 Nested truncation, mammals. Left column drops the heaviest species, right column drops the lightest. Same data, same code, one cut. keep M <= g n b 95% CI | keep M >= g n b 95% CI ------------------------------------------------------------------------------------------------------------ 100 156 0.5959 [0.5125, 0.6794] | 0.3162 422 0.7091 [0.6936, 0.7246] 316.2 224 0.6068 [0.5593, 0.6542] | 1 422 0.7091 [0.6936, 0.7246] 1000 279 0.6316 [0.5996, 0.6637] | 3.162 421 0.7099 [0.6943, 0.7255] 3162 331 0.6492 [0.6253, 0.6731] | 10 397 0.7210 [0.7052, 0.7368] 1e+04 373 0.6676 [0.6479, 0.6872] | 31.62 340 0.7419 [0.7239, 0.7598] 3.162e+04 393 0.6823 [0.6639, 0.7007] | 100 267 0.7673 [0.7445, 0.7901] 1e+05 410 0.7003 [0.6834, 0.7172] | 316.2 198 0.7959 [0.7636, 0.8282] ------------------------------------------------------------------------------------------------------------ Dropping everything above 100 g gives b = 0.5959. Dropping everything below 316 g gives b = 0.7959. Same database, same estimator, same day. The two intervals [0.5125, 0.6794] and [0.7636, 0.8282] do not overlap. ============================================================================== SECTION 9. THE SENSITIVITY CURVE: RANGE AGAINST RESOLVING POWER ============================================================================== To tell 3/4 from 2/3 you need a 95% interval narrower than 0.0833, so a half-width below 0.0417. How much mass range buys that? For each window width W we draw 400 random windows of that width from the mammal mass axis (seeded), fit OLS inside each, and report the spread of the fitted exponent and the mean interval half-width. W dec windows mean n mean b sd of b min b max b mean CIhw P(resolve) ------------------------------------------------------------------------------------------------ 0.50 400 47.1 0.7221 0.2719 -0.1257 1.5139 0.4824 0.000 0.75 400 65.5 0.7015 0.1571 0.2212 1.0784 0.2718 0.000 1.00 400 87.9 0.7026 0.1440 0.2813 1.0501 0.1755 0.000 1.50 400 117.5 0.7125 0.1260 0.3362 0.9045 0.1178 0.000 2.00 400 158.1 0.7288 0.0864 0.5919 0.8701 0.0765 0.318 2.50 400 205.9 0.7386 0.0684 0.6144 0.8391 0.0471 0.677 3.00 400 243.8 0.7461 0.0602 0.6452 0.8239 0.0332 0.790 4.00 400 316.5 0.7377 0.0352 0.6737 0.7964 0.0218 1.000 5.00 400 388.2 0.7188 0.0109 0.7069 0.7430 0.0166 1.000 6.00 400 420.0 0.7071 0.0000 0.7071 0.7071 0.0158 1.000 ------------------------------------------------------------------------------------------------ Read the 'sd of b' column as the answer to the whole ninety-year argument's statistical half. Below about two decades of mass the sampling scatter in the exponent is itself comparable to the quantity being argued over, so a study on a narrow mass range cannot settle it no matter how carefully the respirometry was done. ============================================================================== SECTION 10. CURVATURE ============================================================================== Kolokotrones et al. (2010) reported that mammalian metabolic scaling is convex on log-log axes, so that no single exponent describes it. Test: add a quadratic term and see whether it earns its degree of freedom. Mammalia (n = 422, centred at log10 M = 2.562): linear term 0.6860 +/- 0.0083 quadratic term 0.03646 +/- 0.00544 t = 6.70 p = 6.62e-11 F(1, 419) for adding curvature = 44.93, p = 6.62e-11 local slope d logB/d logM at 10 g : 0.5721 local slope at 1 kg : 0.7179 local slope at 100 kg : 0.8637 Aves (n = 171, centred at log10 M = 1.854): linear term 0.6580 +/- 0.0115 quadratic term -0.00469 +/- 0.00932 t = -0.50 p = 0.616 F(1, 168) for adding curvature = 0.25, p = 0.616 local slope d logB/d logM at 10 g : 0.6660 local slope at 1 kg : 0.6473 local slope at 100 kg : 0.6285 Endotherms (M+A) (n = 593, centred at log10 M = 2.358): linear term 0.6503 +/- 0.0076 quadratic term 0.03527 +/- 0.00481 t = 7.33 p = 7.63e-13 F(1, 590) for adding curvature = 53.72, p = 7.63e-13 local slope d logB/d logM at 10 g : 0.5545 local slope at 1 kg : 0.6956 local slope at 100 kg : 0.8367 The local-slope columns are the point. If curvature is real then asking whether 'the' exponent is 3/4 or 2/3 is asking a question the data decline to answer, because the slope depends on where you stand. ============================================================================== SECTION 11. OUR NUMBERS BESIDE PUBLISHED ONES ============================================================================== Club mammal OLS exponent: 0.7091, SE 0.0079, 95% CI [0.6936, 0.7246], n = 422 published value b our b - theirs in our SEs -------------------------------------------------------------------------------- Kleiber (1932/1947), the canonical 3/4 0.7500 -0.0409 -5.18 Rubner surface law, 2/3 0.6667 0.0424 5.37 White & Seymour (2003), 619 mammals, OLS 0.6900 0.0191 2.42 White & Seymour (2003), refined set n=469 0.6800 0.0291 3.69 Savage et al. (2004), binned mammal BMR 0.7370 -0.0279 -3.53 Hayssen & Lacy (1985), mammals 0.7000 0.0091 1.15 Clarke, Rothery & Isaac (2010), mammals 0.7000 0.0091 1.15 Capellini et al. (2010), phylogenetic 0.7500 -0.0409 -5.18 -------------------------------------------------------------------------------- Those 'in our SEs' figures assume our SE is the right one. It is not: assumption A3 fails, so treat anything under about three as agreement. ============================================================================== SECTION 12. SUMMARY ============================================================================== 1. Mammals, n = 422, 6.18 decades of mass. OLS b = 0.7091, 95% CI [0.6936, 0.7246]. excludes BOTH 3/4 and 2/3 RMA b = 0.7273, 95% CI [0.7120, 0.7430]. excludes BOTH 3/4 and 2/3 The exact midpoint of the two disputed values is 0.708333. Our mammal OLS exponent sits 0.000774 from it, which is 0.098 of one standard error. We did not arrange that and we cannot explain it. 2. Birds, n = 171. OLS b = 0.6549, 95% CI [0.6358, 0.6739]. excludes 3/4, contains 2/3 3. All four classes, n = 627, 6.68 decades. OLS b = 0.7132 [0.6869, 0.7396] excludes BOTH 3/4 and 2/3 RMA b = 0.7883 [0.7623, 0.8151] excludes BOTH 3/4 and 2/3 The two methods put the answer on OPPOSITE SIDES of 3/4 on the same 627 species. The method is not a detail. 4. Slopes differ between classes: F(3, 619) = 3.37, p = 0.0182. Mammal minus bird slope = 0.0542, which is 65% of the 3/4-to-2/3 gap. 5. Restricting the mass range moves the answer by 0.2790 across 19 two-decade windows, which is 3.3 times the quantity in dispute. 6. Neither 3/4 nor 2/3 survives as a universal constant in this compilation. Both survive as a description of some subset of it. Runtime: 16.02 s. numpy 2.4.2, Python 3.12.3 Seed 20260614 (bootstrap and window sampling only). ============================================================================== SECTION 13. NUMBERS BEHIND THE FIGURES ============================================================================== [FIG2] forest plot: group, method, b, lo, hi FIG2 Mammalia OLS 0.70911 0.69358 0.72463 FIG2 Mammalia RMA 0.72735 0.71199 0.74304 FIG2 Aves OLS 0.65488 0.63582 0.67393 FIG2 Aves RMA 0.66679 0.64801 0.68612 FIG2 Reptilia OLS 0.72720 0.58997 0.86442 FIG2 Reptilia RMA 0.76559 0.64056 0.91501 FIG2 Amphibia OLS 0.69332 0.50150 0.88514 FIG2 Amphibia RMA 0.78211 0.61347 0.99711 FIG2 Endotherms (M+A) OLS 0.67783 0.66432 0.69134 FIG2 Endotherms (M+A) RMA 0.69815 0.68477 0.71179 FIG2 Ectotherms (R+A) OLS 0.79200 0.70417 0.87983 FIG2 Ectotherms (R+A) RMA 0.82871 0.74552 0.92118 FIG2 All four classes OLS 0.71324 0.68688 0.73961 FIG2 All four classes RMA 0.78827 0.76235 0.81508 [FIG3] sliding windows: centre, n, b_ols, lo, hi, b_rma FIG3 1.250 194 0.59111 0.53082 0.65141 0.72720 FIG3 1.500 223 0.60828 0.55998 0.65658 0.70905 FIG3 1.750 239 0.61074 0.56897 0.65250 0.69247 FIG3 2.000 254 0.64720 0.61279 0.68160 0.70410 FIG3 2.250 261 0.65089 0.61947 0.68231 0.69971 FIG3 2.500 249 0.68408 0.65312 0.71505 0.72735 FIG3 2.750 235 0.68989 0.65880 0.72098 0.73074 FIG3 3.000 217 0.72446 0.69020 0.75871 0.76797 FIG3 3.250 189 0.73175 0.68954 0.77396 0.78808 FIG3 3.500 169 0.78800 0.73685 0.83914 0.85615 FIG3 3.750 157 0.78728 0.73458 0.83999 0.85448 FIG3 4.000 130 0.84258 0.78262 0.90253 0.90964 FIG3 4.250 109 0.85112 0.78493 0.91732 0.91855 FIG3 4.500 86 0.87008 0.78686 0.95331 0.95087 FIG3 4.750 64 0.77489 0.66139 0.88840 0.89463 FIG3 5.000 49 0.75538 0.60018 0.91058 0.92213 FIG3 5.250 38 0.63861 0.42650 0.85073 0.89533 FIG3 5.500 28 0.62892 0.37127 0.88657 0.89668 FIG3 5.750 20 0.66559 0.40642 0.92477 0.84672 [FIG4] sensitivity: W, sd_b, mean_halfwidth, p_resolve, mean_n FIG4 0.50 0.27186 0.48238 0.0000 47.1 FIG4 0.75 0.15713 0.27181 0.0000 65.5 FIG4 1.00 0.14400 0.17552 0.0000 87.9 FIG4 1.50 0.12595 0.11778 0.0000 117.5 FIG4 2.00 0.08641 0.07651 0.3175 158.1 FIG4 2.50 0.06844 0.04710 0.6775 205.9 FIG4 3.00 0.06022 0.03319 0.7900 243.8 FIG4 4.00 0.03520 0.02175 1.0000 316.5 FIG4 5.00 0.01086 0.01661 1.0000 388.2 FIG4 6.00 0.00000 0.01576 1.0000 420.0 [FIG5] curvature, mammals: local slope at each decade of mass FIG5 0.50 0.53560 0.48262 0.58858 FIG5 0.75 0.55383 0.50596 0.60170 FIG5 1.00 0.57206 0.52924 0.61488 FIG5 1.25 0.59029 0.55244 0.62813 FIG5 1.50 0.60851 0.57552 0.64150 FIG5 1.75 0.62674 0.59843 0.65505 FIG5 2.00 0.64497 0.62105 0.66889 FIG5 2.25 0.66320 0.64321 0.68318 FIG5 2.50 0.68142 0.66457 0.69828 FIG5 2.75 0.69965 0.68462 0.71468 FIG5 3.00 0.71788 0.70288 0.73287 FIG5 3.25 0.73611 0.71935 0.75287 FIG5 3.50 0.75434 0.73448 0.77419 FIG5 3.75 0.77256 0.74880 0.79632 FIG5 4.00 0.79079 0.76265 0.81893 FIG5 4.25 0.80902 0.77621 0.84183 FIG5 4.50 0.82725 0.78958 0.86491 FIG5 4.75 0.84547 0.80284 0.88810 FIG5 5.00 0.86370 0.81602 0.91138 FIG5 5.25 0.88193 0.82914 0.93472 FIG5 5.50 0.90016 0.84222 0.95810 FIG5 5.75 0.91838 0.85526 0.98151 FIG5 6.00 0.93661 0.86828 1.00494 FIG5 6.25 0.95484 0.88128 1.02840 FIG5 6.50 0.97307 0.89426 1.05187 [FIG1] scatter: the full point cloud, class, log10 mass, log10 BMR FIG1N 627 FIG1 Amph 1.3345 -2.0814 FIG1 Amph 1.2833 -2.1561 FIG1 Amph 2.0212 -1.5186 FIG1 Amph 0.8432 -2.3635 FIG1 Amph -0.0458 -2.9788 FIG1 Amph 1.8035 -1.9208 FIG1 Amph 1.7093 -1.8729 FIG1 Amph 1.3345 -2.2396 FIG1 Amph 0.8921 -2.3809 FIG1 Amph 1.1072 -2.3010 FIG1 Amph 0.5065 -2.7077 FIG1 Amph 2.2989 -2.0438 FIG1 Amph 2.5539 -1.5969 FIG1 Amph 1.1492 -2.3862 FIG1 Amph -0.1192 -4.0044 FIG1 Amph 0.6232 -2.7190 FIG1 Amph 1.2810 -2.0670 FIG1 Amph 1.0864 -2.2857 FIG1 Aves 2.1303 -0.0215 FIG1 Aves 3.4771 0.6984 FIG1 Aves 2.8182 0.3243 FIG1 Aves 3.4564 0.7151 FIG1 Aves 2.8142 0.3737 FIG1 Aves 2.6513 0.3516 FIG1 Aves 2.7435 0.5934 FIG1 Aves 2.3979 0.2263 FIG1 Aves 2.8591 0.4554 FIG1 Aves 3.0086 0.6094 FIG1 Aves 2.9562 0.5572 FIG1 Aves 1.6522 -0.3593 FIG1 Aves 0.5051 -1.1061 FIG1 Aves 0.6812 -0.9300 FIG1 Aves 0.5798 -1.1641 FIG1 Aves 0.5682 -1.1641 FIG1 Aves 0.4771 -1.1641 FIG1 Aves 3.3766 0.6052 FIG1 Aves 1.8573 -0.3545 FIG1 Aves 1.5441 -0.8052 FIG1 Aves 2.9805 0.8339 FIG1 Aves 2.9952 0.8339 FIG1 Aves 1.5563 -0.3809 FIG1 Aves 2.6522 0.4639 FIG1 Aves 2.1421 -0.1034 FIG1 Aves 1.9917 -0.0849 FIG1 Aves 3.0000 0.6828 FIG1 Aves 2.6345 0.3535 FIG1 Aves 2.6425 0.4650 FIG1 Aves 2.8814 0.5361 FIG1 Aves 1.9542 -0.0362 FIG1 Aves 2.1951 0.0278 FIG1 Aves 2.6335 0.3345 FIG1 Aves 2.1173 -0.0514 FIG1 Aves 2.6484 0.2292 FIG1 Aves 2.1875 0.0561 FIG1 Aves 2.0899 -0.1331 FIG1 Aves 1.5353 -0.4225 FIG1 Aves 2.0477 -0.0768 FIG1 Aves 2.4544 0.1662 FIG1 Aves 2.3181 0.1144 FIG1 Aves 2.1007 -0.1221 FIG1 Aves 2.8089 0.3783 FIG1 Aves 1.9868 -0.0501 FIG1 Aves 3.0535 0.6952 FIG1 Aves 3.4330 0.7785 FIG1 Aves 2.7536 0.4582 FIG1 Aves 2.6998 0.3381 FIG1 Aves 3.6031 1.0725 FIG1 Aves 2.6149 0.3094 FIG1 Aves 1.3404 -0.5901 FIG1 Aves 1.0334 -0.6925 FIG1 Aves 1.0607 -0.6615 FIG1 Aves 1.0969 -0.5986 FIG1 Aves 0.9494 -0.7011 FIG1 Aves 0.7404 -0.9123 FIG1 Aves 1.5011 -0.1415 FIG1 Aves 1.4150 -0.5040 FIG1 Aves 1.4065 -0.2426 FIG1 Aves 1.8603 -0.0193 FIG1 Aves 1.6128 -0.2871 FIG1 Aves 1.5051 -0.4018 FIG1 Aves 3.0803 0.7455 FIG1 Aves 2.7143 0.5254 FIG1 Aves 2.8062 0.5874 FIG1 Aves 2.5911 0.4221 FIG1 Aves 2.1847 0.1461 FIG1 Aves 2.1673 0.1290 FIG1 Aves 2.2011 0.0777 FIG1 Aves 2.1816 0.1676 FIG1 Aves 1.4281 -0.3605 FIG1 Aves 1.4314 -0.3904 FIG1 Aves 1.2455 -0.5229 FIG1 Aves 1.0128 -0.6668 FIG1 Aves 1.1461 -0.5394 FIG1 Aves 1.3345 -0.4461 FIG1 Aves 1.3222 -0.4283 FIG1 Aves 1.3222 -0.4168 FIG1 Aves 1.4378 -0.4698 FIG1 Aves 1.3096 -0.5075 FIG1 Aves 1.7364 -0.1370 FIG1 Aves 1.1303 -0.5229 FIG1 Aves 1.5955 -0.2168 FIG1 Aves 1.4742 -0.3280 FIG1 Aves 1.1239 -0.6381 FIG1 Aves 1.1461 -0.5224 FIG1 Aves 1.1335 -0.3864 FIG1 Aves 1.2553 -0.5006 FIG1 Aves 1.1335 -0.6330 FIG1 Aves 1.5740 -0.2965 FIG1 Aves 1.6284 -0.1996 FIG1 Aves 1.4314 -0.4168 FIG1 Aves 1.8597 -0.0883 FIG1 Aves 1.2765 -0.5229 FIG1 Aves 1.2945 -0.4698 FIG1 Aves 1.2553 -0.5229 FIG1 Aves 1.1673 -0.5901 FIG1 Aves 1.2455 -0.5513 FIG1 Aves 1.0682 -0.6345 FIG1 Aves 1.1584 -0.6062 FIG1 Aves 1.0128 -0.5986 FIG1 Aves 1.0253 -0.7539 FIG1 Aves 0.9685 -0.8189 FIG1 Aves 0.9138 -0.8953 FIG1 Aves 1.2718 -0.5468 FIG1 Aves 1.1072 -0.7003 FIG1 Aves 1.2788 -0.6113 FIG1 Aves 0.8451 -0.9123 FIG1 Aves 0.9731 -0.7305 FIG1 Aves 1.0792 -0.6668 FIG1 Aves 1.0607 -0.7224 FIG1 Aves 0.9912 -0.7948 FIG1 Aves 0.9912 -0.8085 FIG1 Aves 1.0792 -0.7459 FIG1 Aves 0.8921 -0.8225 FIG1 Aves 1.1399 -0.7457 FIG1 Aves 1.0645 -0.7016 FIG1 Aves 1.2553 -0.5289 FIG1 Aves 1.1732 -0.6757 FIG1 Aves 1.2810 -0.6021 FIG1 Aves 1.6684 -0.1342 FIG1 Aves 1.6405 -0.1735 FIG1 Aves 1.2014 -0.6556 FIG1 Aves 1.3324 -0.5621 FIG1 Aves 1.0755 -0.7111 FIG1 Aves 1.3054 -0.5560 FIG1 Aves 1.4166 -0.4737 FIG1 Aves 1.5224 -0.3418 FIG1 Aves 1.2430 -0.6975 FIG1 Aves 0.9138 -0.7825 FIG1 Aves 1.0294 -0.6819 FIG1 Aves 1.2253 -0.4881 FIG1 Aves 0.7404 -0.7352 FIG1 Aves 2.0390 0.0825 FIG1 Aves 1.8751 -0.0570 FIG1 Aves 1.3945 -0.3809 FIG1 Aves 1.0253 -0.7011 FIG1 Aves 0.9542 -0.6716 FIG1 Aves 1.4393 -0.3737 FIG1 Aves 1.7634 -0.1415 FIG1 Aves 1.7980 -0.1391 FIG1 Aves 1.1430 -0.5891 FIG1 Aves 1.0899 -0.7459 FIG1 Aves 1.5302 -0.4157 FIG1 Aves 1.3345 -0.4623 FIG1 Aves 1.5527 -0.3593 FIG1 Aves 2.7782 0.4379 FIG1 Aves 1.8633 -0.1325 FIG1 Aves 1.3365 -0.4168 FIG1 Aves 3.9101 1.3089 FIG1 Aves 3.4017 0.8729 FIG1 Aves 3.5744 0.9310 FIG1 Aves 1.9395 -0.2891 FIG1 Aves 2.4880 0.1726 FIG1 Aves 2.1367 -0.1439 FIG1 Aves 2.8899 0.5339 FIG1 Aves 2.4293 0.0962 FIG1 Aves 1.4014 -0.5178 FIG1 Aves 1.9053 -0.2849 FIG1 Aves 4.1229 1.4131 FIG1 Aves 2.0934 -0.1844 FIG1 Aves 2.4014 -0.0205 FIG1 Aves 3.3066 0.6258 FIG1 Aves 3.1614 0.7197 FIG1 Aves 2.2201 -0.2321 FIG1 Aves 1.6618 -0.5860 FIG1 Aves 2.7566 0.4273 FIG1 Aves 5.0000 1.8428 FIG1 Aves 3.0170 0.5086 FIG1 Aves 3.1239 0.7489 FIG1 Aves 3.1103 0.7420 FIG1 Mamm 2.0660 -0.1249 FIG1 Mamm 0.8388 -1.3665 FIG1 Mamm 2.0660 -0.4202 FIG1 Mamm 1.6493 -0.5017 FIG1 Mamm 1.6435 -0.6144 FIG1 Mamm 2.6310 -0.2418 FIG1 Mamm 2.8129 -0.1373 FIG1 Mamm 4.5413 1.7073 FIG1 Mamm 5.5403 2.4868 FIG1 Mamm 5.2934 2.3619 FIG1 Mamm 5.0000 2.1730 FIG1 Mamm 3.6325 1.0779 FIG1 Mamm 4.5051 1.6666 FIG1 Mamm 4.8263 2.0595 FIG1 Mamm 3.6232 1.0032 FIG1 Mamm 3.9823 1.3143 FIG1 Mamm 5.1505 2.2792 FIG1 Mamm 5.6096 2.3518 FIG1 Mamm 5.0607 2.1730 FIG1 Mamm 5.5119 2.4577 FIG1 Mamm 4.3324 1.6660 FIG1 Mamm 4.8261 2.0509 FIG1 Mamm 4.7914 2.0915 FIG1 Mamm 4.9294 2.0779 FIG1 Mamm 5.1303 2.0177 FIG1 Mamm 4.3118 1.5207 FIG1 Mamm 3.2083 0.6887 FIG1 Mamm 3.7589 0.6900 FIG1 Mamm 4.0064 1.2883 FIG1 Mamm 3.8876 1.3331 FIG1 Mamm 3.7359 0.9295 FIG1 Mamm 3.9420 1.5186 FIG1 Mamm 3.5563 0.8845 FIG1 Mamm 3.2477 0.6927 FIG1 Mamm 3.2477 0.6944 FIG1 Mamm 3.6609 1.1377 FIG1 Mamm 3.0642 0.4302 FIG1 Mamm 3.3541 0.7009 FIG1 Mamm 4.5849 1.7908 FIG1 Mamm 4.0212 1.2397 FIG1 Mamm 3.5563 0.7183 FIG1 Mamm 3.0052 0.1584 FIG1 Mamm 3.9731 1.3701 FIG1 Mamm 4.9912 1.9758 FIG1 Mamm 4.7024 1.7953 FIG1 Mamm 5.1396 2.1266 FIG1 Mamm 4.5705 1.6931 FIG1 Mamm 3.9243 0.9863 FIG1 Mamm 2.7157 0.3428 FIG1 Mamm 2.7860 0.3518 FIG1 Mamm 2.9294 0.2378 FIG1 Mamm 4.5353 1.5045 FIG1 Mamm 3.8978 1.0631 FIG1 Mamm 2.7952 0.2238 FIG1 Mamm 3.4698 0.8332 FIG1 Mamm 4.4287 1.9933 FIG1 Mamm 4.1038 1.5019 FIG1 Mamm 4.0000 1.3997 FIG1 Mamm 2.9852 0.5538 FIG1 Mamm 2.9638 0.6021 FIG1 Mamm 4.0434 1.2213 FIG1 Mamm 2.0986 0.1059 FIG1 Mamm 2.3522 0.1284 FIG1 Mamm 2.8195 0.4349 FIG1 Mamm 3.9542 1.1779 FIG1 Mamm 3.6304 0.7455 FIG1 Mamm 4.7324 2.0740 FIG1 Mamm 4.4378 1.8650 FIG1 Mamm 3.1072 0.5486 FIG1 Mamm 3.5647 0.8282 FIG1 Mamm 3.5938 0.7475 FIG1 Mamm 3.3651 0.6329 FIG1 Mamm 3.0645 0.4130 FIG1 Mamm 3.6850 1.0182 FIG1 Mamm 4.8258 1.6727 FIG1 Mamm 4.1547 1.1054 FIG1 Mamm 3.3032 0.4893 FIG1 Mamm 3.2302 0.6221 FIG1 Mamm 3.5162 0.7430 FIG1 Mamm 4.4548 2.0321 FIG1 Mamm 0.8921 -1.0915 FIG1 Mamm 2.1004 -0.1945 FIG1 Mamm 1.0374 -0.8386 FIG1 Mamm 1.1239 -0.9318 FIG1 Mamm 1.7853 -0.3979 FIG1 Mamm 1.6637 -0.4449 FIG1 Mamm 1.8457 -0.2204 FIG1 Mamm 1.1732 -0.6198 FIG1 Mamm 1.4683 -0.7122 FIG1 Mamm 0.9823 -0.7852 FIG1 Mamm 1.0682 -1.0862 FIG1 Mamm 1.5250 -0.5735 FIG1 Mamm 1.9253 -0.2526 FIG1 Mamm 1.3404 -0.6021 FIG1 Mamm 1.3304 -0.6253 FIG1 Mamm 1.5705 -0.5817 FIG1 Mamm 2.4947 0.1495 FIG1 Mamm 2.7499 0.2101 FIG1 Mamm 2.7167 0.2090 FIG1 Mamm 2.7767 0.2475 FIG1 Mamm 2.2882 -0.1518 FIG1 Mamm 2.4057 -0.1232 FIG1 Mamm 2.5587 0.1313 FIG1 Mamm 3.0104 0.6519 FIG1 Mamm 2.1644 -0.1649 FIG1 Mamm 1.3424 -0.9830 FIG1 Mamm 1.1239 -0.9469 FIG1 Mamm 0.7634 -1.2924 FIG1 Mamm 0.5682 -1.5686 FIG1 Mamm 1.0755 -1.3979 FIG1 Mamm 1.3979 -0.7011 FIG1 Mamm 0.6990 -1.3279 FIG1 Mamm 1.0086 -1.0862 FIG1 Mamm 3.6087 0.6823 FIG1 Mamm 3.3324 0.4939 FIG1 Mamm 3.0453 0.2322 FIG1 Mamm 3.6571 0.6540 FIG1 Mamm 3.5332 0.6679 FIG1 Mamm 3.9133 0.8389 FIG1 Mamm 4.6550 1.2277 FIG1 Mamm 3.0645 0.0689 FIG1 Mamm 3.2405 0.3408 FIG1 Mamm 1.4116 -0.8508 FIG1 Mamm 1.6675 -0.5986 FIG1 Mamm 1.3979 -0.7235 FIG1 Mamm 1.8254 -0.4547 FIG1 Mamm 1.9590 -0.5850 FIG1 Mamm 2.0000 -0.3575 FIG1 Mamm 3.1228 0.4758 FIG1 Mamm 2.7566 0.1764 FIG1 Mamm 3.2509 0.4972 FIG1 Mamm 2.9756 0.3541 FIG1 Mamm 2.1867 -0.1586 FIG1 Mamm 0.9590 -1.2366 FIG1 Mamm 0.9445 -1.1871 FIG1 Mamm 1.0253 -1.1739 FIG1 Mamm 0.8513 -1.2007 FIG1 Mamm 1.6345 -0.8182 FIG1 Mamm 3.7872 0.9377 FIG1 Mamm 1.2041 -0.9172 FIG1 Mamm 1.3139 -0.8928 FIG1 Mamm 2.6417 -0.0424 FIG1 Mamm 2.5349 0.0770 FIG1 Mamm 2.9734 0.4461 FIG1 Mamm 3.0949 0.5198 FIG1 Mamm 3.4543 0.7242 FIG1 Mamm 2.9096 0.3551 FIG1 Mamm 2.0864 -0.2620 FIG1 Mamm 2.5263 0.0584 FIG1 Mamm 1.9614 -0.4365 FIG1 Mamm 2.0170 -0.4750 FIG1 Mamm 2.8756 0.2755 FIG1 Mamm 1.1461 -1.0757 FIG1 Mamm 1.6464 -0.6882 FIG1 Mamm 1.8116 -0.5072 FIG1 Mamm 3.8426 0.9009 FIG1 Mamm 3.4249 0.6766 FIG1 Mamm 3.6809 0.8910 FIG1 Mamm 4.4720 1.5193 FIG1 Mamm 4.4548 1.4963 FIG1 Mamm 3.4317 0.6716 FIG1 Mamm 2.2201 -0.2411 FIG1 Mamm 2.1116 -0.2865 FIG1 Mamm 3.6284 0.7973 FIG1 Mamm 3.2996 0.5798 FIG1 Mamm 3.6751 0.7592 FIG1 Mamm 3.4502 0.7766 FIG1 Mamm 3.1414 0.5536 FIG1 Mamm 3.0077 0.4958 FIG1 Mamm 3.0193 0.4076 FIG1 Mamm 3.0571 0.5039 FIG1 Mamm 2.9341 0.3560 FIG1 Mamm 1.0000 -0.7905 FIG1 Mamm 4.4369 1.2041 FIG1 Mamm 2.6532 -0.0820 FIG1 Mamm 2.8581 0.4496 FIG1 Mamm 2.9152 0.2871 FIG1 Mamm 3.0840 0.3863 FIG1 Mamm 2.6004 -0.0731 FIG1 Mamm 2.6547 -0.2007 FIG1 Mamm 3.2053 0.5879 FIG1 Mamm 3.3906 0.6950 FIG1 Mamm 2.6355 0.3314 FIG1 Mamm 3.2050 0.8266 FIG1 Mamm 3.3617 0.8642 FIG1 Mamm 3.4792 0.9265 FIG1 Mamm 3.4020 0.8864 FIG1 Mamm 3.3360 0.8689 FIG1 Mamm 2.8369 0.3990 FIG1 Mamm 2.0374 -0.0306 FIG1 Mamm 1.6561 -0.6126 FIG1 Mamm 1.6981 -0.5186 FIG1 Mamm 1.6675 -0.5376 FIG1 Mamm 1.6902 -0.5406 FIG1 Mamm 1.7243 -0.4989 FIG1 Mamm 1.5911 -0.5346 FIG1 Mamm 2.3181 -0.0696 FIG1 Mamm 3.0130 0.2858 FIG1 Mamm 3.4637 0.3668 FIG1 Mamm 4.0737 0.8124 FIG1 Mamm 2.9477 0.3531 FIG1 Mamm 2.6314 -0.0773 FIG1 Mamm 3.1906 0.5054 FIG1 Mamm 2.8555 0.0927 FIG1 Mamm 2.8420 0.2792 FIG1 Mamm 2.8096 0.2463 FIG1 Mamm 3.0953 0.3802 FIG1 Mamm 5.2492 2.2173 FIG1 Mamm 4.2017 0.8403 FIG1 Mamm 3.5608 0.5714 FIG1 Mamm 2.3802 -0.1965 FIG1 Mamm 3.6026 0.5901 FIG1 Mamm 4.4857 1.1627 FIG1 Mamm 3.5893 0.7096 FIG1 Mamm 3.5441 0.7003 FIG1 Mamm 2.9612 0.3978 FIG1 Mamm 3.6693 1.0593 FIG1 Mamm 2.2788 -0.0716 FIG1 Mamm 2.0441 -0.2226 FIG1 Mamm 2.3522 0.1156 FIG1 Mamm 2.9226 0.6463 FIG1 Mamm 3.9370 1.2850 FIG1 Mamm 4.0263 1.2314 FIG1 Mamm 3.4771 0.7751 FIG1 Mamm 4.1028 1.3242 FIG1 Mamm 2.4771 0.0366 FIG1 Mamm 2.4175 0.0810 FIG1 Mamm 2.2304 -0.5452 FIG1 Mamm 2.2343 -0.1169 FIG1 Mamm 1.7952 -0.4295 FIG1 Mamm 2.9972 0.4141 FIG1 Mamm 3.1186 0.5941 FIG1 Mamm 4.8451 1.9179 FIG1 Mamm 3.5250 0.5726 FIG1 Mamm 3.3755 0.6273 FIG1 Mamm 2.3139 -0.1361 FIG1 Mamm 2.4533 -0.1463 FIG1 Mamm 3.0525 0.1772 FIG1 Mamm 2.9861 0.2882 FIG1 Mamm 2.0531 -0.3665 FIG1 Mamm 2.2380 -0.0804 FIG1 Mamm 6.5649 3.3686 FIG1 Mamm 2.8493 0.2769 FIG1 Mamm 2.1183 -0.3788 FIG1 Mamm 1.8971 -0.4559 FIG1 Mamm 2.2856 -0.1959 FIG1 Mamm 1.9469 -0.3665 FIG1 Mamm 1.5478 -0.8928 FIG1 Mamm 3.4200 0.5283 FIG1 Mamm 3.3902 0.6138 FIG1 Mamm 2.8893 0.1711 FIG1 Mamm 2.8056 0.3284 FIG1 Mamm 2.5079 0.1682 FIG1 Mamm 4.4214 1.5653 FIG1 Mamm 2.9036 0.3034 FIG1 Mamm 2.6402 0.1173 FIG1 Mamm 3.8321 1.0262 FIG1 Mamm 1.5763 -0.1707 FIG1 Mamm 1.9763 -0.2125 FIG1 Mamm 0.8573 -1.0223 FIG1 Mamm 2.5626 0.0973 FIG1 Mamm 1.7543 -0.3382 FIG1 Mamm 1.2430 -0.7773 FIG1 Mamm 1.6042 -0.3686 FIG1 Mamm 1.9031 0.0298 FIG1 Mamm 2.0342 -0.1612 FIG1 Mamm 1.3979 -0.4353 FIG1 Mamm 1.3404 -0.4647 FIG1 Mamm 1.6415 -0.3497 FIG1 Mamm 1.5119 -0.4168 FIG1 Mamm 1.6830 -0.3872 FIG1 Mamm 1.5159 -0.2472 FIG1 Mamm 1.5821 -0.3686 FIG1 Mamm 1.3945 -0.5157 FIG1 Mamm 1.8096 -0.1543 FIG1 Mamm 1.2504 -0.5591 FIG1 Mamm 1.3324 -0.5058 FIG1 Mamm 1.4346 -0.4935 FIG1 Mamm 1.4216 -0.2823 FIG1 Mamm 2.2494 -0.1337 FIG1 Mamm 2.4091 0.0615 FIG1 Mamm 2.0512 -0.3372 FIG1 Mamm 1.2900 -0.8210 FIG1 Mamm 2.9896 0.6398 FIG1 Mamm 1.2810 -0.7825 FIG1 Mamm 1.6721 -0.5735 FIG1 Mamm 1.3118 -0.7905 FIG1 Mamm 1.3222 -0.7620 FIG1 Mamm 1.3483 -0.6716 FIG1 Mamm 1.3118 -0.6596 FIG1 Mamm 1.0792 -0.9208 FIG1 Mamm 1.5198 -0.5129 FIG1 Mamm 1.3324 -0.4260 FIG1 Mamm 1.4955 -0.5045 FIG1 Mamm 1.6721 -0.4776 FIG1 Mamm 1.4886 -0.5406 FIG1 Mamm 1.8960 -0.3686 FIG1 Mamm 0.9542 -0.8861 FIG1 Mamm 1.0792 -0.7595 FIG1 Mamm 2.2030 0.0354 FIG1 Mamm 2.0828 -0.2140 FIG1 Mamm 3.9617 1.1853 FIG1 Mamm 3.5853 1.0221 FIG1 Mamm 3.4293 0.9392 FIG1 Mamm 2.9609 0.4478 FIG1 Mamm 1.8751 -0.2882 FIG1 Mamm 2.1430 -0.1107 FIG1 Mamm 1.3385 -0.6576 FIG1 Mamm 1.0000 -0.7471 FIG1 Mamm 1.4166 -0.6596 FIG1 Mamm 2.6972 0.2430 FIG1 Mamm 2.5092 0.0618 FIG1 Mamm 3.5159 0.7095 FIG1 Mamm 3.8370 1.1359 FIG1 Mamm 2.2945 -0.1141 FIG1 Mamm 2.1553 -0.1739 FIG1 Mamm 1.9894 -0.1681 FIG1 Mamm 2.2415 -0.1778 FIG1 Mamm 1.5843 -0.6478 FIG1 Mamm 1.3711 -0.4547 FIG1 Mamm 1.2923 -0.8239 FIG1 Mamm 1.1790 -0.9872 FIG1 Mamm 1.7825 -0.4498 FIG1 Mamm 2.0245 -0.2865 FIG1 Mamm 1.8014 -0.3893 FIG1 Mamm 1.5752 -0.6091 FIG1 Mamm 1.5775 -0.6904 FIG1 Mamm 1.6794 -0.4698 FIG1 Mamm 1.0414 -0.7747 FIG1 Mamm 0.9191 -1.0132 FIG1 Mamm 0.9294 -1.2147 FIG1 Mamm 1.2833 -0.7959 FIG1 Mamm 4.0531 1.1198 FIG1 Mamm 1.6232 -0.5884 FIG1 Mamm 1.7435 -0.6198 FIG1 Mamm 1.3263 -0.4283 FIG1 Mamm 1.4518 -0.4377 FIG1 Mamm 1.3766 -0.5784 FIG1 Mamm 2.3288 -0.0419 FIG1 Mamm 1.8567 -0.3098 FIG1 Mamm 2.0273 0.0069 FIG1 Mamm 1.4728 -0.6253 FIG1 Mamm 1.4728 -0.8894 FIG1 Mamm 1.7505 -0.5834 FIG1 Mamm 2.0354 -0.3429 FIG1 Mamm 2.9542 0.4728 FIG1 Mamm 1.8116 -0.2628 FIG1 Mamm 1.7966 -0.5346 FIG1 Mamm 0.8808 -0.6968 FIG1 Mamm 0.9031 -0.8894 FIG1 Mamm 1.2553 -0.5670 FIG1 Mamm 1.5092 -0.5986 FIG1 Mamm 2.0086 -0.3242 FIG1 Mamm 1.8808 -0.3270 FIG1 Mamm 2.0374 -0.4522 FIG1 Mamm 2.3158 0.1474 FIG1 Mamm 2.0682 -0.1135 FIG1 Mamm 1.5977 -0.7471 FIG1 Mamm 1.8035 -0.5622 FIG1 Mamm 1.9390 -0.3747 FIG1 Mamm 2.1219 -0.3125 FIG1 Mamm 3.2270 0.7799 FIG1 Mamm 1.9722 -0.1506 FIG1 Mamm 1.8331 -0.5622 FIG1 Mamm 1.5740 -0.9788 FIG1 Mamm 2.3002 -0.0186 FIG1 Mamm 2.1878 -0.1457 FIG1 Mamm 2.1011 -0.2510 FIG1 Mamm 3.3617 0.6461 FIG1 Mamm 1.9736 -0.2916 FIG1 Mamm 2.3972 -0.0146 FIG1 Mamm 2.4093 -0.0711 FIG1 Mamm 3.0462 0.3725 FIG1 Mamm 1.7993 -0.2366 FIG1 Mamm 2.0504 -0.2733 FIG1 Mamm 1.8287 -0.3830 FIG1 Mamm 2.3617 -0.1284 FIG1 Mamm 2.2975 -0.0074 FIG1 Mamm 3.6330 0.9358 FIG1 Mamm 3.4232 0.5677 FIG1 Mamm 2.3495 -0.0910 FIG1 Mamm 2.7832 0.3404 FIG1 Mamm 2.7952 0.3806 FIG1 Mamm 2.6435 0.3143 FIG1 Mamm 2.3802 0.1045 FIG1 Mamm 1.7459 -0.3010 FIG1 Mamm 1.6928 -0.4572 FIG1 Mamm 1.9523 -0.0899 FIG1 Mamm 2.3416 0.2082 FIG1 Mamm 2.4679 -0.0991 FIG1 Mamm 2.4254 -0.1035 FIG1 Mamm 2.3802 -0.2013 FIG1 Mamm 2.0973 -0.3002 FIG1 Mamm 2.7230 0.2492 FIG1 Mamm 2.5366 -0.0177 FIG1 Mamm 2.1265 -0.2328 FIG1 Mamm 2.2945 -0.0675 FIG1 Mamm 1.7597 -0.6198 FIG1 Mamm 2.0899 -0.2823 FIG1 Mamm 2.4160 0.0969 FIG1 Mamm 5.2243 1.7405 FIG1 Mamm 5.6532 1.8451 FIG1 Mamm 1.3201 -0.4634 FIG1 Mamm 1.5211 -0.6055 FIG1 Mamm 1.0682 -0.7799 FIG1 Mamm 1.0334 -0.8447 FIG1 Mamm 0.8388 -0.9508 FIG1 Mamm 0.7993 -0.7852 FIG1 Mamm 1.1173 -0.4283 FIG1 Mamm 1.2041 -0.4841 FIG1 Mamm 0.8921 -0.5735 FIG1 Mamm 0.9243 -0.4584 FIG1 Mamm 0.7160 -0.6234 FIG1 Mamm 0.9590 -0.5376 FIG1 Mamm 0.6128 -0.7471 FIG1 Mamm 0.9868 -0.5346 FIG1 Mamm 0.7160 -0.8041 FIG1 Mamm 0.3802 -1.2007 FIG1 Mamm 1.5988 -0.3947 FIG1 Mamm 1.6902 -0.2111 FIG1 Mamm 1.6812 -0.4225 FIG1 Mamm 4.6812 1.5350 FIG1 Rept 3.0330 -0.8128 FIG1 Rept 2.6335 -1.2182 FIG1 Rept 1.4983 -1.8794 FIG1 Rept 3.5480 -0.4450 FIG1 Rept 1.9741 -1.5302 FIG1 Rept 1.8420 -1.3556 FIG1 Rept 1.1959 -2.4724 FIG1 Rept 4.0924 0.1025 FIG1 Rept 4.5051 0.4369 FIG1 Rept 1.4771 -1.6861 FIG1 Rept 2.6928 -0.6722 FIG1 Rept 1.2553 -1.7696 FIG1 Rept 2.6021 -1.6271 FIG1 Rept 2.8573 -0.8517 FIG1 Rept 2.3802 -0.9367 FIG1 Rept 3.7404 -0.2373