Table 1.
Estimates of the slope (b), intercept [log(a)], and SE of the slope estimator in Taylor’s law using the theoretical formulae Eqs. 3–5 and linear regression for six probability distributions
| s() | Quadratic coefficient | |||||||||
| Probability distribution | Formula (analytic) | Formula (numeric) | Regression | Formula (analytic) | Formula (numeric) | Regression | Formula (analytic) | Formula (numeric) | Regression | Regression |
| Poisson (λ = 1) | 1.0000 | 0.9976 (0.9458, 1.0551) | 1.0027 (0.7211, 1.2775) | 0.0000 | −0.0001 (−0.0119, 0.0118) | −0.0043 (−0.0164, 0.0076) | 0.1429 | 0.1424 (0.1357, 0.1508) | 0.1416 (0.1157, 0.1738) | 0.0550 (−4.9482, 4.9072) |
| Negative binomial (r = 5, p = 0.4) | 1.6000 | 1.5972 (1.4860, 1.7213) | 1.6017 (1.0729, 2.1367) | −0.1271 | −0.1250 (−0.2340, −0.0263) | −0.1351 (−0.6023, 0.3312) | 0.2711 | 0.2701 (0.2573, 0.2882) | 0.2703 (0.2214, 0.3322) | 0.2370 (−16.0949, 16.6441) |
| Exponential (λ = 1) | 2.0000 | 1.9929 (1.8709, 2.1518) | 1.9972 (1.6235, 2.3849) | 0.0000 | −0.0001 (−0.0174, 0.0174) | −0.0123 (−0.0288, 0.0042) | 0.2020 | 0.1990 (0.1812, 0.2313) | 0.1920 (0.1560, 0.2352) | 0.0332 (−6.4607, 7.0247) |
| Gamma (α = 4, β = 1) | 2.0000 | 1.9957 (1.8562, 2.1496) | 2.0011 (1.3760, 2.6237) | −0.6021 | −0.5995 (−0.6928, −0.5140) | −0.6096 (−0.9848, −0.2312) | 0.3194 | 0.3180 (0.3019, 0.3411) | 0.3178 (0.2607, 0.3900) | −0.0815 (−22.7731, 22.7344) |
| Lognormal (µ = 1, σ = 1) | 4.7183 | 4.0982 (3.2918, 7.4927) | 3.5991 (3.0485, 4.2296) | −1.0970 | −1.1320 (−3.2884, −0.6054) | −0.8815 (−1.2848, −0.5294) | 0.6660 | 0.4155 (0.2880, 0.9895) | 0.2662 (0.2132, 0.3305) | 3.6911 (−3.7832, 12.3419) |
| Shifted normal [5 + (0,1)] | 0.0000 | −0.0009 (−0.2407, 0.2386) | 0.0011 (−1.4290, 1.4273) | 0.0000 | −0.0006 (−0.1659, 0.1694) | −0.0062 (−1.0024, 0.9936) | 0.7143 | 0.7140 (0.6946, 0.7345) | 0.7249 (0.5933, 0.8843) | −0.1759 (−128.0845, 124.9325) |
Each parameter was first predicted analytically from the corresponding formula using the given distribution parameters [Formula (analytic)], then approximated using the n × N random observations of each distribution from the formulae [Formula (numeric)] and from the regression (Regression) separately. For the last two methods, median and 95% CI of each parameter were calculated from 10,000 random copies of the n × N iid observations (95% CI is given below the associated median value). For each distribution, the median and 95% CI of the quadratic coefficient from the least-squares quadratic regression were similarly calculated from the 10,000 random copies of the n × N iid observations.