Figure 1.
Figure of mean mass remaining versus standard deviation of replicates at each time point for real data.
Mean mass remaining versus standard deviation of replicates at each time point for (A) Long-term Intersite Decomposition Experiment Team (LIDET) data, (B) Hobbie data; (C) EL data; and (D) HG data.
Table 1.
Percent mass remaining at early, mid, late and end stage decomposition for four different decomposition rates (k in d−1).
Figure 2.
Figure of mean mass remaining versus standard deviation of replicates at each time point for 200 simulations with four different error structures: (A) beta errors + normal errors (option 3a; σ = 0.0125, φ = 5); (B) beta errors + normal error (option 3b; σ = 0.05, φ = 5); (C) beta errors (option 2; no 0 or >1 values, φ = 5); and (D) normal error with variable σ (option 1; Var σ2).
Table 2.
Simulations for which ML estimation with beta errors (option 2) failed to converge for 10,000 out of 12,000 generated data sets.
Figure 3.
Simulation results for beta-distributed errors (option 2), k = 0.002.
(A) Percent bias, (B) percent relative error, and (C) average k estimate. Early, mid, late and end are early, mid, late and end stage decomposition simulations. The numbers 2, 5, 7 and 10 are the numbers of measurements used in each simulation. Blue circles = NLS, Red circles = Normal ML, gray/black circles = Beta ML. In most cases, nls = Normal ML so that the red circles cover the blue circles. In panel (A), the gray line shows 0% bias. In panel (C), the gray line shows the true k value, 0.002 d−1.
Figure 4.
Percent bias for beta-distributed errors plus normal errors.
(A) standard deviation (σ) = 0.0125 (option 3a) and SV transformation, (B) σ = 0.0125 (option 3a) and REP transformation, (C) σ = 0.05 (option 3b) and SV transformation, (D) σ = 0.05 (option 3b) and REP transformation. Early, mid, late and end are early, mid, late and end stage decomposition simulations. The numbers 2, 5, 7 and 10 are the numbers of measurements used in each simulation. Blue circles = NLS, Red circles = Normal ML, gray/black circles = Beta ML. In most cases, nls = Normal ML so that the red circles cover the blue circles. Gray lines show 0% bias.
Figure 5.
Percent relative error for beta-distributed errors plus normal errors with different σ and transformations.
(A) σ = 0.0125 (option 3a) and SV transformation, (B) σ = 0.0125 (option 3a) and REP transformation, (C) σ = 0.05 (option 3b) and SV transformation, (D) σ = 0.05 (option 3b) and REP transformation. Early, mid, late and end are early, mid, late and end stage decomposition simulations. The numbers 2, 5, 7 and 10 are the numbers of measurements used in each simulation. Blue circles = NLS, Red circles = Normal ML, gray/black circles = Beta ML. In most cases, nls = Normal ML so that the red circles cover the blue circles.
Figure 6.
Average k estimates for beta-distributed errors plus normal errors with different σ and transformations.
(A) σ = 0.0125 (option 3a) and SV transformation, (B) σ = 0.0125 (option 3a) and REP transformation, (C) σ = 0.05 (option 3b) and SV transformation, (D) σ = 0.05 (option 3b) and REP transformation. Early, mid, late and end are early, mid, late and end stage decomposition simulations. The numbers 2, 5, 7 and 10 are the numbers of measurements used in each simulation. Blue circles = NLS, Red circles = Normal ML, gray/black circles = Beta ML. In most cases, nls = Normal ML so that the red circles cover the blue circles. Gray lines show the true k value of 0.002 d−1.
Figure 7.
Results for simulations with variable normal errors (option 1).
Percent bias using (A) SV and (B) REP transformations and relative error using (C) SV and (D) REP transformations. Early, mid, late and end are early, mid, late and end stage decomposition simulations. The numbers 2, 5, 7 and 10 are the numbers of measurements used in each simulation. Blue circles = NLS, Red circles = Normal ML, gray/black circles = Beta ML. In most cases, nls = Normal ML so that the red circles cover the blue circles. Gray lines in panels (A) and (B) show 0% bias.
Figure 8.
Average k estimates for simulations with variable normal errors (option 1).
(A) SV and (B) REP transformations. Early, mid, late and end are early, mid, late and end stage decomposition simulations. The numbers 2, 5, 7 and 10 are the numbers of measurements used in each simulation. Blue circles = NLS, Red circles = Normal ML, gray/black circles = Beta ML. In most cases, nls = Normal ML so that the red circles cover the blue circles. Gray lines in panels (A) and (B) show the true k value of 0.002 d−1.
Table 3.
Percent of simulations using beta errors (option 2) for which AICc selected maximum likelihood (ML) estimation with beta or normal errors best or found no difference between the two models (Same) from each simulation (k = 0.0002).
Table 4.
Percent of beta error simulations with normal error (σ = 0.0125) added, for which AICc selected maximum likelihood (ML) estimation with beta or normal errors best or found no difference between the models (Same) from each simulation (k = 0.0002).
Table 5.
Percent of beta error simulations with normal error (σ = 0.05) added, for which AICc selected maximum likelihood (ML) estimation with beta or normal errors best or found no difference between the models (Same) from each simulation (k = 0.0002).
Table 6.
Percent of variable normal σ simulations for which AICc selected maximum likelihood (ML) estimation with beta or normal errors best or found no difference between the models (Same) from each simulation (k = 0.0002).
Figure 9.
Daily decomposition rate (k) estimates for the Hobbie (SH) [14], Laliberté and Tylianakis (LT) [12] and Hobbie and Gough (H&G) [10] data compared by error distribution (beta or normal) used to estimate k. (A) untransformed (B) Smithson and Verkuilen (SV) [6] transformed and (C) replacement (zeros = missing data; values ≥1 = 0.9999) transformed data sets. Insets in (b) and (c) show only the SH and H&G data.
Table 7.
Mean k (decomposition rate), fractional bias (FB) and relative bias (RB) produced by each data transformation and error structure using the Hobbie [14], Laliberté and Tylianakis [12] and Hobbie and Gough [10] data sets.