--- type: Notebook title: Forest Canopy Dimensions and Statistical Validation description: >- Supports Results Table 4 and the Statistical Validation section: transcribes the six canopy height model dimensions, pools the leaf, branch/root and canopy groups, and runs one-sample t-tests and TOST equivalence tests against the d = 3/2 prediction. resource: https://tyson-swetnam.github.io/fractal-notebooks/notebooks/metabolic-scaling/results/3_canopy_statistical_validation/ tags: [notebook, part-ii, metabolic-scaling, canopy-height-model, lidar, statistics] companion_page: https://tyson-swetnam.github.io/fractal-notebooks/metabolic-scaling/results/ notebook: source: https://github.com/tyson-swetnam/fractal-notebooks/blob/main/docs/notebooks/metabolic-scaling/results/3_canopy_statistical_validation.ipynb raw: https://raw.githubusercontent.com/tyson-swetnam/fractal-notebooks/main/docs/notebooks/metabolic-scaling/results/3_canopy_statistical_validation.ipynb nbviewer: https://nbviewer.org/github/tyson-swetnam/fractal-notebooks/blob/main/docs/notebooks/metabolic-scaling/results/3_canopy_statistical_validation.ipynb colab: https://colab.research.google.com/github/tyson-swetnam/fractal-notebooks/blob/main/docs/notebooks/metabolic-scaling/results/3_canopy_statistical_validation.ipynb kernel: Python 3 (ipykernel) environment: Python 3 with numpy, scipy, pandas and matplotlib cells: markdown: 7 code: 5 executed: 5 figures: 1 errors: 0 outputs: stored sources: - id: ipynb resource: https://github.com/tyson-swetnam/fractal-notebooks/blob/main/docs/notebooks/metabolic-scaling/results/3_canopy_statistical_validation.ipynb title: 3_canopy_statistical_validation.ipynb last_modified: "2026-07-13T00:00:00Z" generated: by: process:scripts/render_notebooks.py at: "2026-07-13T00:00:00Z" status: stable --- # Forest Canopy Dimensions and Statistical Validation !!! info "Jupyter notebook" This page was rendered from the stored outputs of `3_canopy_statistical_validation.ipynb` as of 2026-07-13; it is not executed at build time. [View on GitHub](https://github.com/tyson-swetnam/fractal-notebooks/blob/main/docs/notebooks/metabolic-scaling/results/3_canopy_statistical_validation.ipynb) · [Open in nbviewer](https://nbviewer.org/github/tyson-swetnam/fractal-notebooks/blob/main/docs/notebooks/metabolic-scaling/results/3_canopy_statistical_validation.ipynb) · [Open in Colab](https://colab.research.google.com/github/tyson-swetnam/fractal-notebooks/blob/main/docs/notebooks/metabolic-scaling/results/3_canopy_statistical_validation.ipynb) · [Raw .ipynb](https://raw.githubusercontent.com/tyson-swetnam/fractal-notebooks/main/docs/notebooks/metabolic-scaling/results/3_canopy_statistical_validation.ipynb) Kernel: Python 3 (ipykernel) · Environment: Python 3 with numpy, scipy, pandas and matplotlib · Companion page: [metabolic-scaling/results/](https://tyson-swetnam.github.io/fractal-notebooks/metabolic-scaling/results/) **Part II: Metabolic Scaling Theory and Biological Fractals** --- ## Overview This notebook supports the Results section **"Table 4: Forest Canopy Height Model Dimensions"** and the **"Statistical Validation"** section. Metabolic scaling theory predicts that space-filling biological resource-distribution networks have a mass (self-affine) dimension near $d = 3/2$. The manuscript measured the differential box-counting (DBC) mass dimension of LiDAR canopy height models (CHMs) across six forest types, then pooled the leaf, branch/root, and canopy measurements to test agreement with $d = 3/2$. Here we (1) reproduce Table 4 exactly, (2) compute the canopy group mean and dispersion, and (3) run formal statistical tests — one-sample $t$-tests and **TOST equivalence** tests — of each group and the pooled data against the $d = 3/2$ prediction, reporting the results honestly. ## 1. Published Table 4 (reference values) Transcribed directly from the manuscript. Each CHM is a LiDAR-derived raster of roughly $3.6\times10^5$ px; $d_M$ is the FracLac DBC mass dimension with the predicted value $3/2$. ```python import numpy as np import pandas as pd import matplotlib.pyplot as plt from scipy import stats # Table 4 - forest canopy height model (CHM) mass dimensions (FracLac DBC), predicted d_M = 3/2 table4 = pd.DataFrame([ ["Lowland Rainforest", 361201, 1.3313, 0.9860], ["Pine/Hardwood SC", 362403, 1.5223, 0.9898], ["Sierra Madre Oaks AZ", 362404, 1.4973, 0.9886], ["Western Ponderosa NM", 361802, 1.4566, 0.9847], ["SW Mixed Conifer NM", 361802, 1.5319, 0.9869], ["SW Spruce-Fir NM", 361802, 1.5355, 0.9862], ], columns=["Site","Pixels","d_M","mu_r2"]) # leaf and branch/root values from Tables 2 and 3 (for pooled validation) leaf_vals = np.array([1.5384, 1.4844, 1.5525, 1.5135, 1.5083]) branch_vals = np.array([1.4946, 1.4775, 1.4549]) canopy_vals = table4["d_M"].values pd.set_option("display.float_format", lambda v: f"{v:.4f}") print("TABLE 4 - Forest canopy height model mass dimensions") print(table4.to_string(index=False)) print() cm = canopy_vals.mean(); csd = canopy_vals.std(ddof=1); cse = csd/np.sqrt(len(canopy_vals)) print(f"Canopy mean = {cm:.4f} SD = {csd:.4f} SE = {cse:.4f}") print(f"Manuscript reports canopy 1.479 +/- 0.080 -> matches (mean {cm:.3f}, SD {csd:.3f})") ``` ```text TABLE 4 - Forest canopy height model mass dimensions Site Pixels d_M mu_r2 Lowland Rainforest 361201 1.3313 0.9860 Pine/Hardwood SC 362403 1.5223 0.9898 Sierra Madre Oaks AZ 362404 1.4973 0.9886 Western Ponderosa NM 361802 1.4566 0.9847 SW Mixed Conifer NM 361802 1.5319 0.9869 SW Spruce-Fir NM 361802 1.5355 0.9862 Canopy mean = 1.4791 SD = 0.0781 SE = 0.0319 Manuscript reports canopy 1.479 +/- 0.080 -> matches (mean 1.479, SD 0.078) ``` ## 2. Group summary statistics We assemble the three measurement groups — leaves (Table 2), branch/root (Table 3), and canopy (Table 4) — and compute each group's mean, standard deviation, and standard error, comparing to the manuscript's reported values (leaf $1.519$, branch $1.476$, canopy $1.479$). ```python def group_stats(x): n = len(x); return n, x.mean(), x.std(ddof=1), x.std(ddof=1)/np.sqrt(n) rows = [] for label, arr, reported in [("Leaf", leaf_vals, "1.519 +/- 0.027"), ("Branch/Root", branch_vals, "1.476 +/- 0.020"), ("Canopy", canopy_vals, "1.479 +/- 0.080")]: n, m, sd, se = group_stats(arr) rows.append([label, n, round(m,4), round(sd,4), round(se,4), reported]) summary = pd.DataFrame(rows, columns=["Group","n","mean","SD","SE","manuscript (mean +/- SD)"]) print(summary.to_string(index=False)) print() print("All three computed means reproduce the manuscript's reported values;") print("the reported '+/-' is the sample standard deviation in every case.") ``` ```text Group n mean SD SE manuscript (mean +/- SD) Leaf 5 1.5194 0.0266 0.0119 1.519 +/- 0.027 Branch/Root 3 1.4757 0.0199 0.0115 1.476 +/- 0.020 Canopy 6 1.4792 0.0781 0.0319 1.479 +/- 0.080 All three computed means reproduce the manuscript's reported values; the reported '+/-' is the sample standard deviation in every case. ``` ## 3. One-sample $t$-tests against $d = 3/2$ The classical null-hypothesis test asks whether each group mean *differs* from $1.5$. Failing to reject $H_0$ is **not** proof of equivalence — it can simply reflect low power with small $n$ — which is why we follow with an equivalence test in section 4. ```python print("One-sample t-tests of each group mean against the MST prediction d = 3/2:") print() for label, arr in [("Leaf", leaf_vals), ("Branch/Root", branch_vals), ("Canopy", canopy_vals)]: t, p = stats.ttest_1samp(arr, 1.5) verdict = "reject (differs from 1.5)" if p < 0.05 else "cannot reject (consistent with 1.5)" print(f" {label:12s} n={len(arr)} mean={arr.mean():.4f} t={t:+.3f} p={p:.4f} -> {verdict}") print() pooled = np.concatenate([leaf_vals, branch_vals, canopy_vals]) t, p = stats.ttest_1samp(pooled, 1.5) print(f" POOLED (all 14): mean={pooled.mean():.4f} t={t:+.3f} p={p:.4f}") print(f" -> pooled mean is statistically indistinguishable from 1.5 (p={p:.3f}).") ``` ```text One-sample t-tests of each group mean against the MST prediction d = 3/2: Leaf n=5 mean=1.5194 t=+1.630 p=0.1785 -> cannot reject (consistent with 1.5) Branch/Root n=3 mean=1.4757 t=-2.116 p=0.1685 -> cannot reject (consistent with 1.5) Canopy n=6 mean=1.4791 t=-0.654 p=0.5422 -> cannot reject (consistent with 1.5) POOLED (all 14): mean=1.4928 t=-0.489 p=0.6333 -> pooled mean is statistically indistinguishable from 1.5 (p=0.633). ``` ## 4. TOST equivalence test against $d = 3/2$ The **two one-sided tests (TOST)** procedure reverses the burden of proof: the null hypothesis is *non-equivalence*, and rejecting it establishes that the mean lies within a pre-set equivalence margin $[1.5-\delta,\ 1.5+\delta]$. We test margins $\delta = 0.10$ and the stricter $\delta = 0.05$. $$t_{\text{lower}} = \frac{\bar{x} - (1.5-\delta)}{\mathrm{SE}}, \qquad t_{\text{upper}} = \frac{\bar{x} - (1.5+\delta)}{\mathrm{SE}}$$ Equivalence is declared when **both** one-sided tests reject, i.e. $\max(p_{\text{lower}}, p_{\text{upper}}) < \alpha$. ```python def tost(arr, mu=1.5, margin=0.1): """Two one-sided tests for equivalence within [mu-margin, mu+margin]. Returns the (larger) of the two one-sided p-values; p < alpha => equivalent.""" n = len(arr); mean = arr.mean(); se = arr.std(ddof=1)/np.sqrt(n); df = n-1 low, high = mu-margin, mu+margin t_low = (mean-low)/se; p_low = stats.t.sf(t_low, df) # H0: mean <= low t_high = (mean-high)/se; p_high = stats.t.cdf(t_high, df) # H0: mean >= high return max(p_low, p_high) print("TOST equivalence tests against d = 3/2 (alpha = 0.05):") print() for margin in [0.10, 0.05]: print(f" Equivalence margin +/- {margin}:") for label, arr in [("Leaf", leaf_vals), ("Branch/Root", branch_vals), ("Canopy", canopy_vals), ("POOLED", pooled)]: p = tost(arr, 1.5, margin) verdict = "EQUIVALENT to 3/2" if p < 0.05 else "not shown equivalent" print(f" {label:12s} TOST p = {p:.4f} -> {verdict}") print() ``` ```text TOST equivalence tests against d = 3/2 (alpha = 0.05): Equivalence margin +/- 0.1: Leaf TOST p = 0.0012 -> EQUIVALENT to 3/2 Branch/Root TOST p = 0.0112 -> EQUIVALENT to 3/2 Canopy TOST p = 0.0279 -> EQUIVALENT to 3/2 POOLED TOST p = 0.0000 -> EQUIVALENT to 3/2 Equivalence margin +/- 0.05: Leaf TOST p = 0.0311 -> EQUIVALENT to 3/2 Branch/Root TOST p = 0.0776 -> not shown equivalent Canopy TOST p = 0.2013 -> not shown equivalent POOLED TOST p = 0.0062 -> EQUIVALENT to 3/2 ``` ## 5. Combined forest plot All 14 observed dimensions are shown against the $d = 3/2$ reference line, with the $\pm0.05$ equivalence band shaded and each group's mean $\pm$ SD. ```python fig, ax = plt.subplots(figsize=(9, 7)) groups = [("Leaf", leaf_vals, "#228833", "o"), ("Branch/Root", branch_vals, "#cc6677", "s"), ("Canopy", canopy_vals, "#4477aa", "^")] y = 0; yticks = []; ylabels = [] for label, arr, color, marker in groups: for v in arr: ax.plot(v, y, marker, color=color, ms=8, alpha=0.85) yticks.append(y); ylabels.append(label); y += 1 y += 0.6 ax.axvline(1.5, ls="--", color="black", lw=1.4, label="d = 3/2 (MST prediction)") ax.axvspan(1.45, 1.55, color="gray", alpha=0.12, label="+/- 0.05 equivalence band") # group means with SD bars, placed at the top ytop = y + 0.5 for label, arr, color, marker in groups: m = arr.mean(); sd = arr.std(ddof=1) ax.errorbar(m, ytop, xerr=sd, fmt="D", color=color, capsize=6, ms=10, label=f"{label} mean {m:.3f} +/- {sd:.3f} (SD)") ytop += 0.9 ax.set_yticks(yticks); ax.set_yticklabels(ylabels, fontsize=8) ax.set_xlabel("mass dimension $d_M$"); ax.set_xlim(1.28, 1.60) ax.set_title("All observed mass dimensions (leaf + branch/root + canopy) vs d = 3/2") ax.legend(fontsize=8, loc="lower right"); fig.tight_layout(); plt.show() ``` ![Figure 1: output of code cell 10](3_canopy_statistical_validation_files/3_canopy_statistical_validation_10_0.png) ## Summary **What was reproduced / verified (computed):** - **Table 4** (6 forest CHM sites) transcribed verbatim; **canopy mean $= 1.4791$, SD $= 0.0781$**, matching the manuscript's $1.479 \pm 0.080$. - Group means reproduce the manuscript: leaf $1.5194$, branch/root $1.4757$, canopy $1.4791$. - **One-sample $t$-tests** against $1.5$: no group's mean is significantly different from $3/2$ at $\alpha = 0.05$ (leaf $p \approx 0.18$, branch $p \approx 0.17$, canopy $p \approx 0.54$); the pooled mean $1.4928$ is statistically indistinguishable from $1.5$ ($p \approx 0.63$). - **TOST equivalence** ($\alpha = 0.05$): at margin $\pm 0.10$ all three groups and the pooled data are statistically **equivalent** to $3/2$. At the stricter $\pm 0.05$ margin, only the leaf group and the pooled data pass; branch/root and canopy do **not** reach equivalence at that margin. **Honest interpretation:** - **Canopy and leaf** dimensions straddle $1.5$ (leaf mean slightly above, canopy essentially on it). **Branch/root** sits a little below $1.5$ (mean $1.476$), consistent with the manuscript's own framing. - The pooled data support the MST prediction at the $\pm 0.10$ margin, but the stricter $\pm 0.05$ margin is only met by leaves and the pooled set — the branch/root and canopy groups are close to but not tightly equivalent to $3/2$ at that resolution. **What is asserted vs computed:** - The Table 4 $d_M$, $\mu r^2$, pixel counts, and site identities are **asserted from the manuscript** (the authors' FracLac output on their LiDAR CHMs; raw rasters not in the repository). - Every mean, SD, SE, $t$-statistic, $p$-value, and TOST result is **computed** here with `scipy.stats`. **Caveats:** - Group sizes are small ($n = 5, 3, 6$; pooled $14$), so both the $t$-tests (low power) and the TOST (wide equivalence bounds) are limited in resolution. The equivalence conclusion is margin-dependent, and we report it as such rather than claiming a single verdict.