Summary of Allometric Models

Summary of Allometric Models

Supplement to

Allometry data and equations for coastal marsh plants

Meng Lu, Joshua S. Caplan, Jonathan D. Bakker, J. Adam Langley, Thomas J. Mozdzer, Bert G. Drake, and J. Patrick Megonigal

• Description of Dataset

– Total number of plants includes only those used in the final analysis (i.e., it excludes outliers).

– Values in tables are calculated from all available data. An implication is that differing rows (e.g., Height vs. Width) may describe slightly different datasets. This is typically because Width data were not collected for plants with Height below 40 cm. In a small number of additional cases, data are missing for other reasons.

• Allometric Models

– Equations will provide estimates oftranformations of Biomass. Resulting values must be reverse- transformed to yield Biomass in original units (grams of dry weight).

– Coefficients that appear in equations are sometimes rounded. The same coefficients with greater numbers of significant digits appear in the corresponding tables that follow.

• Model Evaluation

– Each table row represents a linear model, with values shown for coefficients only if they are included in the model. Intercepts are estimated for all models, including the Height and null models.

– A second table is shown only if the optimal transformation for Biomass in the Height model differs from that in the Height & Width model.

– Although BIC scores are provided, the order of models in the table does not necessarily correspond to ranking of BIC scores.

• Effect of Transformation

– Figures in grey and red show how the Height versus Biomass relationship is altered by transfor- mation(s) of Biomass. Two figures appear if the Height & Width model has the same optimal transformation as the Height model, but an additional figure (in blue) appears otherwise.

– Data points that are designated as outliers appear in orange. These data points are not included in final models and therefore do not appear in diagnostic plots. Note that points may have been designated as outliers because of their value for Width (but not Height), and may not seem unusual in the plots shown.

• Diagnostic Plots

– Plots at the left of each row show the effects of varyingλ, or the transformation on Biomass. The vertical dotted lines denote the 95% confidence interval for the optimal value ofλ, while horizontal dotted lines denote the corresponding log-likelihood values.

– Plots in the center of each row are quantile-quantile (Q-Q) plots for residuals versus normal quantiles, and can be used to evaluate how well residuals comply with the assumption that they are normally distributed. Points will fall along the linear trendline (and within the 95% confidence interval) if they can be considered to have arisen from a random-normal process.

– Plots at the right in each row can be used to evaluate how well residuals adhere to the assumption that they have a constant variance and a mean of zero. After being Studentized (divided by their standard deviation via a leave-one-out process), they will be scattered evenly above and below zero if they comply with the assumption.

Amaranthus cannabinus (L.) Sauer

Other names: Tidalmarsh amaranth, saltmarsh water hemp, saltmarsh pigweed

Description of Dataset

• Year(s) plants collected: 2011 - 2015

• Total records: 232 (7 without Width). Outliers: 3 for Height & Width model, 4 for Height model.

Variable Unit Min Median Max Biomass g 0.031 2.2655 143.61

Height cm 20 101 265

Width mm 1 5.8 37.8

Allometric Models

If Height and Width measurements are both available, the model with the greatest predictive ability is:

ln(Biomass) =−3.0765752 + 0.024084·Height+ 0.4040617·W idth−0.0014429·Height·W idth If only Height measurements are available, the model with the greatest predictive ability is:

ln(Biomass) =−2.3532583 + 0.0305667·Height

Model Evaluation

The following table compares the full model for Height & Width with all nested models. Coefficients are not standardized and only shown when terms are included in a model. The same transformation of Biomass is used in all cases. Note that BIC scores are substantially improved (i.e., >4 smaller) by including Width in models, and for all models with Height or Width compared to the intercept-only (i.e., null) model.

Intercept Height Width Height×Width R² BIC

-3.077 0.02408 0.4041 -0.001443 0.871 431

-1.749 0.01678 0.1254 0.791 533

-2.213 0.0292 0.741 575

-0.7841 0.2488 0.725 589

1.037 0 870

The Height model and corresponding null model are summarized below.

Intercept Height R² BIC

-2.353 0.03057 0.762 583

0.9459 0 905

50 100 150 200 250 0

20 40 60 80 100 120 140

Height (cm)

Biomass (g)

50 100 150 200 250

−4

−2 0 2 4

Height (cm)

ln

(

Biomass

)

Diagnostic Plots: Height & Width Model

−2 −1 0 1 2

−1800

−1600

−1400

−1200

−1000

−800

−600

λ

Log likelihood

95%

λused = 0

−3 −2 −1 0 1 2 3

−1.5

−1.0

−0.5 0.0 0.5 1.0 1.5

Normal quantile

Residual

−2 0 2 4 6

−2

−1 0 1 2

Fitted value

Studentized residual

Diagnostic Plots: Height Model

−2 −1 0 1 2

−1800

−1600

−1400

−1200

−1000

−800

−600

λ

Log likelihood

95%

λused = 0

−3 −2 −1 0 1 2 3

−2

−1 0 1 2

Normal quantile

Residual

−2 0 2 4 6

−2

−1 0 1 2 3

Fitted value

Studentized residual

Atriplex patula L.

Other names: Spear saltbush, common orach, spreading orach

Description of Dataset

• Year(s) plants collected: 2012 - 2015

• Total records: 153 (10 without Width). Outliers: 5 for Height & Width model, 5 for Height model.

Variable Unit Min Median Max

Biomass g 0.1 0.897 27.22

Height cm 18 79 200

Width mm 0.3 1.9 4.1

Allometric Models

If Height and Width measurements are both available, the model with the greatest predictive ability is:

ln(Biomass) =−2.5730397 + 0.0166259·Height+ 0.4796138·W idth+ 8.4056553×10⁻⁴·Height·W idth If only Height measurements are available, the model with the greatest predictive ability is:

ln(Biomass) =−2.6267792 + 0.0291238·Height

Model Evaluation

The following table compares the full model for Height & Width with all nested models. Coefficients are not standardized and only shown when terms are included in a model. The same transformation of Biomass is used in all cases. Note that BIC scores are substantially improved (i.e., >4 smaller) by including Width in models, and for all models with Height or Width compared to the intercept-only (i.e., null) model.

Intercept Height Width Height×Width R² BIC

-2.573 0.01663 0.4796 0.0008406 0.824 188

-2.716 0.01842 0.5529 0.823 183

-2.227 0.02509 0.731 236

-2.171 1.099 0.571 301

-0.04306 0 413

The Height model and corresponding null model are summarized below.

Intercept Height R² BIC

-2.627 0.02912 0.779 258

-0.2382 0 476

50 100 150 200 0

5 10 15 20 25

Height (cm)

Biomass (g)

50 100 150 200

−3

−2

−1 0 1 2 3

Height (cm)

ln

(

Biomass

)

Diagnostic Plots: Height & Width Model

−2 −1 0 1 2

−800

−700

−600

−500

−400

−300

λ

Log likelihood

95%

λused = 0

−2 −1 0 1 2

−0.5 0.0 0.5 1.0

Normal quantile

Residual

−2 −1 0 1 2 3

−2

−1 0 1 2 3

Fitted value

Studentized residual

Diagnostic Plots: Height Model

−2 −1 0 1 2

−900

−800

−700

−600

−500

−400

−300

λ

Log likelihood

95%

λused = 0

−2 −1 0 1 2

−1.5

−1.0

−0.5 0.0 0.5 1.0

Normal quantile

Residual

−2 −1 0 1 2 3

−2

−1 0 1 2

Fitted value

Studentized residual

Iva frutescens L.

Other names: Marsh elder, Jesuit’s bark

Description of Dataset

• Year(s) plants collected: 2012 - 2015

• Total records: 121 (10 without Width). Outliers: 4 for Height & Width model, 2 for Height model.

Variable Unit Min Median Max Biomass g 0.26 12.1 248.11

Height cm 17 77 168

Width mm 1.8 4.6 15.3

Allometric Models

If Height and Width measurements are both available, the model with the greatest predictive ability is:

√3

Biomass= 0.4115549 + 0.015213·Height+ 0.0819737·W idth+ 6.9308044×10⁻⁴·Height·W idth If only Height measurements are available, the model with the greatest predictive ability is:

ln(Biomass) =−1.0819983 + 0.0428543·Height

Model Evaluation

The following table compares the full model for Height & Width with all nested models. Coefficients are not standardized and only shown when terms are included in a model. The same transformation of Biomass is used in all cases. Note that BIC scores are substantially improved (i.e., >4 smaller) by including Width in models, and for all models with Height or Width compared to the intercept-only (i.e., null) model.

Intercept Height Width Height×Width R² BIC

0.4116 0.01521 0.08197 0.0006931 0.907 95

0.0756 0.01899 0.1534 0.902 96

-0.09954 0.03205 0.824 154

0.7993 0.2954 0.797 169

2.511 0 336

The Height model and corresponding null model are summarized below.

Intercept Height R² BIC

-1.082 0.04285 0.814 259

2.259 0 455

50 100 150 0

50 100 150 200 250

Height (cm)

Biomass (g)

50 100 150

1 2 3 4 5 6

Height (cm)

3 Biomass

50 100 150

−1 0 1 2 3 4 5

Height (cm)

ln

(

Biomass

)

Diagnostic Plots: Height & Width Model

−2 −1 0 1 2

−700

−600

−500

−400

−300

−200

λ

Log likelihood

95%

λused = 0.33

−2 −1 0 1 2

−0.5 0.0 0.5 1.0

Normal quantile

Residual

1 2 3 4 5 6

−1 0 1 2 3 4

Fitted value

Studentized residual

Diagnostic Plots: Height Model

−2 −1 0 1 2

−800

−700

−600

−500

−400

−300

λ

Log likelihood

95%

λused = 0

−2 −1 0 1 2

−1.0

−0.5 0.0 0.5 1.0 1.5 2.0

Normal quantile

Residual

0 1 2 3 4 5 6

−2

−1 0 1 2 3

Fitted value

Studentized residual

Kosteletzkya virginica L.

Other names: Seashore mallow, sweat weed, Virginia saltmarsh mallow

Description of Dataset

• Year(s) plants collected: 2011 - 2015

• Total records: 159 (0 without Width). Outliers: 3 for Height & Width model, 3 for Height model.

Variable Unit Min Median Max Biomass g 0.24 7.175 89.54

Height cm 31 106 232

Width mm 1 5.6 14

Allometric Models

If Height and Width measurements are both available, the model with the greatest predictive ability is:

ln(Biomass) =−1.9646317 + 0.0238184·Height+ 0.4211596·W idth−0.0016357·Height·W idth If only Height measurements are available, the model with the greatest predictive ability is:

ln(Biomass) =−0.6811233 + 0.024041·Height

Model Evaluation

The following table compares the full model for Height & Width with all nested models. Coefficients are not standardized and only shown when terms are included in a model. The same transformation of Biomass is used in all cases. Note that BIC scores are substantially improved (i.e., >4 smaller) by including Width in models, and for all models with Height or Width compared to the intercept-only (i.e., null) model.

Intercept Height Width Height×Width R² BIC

-1.965 0.02382 0.4212 -0.001636 0.846 238

-0.9012 0.01381 0.2245 0.811 265

-0.3131 0.3807 0.714 325

-0.7798 0.02477 0.7 332

1.919 0 515

The Height model and corresponding null model are summarized below.

Intercept Height R² BIC

-0.6811 0.02404 0.688 330

1.941 0 507

50 100 150 200 0

20 40 60 80

Height (cm)

Biomass (g)

50 100 150 200

−1 0 1 2 3 4

Height (cm)

ln

(

Biomass

)

Diagnostic Plots: Height & Width Model

−2 −1 0 1 2

−900

−800

−700

−600

−500

−400

−300

λ

Log likelihood

95%

λused = 0

−2 −1 0 1 2

−1.0

−0.5 0.0 0.5 1.0

Normal quantile

Residual

0 1 2 3 4

−2

−1 0 1 2 3

Fitted value

Studentized residual

Diagnostic Plots: Height Model

−2 −1 0 1 2

−900

−800

−700

−600

−500

−400

λ

Log likelihood

95%

λused = 0

−2 −1 0 1 2

−1.5

−1.0

−0.5 0.0 0.5 1.0 1.5

Normal quantile

Residual

0 1 2 3 4 5

−2

−1 0 1 2

Fitted value

Studentized residual

Phragmites australis (Cav.) Trin. ex Steud.

Other names: Phragmites communis Trin., common reed

Note that the plants used in this dataset were haplotpe M, which is invasive in North America.

Description of Dataset

• Year(s) plants collected: 2011 - 2015

• Total records: 311 (51 without Width). Outliers: 1 for Height & Width model, 2 for Height model.

Variable Unit Min Median Max Biomass g 0.85 15.87 68.63

Height cm 33 213 364

Width mm 2 5.9 11.2

Allometric Models

If Height and Width measurements are both available, the model with the greatest predictive ability is:

√3

Biomass= 0.2950557 + 0.0066651·Height+ 0.1605501·W idth−1.0041367×10⁻⁴·Height·W idth If only Height measurements are available, the model with the greatest predictive ability is:

√3

Biomass= 0.6263727 + 0.0091227·Height

Model Evaluation

The following table compares the full model for Height & Width with all nested models. Coefficients are not standardized and only shown when terms are included in a model. The same transformation of Biomass is used in all cases. Note that BIC scores are substantially improved (i.e., >4 smaller) by including Width in models, and for all models with Height or Width compared to the intercept-only (i.e., null) model.

Intercept Height Width Height×Width R² BIC

0.2951 0.006665 0.1606 -0.0001004 0.964 -276

0.4134 0.006158 0.1356 0.963 -278

0.646 0.009054 0.933 -128

0.3367 0.3547 0.86 64

2.478 0 567

The Height model and corresponding null model are summarized below.

Intercept Height R² BIC

0.6264 0.009123 0.932 -178

2.492 0 647

50 100 150 200 250 300 350 0

10 20 30 40 50 60 70

Height (cm)

Biomass (g)

50 100 150 200 250 300 350 1.0

1.5 2.0 2.5 3.0 3.5 4.0

Height (cm)

3 Biomass

Diagnostic Plots: Height & Width Model

−2 −1 0 1 2

−1400

−1200

−1000

−800

−600

−400

λ

Log likelihood

95%

λused = 0.33

−3 −2 −1 0 1 2 3

−0.4

−0.3

−0.2

−0.1 0.0 0.1 0.2

Normal quantile

Residual

1.0 2.0 3.0 4.0

−3

−2

−1 0 1 2

Fitted value

Studentized residual

Diagnostic Plots: Height Model

−2 −1 0 1 2

−1600

−1400

−1200

−1000

−800

−600

−400

λ

Log likelihood

95%

λused = 0.33

−3 −2 −1 0 1 2 3

−0.4

−0.2 0.0 0.2 0.4

Normal quantile

Residual

1.0 2.0 3.0 4.0

−3

−2

−1 0 1 2

Fitted value

Studentized residual

Polygonum hydropiper L.

Other names: Water pepper, marshpepper knotweed

Description of Dataset

• Year(s) plants collected: 2014 - 2015

• Total records: 69 (17 without Width). Outliers: 2 for Height & Width model, 3 for Height model.

Variable Unit Min Median Max

Biomass g 0.06 0.57 9.06

Height cm 9 53 93

Width mm 0.1 1.37 2.9

Allometric Models

If Height and Width measurements are both available, the model with the greatest predictive ability is:

ln(Biomass) =−4.9665869 + 0.0633376·Height+ 0.9745709·W idth−0.0042656·Height·W idth If only Height measurements are available, the model with the greatest predictive ability is:

ln(Biomass) =−3.7329253 + 0.06039·Height

Model Evaluation

The following table compares the full model for Height & Width with all nested models. Coefficients are not standardized and only shown when terms are included in a model. The same transformation of Biomass is used in all cases. Note that BIC scores are substantially improved (i.e., >4 smaller) by including Width in models, and for all models with Height or Width compared to the intercept-only (i.e., null) model.

Intercept Height Width Height×Width R² BIC

-4.967 0.06334 0.9746 -0.004266 0.83 80

-4.548 0.05685 0.6876 0.829 77

-4.28 0.06935 0.76 90

-2.468 1.509 0.453 131

-0.3077 0 157

The Height model and corresponding null model are summarized below.

Intercept Height R² BIC

-3.733 0.06039 0.734 132

-0.6292 0 216

20 40 60 80 0

2 4 6 8

Height (cm)

Biomass (g)

20 40 60 80

−3

−2

−1 0 1 2

Height (cm)

ln

(

Biomass

)

Diagnostic Plots: Height & Width Model

−2 −1 0 1 2

−200

−150

−100

−50

λ

Log likelihood

95%

λused = 0

−2 −1 0 1 2

−0.5 0.0 0.5 1.0

Normal quantile

Residual

−3 −2 −1 0 1 2

−2

−1 0 1 2

Fitted value

Studentized residual

Diagnostic Plots: Height Model

−2 −1 0 1 2

−350

−300

−250

−200

−150

−100

λ

Log likelihood

95%

λused = 0

−2 −1 0 1 2

−1.0

−0.5 0.0 0.5 1.0

Normal quantile

Residual

−3 −2 −1 0 1 2

−2

−1 0 1 2

Fitted value

Studentized residual

Schoenoplectus americanus (Pers.) Volkart ex Schinz & R. Keller

Other names: Scirpus olneyi,Scirpus americanus, chairmaker’s bulrush, Olney’s three-square bulrush

Description of Dataset

• Year(s) plants collected: 1987 - 2016

• Total records: 8430 (5 without Width). Outliers: 71 for Height & Width model, 40 for Height model.

• Records witheCO2: 3290, with N: 370, witheCO2+N: 316

Variable Unit Min Median Max Biomass g 0.003 0.538 3.133

Height cm 3 81 177

Width mm 0.4 3.5 7.3

Allometric Models

If Height and Width measurements are both available, the model with the greatest predictive ability is:

√3

Biomass= 0.0713672 + 0.0058842·Height+ 0.0984255·W idth−2.7383247×10⁻⁴·Height·W idth If only Height measurements are available, the model with the greatest predictive ability is:

√3

Biomass= 0.3093604 + 0.006171·Height

Model Evaluation

The following table compares the full model for Height & Width with all nested models. Coefficients are not standardized and only shown when terms are included in a model. The same transformation of Biomass is used in all cases. Note that BIC scores are substantially improved (i.e., >4 smaller) by including Width in models, and for all models with Height or Width compared to the intercept-only (i.e., null) model.

Intercept Height Width Height×Width R² BIC

0.07137 0.005884 0.09843 -0.0002738 0.926 -25334

0.1467 0.004851 0.07756 0.925 -25168

0.3093 0.006174 0.823 -18044

0.2739 0.1548 0.555 -10313

0.8146 0 -3566

The Height model and corresponding null model are summarized below.

Intercept Height R² BIC

0 50 100 150 0.0

0.5 1.0 1.5 2.0 2.5 3.0

Height (cm)

Biomass (g)

0 50 100 150

0.2 0.4 0.6 0.8 1.0 1.2 1.4

Height (cm)

3 Biomass

Diagnostic Plots: Height & Width Model

−2 −1 0 1 2

−80000

−70000

−60000

−50000

−40000

−30000

λ

Log likelihood

95%

λused = 0.33

−4 −2 0 2 4

−0.15

−0.10

−0.05 0.00 0.05 0.10 0.15

Normal quantile

Residual

0.2 0.6 1.0 1.4

−3

−2

−1 0 1 2 3

Fitted value

Studentized residual

Diagnostic Plots: Height Model

−2 −1 0 1 2

−80000

−70000

−60000

−50000

−40000

−30000

λ

Log likelihood

95%

λused = 0.33

−4 −2 0 2 4

−0.2

−0.1 0.0 0.1 0.2

Normal quantile

Residual

0.4 0.6 0.8 1.0 1.2 1.4

−3

−2

−1 0 1 2 3

Fitted value

Studentized residual

Solidago sempervirens L.

Other names: Seaside goldenrod, saltmarsh goldenrod

Description of Dataset

• Year(s) plants collected: 2013 - 2015

• Total records: 135 (4 without Width). Outliers: 4 for Height & Width model, 2 for Height model.

Variable Unit Min Median Max

Biomass g 0.99 4.96 26.22

Height cm 26 70 132

Width mm 0.5 3.8 9.3

Allometric Models

If Height and Width measurements are both available, the model with the greatest predictive ability is:

ln(Biomass) =−0.4384563 + 0.0163466·Height+ 0.219054·W idth−7.354272×10⁻⁵·Height·W idth If only Height measurements are available, the model with the greatest predictive ability is:

ln(Biomass) =−0.4835933 + 0.0282768·Height

Model Evaluation

The following table compares the full model for Height & Width with all nested models. Coefficients are not standardized and only shown when terms are included in a model. The same transformation of Biomass is used in all cases. Note that BIC scores are substantially improved (i.e., >4 smaller) by including Width in models, and for all models with Height or Width compared to the intercept-only (i.e., null) model.

Intercept Height Width Height×Width R² BIC

-0.4385 0.01635 0.2191 -7.354e-05 0.806 98

-0.416 0.01602 0.2135 0.806 93

-0.4133 0.02744 0.71 139

0.1065 0.3816 0.7 144

1.633 0 292

The Height model and corresponding null model are summarized below.

Intercept Height R² BIC

-0.4836 0.02828 0.706 164

1.599 0 321

40 60 80 100 120 0

5 10 15 20 25

Height (cm)

Biomass (g)

40 60 80 100 120

0 1 2 3

Height (cm)

ln

(

Biomass

)

Diagnostic Plots: Height & Width Model

−2 −1 0 1 2

−350

−300

−250

−200

λ

Log likelihood

95%

λused = 0

−2 −1 0 1 2

−0.5 0.0 0.5 1.0

Normal quantile

Residual

0.5 1.5 2.5

−3

−2

−1 0 1 2 3

Fitted value

Studentized residual

Diagnostic Plots: Height Model

−2 −1 0 1 2

−450

−400

−350

−300

−250

λ

Log likelihood

95%

λused = 0

−2 −1 0 1 2

−0.5 0.0 0.5 1.0

Normal quantile

Residual

0.5 1.5 2.5

−2

−1 0 1 2 3

Fitted value

Studentized residual

Spartina alterniflora Loisel.

Other names: Smooth cordgrass, saltmarsh cordgrass, saltwater cord grass

Description of Dataset

• Year(s) plants collected: 2015

• Total records: 30 (0 without Width). Outliers: 1 for Height & Width model, 2 for Height model.

Variable Unit Min Median Max

Biomass g 1.22 3.02 6.64

Height cm 84 104.5 130

Width mm 0.1 2.8 5.5

Allometric Models

If Height and Width measurements are both available, the model with the greatest predictive ability is:

√3

Biomass²= 0.1184525 + 0.0142013·Height−0.3622746·W idth+ 0.0048323·Height·W idth If only Height measurements are available, the model with the greatest predictive ability is:

√3

Biomass=−0.0549767 + 0.014144·Height

Model Evaluation

The following table compares the full model for Height & Width with all nested models. Coefficients are not standardized and only shown when terms are included in a model. The same transformation of Biomass is used in all cases. Note that BIC scores are substantially improved (i.e., >4 smaller) by including Width in models, and for all models with Height or Width compared to the intercept-only (i.e., null) model.

Intercept Height Width Height×Width R² BIC

0.1185 0.0142 -0.3623 0.004832 0.82 22

-1.319 0.02866 0.1383 0.791 22

-1.995 0.0386 0.7 30

1.333 0.2723 0.535 42

2.057 0 61

The Height model and corresponding null model are summarized below.

Intercept Height R² BIC

-0.05498 0.01414 0.777 -38

1.424 0 0.3

90 100 110 120 130 2

3 4 5 6

Height (cm)

Biomass (g)

90 100 110 120 130 1.5

2.0 2.5 3.0 3.5

Height (cm)

3 Biomass2

90 100 110 120 130 1.2

1.4 1.6 1.8

Height (cm)

3 Biomass

Diagnostic Plots: Height & Width Model

−2 −1 0 1 2

−20

−15

−10

−5

λ

Log likelihood

95%

λused = 0.67

−2 −1 0 1 2

−0.4

−0.2 0.0 0.2 0.4

Normal quantile

Residual

1.5 2.0 2.5 3.0 3.5

−2

−1 0 1 2

Fitted value

Studentized residual

Diagnostic Plots: Height Model

−2 −1 0 1 2

−20

−15

−10

−5

λ

Log likelihood

95%

λused = 0.33

−2 −1 0 1 2

−0.2

−0.1 0.0 0.1 0.2

Normal quantile

Residual

1.2 1.4 1.6 1.8

−2

−1 0 1 2

Fitted value

Studentized residual

Spartina cynosuroides L.

Other names: Big cordgrass, salt reedgrass

Description of Dataset

• Year(s) plants collected: 2014 - 2015

• Total records: 101 (0 without Width). Outliers: 5 for Height & Width model, 6 for Height model.

Variable Unit Min Median Max Biomass g 4.12 17.485 58.36

Height cm 82 210 294

Width mm 1.38 6.15 11.6

Allometric Models

If Height and Width measurements are both available, the model with the greatest predictive ability is:

ln(Biomass) = 0.7104917 + 0.0059962·Height+ 0.0965718·W idth+ 1.9822987×10⁻⁴·Height·W idth If only Height measurements are available, the model with the greatest predictive ability is:

ln(Biomass) = 0.4680203 + 0.011305·Height

Model Evaluation

The following table compares the full model for Height & Width with all nested models. Coefficients are not standardized and only shown when terms are included in a model. The same transformation of Biomass is used in all cases. Note that BIC scores are substantially improved (i.e., >4 smaller) by including Width in models, and for all models with Height or Width compared to the intercept-only (i.e., null) model.

Intercept Height Width Height×Width R² BIC

0.7105 0.005996 0.09657 0.0001982 0.883 -30

0.4919 0.007135 0.1365 0.882 -33

0.5213 0.01118 0.742 37

1.35 0.2353 0.704 50

2.858 0 162

The Height model and corresponding null model are summarized below.

Intercept Height R² BIC

0.468 0.01131 0.809 5

2.845 0 158

100 150 200 250 300 10

20 30 40 50 60

Height (cm)

Biomass (g)

100 150 200 250 300

1.5 2.0 2.5 3.0 3.5 4.0

Height (cm)

ln

(

Biomass

)

Diagnostic Plots: Height & Width Model

−2 −1 0 1 2

−160

−140

−120

−100

−80

−60

λ

Log likelihood

95%

λused = 0

−2 −1 0 1 2

−0.4

−0.2 0.0 0.2 0.4

Normal quantile

Residual

1.5 2.0 2.5 3.0 3.5 4.0

−2

−1 0 1 2

Fitted value

Studentized residual

Diagnostic Plots: Height Model

−2 −1 0 1 2

−200

−180

−160

−140

−120

−100

−80

λ

Log likelihood

95%

λused = 0

−2 −1 0 1 2

−0.4

−0.2 0.0 0.2 0.4

Normal quantile

Residual

1.5 2.0 2.5 3.0 3.5

−2

−1 0 1 2

Fitted value

Studentized residual

Typha angustifolia L.

Other names: Narrowleaf cattail, small reed mace

Description of Dataset

• Year(s) plants collected: 2015

• Total records: 30 (0 without Width). Outliers: 2 for Height & Width model, 0 for Height model.

Variable Unit Min Median Max

Biomass g 2.12 5.71 13.01

Height cm 94 162.5 246

Width mm 2.6 7.55 18.6

Allometric Models

If Height and Width measurements are both available, the model with the greatest predictive ability is:

Biomass= 1.2156641 + 2.4404943×10⁻⁴·Height+ 0.3009887·W idth+ 0.0017654·Height·W idth If only Height measurements are available, the model with the greatest predictive ability is:

ln(Biomass) = 0.3643316 + 0.0086128·Height

Model Evaluation

The following table compares the full model for Height & Width with all nested models. Coefficients are not standardized and only shown when terms are included in a model. The same transformation of Biomass is used in all cases. Note that BIC scores are substantially improved (i.e., >4 smaller) by including Width in models, and for all models with Height or Width compared to the intercept-only (i.e., null) model.

Intercept Height Width Height×Width R² BIC

1.216 0.000244 0.301 0.001765 0.892 90

-0.8202 0.01245 0.6071 0.888 87

0.7286 0.6609 0.869 89

-0.9789 0.04455 0.322 135

6.131 0 142

The Height model and corresponding null model are summarized below.

Intercept Height R² BIC

0.3643 0.008613 0.413 34

1.755 0 46

100 150 200 250 2

4 6 8 10 12 14

Height (cm)

Biomass (g)

100 150 200 250

2 4 6 8 10 12 14

Height (cm)

Biomass

100 150 200 250

1.0 1.5 2.0 2.5

Height (cm)

ln

(

Biomass

)

Diagnostic Plots: Height & Width Model

−2 −1 0 1 2

−25

−20

−15

−10

−5 0 5

λ

Log likelihood

95%

λused = 1

−2 −1 0 1 2

−1.5

−1.0

−0.5 0.0 0.5 1.0 1.5

Normal quantile

Residual

4 6 8 10 12

−2

−1 0 1 2

Fitted value

Studentized residual

Diagnostic Plots: Height Model

−2 −1 0 1 2

−40

−35

−30

−25

−20

λ

Log likelihood

95%

λused = 0

−2 −1 0 1 2

−0.6

−0.4

−0.2 0.0 0.2 0.4 0.6

Normal quantile

Residual

1.2 1.6 2.0 2.4

−2

−1 0 1

Fitted value

Studentized residual

General Graminoid

If a graminoid of interest is not included above, the equations provided here may be used. The underlying dataset is a stratified random selection of records from the five grass, grass-like, and sedge species in the dataset (~30 individuals each).

Description of Dataset

• Year(s) plants collected: 1987 - 2015

• Total records: 147 (0 without Width). Outliers: 3 for Height & Width model, 3 for Height model.

Variable Unit Min Median Max Biomass g 0.03 4.825 59.3

Height cm 28 132.5 315

Width mm 0.1 5.1 18.6

Allometric Models

If Height and Width measurements are both available, the model with the greatest predictive ability is:

√3

Biomass= 0.2009914 + 0.0102512·Height−0.0071438·W idth+ 1.2751998×10⁻⁴·Height·W idth If only Height measurements are available, the model with the greatest predictive ability is:

√3

Biomass= 0.1220092 + 0.011304·Height

Model Evaluation

The following table compares the full model for Height & Width with all nested models. Coefficients are not standardized and only shown when terms are included in a model. The same transformation of Biomass is used in all cases. Note that BIC scores are substantially improved (i.e., >4 smaller) by including Width in models, and for all models with Height or Width compared to the intercept-only (i.e., null) model.

Intercept Height Width Height×Width R² BIC

0.201 0.01025 -0.007144 0.0001275 0.912 17

0.106 0.01095 0.01304 0.912 13

0.122 0.0113 0.91 10

0.9451 0.1636 0.375 282

1.805 0 343

The Height model and corresponding null model are summarized below.

Intercept Height R² BIC

0.122 0.0113 0.91 10

50 100 150 200 250 300 0

10 20 30 40 50 60

Height (cm)

Biomass (g)

50 100 150 200 250 300

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Height (cm)

3 Biomass

Diagnostic Plots: Height & Width Model

−2 −1 0 1 2

−1200

−1000

−800

−600

−400

−200

λ

Log likelihood

95%

λused = 0.33

−2 −1 0 1 2

−0.6

−0.4

−0.2 0.0 0.2 0.4 0.6

Normal quantile

Residual

0.5 1.5 2.5 3.5

−2

−1 0 1 2

Fitted value

Studentized residual

Diagnostic Plots: Height Model

−2 −1 0 1 2

−1200

−1000

−800

−600

−400

−200

λ

Log likelihood

95%

λused = 0.33

−2 −1 0 1 2

−0.6

−0.4

−0.2 0.0 0.2 0.4 0.6

Normal quantile

Residual

0.5 1.5 2.5 3.5

−2

−1 0 1 2

Fitted value

Studentized residual

General Forb

If a forb of interest is not included above, the equations provided here may be used. The underlying dataset is a stratified random selection of records from the five forb species in the dataset (~30 individuals each).

Description of Dataset

• Year(s) plants collected: 2011 - 2015

• Total records: 150 (0 without Width). Outliers: 0 for Height & Width model, 0 for Height model.

Variable Unit Min Median Max Biomass g 0.06 2.3905 61.68

Height cm 18 81.5 200

Width mm 0.1 2.9 13.6

Allometric Models

If Height and Width measurements are both available, the model with the greatest predictive ability is:

ln(Biomass) =−2.2629364 + 0.0219558·Height+ 0.5933913·W idth−0.0025767·Height·W idth If only Height measurements are available, the model with the greatest predictive ability is:

ln(Biomass) =−1.7179881 + 0.0294871·Height

Model Evaluation

The following table compares the full model for Height & Width with all nested models. Coefficients are not standardized and only shown when terms are included in a model. The same transformation of Biomass is used in all cases. Note that BIC scores are substantially improved (i.e., >4 smaller) by including Width in models, and for all models with Height or Width compared to the intercept-only (i.e., null) model.

Intercept Height Width Height×Width R² BIC

-2.263 0.02196 0.5934 -0.002577 0.689 386

-1.389 0.01365 0.2803 0.656 396

-0.6883 0.4113 0.604 412

-1.718 0.02949 0.528 438

0.8657 0 546

The Height model and corresponding null model are summarized below.

Intercept Height R² BIC

-1.718 0.02949 0.528 438

0.8657 0 546

50 100 150 200 0

10 20 30 40 50 60

Height (cm)

Biomass (g)

50 100 150 200

−3

−2

−1 0 1 2 3 4

Height (cm)

ln

(

Biomass

)

Diagnostic Plots: Height & Width Model

−2 −1 0 1 2

−1000

−900

−800

−700

−600

−500

−400

λ

Log likelihood

95%

λused = 0

−2 −1 0 1 2

−1 0 1 2

Normal quantile

Residual

−1 0 1 2 3

−2

−1 0 1 2

Fitted value

Studentized residual

Diagnostic Plots: Height Model

−2 −1 0 1 2

−1000

−900

−800

−700

−600

−500

−400

λ

Log likelihood

95%

λused = 0

−2 −1 0 1 2

−1 0 1 2

Normal quantile

Residual

−1 0 1 2 3 4

−2

−1 0 1 2

Fitted value

Studentized residual