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Table 2 Statistical criteria for model selection applied to biomass estimation of different woody species indigenous of the Tropical Atlantic Rain Forest, Brazil

From: Selection criteria for linear regression models to estimate individual tree biomasses in the Atlantic Rain Forest, Brazil

 

Criterion

Formula

1

Sum of squares of the residuals

\( SSR = \sum\limits_{i - 1}^{n} {e_{i}^{2} } = \sum\limits_{i - 1}^{n} {(w_{i} - \hat{w}_{i} )^{2} } \)

(7)

2

Adjusted coefficient of determination

\( R^{2}_{adj.} = 1 - \frac{(n - 1)}{(n - p)}(1 - R^{2} ) \)

where \( R^{2} = 1 - \frac{{\sum\nolimits_{i - 1}^{n} {e_{i}^{2} } }}{{\sum\nolimits_{i - 1}^{n} {(w_{i} - \bar{w})^{2} } }} \)

(8)

3

Relative standard error of the estimate

\( Syx\% = \frac{syx}{{\bar{y}}}100 \)

where \( Syx = \sqrt {\frac{{\sum\nolimits_{i - 1}^{n} {e_{i}^{2} } }}{n - p}} \)

(10)

4

Akaike information criterion [20]

\( AIC = - 2n\left( {\frac{ - n}{2}\ln \left( {\frac{1}{n}\sum\limits_{i - 1}^{n} {e_{i}^{2} } } \right)} \right) + 2p \)

(11)

5

Akaike information criterion not biased for small samplesa, when (n/p) < 40

\( AIC_{c} = - 2n\left( {\frac{ - n}{2}\ln \left( {\frac{1}{n}\sum\limits_{i - 1}^{n} {e_{i}^{2} } } \right)} \right) + 2p\frac{n}{(n - p - 1)} \)

(12)

6

Schwartz’s information criterion [21]

\( BIC = - 2n\left( {\frac{ - n}{2}\ln \left( {\frac{1}{n}\sum\limits_{i - 1}^{n} {e_{i}^{2} } } \right)} \right) + \ln (n)p \)

(13)

7

Residuals (in %)

\( r_{i} = \frac{{(w_{i} - \hat{w}_{i} )}}{{w_{i} }}100 \)

(14)

  1. \( \hat{w}_{i} \) = estimated biomass. wi = actual biomass. In AIC, AICc and BICp must be increased by 1, which refers to one degree of freedom for variance
  2. aAccording to [11]. Where n = number of data; p = number of parameters of the model (number of coefficients including the intercept + 1)