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The work modeled the possibility of a tree dying in the next year by using logistic function (Equation 16), which is widely applied for tree mortality, tree mortality ranges between 0 and 1. The data used for mortality modelling records for 72 dead and 3004 live trees from 51 plots.

( ) (21) Pi = The probability of tree mortality

b0–bn = Parameters to be estimated x1–xn = Explanatory variables

The candidate variables for the mortality equation were numerous and diverse. As Pedersen (2007) pointed out, “test statistics and a basic understanding of how forest ecosystems func-

tion and how factors contributing to mortality are expressed, play an important role in select- ing‎ the‎ appropriate‎ predictor‎ variables”. Numerous studies have revealed many variables

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that are important for mortality predictions. To develop the natural mortality models, the re- search applied the tree size variables, competition variables and stand level variables.

The tree size variables were: tree diameter at breast height (d1.3) and its transformations (natu-

ral logarithm of d1.3, d1.3-1, d1.32), tree basal area (ba), tree height (h). Additionally, one deriva-

tive variable was checked, h/ d1.3.

The competition variables tested to select the best index for predicting the individual-tree mortality equation included the following indices: basal area of trees larger than the subject tree divided by stand basal area, stand density index, diameter at breast height divided by stand age, basal area of trees larger than the subject tree divided by diameter at breast height, ratio of the tree height and stand top height, squared diameter at breast height divided by stand basal area, basal area of trees larger than the subject tree times squared diameter at breast height, basal area of trees larger than the subject tree, diameter at breast height divided by stand age.

Stand level variables used for the analysis were: stand age, mean stand height (H), quadratic mean diameter (Dg), top height (H100), site index (SI) and basal area of stand per hectare

(BA).

When analyzing each variable, The work conducted a univariate analysis and variables with a significance level lower than 0.25 were used in multivariable analysis (Hosmer and Leme- show, 2000).

The second step, the research verified the importance of each variable included in the equa- tion by applying Wald statistics and comparison of each estimated coefficient with the coeffi- cient from the equation that contains only this variable. Variables that do not meet these crite- ria were removed and new equation was fit.

Third and last step, the study checked correlations among the variables used in the equation. Also, the equation was evaluated by the means of logistic regression analysis:

1) Checking the statistical significance of the model and its estimated parameters. There should be no high inter-correlations (multicollinearity) among the predictors. This can be assessed by a correlation matrix among the predictors. Tabachnick and Fidell (2012) sug- gest that as long correlation coefficients among independent variables are less than 0.90 the assumption is met.

2) The Pearson chi-square is calculated as follows:

( ) ( ) (22)

X2 _{= Value of Pearson‟s chi square statistics}

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= Value of the maximum likelihood function of analyzed model with specified coefficients

Large chi-squared values provided evidence of lack of fit. When the chi-squared values we calculated are less than the critical chi-squared values with significant level p = 0.05, it means there is no significant difference between the probability of observed and predicted dead trees. 3) Estimating the following parameters: log likelihood function values, the Nagelkerkle and

McFadden‟s coefficients of determination and areas under the ROC curves.

The Nagelkerkle R2 and the McFadden R2 are recommended (Allison, 2014), which were pre- sented in this study (Allison, 2014).

McFadden‟s R2 is defined as fllows:

( ) ( ) (23) The Nagelkerkle R2 is :

( ) ( )

(24)

= McFadden‟s coefficients of determination ; q = Number of observations

= Nagelkerkle‟s coefficients of determination

L0 = Value of the maximum likelihood function of logistic regression model with no predictors LM = Value of the maximum likelihood function of analyzed model with specified coefficients

The log-likelihood function is defined to be the natural logarithm of the function.

(𝜃) (𝜃) ∑ 𝑓( ) (25)

L = Value of the maximum likelihood function q = Number of observations 𝜃 = An unknown parameter = Continuous random variable

The log-likelihood function is used throughout various subfields of mathematics, both pure and applied, and has particular importance in fields such as likelihood theory.

The larger the Log likelihood function value the worse adapted is the model. The area under the receiver operating characteristic (ROC) curve was calculated

for the mortality model. It is a threshold independent measure of model discrimination, where a value of 0.5 suggests no discrimination, 0.7–0.8 suggests acceptable discrimination, and 0.8–0.9 suggests excellent discrimination (Hosmer and Lemeshow, 2000).

The predicted and observed mortality were then compared by visually studying deviations over the explicatory variables included in the model. A threshold can be used to assign mor- tality. If the estimated probability of mortality exceeds the threshold then the tree is consid- ered dead. The research tested three cut-points: the first was the overall mortality rate (Monserud and Sterba, 1999), the second was the intersection point of sensitivity and

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specificity (Adame et al, 2010) and the third was a random number which can also be consid- ered by running the random calculation 10 times. The percentage of live and dead trees is the average classification rates (Bravo-Oviedo et al., 2006). In other words, the study implement- ed an individual-tree mortality equation either stochastically by using a random number or deterministically by using overall mortality rate and the intersection point of sensitivity and specificity. For the comparison of the stochastic and deterministic approaches, the simulations were run on a 100-year period using one-year growth steps by running the random calculation

10 times. The average of four different response variables (Quadratic mean diameter, stand

basal area, stand volume and number of trees per hectare) of stochastic approach were com- pared with the four different response variables of deterministic approach, and then, the study used Mann Whitney U Test (Wilcoxon Rank Sum Test) for testing the null hypothesis that the differences between stochastic and deterministic simulations are negligible.