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Stand mortality models play an important role in simulating stand dynamic processes. Periodic stand mortality data from permanent plots tend to be dispersed, and frequently contain an excess of zero counts. Such data have commonly been analyzed using the Poisson distribution and Poisson mixture models, such as the zero-inflated Poisson model (ZIP), and the Hurdle Poisson model (HP). Based on mortality data obtained from sixty Chinese pine (

An important component equation of whole stand models is one that predicts, explicitly or implicitly, stand mortality (trees ha^{-1}) over a period of time (

Mortality is a complicated stochastic process influenced by several stand characteristics and environment factors, which often exist with complex interactions. It is hardly possible to capture all of the observed variability in empirical mortality data. The evolution of the mixed-effects modeling methodology provided a statistical method capable of explicit modeling stochastic structure is a possible approach for solving this problem (

In addition, it has to be considered that a relatively high number of permanent sample plots has no occurrence of mortality, even over periods of several years. Therefore, mortality data are truncated and characteristically exhibit varying degrees of dispersion and skewness in relation to the mean. Moreover, the data often contain an excess number of zero counts. The least squares method implicitly presumes that the data are Gaussian distributed with constant variances, or at least satisfy the Gauss-Markov conditions. If the least squares method is applied to data with a large proportion of zero counts, the estimated results will be biased. Alternatively, if only nonzero mortality observations are used for model development, then mortality will be overestimated (

Although many forest models accounted for mixed effects such as tree diameter growth models (

A systematic sampling of permanent, square-shaped plots (0.067 ha) was carried out by the Inventory Institute of Beijing Forestry with a 5-year re-measurement interval. Overall, sixty plots were sampled in several Chinese pine (

Independent variables characterized by a meaningful biological interpretation were selected among those describing altitude (

Count data models are a subset of discrete-response regression models aimed to describe the number of occurrences or counts of an event. Poisson regression is the simplest regression model for count data, and the probability mass function (PMF) is expressed as follows (

where

ZIP (zero-inflated Poisson) model is a mixed model combining a Poisson distribution with a point mass at zero. In zero-inflated Poisson model, there are two sources of zeros, deriving both from the point mass and from the count component (

where

HP (Hurdle Poisson) model, originally proposed by

Similar to ZIP model, HP model is the combination of a logit regression modeling point mass at zero and a truncated Poisson regression modeling count component. Its PMF is expressed as follows (

The point mass component is obtained by logistic regression using logit(

In this study, a plot-level random-effects parameter was added to the intercept for the Poisson model, and to the intercept of the Poisson component for the estimation of positive mortality counted for ZIP model and HP model. The random-effect parameter was defined as:

All parameter vectors can be estimated through the optimization of the likelihood function, that is, by applying the maximum likelihood method (ML). The unstructured covariance structure (

Performances of the Poisson fixed-effects model, ZIP fixed-effects model, HP fixed-effects model and corresponding mixed-effect models calibrated with the same data set can be compared by using the log-likelihood values [-2L(

The Vuong test is a popular method to compare two non-nested models for count data (

where _{j}(_{i} | _{i}) is the predicted probability of the observed count for the _{i}=0) is expressed as follows (

where

Because of the use of maximum log-likelihoods, AIC, BIC, and Vuong parameters are relative statistics, as they do not ensure that the fit of the “best” model is good. Hence, residual plots (residual values between predicted and observed probabilities plotted against the count class _{j}, between predicted probabilities and observed probabilities was computed as (

where # represents the frequency of observations _{i} in the count class

In this study, a relatively high number of the plots considered has no occurrence of mortality. Mortality data are zero-bounded and characteristically exhibit varying degrees of dispersion around the mean and skewness (

During the processes of estimating all the six models considered, we found that the ZIP mixed-effects model was not convergent, so the following results were obtained from the remnant five models.

Results showed that the statistics -2L(

Poisson fixed-effects and Poisson mixed-effects models were found to largely underestimate the zero-class counts, while ZIP fixed-effects, HP fixed-effects and HP mixed-effects models exactly estimated the zero dead counts (_{j} of the Poisson mixed-effects model were smaller than those obtained for the Poisson fixed-effects model, while the HP mixed-effects model showed lower residuals than those obtained from the HP fixed-effects, ZIP fixed-effects and Poisson mixed-effects model (

Additionally, the Vuong test-statistic

Overall, the best fit was detected for the HP fixed-effects and the HP mixed-effects model, therefore only these models were used in further analysis. Some variables with p>0.05 after the

In a tree mortality context, the use of HP models seems to be the most appropriate when applied to cases with no mortality, as it occurs in a stand before the competition among trees takes place. Once competitive pressures within the stand exceeds a certain threshold, positive mortality occurs consistently and in accordance with a ZIP probability mass function (

In our study, the application of both the above models led to the same qualitative results and gave very similar model fitting performances (

In this study, we used Poisson mixture model with mixed-effects to analyze stand mortality. Random-effects of both the Poisson model and HP model were significant at the 0.05 level. However, the ZIP model was not convergent after the random-effects were incorporated in the intercept of positive count component.

This work was carried out within the frame of the following projects: the Chinese State “TWELVE FIVE” Science and Technology Project (No. 2012BAD22B0201), and the “863” program from the Ministry of Science and Technology of China (No. 2012AA12 A306). The authors are grateful to two anonymous reviewers for their valuable suggestions and comments on an earlier version of the manuscript.

Location of the 60 plantations of Chinese pine (

Histogram of stand mortality for Chinese pine obtained from the sixty plots selected in the study area.

Plots of residuals for the Poisson fixed-effects model (A), Poisson mixed-effects model (B), HP fixed-effects model (C), HP mixed-effects model (D), and ZIP fixed-effects model (E). (_{j}): residuals between predicted and observed probability as computed by using the

Summary statistics of stand-level variables. (SD): standard deviation: (^{0.5}/

Variables | Min | Max | Mean | SD |
---|---|---|---|---|

^{-1}) |
0 | 38 | 2.6 | 6.46 |

12 | 55 | 27 | 8.46 | |

2.5 | 17.4 | 6.4 | 2.83 | |

^{-1}) |
132 | 2164 | 933 | 492.24 |

5.8 | 16.13 | 9.76 | 2.4 | |

^{-1 }m^{-1}) |
0.17 | 3.45 | 0.71 | 0.63 |

140 | 1400 | 449.91 | 268.09 |

Fit statistics of Poisson fixed-effects model, Poisson mixed-effects model, ZIP fixed-effects, HP fixed-effects model, and HP mixed-effects model.

Statistic | Poisson-fixed | Poisson-mixed | ZIP-fixed | HP-fixed | HP-mixed |
---|---|---|---|---|---|

-2L( |
1305.8 | 658.9 | 783.1 | 783 | 513.8 |

AIC | 1319.8 | 670.9 | 795.1 | 795 | 527.8 |

BIC | 1341.4 | 683.8 | 813.6 | 813.6 | 542.8 |

Estimations of parameters of the two model components in the HP fixed-effects and the HP mixed-effects models. (SE): standard error.

ModelComponent | Parameter | HP-fixed | HP-mixed | ||||
---|---|---|---|---|---|---|---|

Estimate | SE | Estimate | SE | ||||

Positive count component | Intercept | 4.0177 | 0.3249 | <0.001 | 6.2386 | 1.0851 | <0.001 |

Rs | -1.5462 | 0.3091 | <0.001 | -3.3797 | 1.1974 | <0.01 | |

Dm | -0.1187 | 0.0244 | <0.001 | -0.3171 | 0.0701 | <0.001 | |

v | - | - | - | 1.764 | 0.5998 | <0.01 | |

Zero component | Intercept | -3.321 | 1.1945 | <0.01 | -3.321 | 1.1945 | <0.01 |

H | 0.1846 | 0.0903 | <0.05 | 0.1846 | 0.0903 | <0.05 | |

Rs | 4.7581 | 1.222 | <0.001 | 4.758 | 1.222 | <0.001 |