Suborna Shekhor Ahmed Department of Forest Resources Management Faculty of Forestry, UBC Western Mensurationists Conference Missoula, MT June 20 to 22, 2010 Modeling Tree Mortality for Large Regions Using Combined Estimators and Meta-Analysis Approaches

Objectives Modelling Tree Mortality Using Meta Modelling Slide-2 Develop mortality models for four target species of the Boreal Forest of Canada (aspen, white spruce, black spruce and jack pine) Data from Alberta, Ontario and Quebec will be used, along with a combined estimator and local mortality models. To select a combined estimator, several estimators will be proposed and tested using the PSP data from Alberta. For this presentation, I will present preliminary results of testing combined estimators using PSP data from Alberta for aspen.

Tree Mortality Tree Mortality Modelling Tree Mortality Using Meta Modelling Slide-3 Tree mortality is an important aspect of stand dynamics and is commonly expressed in terms of loss of volume or basal area per year. Cause and time of death are very important to model the tree mortality. The following variables are considered for modeling tree mortality: Diameter at breast height (DBH), Annual diameter increment during the preceding interval (DIN), Total basal area per hectare at the beginning of the growth interval (BAHA), Site productivity index, Species composition, Length of the growth interval (L), Other measures of competition.

Generalized Logistic Model Generalized Logistic Model Modelling Tree Mortality Using Meta Modelling Slide-4 For repeated measures where the time interval, L, is irregular, a generalized logistic model has been used to model survival where is the annual probability of survival. s are the unknown parameters with explanatory variables. From this, the annual probability of mortality is:

Published Mortality (or Survival) Models for Aspen Modelling Tree Mortality Using Meta Modelling Slide-5 Reference Study location Model Yao et al. (2001) Alberta mixed wood forests Lacerte et al. (2006) Ontario Senecal et al. (2004) Quebec’s boreal forest : white spruce (Picea glauca) species composition as a percentage of BAHA; SPI: site productivity index; canopy position: an ordinal variable of position of the tree within the canopy; growth: corresponds to the last year of radial growth (millimetres); All other variables are previously defined.

Meta Modelling Approaches Meta Modelling Approaches Modelling Tree Mortality Using Meta Modelling Slide-6 Combined Estimator Meta-modeling approaches use observational data to obtain weights for combining existing local scale models may result in improved precision over the naïve approaches. The general approach would be: : kth estimated parameter using the combination of parameter estimates from the r local spatial models; : kth estimated parameter for local scale model j; : weight between 0 and 1 applied to the kth estimated parameters for local scale model j; r : number of local scale models. Sum of the over all regions is 1 for each parameter. Where,

Meta Modelling Approaches Meta Modelling Approaches Modelling Tree Mortality Using Meta Modelling Slide-7 Native Approach 1: One of the native approaches is to use all available data to fit a large scale model Native Approach 2 ( Equal Weights ) : Giving equal weight to each estimate: Where, : weight between 0 and 1 applied to the estimated parameters for local scale model j; r : number of local scale models.

Meta Modelling Approaches Meta Modelling Approaches Modelling Tree Mortality Using Meta Modelling Slide-8 Based on Cochran (Inverse Variances): Weight the parameters by the inverse of their variances, based on Cochran (1977) and extending to r >2: where indicates variance of a particular parameter estimate.

Meta Modelling Approaches Meta Modelling Approaches Modelling Tree Mortality Using Meta Modelling Slide-9 Maximum Likelihood Optimal weights are found that meet a maximum likelihood objective function. Options include having the same weights for all parameters versus having differential weights by parameter.

Meta Modelling Approaches Meta Modelling Approaches Modelling Tree Mortality Using Meta Modelling Slide-10 Stein Rule Estimator (Shrinkage) Where, : vector of estimated parameter for combined model; : weight between 0 and 1 applied for local scale model j; : vector of estimated parameters for local scale model j. : number of observations in the jth region.

Alberta Data Study includes over 1,700 plots measured up to seven times with a variable number of years between measurements, dispersed over the forested land of Alberta. Each plot summarized at each measurement period to obtain explanatory variables for modeling for each species and all species combined. The tree-level variables were then merged with the plot-level variables. The summarized data were considered as census data at the large spatial scale for this research. Competition mortality of trees was taken into account. Plots that have a majority of aspen trees (greater than 30% by basal area per ha in any measurement period) were selected for use. Modelling Tree Mortality Using Meta Modelling Slide-11

Steps for Meta Modelling Using Aspen Data Modelling Tree Mortality Using Meta Modelling Slide-12 Fitted the generalized logistic survival model using all data combined. Split Alberta data into two regions using township and fitted the model separately. Implemented the combined estimators.

Natural Regions of Alberta Map source: Modelling Tree Mortality Using Meta Modelling Slide-13

Results: Fitted Model Results: Fitted Model Modelling Tree Mortality Using Meta Modelling Slide-14 The probability of survival model for the entire data set and for each region is: Where, : probability of survival to the end of the period. DIN: annual diameter increment during the preceding interval. BAL: basal area of all trees larger than the subject tree. L: length of the growth interval. SpOther: percentage of basal area per ha of all trees that were not aspen.

Results: Statistics for Tree- and Plot-Level Variables Results: Statistics for Tree- and Plot-Level Variables Modelling Tree Mortality Using Meta Modelling Slide-15 Data set Number of Plots Number of trees VariableMeanMinimumMaximum All data combined DBH DIN L Region DBH DIN L Region DBH DIN L

Results: Statistics for Tree- and Plot-Level Variables Results: Statistics for Tree- and Plot-Level Variables Modelling Tree Mortality Using Meta Modelling Slide-16 Data set Number of Plots Number of trees VariableMeanMinimumMaximum All data combined DBH DIN L Region DBH DIN L Region DBH DIN L

Results: Statistics for Tree- and Plot-Level Variables Results: Statistics for Tree- and Plot-Level Variables Modelling Tree Mortality Using Meta Modelling Slide-17 Data set Number of Plots Number of trees VariableMeanMinimumMaximum All data combined DBH DIN L Region DBH DIN L Region DBH DIN L

Results: Estimated Parameters (Standard Errors in Brackets) and Fit statistics Results: Estimated Parameters (Standard Errors in Brackets) and Fit statistics Modelling Tree Mortality Using Meta Modelling Slide-18 Parameter/ Fit Statistics All data combinedRegion 1Region 2 (Intercept) (0.0747) (0.0779) (0.2615) (DBH) ( ) ( ) ( ) (DIN) (0.2220) (0.2382) (0.6273) (BAL) ( ) ( ) ( ) (SpOther) ( ) ( ) ( ) -2logL

Results: Estimated Parameters (Standard Errors in Brackets) and Fit statistics Results: Estimated Parameters (Standard Errors in Brackets) and Fit statistics Modelling Tree Mortality Using Meta Modelling Slide-19 Parameter/ Fit Statistics All data combinedRegion 1Region 2 (Intercept) (0.0747) (0.0779) (0.2615) (DBH) ( ) ( ) ( ) (DIN) (0.2220) (0.2382) (0.6273) (BAL) ( ) ( ) ( ) (SpOther) ( ) ( ) ( ) -2logL

Results: Estimated Parameters (Standard Errors in Brackets) and Fit statistics Results: Estimated Parameters (Standard Errors in Brackets) and Fit statistics Modelling Tree Mortality Using Meta Modelling Slide-20 Parameter/ Fit Statistics All data combinedRegion 1Region 2 (Intercept) (0.0747) (0.0779) (0.2615) (DBH) ( ) ( ) ( ) (DIN) (0.2220) (0.2382) (0.6273) (BAL) ( ) ( ) ( ) (SpOther) ( ) ( ) ( ) -2logL

Results: Weights of Parameters Using Meta- Regression for Aspen Results: Weights of Parameters Using Meta- Regression for Aspen Modelling Tree Mortality Using Meta Modelling Slide-21 MethodRegion Weights of parameters Equal weight Region Region Inverse Variances Region Region ML Region Region ML (Differential Weight) Region Region Stein Rule Region Region

Results: Weights of Parameters Using Meta- Regression for Aspen Results: Weights of Parameters Using Meta- Regression for Aspen Modelling Tree Mortality Using Meta Modelling Slide-22 MethodRegion Weights of parameters Equal weight Region Region Inverse Variances Region Region ML Region Region ML (Differential Weight) Region Region Stein Rule Region Region

Results: Weights of Parameters Using Meta- Regression for Aspen Results: Weights of Parameters Using Meta- Regression for Aspen Modelling Tree Mortality Using Meta Modelling Slide-23 MethodRegion Weights of parameters Equal weight Region Region Inverse Variances Region Region ML Region Region ML (Differential Weight) Region Region Stein Rule Region Region

Results: Estimated Parameters Using Meta- Regression for Aspen Results: Estimated Parameters Using Meta- Regression for Aspen Modelling Tree Mortality Using Meta Modelling Slide-24 Method Combined Parameter Estimates Equal weight Inverse Variances ML ML (Differential Weight) Stein Rule

Results: Estimated Parameters Using Meta- Regression for Aspen Results: Estimated Parameters Using Meta- Regression for Aspen Modelling Tree Mortality Using Meta Modelling Slide-25 Method Combined Parameter Estimates Equal weight Inverse Variances ML ML (Differential Weight) Stein Rule

Results: Estimated Parameters Using Meta- Regression for Aspen Results: Estimated Parameters Using Meta- Regression for Aspen Modelling Tree Mortality Using Meta Modelling Slide-26 Method Combined Parameter Estimates Equal weight Inverse Variances ML ML (Differential Weight) Stein Rule

Results: Predicted Annual Probability of Survival and Likelihood Using Meta-Regression and All Data Combined for Aspen Results: Predicted Annual Probability of Survival and Likelihood Using Meta-Regression and All Data Combined for Aspen Modelling Tree Mortality Using Meta Modelling Slide-27 Method Predicted -2logL LiveDead Mean 25% (percentile) 75% (percentile) Mean 25% (percentile) 75% (percentile) Equal weight Inverse Variances ML ML (Differential Weight) Stein Rule All data combined

Results: Predicted Annual Probability of Survival and Likelihood Using Meta-Regression and All Data Combined for Aspen Results: Predicted Annual Probability of Survival and Likelihood Using Meta-Regression and All Data Combined for Aspen Modelling Tree Mortality Using Meta Modelling Slide-28 Method Predicted -2logL LiveDead Mean 25% (percentile) 75% (percentile) Mean 25% (percentile) 75% (percentile) Equal weight Inverse Variances ML ML (Differential Weight) Stein Rule All data combined

Results: Predicted Annual Probability of Survival and Likelihood Using Meta-Regression and All Data Combined for Aspen Results: Predicted Annual Probability of Survival and Likelihood Using Meta-Regression and All Data Combined for Aspen Modelling Tree Mortality Using Meta Modelling Slide-29 Method Predicted -2logL LiveDead Mean 25% (percentile) 75% (percentile) Mean 25% (percentile) 75% (percentile) Equal weight Inverse Variances ML ML (Differential Weight) Stein Rule All data combined

Discussion Discussion Modelling Tree Mortality Using Meta Modelling Slide-30 The simplest meta-regression approach would be to combine existing models using equal weights which resulted in the highest -2logL. Maximum likelihood (differential weight ) to find weights resulted in lower -2logL value than other meta-regression approaches.

Conclusion Conclusion Modelling Tree Mortality Using Meta Modelling Slide-31 Further research will include: Accuracy testing will be done for the estimators. One estimator will be selected for use. Regional models will be fitted for white spruce, black spruce and jack pine using Alberta, Ontario and Quebec PSP data.

 Thanks to Alberta, Ontario and Quebec governments for providing PSP data.  Thanks to my committee members: Dr. Valerie Lemay, UBC. Dr. Steen Magnussen, Nrcan. Dr. Frank Berninger, UQAM. Dr. Peter Marshall, UBC. Dr. Andrew Robinson, U. Melbourne.  Thanks to NSERC and ForValueNet for providing the research funding. Acknowledgements Acknowledgements Modelling Tree Mortality Using Meta Modelling Slide-32

Modeling Tree Mortality for Large Regions Using Combined Estimators and Meta-Analysis Approaches Modelling Tree Mortality Using Meta Modelling Slide-33