Deriving tree diameter distributions using Bayesian model averaging Bronson P. Bullock a, * , Edward L. Boone b,1 a Department of Forestry and Environmental Resources, North Carolina State University, 3102 Jordan Hall, Box 8008, Raleigh, NC 27695, United States b Department of Statistical Sciences and Operations Research, Virginia Commonwealth University, 1001 W. Main Street, Richmond, VA 23284, United States Received 11 April 2006; received in revised form 25 August 2006; accepted 6 January 2007 Abstract Diameter distributions of loblolly pine trees (Pinus taeda L.) were derived using Bayesian methods. The Bayesian derived distributions were compared to traditional maximum likelihood techniques for estimating diameter distributions. The data for this study come from a planting density study established in 1983 at four locations in Virginia and North Carolina. Results of the Bayesian analysis show that different distributional forms had the highest posterior probability at different ages for the diameter distributions. Standard goodness-of-fit statistics for the maximum likelihood estimates did not consistently recommend a distribution for the empirical data. Further, there were cases where no distribution was chosen from the candidate distributions based on the goodness-of-fit statistics. The lack of a clear distribution that fits over the range of ages for the diameter data suggests that a Bayesian model averaging technique would be beneficial for deriving tree diameter distributions with probability proportional to the posterior distribution. A simulation study was conducted to compare the goodness-of-fit statistics from the averaged distributional model and other candidate distributions. Results indicated that the averaged distribution had superior goodness-of-fit values over the candidate models; the exception being when the candidate distribution was an exact fit to the data. # 2007 Elsevier B.V. All rights reserved. Keywords: Loblolly pine; Pinus taeda; BMA; Weibull; Johnson’s S B ; Scaled beta; Maximum likelihood 1. Introduction Loblolly pine (Pinus taeda L.) is the most important commercial tree species in the Southeast US. Models are utilized to forecast the growth and final yield of forest stands based on current stand conditions and site quality estimates (Burkhart et al., 1987; Kangas, 1998; Brooks and Wiant, 2004). Diameter distributions provide a means of characterizing a forest stand and enable prediction of stand yield and size class structures based on a specific probability density function (pdf) (Bailey and Dell, 1973; Cao and Burkhart, 1984; Rennolls et al., 1985; Brooks et al., 1992). The parameters of the pdf for a specific distributional form are predicted or recovered from the current stand characteristics and are used to estimate the population diameter distribution (Hyink and Moser, 1983; Brooks et al., 1992; Bullock and Burkhart, 2005). Many distributional forms have been used to represent diameter distributions, e.g., normal (Bailey, 1980), beta (Strub and Burkhart, 1975; Maltamo et al., 1995), Johnson’s S B (Hafley and Schreuder, 1977; Knoebel and Burkhart, 1991; Parresol, 2003), gamma (Nelson, 1964; Bailey, 1980), two-parameter Weibull (Bailey and Dell, 1973; Bullock and Burkhart, 2005), and three-parameter Weibull (Ek et al., 1975; Matney et al., 1987; Lenhart, 1988; Maltamo, 1997). The Weibull distribution has been used widely in forestry and does a very good job of characterizing unimodal diameter distributions (Bailey and Dell, 1973; Cao and Burkhart, 1984; Matney et al., 1987; Bullock and Burkhart, 2005). The two- parameter Weibull distribution (Weibull-2) can also capture skewness that may be present in a diameter distribution of a forest stand and can be truncated (censored) if the tree diameter data has a minimum recorded value (Zutter et al., 1986). The three-parameter Weibull distribution (Weibull-3) (Lenhart, 1988) and Johnson’s S B distribution (Parresol, 2003) have been considered for parameter recovery techniques. The scaled beta distribution was utilized by Maltamo et al. (1995) for deriving diameter distributions in stands of Scotch pine (Pinus sylvestris) and Norway spruce (Picea abies). The gamma www.elsevier.com/locate/foreco Forest Ecology and Management 242 (2007) 127–132 * Corresponding author. Tel.: +1 919 513 1248; fax: +1 919 515 6193. E-mail addresses: Bronson_Bullock@ncsu.edu (B.P. Bullock), elboone@vcu.edu (E.L. Boone). 1 Tel.: +1 804 828 1301. 0378-1127/$ – see front matter # 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.foreco.2007.01.024