Tree diameter distribution modelling: introducing the logit–logistic distribution Mingliang Wang and Keith Rennolls Abstract: Johnson’s S B distribution is a four-parameter distribution that is transformed into a normal distribution by a logit transformation. By replacing the normal distribution of Johnson’s S B with the logistic distribution, we obtain a new distributional model that approximates S B . It is analytically tractable, and we name it the “logit–logistic” (LL) distribution. A generalized four-parameter Weibull model and the Burr XII model are also introduced for comparison purposes. Using the distribution “shape plane” (with axes skew 2 and kurtosis) we compare the “coverage” properties of the LL, the gen- eralized Weibull, and the Burr XII with Johnson’s S B , the beta, and the three-parameter Weibull, the main distributions used in forest modelling. The LL is found to have the largest range of shapes. An empirical case study of the distribu- tional models is conducted on 107 sample plots of Chinese fir. The LL performs best among the four-parameter models. Résumé : La distribution S B de Johnson est une distribution avec quatre paramètres qui est transformée en une distri- bution normale par une transformation logit. En remplaçant la distribution normale S B de Johnson par la distribution lo- gistique, les auteurs obtenent une nouvelle distribution qui se rapproche de la distribution S B . Ce modèle a une solution analytique et ils l’appelent la distribution “logit logistique”. Un modèle de Weibull généralisé avec quatre paramètres et le modèle de Burr XII sont aussi introduits pour fins de comparaison. En utilisant la distribution “de forme plane” (avec les axes : (dissymétrie) 2 et aplatissement), ils comparent les propriétés de “couverture” du modèle logit logistique, du modèle de Weibull généralisé et du modèle de Burr XII avec celles de la distribution S B de Johnson, de la distribution Bêta et de la distribution de Weibull à trois paramètres, les principales distributions utilisées pour la modélisation en foresterie. La distribution logit logistique a été identifiée comme celle qui avait la plus grande étendue de formes. Une étude de cas empirique des modèles de distribution est effectuée à partir de 107 parcelles-échantillons de sapin chinois. Parmi les modèles avec quatre paramètres, la distribution logit logistique offre la meilleure performance. [Traduit par la Rédaction] Wang and Rennolls 1313 Introduction Forest managers often require information concerning the size-class distribution of a forest stand, often in the form of a tabulation of numbers of trees by diameter class. This diam- eter distribution information is often used to predict volume production, the primary variable that forest managers are in- terested in. Diameter distribution models are also important in forecasting the range of products that might be expected from a stand. A wide range of probability density functions have been used in forestry to model tree diameter distributions (e.g., log-normal: Bliss and Reinker 1964; gamma: Nelson 1964; Weibull: Bailey and Dell 1973; Rennolls et al. 1985; beta: Zohrer 1972; Li et al. 2002), although the three-parameter Weibull and the four-parameter beta and S B models are pos- sibly the most frequently used. Hafley and Schreuder (1977) compared the beta, Johnson’s S B , Weibull, log-normal, gamma, and normal distributions in terms of their coverage in the skewness 2 –kurtosis (the β 1 –β 2 ) plane. They concluded that Johnson’s S B gave the best per- formance in terms of the quality of fitting a variety of sample distributions (tree diameter and height data). Subsequently, S B and its bivariate version have been much used and com- pared with other common distributional models (Schreuder and Hafley 1977; Hafley and Buford 1985; Knoebel and Burkhart 1991; Zhou and McTague 1996; Kamziah et al. 1999; Tewari and Gadow 1997, 1999; Scolforo et al. 2003). Li et al. (2002) conclude that the beta is superior to S B , in apparent contradiction to the conclusion of Hafley and Schreuder (1977). However, these papers have used different parameter estimation methods, different measures of good- ness of fit, and different test data sets. These differences in approach will most likely have contributed to the differing conclusions. Zhang et al. (2003) compared alternative meth- ods of fitting the three-parameter Weibull and S B and used Reynolds’ error index as a comparison criterion. Hence, the current situation is that there is no clear resolu- tion as to which model is the most suitable for tree distribu- tional modelling. There is of course no theoretical reason why there should exist a best model for all situations. It might be that in one case a particular distribution will be found empir- ically to give the best fit, whilst in another case another model will be found to be empirically best. The only way in which it is meaningful to talk about the best distributional model is in terms of the most flexible of models in representational Can. J. For. Res. 35: 1305–1313 (2005) doi: 10.1139/X05-057 © 2005 NRC Canada 1305 Received 30 August 2004. Accepted 16 February 2005. Published on the NRC Research Press Web site at http://cjfr.nrc.ca on 2 July 2005. M. Wang. 1 School of Computing and Mathematical Sciences, University of Greenwich, London, UK; Chinese Academy of Forestry, Wanshou Shan, Beijing 100091, China. K. Rennolls. 1 School of Computing and Mathematical Sciences, University of Greenwich, London, UK. 1 Corresponding authors (e-mail: m.wang@gre.ac.uk and k.rennolls@gre.ac.uk).