Using quadratic mean diameter and relative spacing index to enhance height-diameter and crown ratio models fitted to longitudinal data


The inclusion of quadratic mean diameter (QMD) and relative spacing index (RSI) substantially improved the predictive capacity of height–diameter at breast height (d.b.h.) and crown ratio models (CR), respectively. Data were obtained from 208 permanent plots established in western Arkansas and eastern Oklahoma during 1985–1987 and remeasured for the sixth time (2012–2014). Existing height–d.b.h. and CR estimation models for naturally occurring shortleaf pine forests (Pinus echinata Mill.) were updated and modified for improved performance. Additionally, eight height–d.b.h. relationship models that use only d.b.h. (fundamental local models) were modified using covariates. The model performance was evaluated using fit statistics [root mean square error (RMSE), Fit index and Akaike information criteria (AIC)]. The results showed that the best model form which was an extended non-linear model with autoregressive first order AR(1) structure and power variance function performed better than extended mixed-effects models and predicted well as an ordinary least squares non-linear model. The autocorrelation within individual trees was larger for the height–d.b.h. relationship than for CR estimation. The addition of QMD to mean dominant height (HD) greatly improved height–d.b.h. relationship with a reduction of 8 per cent in RMSE, compared with the use of basal area per hectare. Similarly, multiplying a fundamental local model by using QMD raised to a parametric power reduced RMSE by 16 per cent, improved Fit index by 12 per cent and decreased the AIC value by 7 per cent. D.b.h., HD and RSI best explained the crown ratio relationship with an improved Fit index by 6.7 per cent compared with alternative non-linear models without RSI. The logistic model for CR also provided prediction accuracy similar to that of a commonly used non-linear model. A non-linear model with an application of remedial measures to enhance adherence to modelling assumptions can provide better parameter estimates than mixed effects modelling approach.

  • Citation: Saud, Pradip; Lynch, Thomas B.; K.C., Anup; Guldin, James M. 2016. Using quadratic mean diameter and relative spacing index to enhance height-diameter and crown ratio models fitted to longitudinal data. Forestry. 89: 215-229. 15 p.   10.1093/forestry/cpw004
  • Keywords: mixed-effects model, autocorrelation, height–d.b.h. relationship, crown ratio, quadratic mean diameter, relative spacing index
  • Posted Date: September 12, 2016
  • Modified Date: September 20, 2016
  • Print Publications Are No Longer Available

    In an ongoing effort to be fiscally responsible, the Southern Research Station (SRS) will no longer produce and distribute hard copies of our publications. Many SRS publications are available at cost via the Government Printing Office (GPO). Electronic versions of publications may be downloaded, printed, and distributed.

    Publication Notes

    • This article was written and prepared by U.S. Government employees on official time, and is therefore in the public domain.
    • Our online publications are scanned and captured using Adobe Acrobat. During the capture process some typographical errors may occur. Please contact the SRS webmaster if you notice any errors which make this publication unusable.
    • To view this article, download the latest version of Adobe Acrobat Reader.