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 firstorderAR (1) structure and power variancefunction performed better than extended mixed-effects models and predicted wellas an ordinary leastsquares 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 AICvalue 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 mixedeffects modelling approach.