Year of Award


Document Type

Thesis - Campus Access Only

Degree Type

Master of Science (MS)

Degree Name


Department or School/College

College of Forestry and Conservation

Committee Chair

David Affleck

Commitee Members

Solomon Harrar, John Goodburn


simultaneous equation systems, autocorrelation, heteroskedasticity, variance structure, individual-tree growth models, precommercial thinning


University of Montana


A precommercial thinning study for ponderosa pine (Pinus ponderosa) was initiated by the U.S. Forest Service, the Nez Perce Tribe, and the Spokane Tribe of Indians in 1997/98. The agencies ran plot inventories and measured tagged trees before thinning and then at 1, 3, and 5 years post-thinning. In 2011, at 13/14 years after the original thinning, the Inland Northwest Growth and Yield Cooperative returned to the site to re-measure the tagged trees. This dataset presented an opportunity to quantify the effect of precommercial thinning on ponderosa pine growth in the Inland Northwest, where the effects of this treatment have yet to be rigorously documented. I used this dataset to build density dependent growth models in an attempt to quantify the precise effects of density management on three dimensions of individual tree growth. I first estimated these models using nonlinear least-squares. These models confirmed the results of previous precommercial thinning studies in other regions, that changing densities significantly impact diameter growth and crown dimensions at the individual tree level, and that density management does not significantly affect height. These models captured the mean trends in growth variation quite well, yet because the data exhibited heteroskedasticity and autocorrelation, nonlinear least-squares coeffi- cient estimates were inefficient, and coefficient standard error estimates were biased. I therefore attempted to model the heteroskedasticity and autocorrelation of the data, and reestimated the coefficients, using generalized nonlinear least-squares. These models confirmed the existence of heteroskedasticity and autocorrelation in the data by parameterizing statistically significant parameters for these two conditions. Parame- terizing the covariance structure also resulted in a clarification of residual trends, which led to the addition of another explanatory variable to the basal area increment models. In the height increment and height to crown base models, parameterizing the variance did not appreciably alter the models’ descriptions of mean growth trends, within the context of my analyses. Lastly, I performed parametric bootstrapping simulations to investigate the poten- tial for coefficient biases due to the simultaneous nature of the basal area growth, height growth, and height to crown base models. Simulated data were generated using six different error structures to investigate the consequences of ignoring contemporaneous cross-correlation. I found that ignoring a level of contemporaneous cross-correlation consistent with the empirical levels in this dataset did not result in substantial simul- taneity bias in the model coefficient estimates. However, if the level of contemporaneous cross-correlation between the height and basal area increment residuals was increased, the potential simultaneity bias grew substantial enough to result in meaningful changes in expected predictions.

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